Signal Transmission Through LTI Systems EE 442 Spring 2017 Lecture 3 1 Signal Transmission
Signal Transmission Through LTI Systems EE 442 Spring 2017
Lecture 3
1 Signal Transmission
Signal Transmission 2
Steady-State Response in Linear Time Invariant Network
By steady-state we mean and sinusoidal excitation.
LTI Network H(f)
A sinusoidal signal of frequency f at the input x(t) produces a sinusoidal signal of frequency f at the output y(t). The output y(t) Is given by
x(t) y(t)
( ) ( ) ( )y t H f x t
y(t) will modify input x(t) by a change in magnitude and in phase. However, the frequency f will be unchanged and the output will be causal.
3 Signal Transmission
Pulse Response in Linear Time Invariant Network
We are interested in the pulse response in a given LTI system with a bounded input – bounded output (BIBO).
LTI Network h(t) & H(f)
( )
( )
x t
X f
( ) ( ) ( )
( ) ( ) ( )
y t x t h t
Y f X f H f
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).
( )
( ) ( ( ) ( ))
( ) ( ), ( ) ( ) ( ) ( )
where ( ) is the transfer function of the network.
We can write ( ) ( )
and ( ) ( ) ( )
h
y x h
j f
j f j f j f
x t X f h t H f and y t Y f
H f
H f H f
Y f X f H f
e
e e
by convolution theorem
Lathi & Ding pp. 123-124
Signal Transmission 4
Example: Pulse Response in a LTI Network
( )h t( )x t
This is special case of the transient response of a LTI network.
Signal Transmission 5
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 signal, then the output signal y(t) is required to be y(t) = K·x(t-td) (K is a constant) This means y(t) has it amplitude modified by factor K and it is time shifted by time td. In the frequency domain we have:
The Fourier transform is Y(f) = K·X(f)e-j2ftd
by application of the convolution theorem.
But we have not shown this yet!
6 Signal Transmission
Signal Distortion During Signal Transmission
For distortionless transmission the transfer function H(f) we can write,
2From ( )
( ) and ( ) 2h d
dj ftH f A
H f A f ft
e
Conclusion: Distortionless transmission requires a constant amplitude |H(f)| over frequency and a linear phase response h(f) passing through the origin at f = 0.
|H(f)|
h(f)
f
A
H(f) is transfer function
7 Signal Transmission
All-Pass System versus Distortionless Systems
All-Pass System: Has a constant amplitude response, but doesn’t have a linear phase response.
A distortionless system is always an all-pass system, but the converse is not true in general.
Transmission phase characteristic (if it doesn’t have a constant slope) causes distortion.
In practice, many LTI systems only approximate a linear phase response that passes through f = 0. Hence,
otherwise, the time delay td varies with frequency.
For ES 442: phase distortion is 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).
( )1( ) needs to have a constant slope.
2h
d
d ft f
df
Signal Transmission 8
http://www.slideshare.net/simenli/ch1-2-49340691
Signal Transmission 9
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
2 cycles = 4 radians delay shown
(radians)
(radians/sec)
and
d
d
L
velocity
t
t
1
x
direction of travel
Signal Transmission 10
Phase Delay in Ideal Transmission Line
ZL = ZS = 50
L
For an air-filled coaxial transmission line the time delay is roughly 1 nanosecond (10-9 second) per foot of physical length L. For a transmission line with a polyethylene dielectric the time delay is of the order of 1.5 nanosecond per foot of length.
2 f
dt
Length = L
Linear Phase
Response
11 Signal Transmission
Example (Lathi & Ding – pp. 128-129)
R
C g(t) y(t)
+ + output input
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
11
( ) general form: 1 (2 )(2 )
aRCH f where aa j f RCj f
RC
12 Signal Transmission
Example (Lathi & Ding – pp. 128-129)
2 2
1 1
1
1
1( ) general form:
(2 )(2 )
Therefore, the magnitude ( ) and phase ( ) are given by
1( ) , and
1 2
2( ) tan tan where 2
1
2( )
h
h
h
RC
RC
aH f where a
a j f RCj f
H f f
RCH f
fRC
fbf b f
aRC
f
12 (low frequency)1
ffor f
RCaRC
13 Signal Transmission
Example (Lathi & Ding – pp. 128-129)
The time delay td is derivative of the phase with respect to frequency, hence,
1
2
(tan ) 1From a table of derivatives:
1
d x
dx x
1
1
( ) ( ) ( )1( ) , By definition
(2 ) 2
which is the slope of the phase versus frequency plot.
For our RC low-pass filter example the time delay is
( ) 2( ) tan
(2 ) (2 )
h h hd
hd
d f d f d ft f
d d f df
d f fdt f
d f d f
1tan
RC
d
d a
Signal Transmission 14
Example (Lathi & Ding – pp. 128-129)
1
2
2 22
2 22
1
1
1
( ) 2( ) tan
(2 ) (2 )
21
1 1( )
1(2 ) 1 (2 )21
1( )
1 (2 )1 (2 )
Of course, for very low frequencies:
( )
hd
d
d
d
RC
RC
RC
d f fdt f
d f d f
fd
RCt fd f ff RCRC
RCRCt ffRCf
RC
t f 1 1for 2
RC f aRCa
15 Signal Transmission
Low-Pass Filter Example (Lathi & Ding)
Plot from page 129 of Lathi & Ding:
Slope is - td
Amplitude response within 2% of peak value
1aRC
(constant phase shift asymptote)
17 Signal Transmission
Ideal Filter versus Practical Filter
The signal g(t) is transmitted without distortion, but has a time delay of td.
1
2
2
( ) and ( ) 2 ;2
( ) and2
( ) 2 sinc 2 ( )2
h d
d
d
d
j ft
j ft
fH f f ft
B
fH f
B
fh t F B B t t
B
e
e
Lathi & Ding; Page 130
dtBB
( )H f ( )h t
( ) 2h df ft
1
2B
Violates causality
f
t
1
Brick filter
18 Signal Transmission
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. A good approximation if td is large (td approaches infinity for ideal filter).
Many different practical non-ideal filters exist:
ˆ( ) ( ) ( )h t h t u t
Butterworth Chebyshev I
Chebyshev II Elliptic
Signal Transmission 19
Nth order
Butterworth Filter (Maximally Flat Magnitude)
(nth order filter)
Signal Transmission 20
Butterworth Filter (Maximally Flat Magnitude)
Sallen–Key topology
Cauer topology
22
2
(0)( )
1
n
C
HH j
Unit circle
Re(s)
Im(s)
Signal Transmission 21
Butterworth Filter Impulse Response
( )h t
( )H f
( )h f
Fourth-order filter
Lathi & Ding; Page 133
Signal Transmission 22
Comparing Butterworth, Chebyshev & Bessel Filters
http://www.analog.com/library/analogDialogue/archives/43-09/EDCh%208%20filter.pdf?doc=ADA4666-2.pdf
h(t)
Unit step response
|H(f)| |H(f)|
Unit impulse response
Signal Transmission 23
Phase Delay versus Group (Envelope) Delay
0
0
0
at one frequency
over a frequency band
Phase response of ( ) is ( ), therefore
( )Phase delay ( ) is ,
2
( )Group delay ( )
(2 )
h
hd
hgrp
f
H f f
ft and
f
d ft
d f
Input:
( ) ( ) cos(2 )
Output:
( ) ( ) ( ) cos(2 ( ) )grp d
x t A t ft
y t H f A t t f t t
Envelope
Signal Transmission 24
Group (Envelope) Delay
Group delay is an important way to describe a filter's pass band characteristics.
Consider a simple example of a square wave, which as you know, is composed of a large group of frequency components. A square wave is square only because its frequency components are in proper phase alignment with one another. If we pass a square wave through a network and expect it to remain square, then we need to ensure that the device doesn't misalign these frequency components.
Group delay is:
(1) A measure of a network’s phase distortion. (2) The transit time of a signal through a device versus frequency. (3) The derivative of the device's phase characteristic with respect to frequency (mathematical statement).
Signal Transmission 25
Channel Impairments (Overview)
• Linear distortion caused by impulse response. H(f) attenuates and phase shifts the signal.
• Nonlinear distortion (e.g., such as from clipping)
• Random Noise (independent or signal dependent)
• Interference from other transmissions or sources
• Self interference & ISI (from reflections or multipath)
( ) ( ) ( ) ( ) ( ) ( )y t h t g t Y f H f G f
( )
p
p
x
x t
x
( )
( )
( )
p
p p
p
x t x
x x t x
x t x
( )y t
ISI is intersymbol interference
Signal Transmission 26
Pulse Distortion in a Sine-Squared Pulse
Input pulse
Input pulse Input pulse
Output pulse
Output pulse
Output pulse
(a) Amplitude-frequency distortion and phase-frequency distortion
(b) Amplitude-frequency distortion (c) Phase-frequency distortion
http://users.tpg.com.au/users/ldbutler/Measurement_Distortion.htm
This is typically what pulse distortion looks like in channels.