Signal Transmission Through LTI Systems
ES 442
Spring 2016
Signal Transmission
1
Pulse Response in a LTI Network
We are interested in the pulse response in a given LTI system with a
bounded input – bounded output (BIBO).
y ( t )  x( t )  h ( t )
x( t )

LTI Network
h(t) & H(f)

Lathi & Ding
pp. 123-124
Y ( f )  X ( f )H ( f )
X( f )
by convolution theorem
where x(t) is the input, h(t) is impulse response of the network and
y(t) is the output (Note: the symbol * denotes convolution).
x(t )  X( f ), h(t )  H ( f ) and y(t )  Y ( f )
where H ( f ) is the transfer function of the network.
We can write H ( f )  H ( f ) e
and Y ( f ) e
jy ( f )
jh ( f )
 X( f )  H ( f ) e
Signal Transmission
j(x ( f ) jh ( f ))
2
Signal Distortion During Signal Transmission
In amplifiers and transmission over a channel we want the output
waveform to be a replica of the input waveform. This means we
want distortionless transmission.
Another way to say this: If x(t) is the input waveform, then the output
waveform y(t) should be
y(t) = K·x(t)
(K is a constant)
So in general,
y(t) = K·x(t-td)
The Fourier transform is Y(f) = K·X(f)e-j2ftd
by application of the convolution theorem.
Signal Transmission
3
Signal Distortion During Signal Transmission
For distortionless transmission the transfer function H(f) we can write,
From H ( f )  Ae
 j 2 ftd
H ( f )  A and h ( f )  2 ftd
Conclusion: Distortionless transmission requires a constant amplitude
|H(f)| and a linear phase response h(f) passing through the origin at f = 0.
A
|H(f)|
f
h(f)
Signal Transmission
4
All-Pass versus Distortionless Systems
All-Pass System: Has a constant amplitude but without the linear phase
response.
A distortionless system is always an all-pass system, but the converse is
not true in general.
Transmission phase can cause distortion.
In practice, many systems only approximate a linear phase response that
passes through f = 0. Hence,
td ( f )  
1 dh ( f )
needs to have a constant slope.
2 df
otherwise, time delay td will vary with frequency f.
For ES 442, phase distortion is very important in digital communication
systems because a nonlinear phase characteristic in a channel causes
pulse dispersion (spreading out) and causes pulses to interfere with
adjacent pulses (called interference).
Signal Transmission
5
Time Delay of Ideal Transmission Line
What is the time delay of a coax transmission line of length L?
The delay varies linearly with the length of the transmission line.
ZL = ZS = 50 
L
td 
 (radians)
 (radians/sec)
and
L
td 
velocity

x
2 cycles = 4 radians
Signal Transmission
6
Phase Delay in Ideal Transmission Line
ZL = ZS = 50 
L
With an air filled coaxial transmission line the time delay is roughly
1 nanosecond (10-9 second) per foot of physical length. For a
transmission line with polyethylene dielectric the time delay is
of the order of 1.5 nanosecond per foot of length.
  2 f
Linear
Phase
Response
td  



Length = L1
Signal Transmission
7
Example (Lathi & Ding – pp. 128-129)
input
R
output
+
+
g(t)
C
y(t)
We have an RC low-pass filter, find the transfer function H(f), and sketch
|H(f)|, the phase n(f) and delay td(f).
The transfer function for y(f)/g(f) is
1

RC 
H( f ) 
 1 RC   j(2 f )

a
1 
general
form:
where
a



a

j
(2

f
)
RC


Signal Transmission
8
Example (Lathi & Ding – pp. 128-129)
1


RC
H( f ) 
 1 RC   j(2 f )

a
1 
where a 
 general form:

a

j
(2

f
)
RC


Therefore, the magnitude H ( f ) and phase  h ( f ) are given by
H( f ) 
 1 RC 
 1 RC    2 f 
2

 h ( f )   tan 1 



2 f
h ( f )   
 1
 RC


2 f
1
RC


2
,
and





  

a

for

2 f  1
Signal Transmission
RC

9
Example (Lathi & Ding – pp. 128-129)
The time delay td is derivative of the phase with respect to frequency,
namely,
d h ( f )
d ( f )
1 d h ( f )
 h

,  Definition
d
d(2 f )
2 df
which is the slope of the phase versus frequency plot.
td ( f )  
For our RC low-pass filter example the time delay is

 2 f
d h ( f )
d 
1
td ( f )  

 tan 
d(2 f )
d(2 f ) 
 1
 RC


From a table of derivatives:

 d 
1    
 

tan
 

  d 
 a 

d(tan 1 x )
1

dx
1  x2
Signal Transmission
10
Example (Lathi & Ding – pp. 128-129)

 2 f  
d h ( f )
d 

td ( f )  

 tan 1 
d(2 f )
d(2 f ) 
 1

RC



 2 f 

d
2
1
1


1
1
RC
 RC  
td ( f )  


2
2
d(2 f )
1
 2 f 
 (2 f )2 1 RC
RC

1
 1

 RC 
1
RC
RC
td ( f ) 

2
2
2
1

(2

fRC
)
1
 (2 f )
RC
and for very low frequencies,
1
td ( f )   RC
for 2 f  a  1
RC
a
















Signal Transmission
11
Example (Lathi & Ding – page 129)
Plot from page 129 of Lathi & Ding:
Amplitude response within 2% of peak value
a1
RC
Slope is - td
Signal Transmission
12
Example (Lathi & Ding – page 129)
td ( f )
1
 RC
a
1
a
0
  2 f
a
Signal Transmission
13
Ideal Filter versus Practical Filter
The signal g(t) is transmitted without distortion, but has a time delay of td.
 f 
H( f )   
 and h ( f )  2 ftd ;
2
B


 f   j 2 ftd
 H( f )   
e
and

 2B 
  f   j 2 ftd 
h(t )  F 1  
e
  2 Bsinc  2 B(t  t d ) 
2
B

 

Filter
h(t )
H( f )
1
f
B
B
td
h ( f )  2 ftd
1
2B
t
Lathi & Ding; Page 130
Violates causality
Signal Transmission
14
Ideal Filter versus Practical Filter – II
Impulse response h(t) is response to impulse (t) applied at time t = 0.
The ideal filter on the previous slide is noncausal  unrealizable .
Practical approach to filter design is to cutoff h(t) for t < 0.
hˆ(t )  h(t )  u(t )
A good approximation if td is large (td approaches infinity for ideal filter).
Many different practical non-ideal filters exist:
Butterworth
Chebyshev I
Chebyshev II
Elliptic
Signal Transmission
15
Butterworth Filter (Maximally Flat Magnitude)
Nt
h
(nth order filter)
or
de
r
Signal Transmission
16
Butterworth Filter (Maximally Flat Magnitude)
Im(s)
Cauer topology
2
H ( j ) 
Unit circle
Re(s)
H 2 (0)
 
1


 C
2n
Sallen–Key topology
Signal Transmission
17
Butterworth Filter Impulse Response
Fourth-order filter
h(t )
H( f )
h ( f )
Lathi & Ding; Page 133
Signal Transmission
18
Comparing Butterworth, Chebyshev & Bessel Filters
|H(f)|
|H(f)|
h(t)
Unit step response
http://www.analog.com/library/analogDialogue/archives/43-09/EDCh%208%20filter.pdf?doc=ADA4666-2.pdf
Signal Transmission
19
Phase Delay versus Group (Envelope) Delay
Phase response of H ( f ) is  h ( f ), therefore
h ( f0 )
Phase delay (at one frequency f 0 ) is td  
, and
2 f 0
Group delay (over a frequency band) t grp
Envelope
d

h ( f )
d(2 f )
Input:
x(t )  A(t )  cos(2 ft   )
Output:
y(t )  H ( f )  A(t  t grp )  cos(2 f (t  td )   )
Signal Transmission
20
Channel Impairments (Overview)
• Linear distortion caused by impulse response.
y(t )  h(t )  g(t )  Y( f )  H( f )G( f )
H(f) attenuates (|H(f)|)and phase shifts the signal.
• Nonlinear distortion ( e.g ., clipping)
xp
y(t ) 
x( t )
xp
x(t )  x p
x p  x(t )  x p
x(t )  x p
• Random noise (independent or signal dependent)
• Interference from other transmissions or sources
• Self interference & ISI (reflections or multipath)
Signal Transmission
21
Pulse Distortion in a Sine-Squared Pulse
Input pulse
Output pulse
(a) Amplitude-frequency distortion and phase-frequency distortion
Input pulse
Input pulse
Output pulse
Output pulse
(b) Amplitude-frequency distortion
(c) Phase-frequency distortion
http://users.tpg.com.au/users/ldbutler/Measurement_Distortion.htm
Signal Transmission
22
Inter-Symbol Interference
Path 2
Path 1
http://www.teletopix.org/4g-lte/inter-symbol-interference-in-lte/
Signal Transmission
23
Application of Equalizers
H EQ ( f ) 
1
HC ( f )
EQ ( f )  C ( f )
Signal Transmission
24