UNIT-III
Signal Transmission through
Linear Systems
Introduction



A system can be characterized equally well in the
time domain or the frequency domain, techniques
will be developed in both domains
The system is assumed to be linear and time
invariant.
It is also assumed that there is no stored energy in
the system at the time the input is applied
Impulse Response

The linear time invariant system or network is characterized in
the time domain by an impulse response h (t ),to an input unit
impulse (t)
y (t )  h(t ) when x(t )   (t )

The response of the network to an arbitrary input signal x (t )is
found by the convolution of x (t )with h (t )

y (t )  x(t )  h(t ) 
 x( )h(t   )d



The system is assumed to be causal,which means that there can
be no output prior to the time, t =0,when the input is applied.
The convolution integral can be expressed as:

y (t )   x( )h(t   )d
0
Transfer Function of LTI system

The frequency-domain output signal Y (f )is obtained by taking
the Fourier transform
Y( f ) X( f ) H( f )

Frequency transfer function or the frequency response is
defined as:
Y( f )
H( f ) 
X(f )
H ( f )  H ( f ) e j ( f )

The phase response is defined as:
 ( f )  tan 1
Im{H ( f )}
Re{H ( f )}
Distortion less Transmission



The output signal from an ideal transmission line may have
some time delay and different amplitude than the input
It must have no distortion—it must have the same shape as the
input.
For ideal distortion less transmission:
Output signal in time domain
y(t )  Kx(t  t0 )
Output signal in frequency domain Y ( f )  KX ( f )e j 2 ft0
System Transfer Function
H ( f )  Ke j 2 ft0
What is the required behavior of an ideal
transmission line?





The overall system response must have a constant magnitude
response
The phase shift must be linear with frequency
All of the signal’s frequency components must also arrive with
identical time delay in order to add up correctly
Time delay t0 is related to the phase shift  and the radian
frequency  = 2f by:
t0 (seconds) =  (radians) / 2f (radians/seconds )
Another characteristic often used to measure delay distortion
of a signal is called envelope delay or group delay:
1 d ( f )
( f )  
2 df
Ideal Filter Characteristics


Filter
 A frequency-selective system that is used to limit the
spectrum of a signal to some specified band of frequencies
The frequency response of an ideal low-pass filter condition
 The amplitude response of the filter is a constant inside the
pass band -B≤f ≤B
 The phase response varies linearly with frequency inside the
pass band of the filter
exp(  j 2ft0 ),
H( f )  
0,
B f  B
f B
Ideal Filters

For the ideal band-pass
filter transfer function
Figure1.11 (a) Ideal band-pass filter

For the ideal high-pass filter
transfer function
Figure1.11 (c) Ideal high-pass filter
Bandwidth


Theorems of
communication and
information theory are
based on the
assumption of strictly
band limited channels
The mathematical
description of a real
signal does not permit
the signal to be strictly
duration limited and
strictly band limited.
Different Bandwidth Criteria
(a) Half-power bandwidth.
(b) Equivalent
rectangular or noise
equivalent bandwidth.
(c) Null-to-null bandwidth.
(d) Fractional power
containment
bandwidth.
(e) Bounded power
spectral density.
(f) Absolute bandwidth.
Causality and Stability


h(t )  0, t  0
Causality : It does not respond before the excitation is applied
Stability
 The output signal is bounded for all bounded input signals
(BIBO)
x(t )  M
y(t ) 






h( )x(t   )d

h( )x(t   )d   h( ) x(t   ) d



for all t

y(t)  M  h( ) d

 M  h( ) d
An LTI system to be stable
 The impulse response h(t) must be absolutely integrable
 The necessary and sufficient condition for BIBO stability of a linear

time-invariant system h(t ) dt   (2.100)


Paley-Wiener Criterion

The frequency-domain
equivalent of the
causality requirement
 ( f ) 
  1  f 2 df  

Spectral Density

The spectral density of a signal characterizes the distribution
of the signal’s energy or power in the frequency domain.

This concept is particularly important when considering
filtering in communication systems while evaluating the signal
and noise at the filter output.

The energy spectral density (ESD) or the power spectral
density (PSD) is used in the evaluation.
Energy Spectral Density (ESD)

Energy spectral density describes the signal energy per unit
bandwidth measured in joules/hertz.
Represented as ψx(f), the squared magnitude spectrum

x( f ) X ( f )
According to Parseval’s theorem, the energy of x(t):

2

Ex =

Therefore:


x 2 (t) dt =
-
-

Ex =

2
|X(f)|
df

x
(f) df
-

The Energy spectral density is symmetrical in frequency about
origin and total energy of the signal x(t) can be expressed as:

E x = 2  x (f) df
0
Power Spectral Density (PSD)


The power spectral density (PSD) function Gx(f ) of the
periodic signal x(t) is a real, even, and nonnegative function of
frequency that gives the distribution of the power of x(t) in the
frequency domain.

PSD is represented as:
G x (f ) =  |C n |2 ( f  nf 0 )
n=-

Whereas the average power of a periodic signal x(t) is
T /2

represented as:
1 0
2
2
Px 
x
(t)
dt

|C
|

n
T0 T0 / 2
n=-

Using PSD, the average normalized power of a real-valued
signal is represented as:

Px 
G


x
(f) df  2  G x (f) df
0