1. Lecture 1: Signals & Systems Concepts
(1) Systems, signals, mathematical models. Continuous-time and discrete-time signals and systems. Energy and power signals. Linear systems. Examples for use throughout the course, introduction to Matlab and Simulink tools
Specific Objectives:
Introduce, using examples, what is a signal and what is a system
Why mathematical models are appropriate
What are continuous-time and discrete-time representations and how are they related
Brief introduction to Matlab and Simulink
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2. What is a Signal? A signal is a pattern of variation of some form
Signals are variables that carry information
Examples of signal include:
Electrical signals
Voltages and currents in a circuit
Acoustic signals
Acoustic pressure (sound) over time
Mechanical signals
Velocity of a car over time
Video signals
Intensity level of a pixel (camera, video) over time
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3. How is a Signal Represented?
Mathematically, signals are represented as a function of one or more independent variables.
For instance a black & white video signal intensity is dependent on x, y coordinates and time t f(x,y,t)
On this course, we shall be exclusively concerned with signals that are a function of a single variable: time t
f(t)
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4. Example: Signals in an Electrical Circuitvs + - C vc
The signals vc and vs are patterns of variation over time
Note, we could also have considered the voltage across the resistor or the current as signals
Step (signal) vs at t=1
RC = 1
First order (exponential) response for vc
vs, vc t
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5. Continuous & Discrete-Time Signals
Continuous-Time Signals
Most signals in the real world are continuous time, as the scale is infinitesimally fine.
Eg voltage, velocity,
Denote by x(t), where the time interval may be bounded (finite) or infinite
Discrete-Time Signals
Some real world and many digital signals are discrete time, as they are sampled
E.g. pixels, daily stock price (anything that a digital computer processes)
Denote by x[n], where n is an integer value that varies discretely
Sampled continuous signal x[n] =x(nk) – k is sample time
x(t) t
x[n] n
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6. Signal Properties
On this course, we shall be particularly interested in signals with certain properties:
Periodic signals: a signal is periodic if it repeats itself after a fixed period T, i.e. x(t) = x(t+T) for all t. A sin(t) signal is periodic.
Even and odd signals: a signal is even if x(-t) = x(t) (i.e. it can be reflected in the axis at zero). A signal is odd if x(-t) = -x(t). Examples are cos(t) and sin(t) signals, respectively.
Exponential and sinusoidal signals: a signal is (real) exponential if it can be represented as x(t) = Ceat. A signal is (complex) exponential if it can be represented in the same form but C and a are complex numbers.
Step and pulse signals: A pulse signal is one which is nearly completely zero, apart from a short spike, d(t). A step signal is zero up to a certain time, and then a constant value after that time, u(t).
These properties define a large class of tractable, useful signals and will be further considered in the coming lectures
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7. What is a System?
Systems process input signals to produce output signals
Examples:
A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitor
A CD player takes the signal on the CD and transforms it into a signal sent to the loud speaker
A communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver. The channel typically attenuates and adds noise to the transmitted signal which must be processed by the receiver
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8. How is a System Represented?
A system takes a signal as an input and transforms it into another signal
In a very broad sense, a system can be represented as the ratio of the output signal over the input signal
That way, when we “multiply” the system by the input signal, we get the output signal
This concept will be firmed up in the coming weeks
System
Input signal
x(t)
Output signal
y(t)
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9. Example: An Electrical Circuit Systemvs + - C vc
Simulink representation of the electrical circuit
vs(t)
vc(t)
first order
system
vs, vc t
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10. Continuous & Discrete-Time Mathematical Models of Systems
Continuous-Time Systems
Most continuous time systems represent how continuous signals are transformed via differential equations.
E.g. circuit, car velocity
Discrete-Time Systems
Most discrete time systems represent how discrete signals are transformed via difference equations
E.g. bank account, discrete car velocity system
First order differential equations
First order difference equations
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11. Properties of a System
On this course, we shall be particularly interested in signals with certain properties:
Causal: a system is causal if the output at a time, only depends on input values up to that time.
Linear: a system is linear if the output of the scaled sum of two input signals is the equivalent scaled sum of outputs
Time-invariance: a system is time invariant if the system’s output is the same, given the same input signal, regardless of time.
These properties define a large class of tractable, useful systems and will be further considered in the coming lectures
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12. Introduction to Matlab/Simulink (1)
Command
window
Click on the Matlab icon/start menu initialises the Matlab environment:
The main window is the dynamic command interpreter which allows the user to issue Matlab commands
The variable browser shows which variables currently exist in the workspace
Variable
browser
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13. Introduction to Matlab/Simulink (2)
Type the following at the Matlab command prompt
>> simulink
The following Simulink library should appear
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14. Introduction to Matlab/Simulink (3)
Click File-New to create a new workspace, and drag and drop objects from the library onto the workspace.
Selecting Simulation-Start from the pull down menu will run the dynamic simulation. Click on the blocks to view the data or alter the run-time parameters
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15. How Are Signal & Systems Related (i)?
How to design a system to process a signal in particular ways?
Design a system to restore or enhance a particular signal
Remove high frequency background communication noise
Enhance noisy images from spacecraft
Assume a signal is represented as
x(t) = d(t) + n(t)
Design a system to remove the unknown “noise” component n(t), so that y(t)  d(t)
x(t) = d(t) + n(t)
y(t)  d(t)
System ?
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16. How Are Signal & Systems Related (ii)?
How to design a system to extract specific pieces of information from signals
Estimate the heart rate from an electrocardiogram
Estimate economic indicators (bear, bull) from stock market values
Assume a signal is represented as
x(t) = g(d(t))
Design a system to “invert” the transformation g(), so that y(t) = d(t)
x(t) = g(d(t))
y(t) = d(t) = g-1(x(t))
System ?
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17. How Are Signal & Systems Related (iii)?
How to design a (dynamic) system to modify or control the output of another (dynamic) system
Control an aircraft’s altitude, velocity, heading by adjusting throttle, rudder, ailerons
Control the temperature of a building by adjusting the heating/cooling energy flow.
Assume a signal is represented as
x(t) = g(d(t))
Design a system to “invert” the transformation g(), so that y(t) = d(t)
x(t)
y(t) = d(t)
dynamic
system ?
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18. Lecture 1: Summary
Signals and systems are pervasive in modern engineering courses:
Electrical circuits
Physical models and control systems
Digital media (music, voice, photos, video)
In studying the general properties of signals and systems, you can:
Design systems to remove noise/enhance measurement from audio and picture/video data
Investigate stability of physical structures
Control the performance mechanical and electrical devices
This will be the foundation for studying systems and signals as a generic subject on this course.
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19. Lecture 2: Signals Concepts & Properties
(1) Systems, signals, mathematical models. Continuous-time and discrete-time signals. Energy and power signals. Linear systems. Examples for use throughout the course, introduction to Matlab and Simulink tools
Specific objectives for this lecture include
General properties of signals
Energy and power for continuous & discrete-time signals
Signal transformations
Specific signal types
Representing signals in Matlab and Simulink
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20. Reminder: Continuous & Discrete Signals
Continuous-Time Signals
Most signals in the real world are continuous time, as the scale is infinitesimally fine.
E.g. voltage, velocity,
Denote by x(t), where the time interval may be bounded (finite) or infinite
Discrete-Time Signals
Some real world and many digital signals are discrete time, as they are sampled
E.g. pixels, daily stock price (anything that a digital computer processes)
Denote by x[n], where n is an integer value that varies discretely
Sampled continuous signal
x[n] =x(nk)
x(t) t
x[n] n
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21. “Electrical” Signal Energy & Power
It is often useful to characterise signals by measures such as energy and power
For example, the instantaneous power of a resistor is:
and the total energy expanded over the interval [t1, t2] is:
and the average energy is:
How are these concepts defined for any continuous or discrete time signal?
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22. Generic Signal Energy and Power
Total energy of a continuous signal x(t) over [t1, t2] is:
where |.| denote the magnitude of the (complex) number.
Similarly for a discrete time signal x[n] over [n1, n2]:
By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the average power, P
Note that these are similar to the electrical analogies (voltage), but they are different, both value and dimension.
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23. Energy and Power over Infinite Time
For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). These quantities are therefore defined by:
If the sums or integrals do not converge, the energy of such a signal is infinite
Two important (sub)classes of signals
Finite total energy (and therefore zero average power)
Finite average power (and therefore infinite total energy)
Signal analysis over infinite time, all depends on the “tails” (limiting behaviour)
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24. Time Shift Signal Transformations
A central concept in signal analysis is the transformation of one signal into another signal. Of particular interest are simple transformations that involve a transformation of the time axis only.
A linear time shift signal transformation is given by:
where b represents a signal offset from 0, and the a parameter represents a signal stretching if |a|>1, compression if 0<|a|<1 and a reflection if a<0.
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25. Periodic Signals
An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property
where T>0, for all t.
Examples:
cos(t+2p) = cos(t)
sin(t+2p) = sin(t)
Are both periodic with period 2p
NB for a signal to be periodic, the relationship must hold for all t. 2p
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26. Odd and Even Signals
An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original:
Examples:
x(t) = cos(t)
x(t) = c
An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal
x(t) = sin(t)
x(t) = t
This is important because any signal can be expressed as the sum of an odd signal and an even signal.
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27. Exponential and Sinusoidal Signals
Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals.
A generic complex exponential signal is of the form:
where C and a are, in general, complex numbers. Lets investigate some special cases of this signal
Real exponential signals
Exponential growth
Exponential decay
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28. Periodic Complex Exponential & Sinusoidal Signals
Consider when a is purely imaginary:
By Euler’s relationship, this can be expressed as:
This is a periodic signals because:
when T=2p/w0
A closely related signal is the sinusoidal signal:
We can always use:
cos(1)
T0 = 2p/w0
= p
T0 is the fundamental time period
w0 is the fundamental frequency
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29. Exponential & Sinusoidal Signal Properties
Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power.
Consider energy over one period:
Therefore:
Average power:
Useful to consider harmonic signals
Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency
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30. General Complex Exponential Signals
So far, considered the real and periodic complex exponential
Now consider when C can be complex. Let us express C is polar form and a in rectangular form: So
Using Euler’s relation
These are damped sinusoids
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31. Discrete Unit Impulse and Step Signals
The discrete unit impulse signal is defined:
Useful as a basis for analyzing other signals
The discrete unit step signal is defined:
Note that the unit impulse is the first difference (derivative) of the step signal
Similarly, the unit step is the running sum (integral) of the unit impulse.
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32. Continuous Unit Impulse and Step Signals
The continuous unit impulse signal is defined:
Note that it is discontinuous at t=0
The arrow is used to denote area, rather than actual value
Again, useful for an infinite basis
The continuous unit step signal is defined:
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33. Introduction to Matlab
Simulink is a package that runs inside the Matlab environment.
Matlab (Matrix Laboratory) is a dynamic, interpreted, environment for matrix/vector analysis
User can build programs (in .m files or at command line) C/Java-like syntax
Ideal environment for programming and analysing discrete (indexed) signals and systems
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34. Basic Matlab Operations
>> % This is a comment, it starts with a “%”
>> y = 5*3 + 2^2; % simple arithmetic
>> x = [ ]; % create the vector “x”
>> x1 = x.^2; % square each element in x
>> E = sum(abs(x).^2); % Calculate signal energy
>> P = E/length(x); % Calculate av signal power
>> x2 = x(1:3); % Select first 3 elements in x
>> z = 1+i; % Create a complex number
>> a = real(z); % Pick off real part
>> b = imag(z); % Pick off imaginary part
>> plot(x); % Plot the vector as a signal
>> t = 0:0.1:100; % Generate sampled time
>> x3=exp(-t).*cos(t); % Generate a discrete signal
>> plot(t, x3, ‘x’); % Plot points
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35. Other Matlab Programming Structures
Loops
for i=1:100
sum = sum+i;
end
Goes round the for loop 100 times, starting at i=1 and finishing at i=100
i=1;
while i<=100
i = i+1;
Similar, but uses a while loop instead of a for loop
Decisions
if i==5
a = i*2;
else
a = i*4;
end
Executes whichever branch is appropriate depending on test
switch i
case 5
otherwise
Similar, but uses a switch
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36. Matlab Help! These slides have provided a rapid introduction to Matlab
Mastering Matlab 6, Prentice Hall,
Introduction to Matlab (on-line)
Lots of help available
Type help in the command window or help operator. This displays the help associated with the specified operator/function
Type lookfor topic to search for Matlab commands that are related to the specified topic
Type helpdesk in the command window or select help on the pull down menu. This allows you to access several, well-written programming tutorials.
comp.soft-sys.matlab newsgroup
Learning to program (Matlab) is a “bums on seats” activity. There is no substitute for practice, making mistakes, understanding concepts
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37. Using the Matlab Debugger
Because Matlab is an interpreted language, there is no compile type syntax checking and the likelihood of a run-time error is higher
Run-time debugging can help
Use the debug and breakpoints pull-down menus to determine where to stop program and inspect variables
Step over lines/step into functions to evaluate what happens
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38. Introduction to Simulink
Simulink is a graphical, “drag and drop” environment for building simple and complex signal and system dynamic simulations.
It allows users to concentrate on the structure of the problem, rather than having to worry (too much) about a programming language.
The parameters of each signal and system block is configured by the user (right click on block)
Signals and systems are simulated over a particular time.
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39. Signals in Simulink
Two main libraries for manipulating signals in Simulink:
Sources: generate a signal
Sink: display, read or store a signal
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40. Example: Generate and View a Signal
Copy “sine wave” source and “scope” sink onto a new Simulink work space and connect.
Set sine wave parameters modify to 2 rad/sec
Run the simulation:
Simulation - Start
Open the scope and leave open while you change parameters (sin or simulation parameters) and re-run
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41. Lecture 2: Summary This lecture has looked at signals:
Power and energy
Signal transformations
Time shift
Periodic
Even and odd signals
Exponential and sinusoidal signals
Unit impulse and step functions
Matlab and Simulink are complementary environments for producing and analysing continuous and discrete signals.
This will require some effort to learn the programming syntax and style!
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42. Lecture 3: Signals & Systems Concepts
Systems, signals, mathematical models. Continuous-time and discrete-time signals. Energy and power signals. Linear systems. Examples for use throughout the course, introduction to Matlab and Simulink tools.
Specific objectives:
Introduction to systems
Continuous and discrete time systems
Properties of a system
Linear (time invariant) LTI systems
System implementation in Matlab and Simulink
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43. Linear Systems
A system takes a signal as an input and transforms it into another signal
Linear systems play a crucial role in most areas of science
Closed form solutions often exist
Theoretical analysis is considerably simplified
Non-linear systems can often be regarded as linear, for small perturbations, so-called linearization
For the remainder of the lecture/course we’re primarily going to be considering Linear, Time Invariant systems (LTI) and consider their properties
continuous
time (CT)
x(t)
y(t)
discrete
time (DT)
x[n]
y[n]
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44. Examples of Simple Systems
To get some idea of typical systems (and their properties), consider the electrical circuit example:
which is a first order, CT differential equation.
Examples of first order, DT difference equations:
where y is the monthly bank balance, and x is monthly net deposit
which represents a discretised version of the electrical circuit
Example of second order system includes:
System described by order and parameters (a, b, c)
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45. First Order Step Responses
People tend to visualise systems in terms of their responses to simple input signals (see Lecture 4…)
The dynamics of the output signal are determined by the dynamics of the system, if the input signal has no dynamics
Consider when the input signal is a step at t, n = 1, y(0) = 0
First order CT differential system
First order DT difference system
u(t)
y(t) t
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46. System Linearity
The most important property that a system possesses is linearity
It means allows any system response to be analysed as the sum of simpler responses (convolution)
Simplistically, it can be imagined as a line y x
Specifically, a linear system must satisfy the two properties:
1 Additive: the response to x1(t)+x2(t) is y1(t) + y2(t)
2 Scaling: the response to ax1(t) is ay1(t) where aC
Combined: ax1(t)+bx2(t)  ay1(t) + by2(t)
E.g. Linear y(t) = 3*x(t) why?
Non-linear y(t) = 3*x(t)+2, y(t) = 3*x2(t) why?
(equivalent definition for DT systems)
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47. Bias and Zero Initial Conditions
Intuitively, a system such as:
y(t) = 3*x(t)+2
is regarded as being linear. However, it does not satisfy the scaling condition.
There are several (similar) ways to transform it to an equivalent linear system
Perturbations around operating value x*, y*
Linear System Derivative
Locally, these ideas can also be used to linearise a non-linear system in a small range
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48. Linearity and Superposition
Suppose an input signal x[n] is made of a linear sum of other (basis/simpler) signals xk[n]:
then the (linear) system response is:
The basic idea is that if we understand how simple signals get affected by the system, we can work out how complex signals are affected, by expanding them as a linear sum
This is known as the superposition property which is true for linear systems in both CT & DT
Important for understanding convolution (next lecture)
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49. Definition of Time Invariance
A system is time invariant if its behaviour and characteristics are fixed over time
We would expect to get the same results from an input-output experiment, if the same input signal was fed in at a different time
E.g. The following CT system is time-invariant
because it is invariant to a time shift, i.e. x2(t) = x1(t-t0)
E.g. The following DT system is time-varying
Because the system parameter that multiplies the input signal is time varying, this can be verified by substitution
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50. System with and without Memory
A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent on the output at only that same time (no system dynamics)
e.g. a resistor is a memoryless CT system where x(t) is current and y(t) is the voltage
A DT system with memory is an accumulator (integrator)
and a delay
Roughly speaking, a memory corresponds to a mechanism in the system that retains information about input values other than the current time.
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51. System Causality
A system is causal if the output at any time depends on values of the output at only the present and past times. Referred to as non-anticipative, as the system output does not anticipate future values of the input
If two input signals are the same up to some point t0/n0, then the outputs from a causal system must be the same up to then.
E.g. The accumulator system is causal:
because y[n] only depends on x[n], x[n-1], …
E.g. The averaging/filtering system is non-causal
because y[n] depends on x[n+1], x[n+2], …
Most physical systems are causal
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52. System Stability
Informally, a stable system is one in which small input signals lead to responses that do not diverge
If an input signal is bounded, then the output signal must also be bounded, if the system is stable
To show a system is stable we have to do it for all input signals. To show instability, we just have to find one counterexample
E.g. Consider the DT system of the bank account
when x[n] = d[n], y[0] = 0
This grows without bound, due to 1.01 multiplier. This system is unstable.
E.g. Consider the CT electrical circuit, is stable if RC>0, because it dissipates energy
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53. Invertible and Inverse Systems
A system is said to be invertible if distinct inputs lead to distinct outputs (similar to matrix invertibility)
If a system is invertible, an inverse system exists which, when cascaded with the original system, yields an output equal to the input of the first signal
E.g. the CT system is invertible:
y(t) = 2x(t)
because w(t) = 0.5*y(t) recovers the original signal x(t)
E.g. the CT system is not-invertible
y(t) = x2(t)
because distinct input signals lead to the same output signal
Widely used as a design principle:
Encryption, decryption
System control, where the reference signal is input
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54. System Structures
Systems are generally composed of components (sub-systems).
We can use our understanding of the components and their interconnection to understand the operation and behaviour of the overall system
Series/cascade
Parallel
Feedback
System 1
System 2 x y
System 1
System 2 x y +
System 1 x y +
System 2
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55. Systems In Matlab A system transforms a signal into another signal.
In Matlab a discrete signal is represented as an indexed vector.
Therefore, a matrix or a for loop can be used to transform one vector into another
Example (DT first order system)
>> n = 0:10;
>> x = ones(size(n));
>> x(1) = 0;
>> y(1) = 0;
>> for i=2:11
y(i) = (y(i-1) + x(i))/2;
end
>> plot(n, x, ‘x’, n, y, ‘.’)
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56. System Libraries in Simulink
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57. Example 1: Voltage Simulation
Click File-New to create a new workspace, and drag and drop objects from the library onto the workspace.
Selecting Simulation-Start from the pull down menu will run the dynamic simulation. Click on the blocks to view the data or alter the run-time parameters
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58. Example 2: Mass-Spring Simulation
Mass-spring system demonstration in Simulink
Square wave input signal, oscillatory output signal
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59. Lecture 3: Summary Whenever we use an equation for a system:
CT – differential
DT – difference
The parameters, order and structure represent the system
There are a large class of systems that are linear, time invariant (LTI), these will primarily be studied on this course.
Other system properties such as causality, stability, memory and invertibility will be dealt with on a case by case basis
Matlab and Simulink are standard tools for analysing, designing, simulating complex systems.
Used for system modelling and control design
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60. Lecture 4: Linear Systems and Convolution
2. Linear systems, Convolution (3 lectures): Impulse response, input signals as continuum of impulses. Convolution, discrete-time and continuous-time. LTI systems and convolution
Specific objectives for today:
We’re looking at discrete time signals and systems
Understand a system’s impulse response properties
Show how any input signal can be decomposed into a continuum of impulses
DT Convolution for time varying and time invariant systems
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61. Introduction to Convolution
Definition Convolution is an operator that takes an input signal and returns an output signal, based on knowledge about the system’s unit impulse response h[n].
The basic idea behind convolution is to use the system’s response to a simple input signal to calculate the response to more complex signals
This is possible for LTI systems because they possess the superposition property (lecture 3):
x[n] = d[n]
System
y[n] = h[n]
x[n]
System: h[n]
y[n]
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62. Discrete Impulses & Time Shifts
Basic idea: use a (infinite) set of of discrete time impulses to represent any signal.
Consider any discrete input signal x[n]. This can be written as the linear sum of a set of unit impulse signals:
Therefore, the signal can be expressed as:
In general, any discrete signal can be represented as:
actual value
Impulse, time shifted signal
The sifting property
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63. Example The discrete signal x[n]
Is decomposed into the following additive components
x[-4]d[n+4] +
x[-3]d[n+3] +
x[-2]d[n+2] +
x[-1]d[n+1] + …
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64. Discrete, Unit Impulse System Response
A very important way to analyse a system is to study the output signal when a unit impulse signal is used as an input
Loosely speaking, this corresponds to giving the system a kick at n=0, and then seeing what happens
This is so common, a specific notation, h[n], is used to denote the output signal, rather than the more general y[n].
The output signal can be used to infer properties about the system’s structure and its parameters q.
d[n]
System: q
h[n]
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65. Types of Unit Impulse Response
Causal, stable, finite impulse response
y[n] = x[n] + 0.5x[n-1] x[n-2]
Looking at unit impulse responses, allows you to determine certain system properties
Causal, stable, infinite impulse response
y[n] = x[n] + 0.7y[n-1]
Causal, unstable, infinite impulse response
y[n] = x[n] + 1.3y[n-1]
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66. Linear, Time Varying Systems
If the system is time varying, let hk[n] denote the response to the impulse signal d[n-k] (because it is time varying, the impulse responses at different times will change).
Then from the superposition property (Lecture 3) of linear systems, the system’s response to a more general input signal x[n] can be written as:
Input signal
System output signal is given by the convolution sum
i.e. it is the scaled sum of impulse responses
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67. Example: Time Varying Convolution
x[n] = [0 0 – ]
h-1[n] = [0 0 –1.5 – ]
h0[n] = [ ]
y[n] = [ ]
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68. Linear Time Invariant Systems
When system is linear, time invariant, the unit impulse responses are all time-shifted versions of each other:
It is usual to drop the 0 subscript and simply define the unit impulse response h[n] as:
In this case, the convolution sum for LTI systems is:
It is called the convolution sum (or superposition sum) because it involves the convolution of two signals x[n] and h[n], and is sometimes written as:
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69. System Identification and Prediction
Note that the system’s response to an arbitrary input signal is completely determined by its response to the unit impulse.
Therefore, if we need to identify a particular LTI system, we can apply a unit impulse signal and measure the system’s response.
That data can then be used to predict the system’s response to any input signal
Note that describing an LTI system using h[n], is equivalent to a description using a difference equation. There is a direct mapping between h[n] and the parameters/order of a difference equation such as:
y[n] = x[n] + 0.5x[n-1] x[n-2]
System: h[n]
y[n]
x[n]
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70. Example 1: LTI Convolution
Consider a LTI system with the following unit impulse response:
h[n] = [ ]
For the input sequence:
x[n] = [ ]
The result is:
y[n] = … + x[0]h[n] + x[1]h[n-1] + …
= 0 +
0.5*[ ] +
2.0*[ ] +
= [ ]
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71. Example 2: LTI Convolution
Consider the problem described for example 1
Sketch x[k] and h[n-k] for any particular value of n, then multiply the two signals and sum over all values of k.
For n<0, we see that x[k]h[n-k] = 0 for all k, since the non-zero values of the two signals do not overlap.
y[0] = Skx[k]h[0-k] = 0.5
y[1] = Skx[k]h[1-k] = 0.5+2
y[2] = Skx[k]h[2-k] = 0.5+2
y[3] = Skx[k]h[3-k] = 2
As found in Example 1
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72. Example 3: LTI Convolution
Consider a LTI system that has a step response h[n] = u[n] to the unit impulse input signal
What is the response when an input signal of the form
x[n] = anu[n]
where 0<a<1, is applied?
For n0:
Therefore,
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73. Discrete LTI Convolution in Matlab
In Matlab to find out about a command, you can search the help files or type:
>> lookfor convolution
at the Matlab command line. This returns all Matlab functions that contain the term “convolution” in the basic description
These include:
conv()
To see how this works and other functions that may be appropriate, type:
>> help conv
at the Matlab command line
Example:
>> h = [ ];
>> x = [ ];
>> y = conv(x, h)
>> y = [ ]
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74. Lecture 4: Summary
Any discrete LTI system can be completely determined by measuring its unit impulse response h[n]
This can be used to predict the response to an arbitrary input signal using the convolution operator:
The output signal y[n] can be calculated by:
Sum of scaled signals – example 1
Non-zero elements of h – example 2
The two ways of calculating the convolution are equivalent
Calculated in Matlab using the conv() function (but note that there are some zero padding at start and end)
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75. Lecture 5: Linear Systems and Convolution
2. Linear systems, Convolution (3 lectures): Impulse response, input signals as continuum of impulses. Convolution, discrete-time and continuous-time. LTI systems and convolution
Specific objectives for today:
We’re looking at continuous time signals and systems
Understand a system’s impulse response properties
Show how any input signal can be decomposed into a continuum of impulses
Convolution
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76. Introduction to “Continuous” Convolution
In this lecture, we’re going to understand how the convolution theory can be applied to continuous systems. This is probably most easily introduced by considering the relationship between discrete and continuous systems.
The convolution sum for discrete systems was based on the sifting principle, the input signal can be represented as a superposition (linear combination) of scaled and shifted impulse functions.
This can be generalised to continuous signals, by thinking of it as the limiting case of arbitrarily thin pulses
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77. Signal “Staircase” Approximation
As previously shown, any continuous signal can be approximated by a linear combination of thin, delayed pulses:
Note that this pulse (rectangle) has a unit integral. Then we have:
Only one pulse is non-zero for any value of t. Then as D0
When D0, the summation approaches an integral
This is known as the sifting property of the continuous-time impulse and there are an infinite number of such impulses d(t-t)
dD(t)
1/D D
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78. Alternative Derivation of Sifting Property
The unit impulse function, d(t), could have been used to directly derive the sifting function.
Therefore:
The previous derivation strongly emphasises the close relationship between the structure for both discrete and continuous-time signals
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79. Continuous Time Convolution
Given that the input signal can be approximated by a sum of scaled, shifted version of the pulse signal, dD(t-kD)
The linear system’s output signal y is the superposition of the responses, hkD(t), which is the system response to dD(t-kD).
From the discrete-time convolution:
What remains is to consider as D0. In this case: ^ ^
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80. Example: Discrete to Continuous Time Linear Convolution
The CT input signal (red) x(t) is approximated (blue) by:
Each pulse signal
generates a response
Therefore the DT convolution response is
Which approximates the CT convolution response
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81. Linear Time Invariant Convolution
For a linear, time invariant system, all the impulse responses are simply time shifted versions:
Therefore, convolution for an LTI system is defined by:
This is known as the convolution integral or the superposition integral
Algebraically, it can be written as:
To evaluate the integral for a specific value of t, obtain the signal h(t-t) and multiply it with x(t) and the value y(t) is obtained by integrating over t from – to .
Demonstrated in the following examples
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82. Example 1: CT Convolution
Let x(t) be the input to a LTI system with unit impulse response h(t):
For t>0:
We can compute y(t) for t>0:
So for all t:
In this example
a=1
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83. Example 2: CT Convolution
Calculate the convolution of the following signals
The convolution integral becomes:
For t-30, the product x(t)h(t-t) is non-zero for -<t<0, so the convolution integral becomes:
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84. Lecture 5: Summary
A continuous signal x(t) can be represented via the sifting property:
Any continuous LTI system can be completely determined by measuring its unit impulse response h(t)
Given the input signal and the LTI system unit impulse response, the system’s output can be determined via convolution via
Note that this is an alternative way of calculating the solution y(t) compared to an ODE. h(t) contains the derivative information about the LHS of the ODE and the input signal represents the RHS.
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85. Lecture 6: Linear Systems and Convolution
2. Linear systems, Convolution (3 lectures): Impulse response, input signals as continuum of impulses. Convolution, discrete-time and continuous-time. LTI systems and convolution
Specific objectives for today:
Properties of an LTI system
Differential and difference systems
Rather than concentrate on the mechanics of how to calculate discrete/continuous-time convolution, we’re looking at the impact on the system specification.
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86. LTI Systems and Impulse Response
Any continuous/discrete-time LTI system is completely described by its impulse response through the convolution:
This only holds for LTI systems as follows:
Example: The discrete-time impulse response
Is completely described by the following LTI
However, the following systems also have the same impulse response
Therefore, if the system is non-linear, it is not completely characterised by the impulse response
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87. Commutative Property
Convolution is a commutative operator (in both discrete and continuous time), i.e.:
For example, in discrete-time:
and similar for continuous time.
Therefore, when calculating the response of a system to an input signal x[n], we can imagine the signal being convolved with the unit impulse response h[n], or vice versa, whichever appears the most straightforward.
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88. Distributive Property (Parallel Systems)
Another property of convolution is the distributive property
This can be easily verified
Therefore, the two systems:
are equivalent. The convolved sum of two impulse responses is equivalent to considering the two equivalent parallel system (equivalent for discrete-time systems)
y1(t)
h1(t)
h2(t)
x(t)
y(t)
x(t)
y(t)
h1(t)+h2(t) +
y2(t)
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89. Example: Distributive Property
Let y[n] denote the convolution of the following two sequences:
x[n] is non-zero for all n. We will use the distributive property to express y[n] as the sum of two simpler convolution problems. Let x1[n] = 0.5nu[n], x2[n] = 2nu[-n], it follows that
and y[n] = y1[n]+y2[n], where y1[n] = x1[n]*h[n], y1[n] = x1[n]*h[n].
From Lecture 3 - example 3, and O&W example 2.5
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90. Associative Property (Serial Systems)
Another property of (LTI) convolution is that it is associative
Again this can be easily verified by manipulating the summation/integral indices
Therefore, the following four systems are all equivalent and y[n] = x[n]*h1[n]*h2[n] is unambiguously defined.
This is not true for non-linear systems (y1[n] = 2x[n], y2[n] = x2[n])
h1(t)
x(t)
y(t)
h2(t)
w(t)
x(t)
y(t)
h1(t)*h2(t)
x(t)
x(t)
y(t)
v(t)
y(t)
h2(t)*h1(t)
h2(t)
h1(t)
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91. LTI System Memory
An LTI system is memoryless if its output depends only on the input value at the same time, i.e.
For an impulse response, this can only be true if
Such systems are extremely simple and the output of dynamic engineering, physical systems depend on:
Preceding values of x[n-1], x[n-2], …
Past values of y[n-1], y[n-2], …
for discrete-time systems, or derivative terms for continuous-time systems
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92. System Invertibility
Does there exist a system with impulse response h1(t) such that y(t)=x(t)?
Widely used concept for:
control of physical systems, where the aim is to calculate a control signal such that the system behaves as specified
filtering out noise from communication systems, where the aim is to recover the original signal x(t)
The aim is to calculate “inverse systems” such that
The resulting serial system is therefore memoryless
h(t)
x(t)
y(t)
h1(t)
w(t)
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93. Example: Accumulator System
Consider a DT LTI system with an impulse response
h[n] = u[n]
Using convolution, the response to an arbitrary input x[n]:
As u[n-k] = 0 for n-k<0 and 1 for n-k0, this becomes
i.e. it acts as a running sum or accumulator. Therefore an inverse system can be expressed as:
A first difference (differential) operator, which has an impulse response
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94. Causality for LTI Systems
Remember, a causal system only depends on present and past values of the input signal. We do not use knowledge about future information.
For a discrete LTI system, convolution tells us that
h[n] = 0 for n<0
as y[n] must not depend on x[k] for k>n, as the impulse response must be zero before the pulse!
Both the integrator and its inverse in the previous example are causal
This is strongly related to inverse systems as we generally require our inverse system to be causal. If it is not causal, it is difficult to manufacture!
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95. LTI System Stability Continuous-time system
Remember: A system is stable if every bounded input produces a bounded output
Therefore, consider a bounded input signal
|x[n]| < B for all n
Applying convolution and taking the absolute value:
Using the triangle inequality (magnitude of a sum of a set of numbers is no larger than the sum of the magnitude of the numbers):
Therefore a DT LTI system is stable if and only if its impulse response is absolutely summable, ie
Continuous-time system
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96. Example: System Stability
Are the DT and CT pure time shift systems stable?
Are the discrete and continuous-time integrator systems stable?
Therefore, both the CT and DT systems are stable: all finite input signals produce a finite output signal
Therefore, both the CT and DT systems are unstable: at least one finite input causes an infinite output signal
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97. Differential and Difference Equations
Two extremely important classes of causal LTI systems:
1) CT systems whose input-output response is described by linear, constant-coefficient, ordinary differential equations with a forced response
2) DT systems whose input-output response is described by linear, constant-coefficient, difference equations
Note that to “solve” these equations for y(t) or y[n], we need to know the initial conditions
Examine such systems and relate them to the system properties just described
RC circuit with: y(t) = vc(t), x(t) = vs(t), a = b = 1/RC.
Simple bank account with: a = -1.01, b = 1.
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98. Continuous-Time Differential Equations
A general Nth-order LTI differential equation is
If the equation involves derivative operators on y(t) (N>0) or x(t), it has memory.
The system stability depends on the coefficients ak. For example, a 1st order LTI differential equation with a0=1:
If a1>0, the system is unstable as its impulse response represents a growing exponential function of time
If a1<0 the system is stable as its impulse response corresponds to a decaying exponential function of time
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99. Discrete-Time Difference Equations
A general Nth-order LTI difference equation is
If the equation involves difference operators on y[n] (N>0) or x[n], it has memory.
The system stability depends on the coefficients ak. For example, a 1st order LTI difference equation with a0=1:
If a1>1, the system is unstable as its impulse response represents a growing power function of time
If a1 <1 the system is stable as its impulse response corresponds to a decaying power function of time
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100. Lecture 6: Summary
The standard notions of commutative, associative and distributive properties are valid for convolution operators. These can be used to simplify evaluating convolution, by decomposing the input system/signal into simpler parts, and then solving the transformed problem.
Standard system notations of
Memory
Causality
Invertibility
Stability
Can be given specific definitions in terms of representing a system via its convolution response or in terms of the derivative/differential equation
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101. Lecture 7: Basis Functions & Fourier Series
3. Basis functions (3 lectures): Concept of basis function. Fourier series representation of time functions. Fourier transform and its properties. Examples, transform of simple time functions.
Specific objectives for today:
Introduction to Fourier series (& transform)
Eigenfunctions of a system
Show sinusoidal signals are eigenfunctions of LTI systems
Introduction to signals and basis functions
Fourier basis & coefficients of a periodic signal
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102. Why is Fourier Theory Important (i)?
For a particular system, what signals fk(t) have the property that:
Then fk(t) is an eigenfunction with eigenvalue lk
If an input signal can be decomposed as
x(t) = Sk akfk(t)
Then the response of an LTI system is
y(t) = Sk aklkfk(t)
For an LTI system, fk(t) = est where sC, are eigenfunctions.
x(t) = fk(t)
y(t) = lkfk(t)
System
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103. Why is Fourier Theory Important (ii)?
Fourier transforms map a time-domain signal into a frequency domain signal
Simple interpretation of the frequency content of signals in the frequency domain (as opposed to time).
Design systems to filter out high or low frequency components. Analyse systems in frequency domain.
Invariant to high frequency signals
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104. Why is Fourier Theory Important (iii)?
If F{x(t)} = X(jw) w is the frequency
Then F{x’(t)} = jwX(jw)
So solving a differential equation is transformed from a calculus operation in the time domain into an algebraic operation in the frequency domain (see Laplace transform)
Example
becomes
and is solved for the roots w (N.B. complementary equations):
and we take the inverse Fourier transform for those w.
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105. Introduction to System Eigenfunctions
Lets imagine what (basis) signals fk(t) have the property that:
i.e. the output signal is the same as the input signal, multiplied by the constant “gain” lk (which may be complex)
For CT LTI systems, we also have that
Therefore, to make use of this theory we need: 1) system identification is determined by finding {fk,lk}.
2) response, we also have to decompose x(t) in terms of fk(t) by calculating the coefficients {ak}.
This is analogous to eigenvectors/eigenvalues matrix decomposition
x(t) = fk(t)
y(t) = lkfk(t)
System
x(t) = Sk akfk(t)
y(t) = Sk aklkfk(t)
LTI
System
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106. Complex Exponentials are Eigenfunctions of any CT LTI System
Consider a CT LTI system with impulse response h(t) and input signal x(t)=f(t) = est, for any value of sC:
Assuming that the integral on the right hand side converges to H(s), this becomes (for any value of sC):
Therefore f(t)=est is an eigenfunction, with eigenvalue l=H(s)
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107. Example 1: Time Delay & Imaginary Input
Consider a CT, LTI system where the input and output are related by a pure time shift:
Consider a purely imaginary input signal:
Then the response is:
ej2t is an eigenfunction (as we’d expect) and the associated eigenvalue is H(j2) = e-j6.
The eigenvalue could be derived “more generally”. The system impulse response is h(t) = d(t-3), therefore:
So H(j2) = e-j6!
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108. Example 1a: Phase Shift
Note that the corresponding input e-j2t has eigenvalue ej6, so lets consider an input cosine signal of frequency 2 so that:
By the system LTI, eigenfunction property, the system output is written as:
So because the eigenvalue is purely imaginary, this corresponds to a phase shift (time delay) in the system’s response. If the eigenvalue had a real component, this would correspond to an amplitude variation
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109. Example 2: Time Delay & Superposition
Consider the same system (3 time delays) and now consider the input signal x(t) = cos(4t)+cos(7t), a superposition of two sinusoidal signals that are not harmonically related. The response is obviously:
Consider x(t) represented using Euler’s formula:
Then due to the superposition property and H(s) =e-3s
While the answer for this simple system can be directly spotted, the superposition property allows us to apply the eigenfunction concept to more complex LTI systems.
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110. History of Fourier/Harmonic Series
The idea of using trigonometric sums was used to predict astronomical events
Euler studied vibrating strings, ~1750, which are signals where linear displacement was preserved with time.
Fourier described how such a series could be applied and showed that a periodic signals can be represented as the integrals of sinusoids that are not all harmonically related
Now widely used to understand the structure and frequency content of arbitrary signals
w=1
w=2
w=3
w=4
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111. Fourier Series and Fourier Basis Functions
The theory derived for LTI convolution, used the concept that any input signal can represented as a linear combination of shifted impulses (for either DT or CT signals)
We will now look at how (input) signals can be represented as a linear combination of Fourier basis functions (LTI eigenfunctions) which are purely imaginary exponentials
These are known as continuous-time Fourier series
The bases are scaled and shifted sinusoidal signals, which can be represented as complex exponentials
x(t) = sin(t) +
0.2cos(2t) +
0.1sin(5t)
ejwt
x(t)
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112. Periodic Signals & Fourier Series
A periodic signal has the property x(t) = x(t+T), T is the fundamental period, w0 = 2p/T is the fundamental frequency. Two periodic signals include:
For each periodic signal, the Fourier basis the set of harmonically related complex exponentials:
Thus the Fourier series is of the form:
k=0 is a constant
k=+/-1 are the fundamental/first harmonic components
k=+/-N are the Nth harmonic components
For a particular signal, are the values of {ak}k?
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113. Fourier Series Representation of a CT Periodic Signal (i)
Given that a signal has a Fourier series representation, we have to find {ak}k. Multiplying through by
Using Euler’s formula for the complex exponential integral
It can be shown that
T is the fundamental period of x(t)
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114. Fourier Series Representation of a CT Periodic Signal (ii)
Therefore
which allows us to determine the coefficients. Also note that this result is the same if we integrate over any interval of length T (not just [0,T]), denoted by
To summarise, if x(t) has a Fourier series representation, then the pair of equations that defines the Fourier series of a periodic, continuous-time signal:
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115. Lecture 7: Summary
Fourier bases, series and transforms are extremely useful for frequency domain analysis, solving differential equations and analysing invariance for LTI signals/systems
For an LTI system
est is an eigenfunction
H(s) is the corresponding (complex) eigenvalue
This can be used, like convolution, to calculate the output of an LTI system once H(s) is known.
A Fourier basis is a set of harmonically related complex exponentials
Any periodic signal can be represented as an infinite sum (Fourier series) of Fourier bases, where the first harmonic is equal to the fundamental frequency
The corresponding coefficients can be evaluated
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116. Lecture 8: Fourier Series and Fourier Transform
3. Basis functions (3 lectures): Concept of basis function. Fourier series representation of time functions. Fourier transform and its properties. Examples, transform of simple time functions.
Specific objectives for today:
Examples of Fourier series of periodic functions
Rational and definition of Fourier transform
Examples of Fourier transforms
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117. Example 1: Fourier Series sin(w0t)
The fundamental period of sin(w0t) is w0
By inspection we can write:
So a1 = 1/2j, a-1 = -1/2j and ak = 0 otherwise
The magnitude and angle of the Fourier coefficients are:
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118. Example 1a: Fourier Series sin(w0t)
The Fourier coefficients can also be explicitly evaluated
When k = +1 or –1, the integrals evaluate to T and –T, respectively. Otherwise the coefficients are zero.
Therefore a1 = 1/2j, a-1 = -1/2j
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119. Example 2: Additive Sinusoids
Consider the additive sinusoidal series which has a fundamental frequency w0:
Again, the signal can be directly written as:
The Fourier series coefficients can then be visualised as:
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120. Example 3: Periodic Step Signal
Consider the periodic square wave, illustrated by:
and is defined over one period as:
Fourier coefficients:
NB, these coefficients are real
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121. Example 3a: Periodic Step Signal
Instead of plotting both the magnitude and the angle of the complex coefficients, we only need to plot the value of the coefficients.
Note we have an infinite series of non-zero coefficients
T=4T1
T=8T1
T=16T1
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122. Convergence of Fourier Series
Not every periodic signal can be represented as an infinite Fourier series, however just about all interesting signals can be (note that the step signal is discontinuous)
The Dirichlet conditions are necessary and sufficient conditions on the signal.
Condition 1. Over any period, x(t) must be absolutely integrable
Condition 2. In any finite interval, x(t) is of bounded variation; that is there is no more than a finite number of maxima and minima during any single period of the signal
Condition 3. In any finite interval of time, there are only a finite number of discontinuities. Further, each of these discontinuities are finite.
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123. Fourier Series to Fourier Transform
For periodic signals, we can represent them as linear combinations of harmonically related complex exponentials
To extend this to non-periodic signals, we need to consider aperiodic signals as periodic signals with infinite period.
As the period becomes infinite, the corresponding frequency components form a continuum and the Fourier series sum becomes an integral (like the derivation of CT convolution)
Instead of looking at the coefficients a harmonically –related Fourier series, we’ll now look at the Fourier transform which is a complex valued function in the frequency domain
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124. Definition of the Fourier Transform
We will be referring to functions of time and their Fourier transforms. A signal x(t) and its Fourier transform X(jw) are related by the Fourier transform synthesis and analysis equations
and
We will refer to x(t) and X(jw) as a Fourier transform pair with the notation
As previously mentioned, the transform function X() can roughly be thought of as a continuum of the previous coefficients
A similar set of Dirichlet convergence conditions exist for the Fourier transform, as for the Fourier series (T=(- ,))
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125. Example 1: Decaying Exponential
Consider the (non-periodic) signal
Then the Fourier transform is:
a = 1
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126. Example 2: Single Rectangular Pulse
Consider the non-periodic rectangular pulse at zero
The Fourier transform is:
Note, the values are real
T1 = 1
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127. Example 3: Impulse Signal
The Fourier transform of the impulse signal can be calculated as follows:
Therefore, the Fourier transform of the impulse function has a constant contribution for all frequencies w
X(jw)
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128. Example 4: Periodic Signals
A periodic signal violates condition 1 of the Dirichlet conditions for the Fourier transform to exist
However, lets consider a Fourier transform which is a single impulse of area 2p at a particular (harmonic) frequency w=w0.
The corresponding signal can be obtained by:
which is a (complex) sinusoidal signal of frequency w0. More generally, when
Then the corresponding (periodic) signal is
The Fourier transform of a periodic signal is a train of impulses at the harmonic frequencies with amplitude 2pak
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129. Lecture 8: Summary
Fourier series and Fourier transform is used to represent periodic and non-periodic signals in the frequency domain, respectively.
Looking at signals in the Fourier domain allows us to understand the frequency response of a system and also to design systems with a particular frequency response, such as filtering out high frequency signals.
You’ll need to complete the exercises to work out how to calculate the Fourier transform (and its inverse) and evaluate the frequency content of a signal
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130. Lecture 9: Fourier Transform Properties and Examples
3. Basis functions (3 lectures): Concept of basis function. Fourier series representation of time functions. Fourier transform and its properties. Examples, transform of simple time functions.
Specific objectives for today:
Properties of a Fourier transform
Linearity
Time shifts
Differentiation and integration
Convolution in the frequency domain
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131. Reminder: Fourier Transform
A signal x(t) and its Fourier transform X(jw) are related by
This is denoted by:
For example (1):
Remember that the Fourier transform is a density function, you must integrate it, rather than summing up the discrete Fourier series components
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132. Linearity of the Fourier TransformIf
and
Then
This follows directly from the definition of the Fourier transform (as the integral operator is linear). It is easily extended to a linear combination of an arbitrary number of signals
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133. Time Shifting If Recognising this as
Then
Proof
Now replacing t by t-t0
Recognising this as
A signal which is shifted in time does not have its Fourier transform magnitude altered, only a shift in phase.
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134. Example: Linearity & Time Shift
Consider the signal (linear sum of two time shifted steps)
where x1(t) is of width 1, x2(t) is of width 3, centred on zero.
Using the rectangular pulse example
Then using the linearity and time shift Fourier transform properties
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135. Differentiation & Integration
By differentiating both sides of the Fourier transform synthesis equation:
Therefore:
This is important, because it replaces differentiation in the time domain with multiplication in the frequency domain.
Integration is similar:
The impulse term represents the dc or average value that can result from integration
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136. Example: Fourier Transform of a Step Signal
Lets calculate the Fourier transform X(jw)of x(t) = u(t), making use of the knowledge that:
and noting that:
Taking Fourier transform of both sides
using the integration property. Since G(jw) = 1:
We can also apply the differentiation property in reverse
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137. Convolution in the Frequency Domain
With a bit of work (next slide) it can show that:
Therefore, to apply convolution in the frequency domain, we just have to multiply the two functions.
To solve for the differential/convolution equation using Fourier transforms:
Calculate Fourier transforms of x(t) and h(t)
Multiply H(jw) by X(jw) to obtain Y(jw)
Calculate the inverse Fourier transform of Y(jw)
Multiplication in the frequency domain corresponds to convolution in the time domain and vice versa.
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138. Proof of Convolution Property
Taking Fourier transforms gives:
Interchanging the order of integration, we have
By the time shift property, the bracketed term is e-jwtH(jw), so
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139. Example 1: Solving an ODE
Consider the LTI system time impulse response
to the input signal
Transforming these signals into the frequency domain
and the frequency response is
to convert this to the time domain, express as partial fractions:
Therefore, the time domain response is:
ba
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140. Example 2: Designing a Low Pass Filter
H(jw) wc
-wc
Lets design a low pass filter:
The impulse response of this filter is the inverse Fourier transform
which is an ideal low pass filter
Non-causal (how to build)
The time-domain oscillations may be undesirable
How to approximate the frequency selection characteristics?
Consider the system with impulse response:
Causal and non-oscillatory time domain response and performs a degree of low pass filtering
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141. Lecture 9: Summary
The Fourier transform is widely used for designing filters. You can design systems with reject high frequency noise and just retain the low frequency components. This is natural to describe in the frequency domain.
Important properties of the Fourier transform are:
1. Linearity and time shifts
2. Differentiation
3. Convolution
Some operations are simplified in the frequency domain, but there are a number of signals for which the Fourier transform do not exist – this leads naturally onto Laplace transforms
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142. Lecture 10: Sampling Discrete-Time Systems
4 Sampling & Discrete-time systems (2 lectures): Sampling theorem, discrete Fourier transform
Specific objectives for today:
Sampling of a continuous-time signal
Reconstruction of the signals from its samples
Sampling theorem & Nyquist rate
Reconstruction of a signal, using zero-order holds
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143. What is Discrete Time Sampling?
Sampling is the transformation of a continuous signal into a discrete signal
Widely applied in digital analysis systems
Sample the continuous time signal
Design and process discrete time signal
Convert back to continuous time
x(t),
x[n]
T is the sampling period
t=nT
x(t)
x[n]
y[n]
y(t)
Discrete
Time sampler
Discrete
time system
Signal
reconstruction
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144. Why is Sampling Important?
For many systems (e.g. Matlab, …) designing and processing discrete-time systems is more efficient and more general compared to performing continuous-time system design.
How does Simulink perform continuous-time system simulation?
The signals are sampled and the systems are approximately integrated in discrete time
Mainly due to the dramatic development of digital technology resulting in inexpensive, lightweight, programmable and reproducible discrete-time systems. Widely used for communication
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145. Sampling a Continuous-Time Signal
Clearly for a finite sample period T, it is not possible to represent every uncountable, infinite-dimensional continuous-time signal with a countable, infinite-dimensional discrete-time signal.
In general, an infinite number of CT signals can generate a DT signal.
However, if the signal is band (frequency) limited, and the samples are sufficiently close, it is possible to uniquely reconstruct the original CT signal from the sampled signal
x1(t), x2(t), x3(t),
x[n]
t=nT
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146. Definition of Impulse Train Sampling
We need to have a convenient way in which to represent the sampling of a CT signal at regular intervals
A common/useful way to do this is through the use of a periodic impulse train signal, p(t), multiplied by the CT signal
T is the sampling period
ws=2p/T is the sampling frequency
This is known as impulse train sampling. Note xp(t) is still a continuous time signal T
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147. Analysing Impulse Train Sampling (i)
What effect does this sampling have on the frequency decomposition (Fourier transform) of the CT impulse train signal xp(t)?
By definition:
The signal p(t) is periodic and the coefficients of the Fourier Series are given by:
Therefore, the Fourier transform is given by
One property of the Fourier transform we did not consider is the multiplicative property which says if xp(t) = x(t)p(t), then
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148. Analysing Impulse Train Sampling (ii)
Substituting for P(jw)
Therefore Xp(jw) is a periodic function of w, consisting of a superposition of shifted replicas of X(jw), scaled by 1/T.
|X(jw)|=0: |w|>1
ws=3
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149. Reconstruction of the CT Signal1
1/T T
When the sampling frequency ws is less than twice the band-limited frequency wM, there is no overlaps the spectrum X(jw)
If this is true, the original signal x(t) can be recovered from the impulse sampled xp(t), by passing it through a low pass filter H(jw) with gain T and cutoff frequency between wM and ws-wM.
wM=1
ws=3
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150. Sampling Theorem Let x(t) be a band (frequency)-limited signal
X(jw) = 0 for |w|>wM.
Then x(t) is uniquely determined by its samples {x(nT)} when the sampling frequency satisfies:
where ws=2p/T.
2wM is known as the Nyquist rate, as it represents the largest frequency that can be reproduced with the sample time
The result makes sense because a frequency-limited signal has a limited amount of information that can be fully captured with the sampled sequence {x(nT)}
X(jw) ws
-wM wM w
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151. Zero Order Hold Sampling
A zero order hold is a common method for bridging CT-DT signals
A zero order hold samples the current signal and holds that value until the next sample
In most systems, it is difficult to generate and transmit narrow, large-amplitude pulses (impulse train sampling) xp(t)
We can often use a variety of filtering/interpolation techniques to reconstruct the original time-domain signal, however often the zero-order hold signal x0(t) is sufficiently accurate
xp(t)
x0(t)
x(t) t
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152. Lecture 10: Summary
The sample time for converting a continuous time signal into a sampled, discrete time signal is determined by the Nyquist rate, amongst other things.
The signal must satisfy the relationship:
If the signal is to be preserved exactly. Information in frequencies higher than this will be lost when the signal is sampled.
A continuous time signal is often sampled and communicated using a zero order hold
Often this is enough to be considered as the re-constructed continuous time signal, but sometimes approximate methods for re-constructing the signal are used
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153. Lecture 11: Discrete Fourier Transform
4 Sampling Discrete-time systems (2 lectures): Sampling theorem, discrete Fourier transform
Specific objectives for today:
Discrete Fourier transform
Examples
Convergence & properties
Convolution
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154. Reminder: CT Fourier Transform
CT Fourier transform maps a time domain frequency signal to the frequency domain via
The Fourier transform is used to
Analyse the frequency content of a signal
Design a system/filter with particular properties
Solve differential equations in the frequency domain using algebraic operators
Note that the transform/integral is not defined for some signals (infinite energy)
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155. Derivation of the DT Fourier Transform
By analogy with the CT Fourier transform, we might “guess”
This is because tn, & the integral operator represents the limit of a sum as the sum’s width tends to zero.
X(ejw) is periodic of period 2p, so is ejwn. Try substituting into the inverse Fourier transform with integral over 2p:
which is the original DT signal.
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156. Discrete Time Fourier Transform
The DT Fourier transform analysis and synthesis equations are therefore:
The function X(ejw) is referred to as the discrete-time Fourier transform and the pair of equations are referred to as the Fourier transform pair
X(ejw) is sometimes referred to as the spectrum of x[n] because it provides us with information on how x[n] is composed of complex exponentials at different frequencies
It converges when the signal is absolutely summable
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157. Example 1: 1st Order System, Decay Power
Calculate the DT Fourier transform of the signal:
Therefore:
stable system
a=0.8
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158. Example 2: Rectangular Pulse
Consider the rectangular pulse
and the Fourier transform is
N1=2
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159. Example 3: Impulse Signal
Fourier transform of the DT impulse signal is
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160. Properties: Periodicity, Linearity & Time
The DT Fourier transform is always periodic with period 2p, because
X(ej(w+2p)) = X(ejw)
It is relatively straightforward to prove that the DT Fourier transform is linear, i.e.
Similarly, if a DT signal is shifted by n0 units of time
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161. Convolution in the Frequency Domain
Like continuous time signals and systems, the time-domain convolution of two discrete time signals can be represented as the multiplication of the Fourier transforms
If x[n], h[n] and y[n] are the input, impulse response and output of a discrete-time LTI system so, by convolution,
y[n] = x[n]*h[n]
Then
Y(ejw) = X(ejw)H(ejw)
The proof is analogous to proof used for the convolution of continuous time Fourier transforms
Convolution in the discrete time domain is replaced by multiplication in the frequency domain.
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162. Example: 1st Order System
Consider an LTI system with impulse response
h[n]=anu[n], |a|<1
and the system input is
x[n]=bnu[n], |b|<1
The DT Fourier transforms are: So
Expressing as partial fractions, assuming ab:
and spotting the inverse Fourier transform
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163. Lecture 11: Summary
Apart from a slightly difference, the Fourier transform of a discrete time signal is equivalent to the continuous time formulae
They have similar properties to the continuous time Fourier transform for linearity, time shifts, differencing and accumulation
The main result is that like continuous time signals and systems, convolution in the time domain is replaced by multiplication in the frequency domain.
Y(ejw) = X(ejw)H(ejw)
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164. Lecture 12: Laplace Transform
5 Laplace transform (3 lectures):
Laplace transform as Fourier transform with convergence factor. Properties of the Laplace transform
Specific objectives for today:
Introduce Laplace transform
Understand the relationship to Fourier transform
Investigate Laplace transform of exponential signals
Derive region of convergence of Laplace transform
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165. Introduction to the Laplace Transform
Fourier transforms are extremely useful in the study of many problems of practical importance involving signals and LTI systems.
purely imaginary complex exponentials est, s=jw
A large class of signals can be represented as a linear combination of complex exponentials and complex exponentials are eigenfunctions of LTI systems.
However, the eigenfunction property applies to any complex number s, not just purely imaginary (signals)
This leads to the development of the Laplace transform where s is an arbitrary complex number.
Laplace and z-transforms can be applied to the analysis of un-stable system (signals with infinite energy) and play a role in the analysis of system stability
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166. The Laplace Transform
The response of an LTI system with impulse response h(t) to a complex exponential input, x(t)=est, is
where s is a complex number and
when s is purely imaginary, this is the Fourier transform, H(jw)
when s is complex, this is the Laplace transform of h(t), H(s)
The Laplace transform of a general signal x(t) is:
and is usually expressed as:
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167. Laplace and Fourier Transform
The Fourier transform is the Laplace transform when s is purely imaginary:
An alternative way of expressing this is when s = s+jw
The Laplace transform is the Fourier transform of the transformed signal x’(t) = x(t)e-st. Depending on whether s is positive/negative this represents a growing/negative signal
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168. Example 1: Laplace Transform
Consider the signal
The Fourier transform X(jw) converges for a>0:
The Laplace transform is:
which is the Fourier Transform of e-(s+a)tu(t) Or
If a is negative or zero, the Laplace Transform still exists
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169. Example 2: Laplace Transform
Consider the signal
The Laplace transform is:
Convergence requires that Re{s+a}<0 or Re{s}<-a.
The Laplace transform expression is identical to Example 1 (similar but different signals), however the regions of convergence of s are mutually exclusive (non-intersecting).
For a Laplace transform, we need both the expression and the Region Of Convergence (ROC).
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170. Example 3: sin(wt)u(t)
The Laplace transform of the signal x(t) = sin(wt)u(t) is:
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171. Fourier Transform does not Converge …
It is worthwhile reflecting that the Fourier transform does not exist for a fairly wide class of signals, such as the response of an unstable, first order system, the Fourier transform does not exist/converge
E.g. x(t) = eatu(t), a>0
does not exist (is infinite) because the signal’s energy is infinite
This is because we multiply x(t) by a complex sinusoidal signal which has unit magnitude for all t and integrate for all time. Therefore, as the Dirichlet convergence conditions say, the Fourier transform exists for most signals with finite energy
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172. Region of Convergence
The Region Of Convergence (ROC) of the Laplace transform is the set of values for s (=s+jw) for which the Fourier transform of x(t)e-st converges (exists).
The ROC is generally displayed by drawing separating line/curve in the complex plane, as illustrated below for Examples 1 and 2, respectively.
The shaded regions denote the ROC for the Laplace transform Re Im -a Re Im -a
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173. Example 4: Laplace Transform
Consider a signal that is the sum of two real exponentials:
The Laplace transform is then:
Using Example 1, each expression can be evaluated as:
The ROC associated with these terms are Re{s}>-1 and Re{s}>-2. Therefore, both will converge for Re{s}>-1, and the Laplace transform:
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174. Lecture 12: Summary
The Laplace transform is a superset of the Fourier transform – it is equal to it when s=jw i.e. F{x(t)} = X(jw)
Laplace transform of a continuous time signal is defined by:
And can be imagined as being the Fourier transform of the signal x’(t) = x(t)est, when s=s+jw
The region of convergence (ROC) associated with the Laplace transform defines the region in s (complex) space for which the Laplace transform converges.
In simple cases it corresponds to the values for s (s) for which the transformed signal has finite energy
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175. Lecture 13: Inverse Laplace Transform
5 Laplace transform (3 lectures):
Laplace transform as Fourier transform with convergence factor. Properties of the Laplace transform
Specific objectives for today:
Poles and zeros of a Laplace transfer function
Rational polynomial transfer functions
Inverse Laplace transform
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176. Reminder: Laplace Transforms
Equivalent to the Fourier transform when s=jw
There is an associated region of convergence for s where the (transformed) signal has finite energy. The Laplace transform is only defined for these values
Laplace transform is linear (easy!)
Examples for the Laplace transforms include
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177. Ratio of Polynomials
In each of these examples, the Laplace transform is rational, i.e. it is a ratio of polynomials in the complex variable s.
where N and D are the numerator and denominator polynomial respectively.
In fact, X(s) will be rational whenever x(t) is a linear combination of real or complex exponentials. Rational transforms also arise when we consider LTI systems specified in terms of linear, constant coefficient differential equations.
We can mark the roots of N and D in the s-plane along with the ROC
Example 3: Re Im -1 1 x -2
s-plane
x – roots of D(s)
– roots of N(s)
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178. Poles and Zeros
The roots of N(s) are known as the zeros. For these values of s, X(s) is zero.
The roots of D(s) are known as the poles. For these values of s, X(s) is infinite, the Region of Convergence for the Laplace transform cannot contain any poles, because the corresponding integral is infinite
The set of poles and zeros completely characterise X(s) to within a scale factor (+ ROC for Laplace transform)
The graphical representation of X(s) through its poles and zeros in the s-plane is referred to as the pole-zero plot of X(s)
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179. Example: Poles and Zeros
Consider the signal:
By linearity (& last lecture) we can evaluate the second and third terms
The Laplace transform of the impulse function is:
which is valid for any s. Therefore, Re Im 1 2 x -1
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180. ROC Properties for Laplace Transform
Property 1: The ROC of X(s) consists of strips parallel to the jw-axis in the s-plane
Because the Laplace transform consists of s for which x(t)e-st converges, which only depends on Re{s} = s
Property 2: For rational Laplace transforms, the ROC does not contain any poles
Because X(s) is infinite at a pole, the integral must not converge.
Property 3: if x(t) is finite duration and is absolutely integrable then the ROC is the entire s-plane.
Because x(t) is magnitude bounded, multiplication by any exponential over a finite interval is also bounded. Therefore the Laplace integral converges for any s.
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181. Inverse Laplace Transform
The Laplace transform of a signal x(t) is:
We can invert this relationship using the inverse Fourier transform
Multiplying both sides by est:
Therefore, we can recover x(t) from X(s), where the real component is fixed and we integrate over the imaginary part, noting that ds = jdw
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182. Inverse Laplace Transform Interpretation
Just about all real-valued signals, x(t), can be represented as a weighted, X(s), integral of complex exponentials, est.
The contour of integration is a straight line (in the complex plane) from s-j to s+j (we won’t be explicitly evaluating this, just spotting known transformations)
We can choose any s for this integration line, as long as the integral converges
For the class of rational Laplace transforms, we can express X(s) as partial fractions to determine the inverse Fourier transform.
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183. Example 1: Inverting the Laplace Transform
Consider when
Like the inverse Fourier transform, expand as partial fractions
Pole-zero plots and ROC for combined & individual terms Re Im -1 x -2
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184. Example 2
Consider when
Like the inverse Fourier transform, expand as partial fractions
Pole-zero plots and ROC for combined & individual terms Re Im -1 x -2
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185. Lecture 13: Summary
For many signals that are made up of a linear combination of complex exponentials and CT LTI systems that are described by differential equations, the Laplace transform is rational, i.e. it is a ratio of polynomials in s: N(s)/D(s)
The roots of N(s) and D(s) are known as the zeros and poles of the transfer function, respectively.
The Region of Convergence does not contain any poles
The inverse Laplace transform is given by
It is usually calculated by expressing the Laplace transform as partial fractions, and then spotting known relationships (rather than directly evaluating the inverse transform)
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186. Lecture 14: Laplace Transform Properties
5 Laplace transform (3 lectures):
Laplace transform as Fourier transform with convergence factor. Properties of the Laplace transform
Specific objectives for today:
Linearity and time shift properties
Convolution property
Time domain differentiation & integration property
Transforms table
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187. Reminder: Laplace Transforms
Equivalent to the Fourier transform when s=jw
Associated region of convergence for which the integral is finite
Used to understand the frequency characteristics of a signal (system)
Used to solve ODEs because of their convenient calculus and convolution properties (today)
Laplace transform
Inverse Laplace transform
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188. Linearity of the Laplace TransformIf
and
Then
This follows directly from the definition of the Laplace transform (as the integral operator is linear). It is easily extended to a linear combination of an arbitrary number of signals
ROC=R1
ROC=R2
ROC= R1R2
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189. Time Shifting & Laplace Transforms
Then
Proof
Now replacing t by t-t0
Recognising this as
A signal which is shifted in time may have both the magnitude and the phase of the Laplace transform altered.
ROC=R
ROC=R
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190. Example: Linear and Time Shift
Consider the signal (linear sum of two time shifted sinusoids)
where x1(t) = sin(w0t)u(t).
Using the sin() Laplace transform example
Then using the linearity and time shift Laplace transform properties
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191. Convolution
The Laplace transform also has the multiplication property, i.e.
Proof is “identical” to the Fourier transform convolution
Note that pole-zero cancellation may occur between H(s) and X(s) which extends the ROC
ROC=R1
ROC=R2
ROCR1R2
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192. Example 1: 1st Order Input & First Order System Impulse Response
Consider the Laplace transform of the output of a first order system when the input is an exponential (decay?)
Taking Laplace transforms
Laplace transform of the output is
Solved with Fourier transforms when a,b>0
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193. Example 1: Continued … Splitting into partial fractions
and using the inverse Laplace transform
Note that this is the same as was obtained earlier, expect it is valid for all a & b, i.e. we can use the Laplace transforms to solve ODEs of LTI systems, using the system’s impulse response
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194. Example 2: Sinusoidal Input
Consider the 1st order (possible unstable) system response with input x(t)
Taking Laplace transforms
The Laplace transform of the output of the system is therefore
and the inverse Laplace transform is
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195. Differentiation in the Time Domain
Consider the Laplace transform derivative in the time domain
sX(s) has an extra zero at 0, and may cancel out a corresponding pole of X(s), so ROC may be larger
Widely used to solve when the system is described by LTI differential equations
ROC=R
ROCR
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196. Example: System Impulse Response
Consider trying to find the system response (potentially unstable) for a second order system with an impulse input x(t)=d(t), y(t)=h(t)
Taking Laplace transforms of both sides and using the linearity property
where r1 and r2 are distinct roots, and calculating the inverse transform
The general solution to a second order system can be expressed as the sum of two complex (possibly real) exponentials
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197. Lecture 14: Summary
Like the Fourier transform, the Laplace transform is linear and represents time shifts (t-T) by multiplying by e-sT
Convolution
Convolution in the time domain is equivalent to multiplying the Laplace transforms
Laplace transform of the system’s impulse response is very important H(s) = h(t)e-stdt. Known as the transfer function.
Differentiation
Very important for solving ordinary differential equations
ROCR1R2
ROCR
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198. Lecture 15: Continuous-Time Transfer Functions
6 Transfer Function of Continuous-Time Systems (3 lectures): Transfer function, frequency response, Bode diagram. Physical realisability, stability. Poles and zeros, rubber sheet analogy.
Specific objectives for today:
Transfer function of a system and examples
Transient and steady-state behaviour
Frequency response
System gain/amplitude and phase margin
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199. System/Transfer Functions
The system/transfer function, H(s), is defined as the Laplace transform of the system’s impulse response
When s=jw, this is the Fourier transform and more generally, this is the Laplace transform
The transfer function is very important because
the unknown system output (Laplace transform) is given by the multiplication X(s) and H(s)
x(t)=d(t)
y(t)=h(t)
H(s)
x(t)
y(t)
H(s)
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200. Example 1: First Order System
Consider a general LTI, first order, differential equation with an impulse input
Taking Laplace transforms
which gives the system’s transfer function, H(s). This can be solved to show that (see earlier examples)
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201. Example 2: Second Order System
Consider a general LTI, second order, differential equation with an impulse input
Taking Laplace transforms
which gives the system’s transfer function, H(s). This can be solved, using partial fractions, to show that
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202. Transfer Functions & System Eigenfuctions
Remember that x(t)=est is an eigenfuction of an LTI system with corresponding eigenvalue H(s)
In addition, by Laplace theory, most input signals x(t) can be expressed as a linear combination of basis signals est (inverse Laplace transform):
Therefore the system’s output can be expressed as
which is why Y(s)=H(s)X(s), again using the transfer function
x(t)=est
y(t)=H(s)est
H(s)
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203. Frequency Response Analysis
Of particular interest is frequency response analysis. This corresponds to input signals of the form
and the corresponding transfer function is
In this case s=jw, the transfer function corresponds to the Fourier transform. It is a complex function of frequency.
Example when x(t) is periodic, with fundamental frequency w0, the Fourier transform is given by:
The system’s response is given by
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204. Example: 1st Order System and cos() Input
The transfer function of a 1st order system is given by:
The input signal x(t)=cos(w0t), which has fundamental frequency w0 is:
The (stable) system’s output is:
Assume a=1
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205. Gain and Phase Transfer Function Analysis
For a complex number/function, we can represent it in polar form by calculating the magnitude (gain) and angle (phase):
H(jw) = |H(jw)|ejH(jw)
In filter/system analysis and design, we’re interested in how certain frequencies are magnified or suppressed -
These are of particular interest by plotting the system properties (amplitude/phase) against frequency:
Frequency shaping
Frequency selection
Low pass
High pass
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206. Frequency Shaping Filters
It is often necessary to change the relative magnitude of a signal at different frequencies, which is referred to as filtering. LTI systems that change the shape of the spectrum are referred to as frequency shaping filters.
Audio systems often contain frequency shaping filters (LTI systems) to change the relative amount of bass (low frequency) and treble (high frequency). Real valued and often plotted on log scaling (dB = 20log10|H(jw)|)
Complex differentiating filters are defined by
H(jw) = jw
which are useful for enhancing rapid variations in a signal. Both the magnitude and phase are plotted against frequency w
H(jw) w
|H(jw)| w
H(jw)
-p/2
p/2
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207. Frequency Selection Filters
Frequency selection filters are specially designed to accept some frequencies and reject others. Noise in an audio recording can be removed by low pass filtering, multiple communication signals can be encoded at different frequencies and then recovered by selecting particular frequencies
Low pass filters are designed to reject/attenuate high frequency “noise” while passing on the low frequencies
High pass filters are designed to reject/attenuate low frequency signal components while passing on high frequency w
H(jw) wc
-wc 1 w
H(jw) wc
-wc 1
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208. Electrical Low Pass Filter+ -
vs(t) R C
vc(t)
Differential equation for the LTI system
The frequency response transfer function H(jw) can be determined using the eigensystem property or using its impulse response definition
Magnitude-phase plot shown right
Step response:
High RC – good frequency selection
Low RC – fast time response
Inevitable time/frequency design compromise
RC=1
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209. Electrical High Pass Filter+ -
vs(t) R C
vr(t)
Differential equation for the LTI system
The frequency response transfer function H(jw) can be determined using eigenfunction property or impulse response
Magnitude-phase plot shown right
Step response:
High RC – good frequency selection
Low RC – fast time response
Inevitable time/frequency design compromise
RC=1
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210. Summary
The system’s transfer function H(s) uniquely determines an LTI because, using convolution, it is possible to determine the output Fourier/Laplace transform, Y(s), given the input Fourier/Laplace transform X(s)
When the system is a LTI ODE, the transfer function is a rational function of s, and it can be inverted (solving the ODE) by expressing the transfer function as partial fractions
In this case, the impulse response of an ODEs is a sum of complex exponentials
Frequency response analysis, by investigating the magnitude and phase at different frequencies, is a standard method for system design
ODEs can be used to implement approximate high pass and low pass filters
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211. Lecture 16: Continuous-Time Transfer Functions
6 Transfer Function of Continuous-Time Systems (3 lectures): Transfer function, frequency response, Bode diagram. Physical realisability, stability. Poles and zeros, rubber sheet analogy.
Specific objectives for today:
Transfer functions and frequency response
Bode diagrams
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212. Introduction: Transfer Functions & Frequency Response
H(s)
x(t)
y(t)
We can use the Fourier (Laplace) transfer function H(jw) (H(s)) in a variety of ways:
Design a system/filter with appropriate frequency domain characteristics
Calculate the system’s time domain response using Y(jw)=H(jw)X(jw) and taking the inverse Fourier transform
However, we can also get a lot of information from studying H(jw) directly and representing it in polar fashion as
H(jw) = |H(jw)|ejH(jw)
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213. Example: 1st Order System and Cos Input
The 1st order system transfer function is: (a>0, h(t)=e-atu(t))
The input signal x(t)=cos(w0t), which has fundamental frequency w0, has Fourier transform:
The (stable) system’s output is:
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214. System Transient & Steady State Response
Compare with the example from lecture 14
which was solved using the Laplace transform
This is composed of two parts:
Transient (blue) and steady state/natural (green -Fourier) responses
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215. System Gain and Phase Shift
In the frequency domain, the effect of the system on the input signal for the frequency component w is:
Y(jw) = |H(jw)|ejH(jw) |X(jw)|ejX(jw)
|Y(jw)| = |H(jw)||X(jw)|
Y(jw) = H(jw) + X(jw)
The effect of a system, H(jw), has on the Fourier transform of an input signal is to:
Scale the magnitude by |H(jw)|. This is commonly referred to as the system gain.
Shift the phase of the input signal by adding H(jw) to it. This is commonly referred to as the phase shift.
These modifications (magnitude and phase distortions) may be desirable/undesirable and must be understood in system analysis and design.
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216. Example: Cos Input to a 1st Order System
Consider a sinusoidal input signal to a first order, LTI, stable system
When w0 is close to zero, its magnitude is passed on scaled by 1/a
When the |w0| is high, the signal is substantially suppressed
i.e. it is a low pass filter …
Magnitude plot
(even)
Phase plot
(odd)
We deduce the properties solely by looking at the transfer function in the frequency domain
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217. The Effect of Phase …
The effect of the transfer function’s magnitude is fairly easy to see – it magnifies/suppresses the input signal
The effect of the change in phase is a bit less obvious to imagine.
Consider when the phase shift is a linear function of w:
This system corresponds to a pure time shift of the input (see lectures 7,9,14)
y(t) = x(t-t0)
Slope of the phase corresponds to the time delay
When the phase is not a linear function, it is slightly more complex
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218. Log-Magnitude and Phase Plots
When analysing system responses, it is typical to use a log scaling for the magnitude
log(|Y(jw)|) = log(|H(jw)|) + log(|X(jw)|)
So the gain effect is additive: 0 means “no change”
If the log magnitude is plotted, the effect can be interpreted as adding each individual component (like the time-delayed phase)
Often units are decibels (dB) 20log10
Similarly, taking logs of frequency allows us to view detail over a much greater range (which is important for frequency selective filters)
Note that taking a log of the frequency, we typically only consider positive frequency values (as the magnitude is even, and the phase is odd)
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219. Bode Plots
A Bode Plot for a system is simply plots of log magnitude and phase against log frequency
Both the log magnitude and phase effects are now additive
Widely used for analysis and design of filters and controllers
Example
Low pass, unity filter
Log mag v log freq
Phase v log freq
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220. Example 1: Bode Plot 1st Order System
Consider a LTI first order system described by:
Fourier transfer function is:
the impulse response is:
and the step response is:
Bode diagrams are shown as log/log plots on the x and y axis with t=2.
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221. Example 2: Bode Plot 2nd Order System
The LTI 2nd order differential equation
which can represent the response of mass-spring systems and RLC circuits, amongst other things
wn is the undamped natural frequency
z is the damping ratio
step response(t)
wn=1
z=[ ]
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222. Lecture 16: Summary
A frequency domain analysis of the transfer function/Fourier transform is an important design/analysis concept
It can be understood in terms of
|H(jw)| - magnitude of the Fourier transform of the impulse response (transfer function)
H(jw) – phase of the Fourier transform of the impulse response (transfer function)
Bode plots are plots of log magnitude and phase against log frequency.
Used to plot a greater range of frequencies
Used to plot decibel-type information
Transfer function is now “additive”
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223. Lecture 17: Continuous-Time Transfer Functions
6 Transfer Function of Continuous-Time Systems (3 lectures): Transfer function, frequency response, Bode diagram. Physical realisability, stability. Poles and zeros, rubber sheet analogy.
Specific objectives for today:
System causality & transfer functions
System stability & transfer functions
Structures of sub-systems – series and feedback
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224. Review: Transfer Functions, Frequency Response & Poles and Zeros
H(jw)
The system’s transfer function is the Laplace (Fourier) transform of the system’s impulse response H(s) (H(jw)).
The transfer function’s poles and zeros are H(s)Pi(s-zi)/Pj(s-pi).
This enables us to both calculate (from the differential equations) and analyse a system’s response
Frequency response magnitude/phase decomposition
H(jw) = |H(jw)|ejH(jw)
Bode diagrams are a log/log plot of this information
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225. System Causality & Transfer Functions
Remember, a system is causal if y(t) only depends on x(t), dx(t)/dt,…,x(t-T) where T>0
This is equivalent to saying that an LTI system’s impulse is h(t) = 0 whenever t<0.
Theorem The ROC associated with the (Laplace) transfer function of a causal system is a right-half plane
Note the converse is not necessarily true (but is true for a rational transfer function)
Proof By definition, for a causal system, s0ROC:
If this converges for s0, then consider any s1>s0
so s1ROC
s-plane Im x Re
s=jw
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226. Examples: System Causality
Consider the (LTI 1st order) system with an impulse response
This has a transfer function (Laplace transform) and ROC
The transfer function is rational and the ROC is a right half plane. The corresponding system is causal.
Consider the system with an impulse response
The system transfer function and ROC
The ROC is not the right half plane, so the system is not causal
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227. System Stability
Remember, a system is stable if , which is equivalent to bounded input signal => bounded output
This is equivalent to saying that an LTI system’s impulse is |h(t)|dt<.
Theorem An LTI system is stable if and only if the ROC of H(s) includes the entire jw axis, i.e. Re{s} = 0.
Proof The transfer function ROC includes the “axis”, s=jw along which the Fourier transform has finite energy
Example The following transfer function is stable
s-plane Im x Re
s=jw
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228. Causal System Stability
Theorem A causal system with rational system function H(s) is stable if and only if all of the poles of H(s) lie in the left-half plane of s, i.e. they have negative real parts
Proof Just combine the two previous theorems
Example
Note that the poles of H(s) correspond to the powers of the exponential response in the time domain. If the real part is negative, they exponential responses decay => stability. Also, the Fourier transform will exist and the imaginary axis lies in the ROC
s-plane Im x x -2 -1 Re
s=jw
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229. LTI Differential Equation Systems
Physical and electrical systems are causal
Most physical and electrical systems dissipate energy, they are stable. The natural state is “at rest” unless some input/excitation signal is applied to the system
When performing analogue (continuous time) system design, the aim is to produce a time-domain “differential equation” which can then be translated to a known system (electrical circuit …)
This is often done in the frequency domain, which may/may not produce a causal, stable, time-domain differential equation.
Example: low pass filter w
H(jw) wc
-wc
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230. Structures of Sub-Systems
How to combine transfer functions H1(s) and H2(s) to get input output transfer function Y(s) = H(s)X(s)?
Series/cascade
Design H2() to cancel out the effects of H1()
Feedback
Design H2() to regulate y(t) to x(t), so H()=1
System 1
System 2 x y
System 1 x y + -
System 2
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231. Series Cascade & Feedback Proofs
Proof of Series Cascade transfer function
Proof of Feedback transfer function x w y
H1(s)
H2(s) y x +
H1(s) - w
H2(s)
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232. Example: Cascaded 1st Order Systems
Consider two cascaded LTI first order systems
The result of cascading two first order systems is a second order system. However, the roots of this quadratic are purely real (assuming a and b are real), so the output is not oscillatory, as would be the case with complex roots. x w y
H1(s)
H2(s)
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233. Example: Feedback Control
The idea of feedback is central for control (next semester)
The aim is to design the controller C(s), such that the closed loop response, Y(s), has particular characteristics
The plant P(s) is the physical/electrical system (transfer function of differential equation) that must be controlled by the signal u(t)
The aim is to regulate the plant’s response y(t) so that it follows the demand signal x(t)
The error e(t)=x(t)-y(t) gives an idea of the tracking performance
Real-world example
Control an aircraft’s ailerons so that it follows a particular trajectory
C(s)
x(t)
P(s)
e(t)
u(t)
y(t) + -
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234. Example Continued … High Gain Feedback
Simple control scheme (high gain feedback),
C(s)=k>>0
u(t) = ke(t)
For this controller, the system’s response
as desired, when k is extremely large
The controller can be an operational amplifier
While this is a simple controller, it can have some disadvantages.
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235. Lecture 17: Summary
System properties such as stability, causality, … can be interpreted in terms of the time domain (lecture 3), impulse response (lecture 6) or transfer function (this lecture).
For system causality the ROC must be a right-half plane
For system stability, the ROC must include the jw axis
For causal stability, the ROC must include Re{s}>-e
We can use the block transfer notation to calculate the transfer functions of serial, parallel and feedback systems.
Often the aim is to design a sub-system so that the overall transfer function has particular properties
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236. Lecture 18: Discrete-Time Transfer Functions
7 Transfer Function of a Discrete-Time Systems (2 lectures): Impulse sampler, Laplace transform of impulse sequence, z transform. Properties of the z transform. Examples. Difference equations and differential equations. Digital filters.
Specific objectives for today:
z-transform of an impulse response
z-transform of a signal
Examples of the z-transform
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237. Reminder: Laplace Transform
The continuous time Laplace transform is important for two reasons:
It can be considered as a Fourier transform when the signals had infinite energy
It decomposes a signal x(t) in terms of its basis functions est, which are only altered by magnitude/phase when passed through a LTI system.
Points to note:
There is an associated Region of Convergence
Very useful due to definition of system transfer function H(s) and performing convolution via multiplication Y(s)=H(s)X(s)
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238. Discrete Time EigenFunctions
Consider a discrete-time input sequence (z is a complex number):
x[n] = zn
Then using discrete-time convolution for an LTI system:
But this is just the input signal multiplied by H(z), the z-transform of the impulse response, which is a complex function of z.
zn is an eigenfunction of a DT LTI system
Z-transform of the impulse response
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239. z-Transform of a Discrete-Time Signal
The z-transform of a discrete time signal is defined as:
This is analogous to the CT Laplace Transform, and is denoted:
To understand this relationship, put z in polar coords, i.e. z=rejw
Therefore, this is just equivalent to the scaled DT Fourier Series:
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240. Geometric Interpretation & Convergence
The relationship between the z-transform and Fourier transform for DT signals, closely parallels the discussion for CT signals
The z-transform reduces to the DT Fourier transform when the magnitude is unity r=1 (rather than Re{s}=0 or purely imaginary for the CT Fourier transform)
For the z-transform convergence, we require that the Fourier transform of x[n]r-n converges. This will generally converge for some values of r and not for others.
In general, the z-transform of a sequence has an associated range of values of z for which X(z) converges.
This is referred to as the Region of Convergence (ROC). If it includes the unit circle, the DT Fourier transform also converges.
Re(z)
Im(z) 1 w
z-plane r
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241. Example 1: z-Transform of Power Signal
Consider the signal x[n] = anu[n]
Then the z-transform is:
For convergence of X(z), we require
The region of convergence (ROC) is
and the Laplace transform is:
When x[n] is the unit step sequence a=1
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242. Example 1: Region of Convergence
The z-transform is a rational function so it can be characterized by its zeros (numerator polynomial roots) and its poles (denominator polynomial roots)
For this example there is one zero at z=0, and one pole at z=a.
The pole-zero and ROC plot is shown here
For |a|>1, the ROC does not include the unit circle, for those values of a, the discrete time Fourier transform of anu[n] does not converge.
Re(z)
Im(z) 1
Unit
circle a x
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243. Example 2: z-Transform of Power Signal
Now consider the signal x[n] = -anu[-n-1]
Then the Laplace transform is:
If |a-1z|<1, or equivalently, |z|<|a|, this sum converges to:
The pole-zero plot and ROC is shown right for 0<a<1
Re(z)
Im(z) 1
Unit
circle a x
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244. Example 3: Sum of Two Exponentials
Consider the input signal
The z-transform is then:
For the region of convergence we require both summations to converge |z|>1/3 and |z|>1/2, so
|z|>1/2
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245. Lecture 18: Summary
The z-transform can be used to represent discrete-time signals for which the discrete-time Fourier transform does not converge
It is given by:
where z is a complex number. The aim is to represent a discrete time signal in terms of the basis functions (zn) which are subject to a magnitude and phase shift when processed by a discrete time system.
The z-transform has an associated region of convergence for z, which is determined by when the infinite sum converges.
Often X(z) is evaluated using an infinite sum.
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246. Lecture 19: Discrete-Time Transfer Functions
7 Transfer Function of a Discrete-Time Systems (2 lectures): Impulse sampler, Laplace transform of impulse sequence, z transform. Properties of the z transform. Examples. Difference equations and differential equations. Digital filters.
Specific objectives for today:
Properties of the z-transform
z-transform transform equations
Solving difference equations
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247. Introduction to Discrete Time Transfer Fns
A discrete-time LTI system can be represented as a (first order) difference equation of the form:
This is analogous to a sampled CT differential equation
This is hard to solve analytically, and we’d like to be able to perform some form of analogous manipulation like continuous time transfer functions, i.e.
Y(s) = H(s)X(s)
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248. Linearity of the z-TransformIf
and
Then
This follows directly from the definition of the z-transform (as the summation operator is linear, see Example 3). It is easily extended to a linear combination of an arbitrary number of signals
ROC=R1
ROC=R2
ROC= R1R2
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249. Time Shifting & z-Transforms
Then
Proof
This is very important for producing the z-transform transfer function of a difference equation which uses the property:
ROC=R
ROC=R
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250. Example: Linear & Time Shift
Consider the input signal
We know that So
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251. Discrete Time Transfer Function
Consider a first order, LTI differential equation such as:
Then the discrete time transfer function is the z-transform of the impulse response, H(z)
As Z{d[n]} = 1, taking the z-transform of both sides of the equation (linearity we get), for the impulse response
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252. Discrete Time Transfer Function
The discrete-time transfer function of an LTI system is a rational polynomial in z. (This is equivalent to the transfer function of a continuous time differential system which is a rational polynomial in s)
As usual the z-transform transfer function can computed by either:
If the difference equation is known, take the z-transform of each sides when the input signal is an impulse d[n]
If the discrete-time impulse response signal is known, calculate the z-transform of the signal h[n].
In either case, the same result will be obtained.
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253. Convolution using z-Transforms
The z-transform also has the multiplication property, i.e.
Proof is “identical” to the Fourier/Laplace transform convolution and follows from eigensystem property
Note that pole-zero cancellation may occur between H(z) and X(z) which extends the ROC
While this is true for any two signals, it is particularly important as H(z) represents the transfer function of discrete-time LTI system
ROC=R1
ROC=R2
ROCR1R2
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254. Example 1: First Order Difference Equation
Calculate the output of a first order difference equation of a input signal x[n] = 0.5nu[n]
System transfer function (z-transform of the impulse response)
The (z-transform of the) output is therefore:
ROC |z|>0.8
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255. Example 2: 2nd Order Difference Equation
Consider the discrete time step input signal
to the 2nd order difference equation
To calculate the solution, multiply and express as partial fractions
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256. Lecture 19: Summary The z-transform is linear
There is a simple relationship for a signal time-shift
This is fundamental for deriving the transfer function of a difference equation which is expressed in terms of the input-output signal delays
The transfer function of a discrete time LTI system is the z-transform of the system’s impulse response
It is a rational polynomial in the complex number z.
Convolution is expressed as multiplication
and this can be solved for particular signals and systems
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