ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 9
Chapter 9:
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
ECE 6640
Digital Communication Through Band-Limited Channels
Characterization of Band-Limited Channels
Signal Design for Band-Limited Channels
Optimum Receiver for Channels with ISI and AWGN
Linear Equalization
Decision-Feedback Equalization
Reduced Complexity ML Detectors
Iterative Equalization and Decoding—Turbo Equalization
Bibliographical Notes and References
Problems
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
597
598
602
623
640
661
669
671
673
674
2
Chapter Content
• In this chapter, we consider the problem of signal design when the
channel is bandlimited to some specified bandwidth of W Hz. Under
this condition, the channel may be modeled as a linear filter having an
equivalent lowpass† frequency response C( f ) that is zero for | f | > W.
• The first topic that is treated is the design of the signal pulse g(t) in a
linearly modulated signal, represented as
vt    I n  g t  n  T 
n
that efficiently utilizes the total available channel bandwidth W.
• The second topic that we consider is the design of the receiver in the
presence of intersymbol interference and AWGN. The solution to the
ISI problem is to design a receiver that employs a means for
compensating or reducing the ISI in the received signal. The
compensator for the ISI is called an equalizer.
ECE 6640
3
Band Limited Channels
•
For our purposes, a bandlimited channel such as a telephone channel
will be characterized as a linear filter having an equivalent lowpass
frequency-response characteristic C( f ). Its equivalent lowpass
impulse response is denoted by c(t). Then, if a signal of the form
s t   Revt   exp j  2  f c  t 
is transmitted over a bandpass telephone channel, the equivalent lowpass received signal is

r t    v   ct     d  nt 

where c(t) is a filter representing a complex channel response.
• Equalization will be about compensating for (or whitening) the channel.
– whitening is an attempt to make the channel uniform in power spectral density.
similar to the psd of white noise.
ECE 6640
4
Frequency Limitation
• The first assumption is that only a limited frequency band
is available and can be used by the signal of interest.
– For a low pass band
C  f   0,
for f  W
– For a bandpass signal
0,
for 0  f  f c  W
0,
for f c  W  f  

C f   
ECE 6640
5
Channel Frequency Response
• Within the channel, there is a complex frequency response
that can be represented in magnitude and phase.
– For low pass signals
0,
for f  W
 C  f   exp j    f ,
for f  W

C f   
– For bandpass signals
0,

C  f    C  f   exp j    f ,

0,
ECE 6640
for 0  f  f c  W
for f c  W  f  f c  W
for f c  W  f  
6
Ideal, Non-distorting Channel
• The amplitude is a constant across the frequency band.
– Based on RF path loss at center of frequency band.
– Not path loss is frequency/wavelength dependent. Therefore this is
an ideal assumption!
• The phase must be a linear function.
– A time delay exists in the signal. This directly defines a linear
phase delay in frequency.

C  f     t  Td   exp j  2  f  t   dt  exp j  2  f  Td 

ECE 6640
  f   2  f  Td
– “Envelope delay” is defined as
1   f 


 Td
2
f
7
Distorted Channels
• The magnitude will vary with frequency.
– amplitude distortion
• The phase will not be linear.
– phase distortion
ECE 6640
8
Channel - Linear Envelop Delay
PAM Nyquist Pulse
Channel Input:
Zeros at adjacent
symbol locations
ECE 6640
Linear Delay Channel
Channel Output:
Distortion will cause ISI
9
Medium Range Telephone Channel
Wired network
with switching
Usable Bandwidth:
300Hz to 3000 Hz
Impulse response
duration: 10 msec
ECE 6640
Telephone lines also
have 50-60 Hz power
line interference present.
10
Additional Impairments
• Impulse Noise
– additive disturbances from impulsive sources (e.g. lightening,
switching power transients)
– Popcorn noise or Laplacian noise
• Thermal Noise
– Gaussian or white noise from thermal noise sources
• Phase Jitter
– FM caused by low-frequency harmonic interference
• Frequency Offset
– carrier frequency mismatch between transmitter, relay equipment,
and receiver
• Multipath
ECE 6640
– wireless signals being reflected by near-path objects
11
For More on Channels
•
Wikipedia
– https://en.wikipedia.org/wiki/Channel_%28communications%29
•
PDF Presentation: V. Arun, Associate Professor, Department of
Computer Science, University of Massachusetts Amherst,
CS653: Advanced Computer Networks
– https://people.cs.umass.edu/~arun/653/home.html
– https://people.cs.umass.edu/~arun/653/lectures/channel_models.pdf
•
GA Tech Ph.D. Thesis, Chirag S. Patel, “Wireless Channel Modeling,
Simulation, and Estimation”, 2006.
– https://smartech.gatech.edu/bitstream/handle/1853/10480/patel_chirag_s_
200605_phd.pdf
• IEEE 802.16 Broadband Wireless Access Working Group,
“Channel Models for Fixed Wireless Applications”, 2001-07-17.
– http://www.ieee802.org/16/tg3/contrib/802163c-01_29r4.pdf
ECE 6640
• Web search provides other examples
12
Section 9.2 Signal Design
• The Nyquist criteria defines how to achieve zero ISI for a
“simple” channel.
– Nyquist (p. 607 & Equ. 9.2-26) and square root Nyquist filters are
typically used.
– This is one of the filters you applied in Exam #1. The tri2 and sinc
filters caused ISI which is why the performance degraded!
• This criteria can be used to design “optimal” transmit and
receive filters for a “simple” channel.
ECE 6640
13
Filtering Comm. Symbols
• General Filter Concept
st    d n   ht  n  T 
n
The scaling terms d(n), are selected from a small finite alphabet such as
for BPSK {-1, +1} or for ASK {-1,-1/3, +1/3, +1} in accord with a specified
mapping scheme between input symbol (bits) and output levels.
The signal s(t) is sampled at equally spaced time increments identified by
a timing recovery process in the receiver to obtain output samples as
shown in Eqn. (4.2).
sm  T    d n   hm  T  n  T 
n
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
14
Filter Concept
We can partition this sum as shown in Eqn. (4.3), to emphasize the
desired and the undesired components of the measurement. Here
the desired component is d(m) and the undesired component is the
remainder of the sum which if non-zero, is the ISI.
sm  T   d m   h0    d n   hm  T  n  T 
nm
How do we eliminate the intersymbol interference (ISI) ?
Let the time/sample representation of the filter be.
0,
hn  T   
1,
ECE 6560
n0
n0
“Perfect time
sampling is implied”
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
15
Possible Filters
“Function/filter must
be zero at all integer
values except n = 0”
• They could be multiple symbols in length if they are zero at all integer
values except n = 0
• Do we already know of a filter with this characteristic?
(What about a time domain Sinc ?!)
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
16
Sinc as a Zero ISI Filter
Considering the spectral and time domain requirements, we can also use
t 

sin  2 

2 T 
ht   
t
2 
2 T
ECE 6560
n T 

sin  2 

2  T  sin   n 
hn  T   

n T
n
2 
2 T
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
17
Sinc Function and “Reconstruction”
• The “convolution” of the sinc function with sampled time
waveform “impulse samples” is how perfect band-limited
signal reconstruction is performed.
– The continuous time sinc is the time-domain transform of the
perfect frequency-domain “brick-wall low pass filter”
• For symbols, we only need to “reconstruct” the symbol
value without ISI at one time instant during the symbol
period.
– Nominally select the center of the symbol.
– The “reconstructed” continuous time signal need not look like the
original symbol waveform (they have significantly different
frequency spectra and bandwidth!)
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris,
Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13146511-2.
18
Rough Example
• See SincEye.m amd SincEyev2.m
– Each Symbol represented by a multi-cycle sinc function
– The nulls of the sinc function occur at the “optimal” symbol
sample point. All other sample points would be required to sum
the signals levels from the other symbols (symbol interference).
– Therefore, to limit ISI, you must
1. Properly filter
2. Properly (perfectly) sample in time
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
19
Optimal Filter for Pulse Detection (1)
• If we want to detect a transmitted pulse with maximum SNR,
the following applies
SNR 
PSignal
PNoise
 
E s t 

N 0  BEQ
2
filtered to
s t   nt 
so t   no t 


so t   no t   h   s t     nt     d
0
2
 
 

 
E  h   st     d  
 
 0
 


SNRout 

1
N o   ht 2  dt
2

0
ECE 6560
Notes
Methods
Notesand
andfigures
figuresare
arebased
basedon
onorortaken
takenfrom
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textbook:Probabilistic
fredric j. harris,
of
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and
System
Analysis
(3rd
ed.)
by
George
R.
Cooper
and
Clare
D.
McGillem;
Oxford
Press,
Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-131999. ISBN: 0-19-512354-9.
146511-2.
20
Optimal Filter for Pulse Detection (2)
• Applying Schwartz’s Inequality to the output SNR
2



 




 

2
2














h

s

d

h

d

s

d




 




 

0

0
 0




• The upper bounds on the SNR may be defined as
SNRout

 

2
  h   d   E   s t   2  d 

 



2
2
0
0













E
s
t

d





No  0
1
2

N o    ht   dt
2 0
• But we can also define a condition for “equality”
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
and
figures
areAnalysis
based on(3rd
or taken
from
materials
in the and
course
textbook:
fredric Oxford
j. harris,Press,
ofNotes
Signal
and
System
ed.) by
George
R. Cooper
Clare
D. McGillem;
Multirate
Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-131999.
ISBN:
0-19-512354-9.
146511-2.
21
Optimal Filter for Pulse Detection (3)
• For equality to exist
2



 







2
2
















h

s
t

d

h

d

s
t

d




 




 

0

0
 0




• A possible solution is
h   K  st     u  
• This is an “optimal inverse-time filter”
– The filter is the inverse time response of the transmitted pulse s(t)!
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
and
figures
areAnalysis
based on(3rd
or taken
from
materials
in the and
course
textbook:
fredric Oxford
j. harris,Press,
ofNotes
Signal
and
System
ed.) by
George
R. Cooper
Clare
D. McGillem;
Multirate
Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-131999.
ISBN:
0-19-512354-9.
146511-2.
22
Optimal Filter for Pulse Detection (4)
• Continuing for completeness
– The desired impulse response is simply the time inverse of the signal
waveform at time t, a fixed moment chosen for optimality. If this is
done, the maximum filter power (with K=1) can be computed as


0
h 2  d 


st   2  d 
0
t

s  2  d   t 

• And the maximum output SNR becomes
maxSNRout  
2
  t 
No
• The filter is commonly called a “Matched Filter”
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
and
figures
areAnalysis
based on(3rd
or taken
from
materials
in the and
course
textbook:
fredric Oxford
j. harris,Press,
ofNotes
Signal
and
System
ed.) by
George
R. Cooper
Clare
D. McGillem;
Multirate
Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-131999.
ISBN:
0-19-512354-9.
146511-2.
23
An Approach to Generating Filters
1. Defined the desired/required/stuck-with symbol spectrum
or “time pulse” with a finite duration (less than or equal to
the symbol period).
2. Multiply by the sinc in the time domain
–
–
–
Convolve in the frequency domain
Infinite time /non-causal nature still a problem
This enforces the h(nT) requirement!
3. Apply a time domain window after spectral filter design
–
ECE 6560
Modify passband and stopband ripple and edges as needed
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
24
Spectral Convolutions
Zero Time Requirements
Infinite Sinc for ISI
Window Function
Spectral Convolution
Symbol Time Window,
finite time, small BW penalty
/T for 0.1<<0.5
Windowed, zero ISI filter
Note: frequency domain shown as
two-sided spectrum widths
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
25
Using Previously Defined Windows
• Start with the time-domain sinc function
– Determine the filter length based on the number of sinc cycles to
be maintained (null-to-null samples from +/-1st, +/-2nd, +/-3rd, etc.
– The Fourier transform has a sin(n)/sin() shape with frequency
periodicity based on the number of sinc cycles.
• Generate a window of the same number of samples
– Multiply in time domain, convolve in frequency domain.
– Removes the Gibbs phenomenon peaks and reduce the passband
ripple.
• Is there a preferred window/filter? (Yes, raised Cosine)
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
26
Web References
• Wikipedia
– InterSymbol Interference (ISI)
• http://en.wikipedia.org/wiki/Intersymbol_interference
– Nyquist ISI Criterion
• http://en.wikipedia.org/wiki/Nyquist_ISI_criterion
– Inter Symbol Interference (ISI) and Raised cosine filtering
• http://complextoreal.com/wp-content/uploads/2013/01/isi.pdf
• From C. Langton “Complex to Real” web site
– A windowed sinc function will be used for ISI
• The window often applied is the raised cosine
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
27
Nyquist Filtering with Raised Cosine
• The Nyquist pulse is the wave shape required to
communicate over band-limited channels with no ISI.
– It is generated as a raised cosine frequency spectrum window
• Even symmetric spectral window.
• Finite frequency width that is a fraction of the perfect
reconstruction width. (i.e. /T)
– Preference to limit the time response to a length 4T/
– Truncated window (window length) and infinite sinc
• With convolution in the frequency domain, the spectrum
becomes a width of (1+)/T
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
28
Spectral Convolutions (1)
• The term  is called the roll-off factor and is typically on
the order of 0.1 to 0.5 with many systems using values of
 = 0.2.
– The transition bandwidth caused by the convolution is seen to
exhibit odd symmetry about the half amplitude point of the original
rectangular spectrum.
– This is a desired consequence of requiring even symmetry for the
convolving spectral mass function. When the windowed signal is
sampled at the symbol rate 1/T Hz, the spectral component
residing beyond the 1/T bandwidth folds about the frequency
±1/2T into the original bandwidth.
– This folded spectral component supplies the additional amplitude
required to bring the spectrum to the constant amplitude of H(f).
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
29
Spectral Convolutions (2)
• We also note that the significant amplitude of the
windowed wave shape is confined to an interval of
approximate width 4T/  so that a filter with  = 0.2 spans
approximately 20T, or 20 symbol durations!
– We can elect to simply truncate the windowed impulse response to
obtain a finite support filter, and often choose the truncation points
at ± 2T/  or 10 symbols.
– A second window, a rectangle, performs this truncation. The result
of this second windowing operation is a second spectral
convolution with its transform. This second convolution induces
pass-band ripple and out-of-band side lobes in the spectrum of the
finite support Nyquist filter. (Nothing is perfect ….)
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
30
Symbol Periods in Communications
“Let’s do the math”
A communication system can be modeled most simply by the
signal flow shown in Figure 4.4. Here d(n) represents the sequence
of symbol amplitudes presented at symbol rate to the shaping filter
h1(t).
Perfect time sampling provides the detected symbol output ….
the equivalent of filter-decimation!
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
31
Symbol Periods Math (1)
st  

1
m  
Filter received signal
y t   r t   h2 t 

y t    r t     h2    d 

r t   s t   nt 
 d m h t  m  T 


 st     nt    h   d
2



 
y t      d m   h1 t    m  T   h2    d   nt     h2    d

  m  



y t    d m    h1 t    m  T  h2    d   n2 t 
m  
 


ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
32
Symbol Periods Math (2)
Filter received signal



y t    d m    h1 t    m  T  h2    d   n2 t 
m  


Define the convolution of the transmitter and receiver filters
g t  

 h t    h   d
1
2

The received signal is then
y t  

 d mg t  m  T   n t 
m  
ECE 6560
2
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
33
Symbol Periods Math (3)
Minimally sampling the symbol output
t  n T
y n  T  

 d mg n  m T   n n  T 
2
m  
Expanding
y n  T   dˆ n   d n   g 0   d m  g n  m   T   n2 n  T 
mn
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
34
Discrete Samples at the Symbol Rate
Interpretation
y n  T   d n   g 0   d m  g n  m   T   n2 n  T 
m n
1. Desired Signal (convolved filter, prefer a matched filter)
2. Band limited Noise (n2(t)) filtered by h2(t)
(typically filtered white noise, for power use BWeqn
3. Combined ISI (remove with Nyquist g(t))
We want g(nT) to be a Nyquist filter to remove ISI !!
G w  H1 w  H 2 w  H Nyquist w  e
ECE 6560
 j wTdelay
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
35
Transmit and Receive Filters
• We want the result to be a Nyquist Filter with time delay
H1 w  H 2 w  H Nyquist  f   exp j  w  Tdelay 
• Using a matched filter for H1 and H2 should provide
maximum outputs. A Square-root Nyquist filter!
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
36
One Approach:
Square Root Nyquist Concept
To maximize the signal-to-noise (SNR) in (4.10), the receiver filter must be
matched to the transmitter-shaping filter. The matched filter is a timereversed and delayed version of the shaping filter, which is described in
the frequency domain as shown in (4.11). Let
H 2 w  conjH1 w exp j  w  Tdelay 
H1 w  H 2 w  H1 w  exp j  w  Tdelay   H Nyquist w  exp j  w  Tdelay 
2
H1 w  H Nyquist w
2
H1 w  H Nyquist w
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
37
Square Root Nyquist
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
38
Second Approach:
Square Root Nyquist Concept
Use the equivalent of a ZOH output ….
h1 t  
1
Tsymbol
 t 

 rect 
T

symbol


H1  f   sinc f  Tsymbol 
H1  f   H 2  f   exp j  2  f  Tdelay   H Nyquist  f   exp j  2  f  Tdelay 
Note that the Nyquist filter is formed from a sinc basis, therefore
the nulls appear at the same locations as the Nyquist filter! Let,
H2 f  
ECE 6560
H Nyquist  f 
H1_ compensation  f 
 exp j  2  f  Tdelay 
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
39
Nyquist and Square Root Nyquist
• Well defined time and spectral responses …
• Wikipedia Raised-Cosine Filter
– http://en.wikipedia.org/wiki/Raised-cosine_filter
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
40
Nyquist Filter
The most common spectral mass selected for communication systems
is the half cosine of width *fSYM. The half cosine convolved with the
spectral rectangle forms the spectrum known as the cosine-tapered
Nyquist pulse with roll-off  .
The description of this band-limited spectrum normalized to unity passband gain is presented in (4.14).

w
 1   
1
for




 

H Nyq w  0.5  1  cos 

 2  




0

wSym
 w
 
w


 1    for 1    
 1   
w
 
w
Sym
 Sym
 
w
for 1    
wSym
 sin   f Sym  t   cos    f Sym  t  
hNyq t   f Sym  

2

f
t




1
2

f
t





 
Sym

Sym
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
41
Discrete Time Filter
•
•
Let fsample = M*fsymbol so that fsymbol*t is replaced by
fsymbol*n/(M * fsymbol) or n/M.
It is common to operate the filter at M= 4 or 8 samples per
symbol

hNyq 


 
  

n
n
   cos     f Sym 

 sin    f Sym 


M  f Sym    
M  f Sym  
n 
 
 f Sym  
2

n
f sample 


n  
   f Sym 
 1   2    f 
M  f Sym   
Sym

M  f Sym  

  
 
n
sin



1   M
hNyq n  

M   n

M
ECE 6560
n 
  
   cos      
M 
   
2
  
n  
 1   2    M  
 
 
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
42
Discrete Time Filter
•
The filter described in Eqn. (4.16) has a two-sided bandwidth that is
approximately 1/Mth of the sample rate. A digital filter exhibits a
processing gain proportional to the ratio of input sample rate to output
bandwidth, in this case a factor of M. The 1/M scale factor in Eqn.
(4.16) cancels this processing gain to obtain unity gain. When the filter
is used for shaping and up sampling, as it is at the transmitter, we
remove the 1/M scale factor since we want the impulse response to
have unity peak value rather than unity processing gain.
 
n
sin



1   M
hNyq n  

M   n

M
ECE 6560
n 
  
cos





  
M 
   
2
  
n  
 1   2    M  
 
 
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
43
Square Root Nyquist Filter
•
The square root of the cosine-tapered Nyquist filter results in a
quarter cycle cosine tapered filter. This description is normally
confined to square-root raised cosine or root raised cosine
Nyquist filter. The description of this band-limited spectrum
normalized to unity pass-band gain is shown in (4.17).



  
H Sqrt  Nyq w  cos 
  4  



1
for
 w



 1   
w

 Sym

w
wSym
 1   
for 1    
for 1    
0
w
wSym
 1   
w
wSym
Textbook
sign error!
 4    f Sym  t  cos  1     f Sym  t   sin   1     f Sym  t 
hSqrt  Nyq t   f Sym  

2
1  4    f Sym  t     f Sym  t 



ECE 6560

Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
44
Discrete Time Filter
•
Let fsample = M*fsymbol so that fsymbol*t is replaced by
fsymbol*n/(M * fsymbol) or n/M.

hSqrt  Nyq 








n
n
n 
  4    f 
  sin    1     f Sym 

 cos   1     f Sym 
Sym









M
f
M
f
M
f
n 
Sym 
Sym 
Sym 



 f Sym   
2


f Sample 
 

n 
n   


1   4    f Sym 
   f Sym 
M  f Sym 
M  f Sym   


 





  4    n
1 
M
hSqrt  Nyq n  

M



ECE 6560
n 
n



  cos   1       sin    1    
M
M



2
 
n   
n 
1   4          
M    M 
 








Textbook errors!
Sign and extra n.
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal
Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.
45
Design with Controlled ISI
• Partial Response Signaling
• If ISI can not be avoided, define a signaling system hat
allows successive symbols to be combined and correctly
decoded. A “controlled” or deterministic amount of ISI.
– The “wave shape” can be used to define successive symbols.
– By decoding symbol pairs successively, correct symbol detection
can be performed.
– A trellis structure can be used!
• Examples and concept presented in textbook.
– Signal Generation Section 9.2-2: p. 609-612
– Signal Detection Section 9.2-3: p. 613-619
ECE 6640
46
Section 9.3
• When the channel is not simple, the following model
applies
• The received filter seeks to be an optimally matched filter
for the transmission AND remove the channel effects
introduced by the effective channel filter.
– It is a combination of the previously defined filter and an equalizer,
where the equalizer is equivalent to a reciprocal filter that “flattens
the channel” so that the channel and equalizer act like the “simple”
channel (time delay only).
ECE 6640
47
Section 9.3-1 Optimum
Maximum-Likelihood Receiver
• Following up on the concept of partial response signaling.
• If ISI is going to be present, can a Viterbi like decoder use
the ISI that exists over a limited number of symbols to
perform detection?
ECE 6640
48
Discrete-Time Model for Channel
•
•
•
In dealing with band-limited channels that result in ISI, it is convenient to develop an
equivalent discrete-time model for the analog (continuous-time) system.
Since the transmitter sends discrete-time symbols at a rate of 1/T symbols/s and the
sampled output of the matched filter at the receiver is also a discrete-time signal with
samples occurring at a rate of 1/T per second, it follows that the cascade of the analog
filter at the transmitter with impulse response g(t), the channel with impulse response
c(t), the matched filter at the receiver with impulse response h∗(−t), and the sampler can
be represented by an equivalent discrete-time transversal filter having tap gain
coefficients {xk}.
Consequently, we have an
equivalent discrete-time
transversal filter that spans a
time interval of 2LT seconds..
ECE 6640
49
Discrete Time Model Concern
•
•
The major difficulty with this discrete-time model occurs in the
evaluation of performance of the various equalization or estimation
techniques that are discussed in the following sections. The difficulty
is caused by the correlations in the noise sequence {νk} at the output
of the matched filter.
While many systems ignore this in simulation, a technique to eliminate
the correlation is developed in Section 9.3-1.
ECE 6640
50
Discrete Time Channel Characteristics
ECE 6640
51
General Equalizer Discussion
•
Wikipedia
– https://en.wikipedia.org/wiki/Equalization_%28communications%29
• See Digital Equalizer Types, in particular
– MMSE equalizer,
• Does not completely remove ISI
– Zero Forcing Equalizer
• Minimizes ISI but may increase noise
• Exclusively discussed in textbook
– Decision Feedback Equalizer
• previously mentioned based on an adaptive system
• Feedback attempts to remove ISI from previous symbols
– Blind Equalizer
ECE 6640
• based on symbol statistics alone
52
Decision Feedback Equalizer
ECE 6640
53
Equalization Notes from ECE 6950
• The following was derived for the adaptive systems class.
• Adaptive System Demonstration
– Matlab Adaptive_TB, chap5, equalizer
– Other examples developed for exam2 and the final.
ECE 6640
54