1. Chapter 2. Signals and Linear Systems
Essentials of Communication Systems Engineering

2. 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

3. 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

4. Sinusoidal Signal & Complex Exponential Signal
Sinusoidal signals
Definition :
A : Amplitude
f0 : Frequency
 : Phase
Period : T0 = 1/f0
Complex exponential signal

5. Unit Step, Rectangular & Triangular Signal
Unit step signal
Definition
Rectangular pulse
Triangular Signal

6. 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, 
Sign or Signum signal
Definition :
Sign of the independent variable t
Can be expressed as the limit of the signal xn(t) when n  

7. Impulse or Delta Signal
Definition of the impulse signal
1. (t) = 0 for all t  0 and (0) =  2.
Properties
1. x(t)(t-t0) = x(t0)(t-t0) 3. 4.

8. Fourier Series LTI systems
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
Convolution integral : Input and output relation of an LTI system :
where h(t) : Impulse response of the system.

9. Fourier Series Another approach to analyzing LTI systems Basic idea
Expand the input as a linear combination of some basic signals whose output can be easily obtained
Employ the linearity properties of the system to obtain the corresponding output
Easier than a direct computation of the convolution integral
Provide better insight into the behavior of LTI systems

10. Fourier Series Response of an LTI system to a complex exponential
A complex exponential with the same frequency with a change in amplitude and phase (p.45,Example )
Which signals can be expanded in terms of complex exponentials?
Answer: periodic signals which satisfy Dirichlet conditions

11. Fourier Series Fourier series Dirichlet conditions
1. x(t) is absolutely integrable over its period, i.e.,
2. The number of maxima and minima of x(t) in each period is finite
3. The number of discontinuities of x(t) in each period is finite
Fourier series
for some arbitrary  (usually,  = 0 or )

12. Fourier Series Observations concerning Fourier series
xn : Fourier-series coefficients of the signal x(t)
Dirichlet conditions are only sufficient conditions for the existence of the Fourier series
For some signals that do not satisfy these conditions, we can still find the Fourier series expansion
The quantity f0 = 1/T0 is called the fundamental frequency of the signal x(t)
The frequencies of the complex exponential signals are multiples of this fundamental frequency
The nth multiple of f0 is called the nth harmonic

13. Example 2.2.1
x(t) : Periodic signal depicted in Figure 2.25 and described analytically by
 : A given positive constant (pulse width)
Determine the Fourier series coefficient

14. Example 2.2.1 Solution Period of the signal is T0 and
For n = 0, the integration is very simple and yields

15. Fourier Series for Real Signals
Real signal x(t)
The positive and negative coefficients are conjugates
|xn| : Even symmetry (|xn| = |x-n| )
xn : Odd symmetry (xn = - x-n) with respect to the n = 0 axis

16. Response of LTI Systems to Periodic Signals
If h(t) is the impulse response of the system, that the response to the exponential ej2f0t is H( f0) ej2f0t (From ex with A= 0 & )
x(t) , the input to the LTI system, is periodic with period To and has a Fourier-series representation
Response of LTI systems

17. Response of LTI Systems to Periodic Signals
If the input to an LTI system is periodic with period To, then the output is also periodic. The output has a Fourier-series expansion given by
where

18. Parseval's Relation
The power content of a periodic signal is the sum of the power contents of its components in the Fourier-series representation of that signal
The left-hand side of this relation is Px, the power content of the signal x(t)
|xn|2 is the power content of , the nth harmonic
Parseval's relation says that the power content of the periodic signal is the sum of the power contents of its harmonics