“ Signals and Systems” - Schaum’s Outlines

“ Signals and Systems” - Schaum’s Outlines
H P Hsu, Rakesh Ranjan

Introduction Chapter 1

Signals A signal is a function of time, e.g., notation:
f is the force on some mass vout is the output voltage of some circuit p is the acoustic pressure at some point notation: f, vout, p or f(.), vout(.), p(.) refer to the whole signal or function f(t), vout(1.2), p(t + 2) refer to the value of the signals at times t, 1.2, and t + 2, respectively for times we usually use symbols like t, t , t1, . . .

Signal Example

Real Signals AM radio signal FM radio signal cable TV signal
audio signal NTSC video signal 10BT Ethernet signal telephone signal

System a system transforms input signals into output signals
a system is a function mapping input signals into output signals we concentrate on systems with one input and one output signal, i.e., single-input, single-output (SISO) systems notation: y = S(u) means the system S acts on input signal u to produce output signal y

Block System systems often denoted by block diagram
boxes denote systems; arrows show inputs & outputs lines with arrows denote signals (not wires) special symbols for some systems

System Example

Signals and Systems Modeling the physical world
Physical system (e.g., LRC circuit) – using mathematical equation Input/output signal – using mathematical function

Signals and Systems Example: LRC
LRC represented by a mathematical Equation ordinary diff. eqn. No sampling (continuous time system) V(i) is a mathematical function

Signals and Systems - Examples
Different systems can be MODELED using the same mathematical function

Signals and Systems - Examples
Human speech production system — anatomy and block diagram

Signals and System Categorizations
Continuous time (analog) Discrete time (digital)

Systems Described in Differential Equations
Many systems are described by a linear constant coefficient ordinary differential equation (LCCODE)

Second Order Continuous System
Second-order RC circuit Closed loop system Find the mathematical relationship in terms of input & output Remember: v1-y = iR2 v1=iR2+y and i(t) =C dv/dt Substitute: The 2nd order diff eqn can be solved using characteristic equation or auxiliary equation

Continuous System Example
A digital player/recorder Processor Analog/Digital Converter Digital/Analog Converter Reconstructed Digital Signal Sampling Signal Digital Output Analog Input

Sample Matlab Code To Generate Signal on the Soundcard!
%%%%%%% % The following program will send a 500 Hz sine wave to analog % output channel 1 for one second. %%Open the analog device and channels AO = analogoutput('winsound',0); chan = addchannel(AO,1); %% Set the sample rate and how long we will send data for %% 44,100 Hz, 1 seconds of data duration = 1; %in seconds frequency = 500 %in Hz SampleRate = 44100; set(AO,'SampleRate',SampleRate) set(AO,'TriggerType','Manual') NumSamples = SampleRate*duration; %% Create a signal that we would like to send, 500 Hz sin wave x = linspace(0,2*pi*frequency,NumSamples); y = tan(sin(1*x))' - sin(tan(1*x))'; %y = sin(x)'; %data = y data = awgn(y,10,'measured'); % wite noise %% Put the data in the buffer, start the device, and trigger putdata(AO,data) start(AO) trigger(AO) %% clean up, close down waittilstop(AO,5) delete(AO) clear AO %% Now let's plot the function for 5 cycles x = 0:.1:2*pi*5; data = tan(sin(x)) - sin(tan(x)); plot(x,data) %% Now let's add random noise %y = awgn(data,10,'measured'); % Add white Gaussian noise. y = sin(x)'; plot(x,data,x,y) % Plot both signals. legend('Original signal','Signal with AWGN');

Chapter 2 Updated 7/1/2018

Time reversal: X(t) Y=X(-t) Time Reversal
Example: Playing a tape recorder backward Beatle Revolution 9 is said be played backward!

Time scaling Example: Given x(t), find y(t) = x(2t). This SPEEDS UP x(t) (the graph is shrinking) The period decreases! X(t) Y=X(at) Time Scaling Example: fast forwarding / slow playing What happens to the period T? The period of x(t) is 2 and the period of y(t) is 1, a>1  Speeds up  Smaller period  Graph shrinks! a<1  slows down  Larger period  Graph expands

Time scaling Given y(t), find w(t) = y(3t) v(t) = y(t/3).

Time Shift: y(t)=x(t-to)
Time Shifting X(t) Y=X(t-to) Time Shifting The original signal x(t) is shifted by an amount t0 . Time Shift: y(t)=x(t-to) X(t)X(t-to) // to>0 Signal Delayed Shift to the right X(t)X(t+to) // to<0 Signal Advanced Shift to the left Example Delay – at the air port / radio stations / LRC circuits Time advancing is not physically realizble!

Time Shifting Example Given x(t) = u(t+2) -u(t-2), find x(t-t0)=
Answer: x(t-t0)= u(t-to+2) -u(t-to-2), x(t+t0)= u(t+to+2) -u(t+to-2),

Note: Unit Step Function
Step unit function (a discontinuous continuous-time signal):

Draw x(t) = u(t+1)- u(t-2) t=0 u(t+1)- u(t-2)

Time Shifting Determine x(t) + x(2-t) , where x(t) = u(t+1)- u(t-2)
Which is x(t): find x(2-t): Advance, then reverse in time. t=0 u(t+1)- u(t-2) Add the two functions: x(t) + x(2-t)

Summary See This is really: Notes X(-(t+1)) Delayed/ Moved right
shift to the left of t=0 by two units! Shifting to the right; increasing in time  Delaying the signal! Delayed/ Moved right Advanced/ Moved left Reversed & Delayed Or rewrite as: X[-(t+1)] Hence, reverse the signal in time. Then shift to the left of t=0 by one unit! Or rewrite as: X[-(t-2)] Hence, reverse the signal in time. Then shift to the right of t=0 by two units! See Notes This is really: X(-(t+1))

Amplitude Operations In general: y(t)=Ax(t)+B B>0  Shift up
B<0  Shift down |A|>1 Gain |A|<1 Attenuation A>0NO reversal A<0 reversal Reversal Scaling Scaling

Signals can be added or multiplied
Amplitude Operations Given x2(t), find 1 - x2(t). Ans. Remember: This is y(t) =1 Multiplication of two signals:x2(t)u(t) Ans. Step unit function Signals can be added or multiplied

Amplitude Operations Given x2(t), find 1 - x2(t).
Note: You can also think of it as X2(t) being amplitude revered and then shifted by 1. Given x2(t), find 1 - x2(t). Remember: This is y(t) =1 Multiplication of two signals:x2(t)u(t) Step unit function Signals can be added or multiplied  e.g., we can filter parts of a signal!

Signal Characteristics
Even and odd signals X(t) = Xe(t) + Xo(t) X(-t) = X(t)  Even X(-t) = -X(t)  Odd Properties Represent Xo(t) in terms of X(t) only! Represent Xe(t) in terms of X(t) only! Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo

Even and odd signals X(t) = Xe(t) + Xo(t) X(-t) = X(t)  Even X(-t) = -X(t)  Odd Properties Xe * Ye = Ze Xo * Yo = Ze Xe * Yo = Zo Xe + Ye = Ze Xo + Yo = Zo Xe + Yo = Ze + Zo Know These!

Proof Examples Prove that product of two even signals is even.
Prove that product of two odd signals is odd. What is the product of an even signal and an odd signal? Prove it! Change t -t

Given:

Anti-symmetric across the vertical axis Symmetric across the vertical axis

Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 2___ 5 5

Example Given x(t) find xe(t) and xo(t) 4___ 5 2___ 2___ -5 5 -5 5

Example Given x(t) find xe(t) and xo(t) 4___ 4e-0.5t 2___ 2___ 2___
-2___

Periodic and Aperiodic Signals
Given x(t) is a continuous-time signal X (t) is periodic iff X(t) = x(t+nT) for any T and any integer n Example Is X(t) = A cos(wt) periodic? X(t+nT) = A cos(w(t+Tn)) = A cos(wt+w2np)= A cos(wt) Note: f0=1/T0; wo=2p/To T0 is fundamental period; T0 is the minimum value of T that satisfies X(t) = x(t+T)

Periodic signals Example: Is tesin(t) periodic?
Show that sin(t) is in fact a periodic signal. Use a graph Show it mathematically What is the period? Is this an even or odd signal? Is tesin(t) periodic? X(t) = x(t+T)?

Sum of periodic Signals
X(t) = x1(t) + X2(t) X(t+T) = x1(t+m1T1) + X2(t+m2T2) m1T1=m2T2 = To = Fundamental period Example: cos(tp/3)+sin(tp/4) T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8; T1/T2=6/8 = ¾ = (rational number) = m2/m1 m1T1=m2T2  Find m1 and m2 6.4 = 3.8 = 24 = To

X(t) = x1(t) + X2(t) X(t+T) = x1(t+m1T1) + X2(t+m2T2) m1T1=m2T2 = To=Fundamental period Example: cos(tp/3)+sin(tp/4) T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8; T1/T2=6/8 = ¾ m1T1=m2T2 = 6.4 = 3.8 = 24 = To Note that T1/T2 must be RATIONAL (ratio of integers) An irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero. Read about Irrational Numbers:

Product of periodic Signals
X(t) = xa(t) * Xb(t) = X(t) = 2sin[t/2(7p/12)]* cos[t/2(p/12)]; find the period of x(t) Using sin(A+B)=2sin(t/2(A+B))cos(t/2(A-B)) x(t) = sin(tp/3)+sin(tp/4) Thus, To=24 , as before!

Sum of periodic Signals – may not always be periodic!
T1=(2p)/(1)= 2p; T2 =(2p)/(sqrt(2)); T1/T2= sqrt(2); Note: T1/T2 = sqrt(2) is an irrational number X(t) is aperiodic Note that T1/T2 is NOT RATIONAL (ratio of integers) In this case!

Example X1(t) = cos(3.5t) X2(t) = sin(2t) X3(t) = 2cos(t7/6) Is v(t) = x1(t) + x2(t) + x3(t) periodic? What is the fundamental period of v(t)? Find even and odd parts of v(t). notes

Sum of periodic Signals (cont.)
X1(t) = cos(3.5t)  f1 = 3.5/2p  T1 = 2p /3.5 X2(t) = sin(2t)  f2 = 2/2p  T2 = 2p /2 X3(t) = 2cos(t7/6)  f3 = (7/6)/2p  T3 = 2p /(7/6)  T1/T2 = 4/7 Ratio or two integers  T1/T3 = 1/3 Ratio or two integers  Summation is periodic m1T1 = m2T2 = m3T3 = To ; Hence we find To The question is how to choose m1, m2, m3 such that the above relationship holds We know: 7(T1) = 4(T2) & 3(T1) = 1(T3) ;  m1(T1)=m2(T2) Hence: 21(T1) = 12(T2)= 7(T3); Thus, fundamental period: To = 21(T1) = 21(2p /3.5)=12(T2)=12p Find even and odd parts of v(t).

Important Engineering Signals
Euler’s Formula (polar form or complex exponential form) Remember: ejF = 1|_F and arg [ ejF ] = F (can you prove these?) Unit Step Function (Singularity Function) Use Unit Step Function to express a block function (window) notes What is sin(A+B)? Or cos(A+B)? notes notes -T/t T/t

Important Engineering Signals
Euler’s Formula Unit Step Function (Singularity Function) Can you draw x(t) = cos(t)[u(t) – u(t-2p)]? Use Unit Step Function to express a block function (window) notes notes Next -T/2 T/2

Unit Step Function Properties
Examples: Note: U(-t+3)=1-u(t-3)

More on Unit Step Functions u(t)
Unit ramp function can be achieved by: 1 notes to to+1 Non-zero only for t>t0

Example

Example Plot t<-2  f(t)=0 -2<t<-1  f(t)=3[t+2]

Block Function (window)
rect(t/T) Can be expressed as u(T/2-t)-u(-T/2-t) Draw u(t+T/2) first; then reverse it! Can be expressed as u(t+T/2)-u(t-T/2) Can be expressed as u(t+T/2).u(T/2-t) 1 -T/2 T/2 1 -T/2 T/2 1 -T/2 T/2 -T/2 T/2

Unit Impulse Function d(t)
Not real (does not exist in nature – similar to i=sqrt(-1) Also known as Dirac delta function Generalized function or testing function The Dirac delta can be loosely thought of as a function of the real line which is zero everywhere except at the origin, where it is infinite Note that impulse function is not a true function – it is not defined for all values It is a generalized function = f(0) Mathematical definition Mathematical definition d(t) d(t-to) to

Unit Impulse Function d(t)
Also note that Also

Unit Impulse Properties
Scaling Kd(t) Area (or weight) under = K Multiplication X(t) d(t) X(0) d(t) = Area (or weight) under Time Shift X(t) d(t-to) X(to) d(t-to) Draw 3x(t-1) d(t-3/2) where x(t)=sin(t) Using X(t) d(t-to) X(to) d(t-to) ; 3x(3/2-1) d(t-3/2)=3sin(1/2) d(t-3/2)

Integration of a test function Example Other properties: Sin^2(a/b) Check out how to prove these!

Example: Verify Evaluate the following notes notes Remember:

Continuous-Time Systems
A system is an operation for which cause-and-effect relationship exists Can be described by block diagrams Denoted using transformation T[.] System behavior described by mathematical model X(t) y(t) T [.]

System - Example Consider an RL series circuit
Using a first order equation: R L V(t)

Interconnected Systems
Parallel Serial (cascaded) Feedback notes R R L L V(t)

Interconnected System Example
Consider the following systems with 4 subsystem Each subsystem transforms it input signal The result will be: y3(t)=y1(t)+y2(t)=T1[x(t)]+T2[x(t)] y4(t)=T3[y3(t)]= T3(T1[x(t)]+T2[x(t)]) y(t)= y4(t)* y5(t)= T3(T1[x(t)]+T2[x(t)])* T4[x(t)]

Feedback System Used in automatic control
Example: The following system has 3 subsystems. Express the equation denoting interconnection for this system - (mathematical model will depend on each individual subsystem) e(t)=x(t)-y3(t)= x(t)-T3[y(t)]= y(t)= T2[m(t)]=T2(T1[e(t)])  y(t)=T2(T1[x(t)-y3(t)])= T2(T1( [x(t)] - T3[y(t)] ) ) = =T2(T1([x(t)] –T3[y(t)])) Find this first Then, calculate this

System Properties

Continuous-Time Systems - Properties
System with Memory (Dynamic) Examples – Memoryless (static)? Has memory if output depends on inputs other than the one defined at current time

Invertible Examples Noninvertible Systems Thermostat Example! (notes) Each distinct input  distinct output

If a system is invertible it has an Inverse System Example: y(t)=2x(t) System is invertible must have inverse, that is: For any x(t) we get a distinct output y(t) Thus, the system must have an Inverse x(t)=1/2 y(t)=z(t) System Inverse System x(t) y(t) x(t) System (multiplier) Inverse System (divider) x(t) y(t)=2x(t) x(t) If the system is not invertible it does not have an INVERSE!

Continuous-Time Systems - Causality
Causality (non-anticipatory system) - A System can be causal with non-causal components! Examples Remember: Reverse is not TRUE! ?? ?? What it t<0? Depends on cause-and-effect  no future dependency

Continuous-Time Systems - Stability
Stability has Many different definitions Bounded-input-bounded-output Example: An ideal amplifier y(t) = 10 x(t)  B2=10 B1 Square system: y(t)=x^2  B2=B1^2 A system can be unstable or marginally stable

Continuous-Time Systems – Time Invariance
Testing for Time Invariance notes To test: y(t)|t-to  y(t)|x(t-to) Example: y(t) = e^x(t): y(t)|t-to  e^x(t-to) y(t)|x(t-to)  e^x(t-to)  Time Invariant What if the system is time reversal? (next slide) Time-shift in input results in time-shift in output  system always acts the same way (Fix System)

Example of a system: U(1-t) Draw the First output! Draw the Second output! Are they the same?

Continuous-Time Systems – Time Reversal & Time Invariance
Example of a system: U(1-t) Pay attention! Due to time - reversal Time reversal operation is NOT time invariant!

Examples: Do these yourself!

Continuous-Time Systems – Linearity
A linear system must satisfy superposition condition (additive and homogeneity hoh-muh-juh-nee-i-tee) Note: for a linear system a zero input always generates an output zero Example notes

Chapter 3 CT LTI Systems

A Continuous-Time System
How do we know the output? X(t) y(t) System

What is the input-output relationship for LTI-CT Systems?
LTI Systems Time Invariant X(t) y(t) & x(t-to)  y(t-to) Linearity a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t) a1y1(t)+ a2y2(t)= T[a1x1(t)+a2x2(t)] Meet the description of many physical systems They can be modeled systematically Non-LTI systems typically have no general mathematical procedure to obtain solution What is the input-output relationship for LTI-CT Systems?

Convolution Integral An approach (available tool or operation) to describe the input-output relationship for LTI Systems In a LTI system d(t) h(t) Remember h(t) is T[d(t)] Unit impulse function  the impulse response It is possible to use h(t) to solve for any input-output relationship One way to do it is by using the Convolution Integral X(t)=d(t) y(t)=h(t) LTI System X(t) y(t) LTI System: h(t)

? Convolution Integral Remember So what is the general solution for
X(t)=Ad(t-kto) y(t)=Ah(t-kto) LTI System X(t) y(t) LTI System ?

tot and integrate by dt
Convolution Integral Any input can be expressed using the unit impulse function Proof: Sifting Property tot and integrate by dt X(t) y(t) LTI System

Convolution Integral Given We obtain Convolution Integral
That is: A system can be characterized using its impulse response: y(t)=x(t)*h(t) X(t) y(t) LTI System X(t) y(t) LTI System: h(t) By definition Do not confuse convolution with multiplication! y(t)=x(t)*h(t)

Convolution Integral X(t) y(t) LTI System: h(t)

Convolution Integral - Properties
Commutative Associative Distributive Thus, using commutative property: Draw the block diagram representation!

Convolution Integral - Properties
Commutative Associative Distributive

Simple Example What if a step unit function is the input of a LTI system? S(t) is called the Step Response u(t) y(t)=s(t) LTI System Step response can be obtained by integrating the impulse response! Impulse response can be obtained by differentiating the step response

Example 1 Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^(-at) for all a>0 and t>0 and input x(t)=u(t). Find the output. u(t) y(t) h(t)=e^-at Because t>0 Draw x(t), h(t), h(t-t),etc.  next slide The fact that a>0 is not an issue!

* Example – Cont. y(t) U(-(t-t)) U(-(t-t)) t>0 t<0
Remember we are plotting it over t and t is the variable

Draw x(t), h(t), h(t-t),etc.  next slide
Example 2 Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^-at for all a>0 and t>0 and input x(t)= e^at u(-t). Find the output. x(t) y(t) h(t)=e^-at Note that for t>0; x(t) =0; so the integration can only be valid up to t=0 Draw x(t), h(t), h(t-t),etc.  next slide

Example – Cont. * x(t)= e^at u(-t) h(t)=e^-at u(t) ?

Another Example notes

Properties of CT LTI Systems
When is a CT LTI system memory-less (static) When does a CT LTI system have an inverse system (invertible)? When is a CT LTI system considered to be causal? Assuming the input is causal: When is a CT LTI system considered to be Stable? notes

Example Is this an stable system? What about this? notes

Differential-Equations Models
This is a linear first order differential equation with constant coefficients (assuming a and b are constants) The general nth order linear DE with constant equations is

Is the First-Order DE Linear?
Consider Does a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)? Is it time-invariant? Does input delay results in an output delay by the same amount? Is this a linear system? notes notes + e(t) Y(t) Sum Integrator X(t) - a

Example Is this a time invariant linear system? R L V(t)
Ldi(t)/dt + Ri(t)= v(t) a= -R/L b=1/L y(t)=i(t) x(t) = V(t)

Solution of DE A classical model for the solution of DE is called method of undermined coefficients yc(t) is called the complementary or natural solution yp(t) is called the particular or forced solution

Solution of DE Thus, for x(t) =constant  yp(t)=P
x(t) =Ce^-7t  yp(t)= Pe^-7t x(t) =2cos(3t)  yp(t)=P1cos(3t)+P2sin(3t)

Example Solve Assume x(t) = 2 and y(0) = 4 What happens if notes
yc(t) = Ce^-2t; yp(t) = P; P = 1 y(t) = Ce^-2t + 1 y(0) = 4  C = 3  y(t) = 3e^-2t + 1

Fourier Series & Transforms
Chapter 4 Fourier Series & Transforms

Taylor Series Complex signals are often broken into simple pieces
Signal requirements Can be expressed into simpler problems The first few terms can approximate the signal Example: The Taylor series of a real or complex function ƒ(x) is the power series

Square Wave S(t)=sin(2pft) S(t)=1/3[sin(2p(3f)t)]
S(t)= 4/p{sin(2pft) +1/3[sin(2p(3f)t)]} Fourier Expansion

Square Wave K=1,3,5 K=1,3,5, 7 Frequency Components of Square Wave
Fourier Expansion K=1,3,5, 7, 9, …..

Periodic Signals A Periodic signal/function can be approximated by a sum (possibly infinite) sinusoidal signals. Consider a periodic signal with period T A periodic signal can be Real or Complex The fundamental frequency: wo Example:

Fourier Series We can represent all periodic signals as harmonic series of the form Ck are the Fourier Series Coefficients; k is real k=0 gives the DC signal k=+/-1 indicates the fundamental frequency or the first harmonic w0 |k|>=2 harmonics

Fourier Series Coefficients
Fourier Series Pair We have For k=0, we can obtain the DC value which is the average value of x(t) over one period Series of complex numbers Defined over a period of x(t)

Euler’s Relationship Review Euler formulas notes

Examples Find Fourier Series Coefficients for C1=1/2; C-1=1/2; No DC
C1=1/2j; C-1=-1/2j; No DC notes

Different Forms of Fourier Series
Fourier Series Representation has three different forms Also: Complex Exp. Also: Harmonic Which one is this? What is the DC component? What is the expression for Fourier Series Coefficients

Examples Find Fourier Series Coefficients for Remember:

Examples notes textbook
Find the Complex Exponential Fourier Series Coefficients notes textbook

Example Find the average power of x(t) using Complex Exponential Fourier Series – assuming x(t) is periodic This is called the Parseval’s Identity

Example Consider the following periodic square wave
Express x(t) as a piecewise function Find the Exponential Fourier Series of representations of x(t) Find the Combined Trigonometric Fourier Series of representations of x(t) Plot Ck as a function of k X(t) V To/2 To -V Use a Low Pass Filter to pick any tone you want!! 2|Ck| |4V/p| |4V/3p| |4V/5p| notes w0 3w0 5w0

Practical Application
Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?

Practical Application
Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies? Square Signal @ wo Level Shifter [kwo] Sinusoidal waveform X(t) 1 To/2 @ [kwo] To X(t) 0.5 To/2 To -0.5 kwo B changes depending on k value

Demo Ck corresponds to frequency components In the signal.

Example Only a function of freq.
Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. 1 Note: sinc (infinity)  1 & Max value of sinc(x)1/x Sinc Function Note: First zero occurs at Sinc (+/-pi) Only a function of freq.

Use the Fourier Series Table (Table 4.3)
Consider the following periodic square wave Find the Exponential Fourier Series of representations of x(t) X0V X(t) V To/2 To -V 2|Ck| |4V/p| |4V/3p| |4V/5p| w0 3w0 5w0

Fourier Series - Applet

Using Fourier Series Table
Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular wave) X01 C0=T/To T/2=T1T=2T1 Ck=T/T0 sinc (Tkw0/2) Same as before Note: sinc (infinity)  1 & Max value of sinc(x)1/x

Using Fourier Series Table
Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To

Fourier Series Transformation
Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To From the table: Xo/2 -Xo/2 To

Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To From the table: Xo/2 -Xo/2 To Only DC value changed!

Express the Fourier Series for a sawtooth waveform? Express the Fourier Series for this sawtooth waveform? Xo To From the table: Xo 1 To -3

Express the Fourier Series for a sawtooth waveform? Express the Fourier Series for this sawtooth waveform? We are using amplitude transfer Remember Ax(t) + B Amplitude reversal A<0 Amplitude scaling |A|=4/Xo Amplitude shifting B=1 Xo To From the table: Xo 1 To -3

Example

Fourier Series and Frequency Spectra
We can plot the frequency spectrum or line spectrum of a signal In Fourier Series k represent harmonics Frequency spectrum is a graph that shows the amplitudes and/or phases of the Fourier Series coefficients Ck. Amplitude spectrum |Ck| Phase spectrum fk The lines |Ck| are called line spectra because we indicate the values by lines

Chapter 5 The Fourier Transform

Basic Idea We covered the Fourier Transform which to represent periodic signals We assumed periodic continuous signals We used Fourier Series to represent periodic continuous time signals in terms of their harmonic frequency components (Ck). We want to extend this discussion to find the frequency spectra of a given signal

Basic Idea The Fourier Transform is a method for representing signals and systems in the frequency domain We start by assuming the period of the signal is T= INF All physically realizable signals have Fourier Transform For aperiodic signals Fourier Transform pairs is described as Fourier Transforms of f(t) Inverse Fourier Transforms of F(w) Remember: notes

Example – Rectangular Signal
Compute the Fourier Transform of an aperiodic rectangular pulse of T seconds evenly distributed about t=0. Remember this the same rectangular signal as we worked before but with T0 infinity! V -T/2 T/2 notes All physically realizable signals have Fourier Transforms

Fourier Transform of Unit Impulse Function
Example: Plot magnitude and phase of f(t)

Fourier Series Properties
Make sure how to use these properties!

Fourier Series Properties - Linearity
Find F(w)

Fourier Series Properties - Linearity
Due to linearity

Fourier Series Properties - Time Scaling
rect(t/T) rect(t/(T/2)) Due to Time Scaling Property Remember: sinc(0)=1; sinc(2pi)=0=sinc(pi)

Fourier Series Properties - Duality or Symmetry
Example: Find the time-domain waveform for Arect(w/2B) Refer to FTP Table Remember we had: FTP: Fourier Transfer Pair

Example: find the frequency response Of y(t)

Example: find the frequency response Of y(t) We know Using Fourier Transform Pairs Using duality

Fourier Series Properties - Convolution
Proof Proof

Example: Find the Fourier Transform of x(t)=sinc2(t) In this case we have B=1, A=1 w w X1(w) X2(w) Refer to Schaum’s Prob. 2.6

Example: Find the Fourier Transform of x(t)= sinc2(t) sinc(t) We need to find the convolution of a rect and a triangle function: w Refer to Schaum’s Prob. 2.6

Fourier Series Properties - Frequency Shifting
Example: Find the Fourier Transform of g3(t) if g1(t)=2cos(200pt), g2(t)=2cos(1000pt); g3(t)=g1(t).g2(t) ; that is [G3(w)] Remember: cosa . cosb=1/2[cos(a+b)+cos(a-b)]

Fourier Series Properties - Time Differentiation
Example:

Chapter 7 Laplace Transforms

Applications of Laplace Transform
Easier than solving differential equations Used to describe system behavior We assume LTI systems Uses S-domain instead of frequency domain Applications of Laplace Transforms/ Circuit analysis Provides the general solution to any arbitrary wave (not just LRC) Transient Sinusoidal steady-state-response (Phasors) Signal processing Communications Definitely useful for Interviews! notes

Building the Case…

Laplace Transform

Laplace Transform We use the following notations for Laplace Transform pairs – Refer to the table!

Laplace Transform Convergence
The Laplace transform does not converge to a finite value for all signals and all values of s The values of s for which Laplace transform converges is called the Region Of Convergence (ROC) Always include ROC in your solution! Example: 0+ indicates greater than zero values Remember: e^jw is sinusoidal; Thus, only the real part is important!

Example of Bilateral Version
Find F(s): ROC S-plane Re(s)<a a Find F(s): Remember These! Note that Laplace can also be found for periodic functions

Example – RCO may not always exist!
Note that there is no common ROC  Laplace Transform can not be applied!

Example – Unilateral Version
Find F(s): Find F(s):

Example

Properties The Laplace Transform has many difference properties
Refer to the table for these properties

Linearity

Scaling & Time Translation
Do the time translation first! Time Translation b=0

Shifting and Time Differentiation
Shifting in s-domain Differentiation in t Read the rest of properties on your own!

Examples Note the ROC did not change!

Example – Application of Differentiation
Matlab Code: Read Section 7.4 Read about Symbolic Mathematics: And

Example What is Laplace of t^3? Find the Laplace Transform:
From the table: 3!/s^4 Re(s)>0 Find the Laplace Transform: Time transformation Note that without u(.) there will be no time translation and thus, the result will be different: Assume t>0

A little about Polynomials
Given Laplace find f(t)! Consider a polynomial function: A rational function is the ratio of two polynomials: A rational function can be expressed as partial fractions A rational function can be expressed using polynomials presented in product-of-sums Has roots and zeros; distinct roots, repeated roots, complex roots, etc.

Finding Partial Fraction Expansion
Given a polynomial Find the POS (product-of-sums) for the denominator: Write the partial fraction expression for the polynomial Find the constants If the rational polynomial has distinct poles then we can use the following to find the constants:

Application of Laplace
Consider an RL circuit with R=4, L=1/2. Find i(t) if v(t)=12u(t). Matlab Code Given Partial fraction expression

Application of Laplace
What are the initial [i(0)] and final values: Using initial-value property: Using the final-value property Note that Initial Value: t=0, then, i(t) 3-3=0 Final Value: t INF then, i(t) 3 Note: using Laplace Properties

Using Simulink v(t) H(s) i(t)

Actual Experimentation
Note how the voltage looks like: Output Voltage: Input Voltage:

Partial Fraction Expansion (no repeated Poles/Roots) – Example
Using Matlab: Matlab code: b=[ ]; a=[ ]; [r,p,k]=residue(b,a) We can also use ilaplace (F); but the result may not be simplified!

Finding Poles and Zeros
Express the rational function as the ratio of two polynomials each represented by product-of-sums Example: Pole S-plane zero

H(s) Replacing the Impulse Response
x(t) h(t) y(t) X(s) H(s) Y(s) convolution multiplication

H(s) Replacing the Impulse Response
x(t) h(t) y(t) X(s) H(s) Y(s) convolution multiplication h(t) Example: Find the output X(t)=u(t); h(t) 1 1 e^-sF(s) y(t) 1 1 This is commonly used in D/A converters!

Dealing with Complex Poles
Given a polynomial Find the POS (product-of-sums) for the denominator: Write the partial fraction expression for the polynomial Find the constants The pole will have a real and imaginary part: P=|k|f When we have complex poles {|k|f} then we can use the following expression to find the time domain expression:

Laplace Transform Characteristics
Assumptions: Linear Continuous Time Invariant Systems Causality No future dependency If unilateral: No value for t<0; h(t)=0 Stability System mode: stable or unstable We can tell by finding the system characteristic equation (denominator) Stable if all the poles are on the left plane Bounded-input-bounded-output (BIBO) Invertability H(s).Hi(s)=1 Frequency Response H(w)=H(s);sjw=H(s=jw) We need to add control mechanism to make the overall system stable

Frequency Response – Matlab Code

Inverse Laplace Transform

Example of Inverse Laplace Transform

Bilateral Transforms Laplace Transform of two different signals can be the same, however, their ROC can be different:  Very important to know the ROC. Signals can be Right-sided  Use the bilateral Laplace Transform Table Left-sides Have finite duration How to find the transform of signals that are bilateral! See notes

How to Find Bilateral Transforms
If right-sided use the table for unilateral Laplace Transform Given f(t) left-sided; find F(s): Find the unilateral Laplace transform for f(-t) laplace{f(-t)}; Re(s)>a Then, find F(-s) with Re(-s)>a Given Fb(s) find f(t) left-sided : Find the unilateral Inverse Laplace transform for F(s)=fb(t) The result will be f(t)=–fb(t)u(-t) Example

Examples of Bilateral Laplace Transform
Find the unilateral Laplace transform for f(-t) laplace{f(-t)}; Re(s)>a Then find F(-s) with Re(-s)>a Alternatively: Find the unilateral Laplace transform for f(t)u(-t) (-1)laplace{f(t)}; then, change the inequality for ROC.

Feedback System Find the system function for the following feedback system: F(s) X(t) + Sum e(t) y(t) + r(t) G(s) Equivalent System H(s) X(t) y(t)

Discrete-Time Linear Time-Invariant Systems
Chapter 10 Discrete-Time Linear Time-Invariant Systems

Representation of Discrete-Time Signals
We assume Discrete-Time LTI systems The signal X[n] can be represented using unit sample function or unit impulse function: d[n] Remember: Notations: notes

Representation of Discrete-Time Signals - Example

Convolution for Discrete-Time Systems
LTI system response can be described using: For time-invariant: d[n-k]h[n-k] For a linear system: x[k]d[n-k]x[k]h[n-k] Remember: Thus, for LTI: We call this the convolution sum System d[n] h[n] Impulse Response of a System

Convolution for Discrete-Important Properties
By definition Remember (due to time-invariance property): Multiplication

Properties of Convolution
Commutative Associative Distributive

Example Figure 10.3 Figure 10.3 Given the following block diagram
Find the difference equation Find the impulse response: h[n]; plot h[n] Is this an FIR (finite impulse response) or IIR system? Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n Plot y[n] vs. n using Matlab Difference equation To find h[n] we assume x[n]=d[n], thus y[n]=h[n] Thus: h[0]=h[1]=h[2]=1/3 Since h[n] is finite, the system is FIR In terms of inputs: Figure 10.3 Figure 10.3 FIR system contains finite number of nonzero terms

Try for different values of n
Example – cont. Given the following block diagram Find the difference equation Find the impulse response: h[n]; plot h[n] Is this an FIR (finite impulse response) or IIR system? Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n Plot y[n] vs. n using Matlab In terms of inputs: Calculate for n=0, n=1, n=2, n=3, n=4, n=5, n=6 n=0; y[0]=0 n=1; y[n]=1 n=2; y[2]=2.5 n=3; y[2\3]=4.5 n=4; y[4]=3.5 n=5; y[5]=2 n=6; y[6]=0 Figure 10.3 Try for different values of n

Example – cont. (Graphical Representation)
X[m] X[n-k] h[0]=h[1]=h[2]=1/3 x[1]=3, x[2]=4.5, x[3]=6

Example Consider the following difference equation:y[n]=ay[n-1]+x[n]
Draw the block diagram of this system Find the impulse response: h[n] Is it a causal system? Is this an IIR or FIR system?

Example Consider the following difference equation:y[n]=ay[n-1]+x[n];
Draw the block diagram of this system Find the impulse response: h[n] Is this an IIR or FIR system? We assume x[n]=d[n] y[n]=h[n]=ah[n-1]+d[n]; y[0]=h[0]=1 y[1]=h[1]=a y[2]=h[2]=a^2 y[3]=h[3]= a^3 h[n]=a^n ; n>=0 It is IIR (unbounded) Causal system (current and past)

Example Assume h[n]=0.6^n*u[n] and x[n]=u[n]
Find the expression for y[n] Plot y[n] Plot y[n] using Matlab h[n] x[n] y[n] y[0]=1 y[1]=1.6 ….. y(100)=2.5  Steady State Value is 2.5

Remember These Geometric Series:

Properties of Discrete-Time LTI Systems
Memory: A memoryless system is a pure gain system: iff h[n]=Kd[n]; K=h[0] = constant & h[n]=0 otherwise Causality y[n] has no dependency on future values of x[n]; thus h[n]=0 for n<0 (note h[n] is non-zero only for d[n=0]. Note that if k<0depending on future; Thus h[k] should be zero to remove dependency on the future.

Properties of Discrete-Time LTI Systems
Stability BIBO: |x[n]|< M Absolutely summable: Invertibility: If the input can be determined from output It has an inverse impulse response Invertible if there exists: hi[n]*h[n]=d[n]

Example 1 Assume h[n]= u[n] (1/2)^n Memoryless? Casual system? Stable?
Has memory (dynamic): h[n] is not Kd[n] (not pure gain); h[n] is non-zero Is causal: h[n]=0 for n<0 Stable: h[n] x[n] y[n]

Example 2 Assume h[n]= u[n+1] (1/2)^n Memoryless? Casual system?
Stable? Has memory (dynamic): h[n] is not Kd[n] (not pure gain) Is NOT causal: h[n] not 0 for n<0; h[-1]=2 Stable: h[n] x[n] y[n]

Example 3 Assume h[n]= u[n] (2)^n Memoryless? Casual system? Stable?
Has memory (dynamic): h[n] is not Kd[n] (not pure gain) Is causal: h[n]=0 for n<0 Not Stable: h[n] x[n] y[n]

Fourier Transforms of Discrete Signals
Chapter 12 Fourier Transforms of Discrete Signals

Sampling Continuous signals are digitized using digital computers
When we sample, we calculate the value of the continuous signal at discrete points How fast do we sample What is the value of each point Quantization determines the value of each samples value

Sampling Periodic Functions
- Note that wb = Bandwidth, thus if then aliasing occurs (signal overlaps) To avoid aliasing According sampling theory: To hear music up to 20KHz a CD should sample at the rate of 44.1 KHz

Discrete Time Fourier Transform
In likely we only have access to finite amount of data sequences (after sampling) Recall for continuous time Fourier transform, when the signal is sampled: Assuming Discrete-Time Fourier Transform (DTFT):

Discrete Time Fourier Transform
Discrete-Time Fourier Transform (DTFT): A few points DTFT is periodic in frequency with period of 2p X[n] is a discrete signal DTFT allows us to find the spectrum of the discrete signal as viewed from a window

Example D See map!

Example of Convolution
We can write x[n] (a periodic function) as an infinite sum of the function xo[n] (a non-periodic function) shifted N units at a time This will result Thus See map!

Finding DTFT For periodic signals
Starting with xo[n] DTFT of xo[n] Example

Example A & B notes X[n]=a|n|, 0 < a < 1. notes

DT Fourier Transforms W is in radian and it is between 0 and 2p in each discrete time interval This is different from w where it was between – INF and + INF Note that X(W) is periodic

Properties of DTFT Remember:
For time scaling note that m>1  Signal spreading

Fourier Transform of Periodic Sequences
Check the map~~~~~ See map!

Discrete Fourier Transform
We often do not have an infinite amount of data which is required by DTFT For example in a computer we cannot calculate uncountable infinite (continuum) of frequencies as required by DTFT Thus, we use DTF to look at finite segment of data We only observe the data through a window In this case the xo[n] is just a sampled data between n=0, n=N-1 (N points)

Discrete Fourier Transform
It turns out that DFT can be defined as Note that in this case the points are spaced 2pi/N; thus the resolution of the samples of the frequency spectrum is 2pi/N. We can think of DFT as one period of discrete Fourier series

A short hand notation remember:

Inverse of DFT We can obtain the inverse of DFT Note that

Using MATLAB to Calculate DFT
Example: Assume N=4 x[n]=[1,2,3,4] n=0,…,3 Find X[k]; k=0,…,3 or

Example of DFT Find X[k] We know k=1,.., 7; N=8

Example of DFT

Example of DFT Polar plot for Time shift Property of DFT
Other DFT properties:

Example of DFT

Using the shift property!
Example of DFT Summation for X[k] Using the shift property!

Example of IDFT Remember:

Fast Fourier Transform Algorithms
Consider DTFT Basic idea is to split the sum into 2 subsequences of length N/2 and continue all the way down until you have N/2 subsequences of length 2 Log2(8) N

Radix-2 FFT Algorithms - Two point FFT
We assume N=2^m This is called Radix-2 FFT Algorithms Let’s take a simple example where only two points are given n=0, n=1; N=2 Butterfly FFT y0 y0 y1 Advantage: Less computationally intensive: N/2.log(N)

General FFT Algorithm First break x[n] into even and odd
Let n=2m for even and n=2m+1 for odd Even and odd parts are both DFT of a N/2 point sequence Break up the size N/2 subsequent in half by letting 2mm The first subsequence here is the term x[0], x[4], … The second subsequent is x[2], x[6], …

Example Let’s take a simple example where only two points are given n=0, n=1; N=2 Same result

FFT Algorithms - Four point FFT
First find even and odd parts and then combine them: The general form:

FFT Algorithms - 8 point FFT
Applet:

A Simple Application for FFT
t = 0:0.001:0.6; % 600 points x = sin(2*pi*50*t)+sin(2*pi*120*t); y = x + 2*randn(size(t)); plot(1000*t(1:50),y(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('time (milliseconds)') Taking the 512-point fast Fourier transform (FFT): Y = fft(y,512) The power spectrum, a measurement of the power at various frequencies, is Pyy = Y.* conj(Y) / 512; Graph the first 257 points (the other 255 points are redundant) on a meaningful frequency axis: f = 1000*(0:256)/512; plot(f,Pyy(1:257)) title('Frequency content of y') xlabel('frequency (Hz)') ML Help! This helps us to design an effective filter!

Example Express the DFT of the 9-point {x[0], …,x[9]} in terms of the DFTs of 3-point sequences {x[0],x[3],x[6]}, {x[1],x[4],x[7]}, and {x[2],x[5],x[8]}. Later

“ Signals and Systems” - Schaum’s Outlines

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