1. Chapter 2. Signals and Linear Systems
Essentials of Communication Systems Engineering
John G. Proakis and Masoud Salehi

2. Signals and Linear Systems
Review of the basics of signals and linear systems
The basic role in modeling various types of communication systems
Information-bearing signals
Examples of information-bearing signals
Speech signals, video signals, and the output of an ASCII terminal
Signals are used to transmit information over a communication channel
The shape of the signal is changed, or distorted, by the channel
Received signal
The output of the communication channel
Not an exact replica of the channel input due to many factors, including channel distortion
Communication channel : Example of a system
Entity that produces an output signal when excited by an input signal
A large number of communication channels can be modeled by a subclass of systems called linear systems
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

3. Basic Operations on Signals
x(t) : A signal
Time shifting or delaying
y(t) : Time shifted version of x(t) : , t0 : Time shift
When t0 > 0, x(t) is shifted to the right : time delay
When t0 < 0, x(t) is shifted to the left : time advance
Time reversal, or flipping
y(t) : Signal by replacing time t by –t :
Reflected version of x(t) about t = 0
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

4. Basic Operations on Signals
Time scaling
Expands (stretches) or contracts (compresses) a signal along the time axis
y(t) : Signal by scaling the independent variable, time t, by a factor b
If b>1, the signal y(t) is a compressed version of x(t)
If 0<b<1, the signal y(t) is an expanded version of x(t)
Combination of these operation
x(-2t) = Combination of flipping the signal then contracting it by a factor of 2
x(2t-3) : x[2(t-1.5)] = Contract the signal by a factor of 2 and then shifting it to the right by 1.5
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

5. Continuous-Time and Discrete-Time Signals
Based on the range of the independent variable
Continuous-time signal
Notation : x(t)
The independent variable t takes real number
Discrete-time signal
Notation : x[nT] or x[n]
The independent variable n takes its values in the set of integers
By sampling a continuous-time signal x(t) at time instants separated by T0, we can define the discrete-time signal x[n] = x(nT0)
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

6. Continuous-Time and Discrete-Time Signals
Example & (Figure 2.6 & 2.7)
Sinusoidal signal
Example of a CT signal
Example of a DT signal
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

7. Real and Complex Signals
Real signals
Takes its values in the set of real numbers
x(t)  R
Complex signals
Takes its values in the set of complex numbers
x(t)  C
In communications
Used to model signals
Convey amplitude and phase information
Represented by two real signals
Real and imaginary parts
Absolute value (or modulus or magnitude) and phase
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

8. Representation of Complex signals – Example 2.1.3
The signal :
Complex signals
From Euler’s relation :
Its real part :
Its imaginary part :
Absolute value of x(t) :
Its phase :
Figure 2.8
Representation of complex signals
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

9. Representation of Complex signals – Example 2.1.3
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

10. Deterministic and Random Signals
Deterministic signals
At any time instant t, the value of x(t) is given as a real or a complex number
Model : Completely specified functions of time
Random (Stochastic) signals
At any time instant t, the value of x(t) is a random variable
Defined by a probability density function
Model : Probability model
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

11. Periodic and Nonperiodic Signals
The CT signal x(t) is periodic if and if only (iff) there exists a T0 >0 such that
T0 : period of the signals (positive real number)
The DT signal x(n) is periodic if and if only (iff) there exists a N0>0 such that
N0 : period of the signals (positive integer)
A signal that does not satisfy the conditions of periodicity is called nonperiodic
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

12. Causal and Noncausal Signals
Causality
Close relationship to the realizability of a system
Causal signal
A signal x(t) is called causal if for all t < 0, we have x(t) = 0; otherwise, the signal is noncausal
A discrete-time signal is a causal signal if it is identically equal to zero for n < 0.
Anticausal signals
Signals whose time inverse is causal
An anticausal signal is identically equal to zero for t > 0
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

13. Even and Odd Signals Even Signals Odd Signals
A CT signal x(t) is said to be an even signals if
Odd Signals
A CT signal x(t) is said to be an odd signals if
Similar remarks apply to DT signals
In general, any signal x(t) can be written as the sum of its even and odd part as
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

14. Hermitian Symmetry for Complex Signals
Another form of symmetry for complex signals
A complex signal x(t) is called Hermitian if its real part is even and its imaginary part is odd
Its magnitude is even and its phase is odd
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

15. Energy-Type and Power-Type Signals
Energy content
For any signal x(t), the energy content of the signal is defined by
Power content
For any signal x(t), the power content of the signal is defined by
For real signal, is replaced by
A signal is an energy-type signal if and only if Ex is finite
A signal is an power-type signal if and only if Px satisfies
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

16. Example 2.1.10 The energy content of
Therefore, this signal is not an energy-type signal
However, the power of this signal is
Hence, x(t) is a power-type signal and its power is
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

17. Example 2.1.11 For any periodic signal with period T0, the energy is
Therefore, periodic signals are not typically energy type
The power content of any periodic signal is
This means that the power content of a periodic signal is equal to the average power in one period
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

18. Sinusoidal Signal & Complex Exponential Signal
Sinusoidal signals
Definition :
A : Amplitude
f0 : Frequency
 : Phase
Period : T0 = 1/f0
Figure 2.6
Complex exponential signal
Figure 2.8
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

19. Unit Step, Rectangular & Triangular Signal
Unit step signal
Definition
The unit step multiplied by any signal produces a “causal version” of the signal
Note that for positive a, we have
Figure 2.9
Rectangular pulse
Figure 2.13
Triangular Signal
Figure 2.15
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

20. Sinc & Sign or Signum Signal
Sinc signal
Definition
The sinc signal achieves its maximum of 1 at t = 0.
The zeros of the sinc signal are at t = 1, 2, 3, 
Figure 2.17
Sign or Signum signal
Definition :
Sign of the independent variable t
Can be expressed as the limit of the signal xn(t) when n  
Figure 2.18
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

21. Impulse or Delta Signal
Not a function (or signal)
A distribution or a generalized function
A distribution is defined in terms of its effect on another function (usually called the “test function”) under the integral sign
Definition of the impulse distribution (or signal) by the relation
Sifting property : Effect of the impulse distribution on the “test function” (t), assumed to be continuous at the origin
Helpful to visualize (t) as the limit of certain known signals
Figure 2.19
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

22. Impulse Signal – Properties
Figure 2.20 : Schematic representation of the impulse signal
(t) = 0 for all t  0 and (0) = 
x(t)(t-t0) = x(t0)(t-t0)
For any (t) continuous at t0,
For all a  0,
The result of the convolution of any signal with the impulse signal is the signal it self :
The unit step signal is the integral of the impulse signal, and the impulse signal is the generalized derivative of the unit step signal
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

23. Impulse Signal – Properties
Similar to the way we defined (t) we can `(t), ``(t) , …, (n)(t), the generalized derivatives of (t), by the following equation
The result of convolution of any signal with nth derivative of x(t) is the nth derivative of x(t)
The result of the convolution of any signal x(t) with the unit step signal is the integral of the signal x(t)
For even values of n, (n)(t) is even; for odd values of n, it is odd. In particular, (t) is even and `(t) is odd
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

24. Classification of Systems
An interconnection of various elements or devices that behave as a whole.
Communication point of view
A entity that is excited by an input signal and , as a result of this excitation, produces an output signal
Communication engineer’s point of view
A law that assigns output signals to various input signals
Its output must be uniquely defined for ant legitimate input
x(t) : Input
y(t) : Output
 : Operation performed by the system
Characteristics
The operation that describes the system : Use the operator  to denote the operation
The set of legitimate input signals : Use the operator H 
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

25. Discrete-Time and Continuous-Time Systems
Systems can accept either discrete-time or continuous-time signals as their inputs and outputs
Discrete-time system
Accepts discrete-time signals as the input
Produces discrete-time signals at the output
Continuous-time system
Both input and output signals are continuous-time signals
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

26. Linear and Nonlinear Systems
Systems for which the superposition property is satisfied
The system response to a linear combination of the inputs is the combination of the response to the corresponding inputs
A system  is linear if and only if, for any two input signals x1(t) and x2(t) and for any two scalars  and , we have
A system that does not satisfy this relationship is called nonlinear
Linearity : Additive + homogeneous
Output of a linear system
Decompose the input into a linear combination of some fundamental signals
Linear combination of the corresponding outputs
L : Operation of linear systems
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

27. Time-Invariant and Time-Varying Systems
Time-invariant system
A system is called time invariant if its input-output relationship does not change with time
A delayed version of an input results in a delayed version of the output
A system is time-invariant if and only if, for all x(t) and all values of t0, its response to x(t-t0) is y(t-t0) , y(t) is the response of the system to x(t)
Linear time-invariant systems
The response of these systems to inputs can be derived simply by finding the convolution of the input and the impulse response of the system
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

28. Causal and Noncausal Systems
Causality
Deals with the physical realizability of systems
In a physically realizable system, the output at any time depends on the values of the input signal up to that time and does not depend on the future values of the point
A system is causal if its output at any time t0 depends on the input at times prior to t0,
A necessary and sufficient condition for an LTI system to be causal
Its impulse response h(t) (i.e., the output when the input is (t)) must be a causal signal
For t < 0, h(t) = 0
Noncausal systems
The value of the output at t0 also depends on the values of the input at times after t0
Encountered in situations where signals are not processed in real time
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

29. Analysis of LTI Systems in the Time Domain
Impulse response
The impulse response h(t) of a system is the response of the system to a unit impulse input (t)
h(t, ) : Response of the system to a unit impulse applied at time , i.e., (t-)
h(t, ) = h(t-) for time-invariant systems
The convolution Integral
Objective : Drive the output y(t) of an LTI system to any input signal x(t)
Response of the LTI system to the input x(t) : y(t)
Convolution of x(t) and the impulse response h(t)
The impulse response completely characterizes the system
The impulse response contains all the information we need to describe the system behavior
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

30. Example
Let a linear time-invariant system have the impulse response h(t)
Assume this system has a complex exponential signal as input,
The response to the input
The response of an LTI system to the complex exponential with frequency f0 is a complex exponential with the same frequency
Amplitude of the response : Multiplying the amplitude of the input by
Phase response : Adding to the input phase
: A function of the impulse response and the input frequency
Complex exponential
Eigenfunctions of the class of linear time-invariant systems
Eigenfunctions of a system : Set of inputs for which the output is a scaling of the input
Simple to find the response of LTI system to the class of complex exponential signals
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi

31. Fourier Series LTI systems Main objective
Model of a large number of building blocks in a communication system
Good and accurate models for a large class of communication channels
Some basic components of transmitters and receivers
Such as filters, amplifiers, and equalizers
Main objective
Develop methods and tools necessary to analyze LTI systems
Determine the output corresponding to a given input
Provide insight into the behavior of the system
Convolution integral : Input and output relation of an LTI system :
h(t) : Impulse response of the system
Basic tool for analyzing LTI systems
Essentials of Communication Systems Engineering by John G. Proakis and Masoud Salehi