Symbol Filtering
Based on Notes from ECE 6560
Multirate Signal Processing
Chapter 4
Dr. Bradley J. Bazuin
Western Michigan University
College of Engineering and Applied Sciences
Department of Electrical and Computer Engineering
1903 W. Michigan Ave.
Kalamazoo MI, 49008-5329
Chapter 4: Useful Classes of Filters
4.1 Nyquist Filter and Square-Root Nyquist Filter
4.2 The Communication Path
4.3 The Sampled Cosine Taper
4.3.1 Root-raised Cosine Side-lobe Levels
4.3.2 Improving the Stop-band Attenuation
4.4 Half-band Filters
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.
82
86
89
91
92
97
2
A Communication System
• Digital communication systems transmit a sequence of
symbols. Due to filtering the receiver filter response from
one symbol may overlaps that of another symbol, resulting
in intersymbol interference (ISI).
– ISI is a coherent error term that directly degrades our ability to
resolve the current symbol.
• The goal is to define digital filters that, when sampled at
the appropriate time, will zero any ISI.
– if not correctly sampled, there will be ISI.
– See: http://complextoreal.com/ by Charan Langton, Tutorial #14
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.
3
Demonstration of MPSK and MQAM
with Square Root Nyquist Simulations
• Advanced Digital Communication Tool Demo
– BER_Test_NyquistFilter
– BER_Test_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.
4
CW Communication with Noise
Model of a CW communication system with noise: Figure 10.1-1
x t   At   cos2  f c  t  t 
x c t  
vt  
A t 
 cos2  f c  t  t 
L
At 
 cos2  f c  t  t   n t 
L
 At 

 cos2  f c  t   t   nt   hR t 
Pr eDt   
 L

ECE 6560
5
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.
Digital Formatting and Transmission
ECE 6560
6
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.
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.
7
Regeneration of a unipolar signal
(a) signal plus noise (b) S/H output (c) comparator output: Figure 11.2-2
ECE 6560
Notes and figures are based on or taken from materials in the course textbook: fredric j. harris,
Bruce Carlson,
P.B. Crilly, Communication
Systems,
5th
ed., 2004. ISBN 0-13Multirate SignalA.Processing
for Communication
Systems, Prentice
Hall
PTR,
McGraw-Hill, 2010. 146511-2.
ISBN: 978-0-07-338040-7.
8
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.
9
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.
10
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.
11
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 0-13-146511-2.
12
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.
13
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
s t   nt 
filtered to
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
figures
are based
or taken
from materials
in thetextbook:
course textbook:
Probabilistic
Notes
andand
figures
are based
on oron
taken
from materials
in the course
fredric j. harris,
MultirateMethods
Signal
of Signal
and System
Analysis (3rd ed.)
by George
R.Hall
Cooper
Clare
D.0-13-146511-2.
McGillem; Oxford Press,
Processing
for Communication
Systems,
Prentice
PTR,and
2004.
ISBN
1999. ISBN: 0-19-512354-9.
14
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”
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
ECE 6560
Notes
and figures
are basedAnalysis
on or taken
in the
textbook:
fredric
j. harris, Multirate
of Signal
and System
(3rdfrom
ed.)materials
by George
R. course
Cooper
and Clare
D. McGillem;
OxfordSignal
Press,
Processing
for
Communication
Systems,
Prentice
Hall
PTR,
2004.
ISBN
0-13-146511-2.
1999. ISBN: 0-19-512354-9.
15
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)!
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
ECE 6560
Notes
and figures
are basedAnalysis
on or taken
in the
textbook:
fredric
j. harris, Multirate
of Signal
and System
(3rdfrom
ed.)materials
by George
R. course
Cooper
and Clare
D. McGillem;
OxfordSignal
Press,
Processing
for
Communication
Systems,
Prentice
Hall
PTR,
2004.
ISBN
0-13-146511-2.
1999. ISBN: 0-19-512354-9.
16
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”
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods
ECE 6560
Notes
and figures
are basedAnalysis
on or taken
in the
textbook:
fredric
j. harris, Multirate
of Signal
and System
(3rdfrom
ed.)materials
by George
R. course
Cooper
and Clare
D. McGillem;
OxfordSignal
Press,
Processing
for
Communication
Systems,
Prentice
Hall
PTR,
2004.
ISBN
0-13-146511-2.
1999. ISBN: 0-19-512354-9.
17
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.
18
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.
19
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.
20
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.
21
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.
22
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.
23
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.
24
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.
25
Symbol Periods Math (1)
st  

r t   s t   nt 
 d m h t  m  T 
1
m  
Filter received signal
y t   r t   h2 t 

y t    r t     h2    d 



 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.
26
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.
27
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.
28
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.
29
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.
30
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.
31
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.
32
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.
33
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.
34
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.
35
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.
36
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.
37
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.
38
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.
39
Generating Filter in Matlab
• See Chap4_1.m
• See Chap4_2.m
• See Chap4_3.m
• See Chap4_4.m
• See Chap4_5.m
ECE 6560
introduce nyquistfilt.m
introduce sqnyquistfilt2.m
nyquistfilt.m vs. firrcos.m
(vary alpha & length)
nyquistfilt.m vs. firrcos.m vs. rcosfir.m
(vary alpha & fixed length)
sqnyquistfilt2.m vs. square root firrcos.m
vs. square root rcosfir.m
(vary alpha & fixed length)
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
Contemplating the passed variables
• See Chap4_4.m for an example
• nyquistfilt
– Alpha, fsample/fsymbol (samples per symbol), 2*k symbols is
length of the filter (+1 so it is odd length)
• Firrcos
– N+1 filter length, alpha, fsample frequency, fsymbol/2 cutoff
frequency
• Rcosfir
– R=Alpha rolloff, T=1/fsample, rate = fsample/fsymbol (samples
per symbol), 2*k symbol length filter (+1 so it is odd)
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
Root-raised Cosine Side-lobe Levels
We commented earlier that when we implement the SQRT Nyquist
filter, we actually apply two windows; the first window is a smooth
continuous function used to control the transition bandwidth and the
second is a rectangle used to limit the impulse response to a finite
duration.
This second windowing forces side lobes in the spectrum of the
SQRT Nyquist filter. These side lobes are quite high, on the order
of 24 to 46 dB below the pass band gain depending on roll-off
factor and the length of the filter in number of symbols.
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.
42
Root-raised Cosine Side-lobe Levels
Chap4_4.m gets
different values?!
2*k*M filter+1 length (fsample=M*fsymbol and k symbols)
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.
43
Response Problems
•
The reason for the poor side-lobe response is the discontinuous first
derivative at the boundary between the half-cosine transition edge and
the start of the stop band. Consequently the envelope of the time
function falls off, as seen in Eqn. (4.18), as 1/t^2 enabling a significant
time discontinuity when the rectangle window is applied to the filter
impulse response.
•
“In retrospect, the cosine tapered Nyquist pulse was a poor choice for
the shaping and matched filter in communication systems.” p. 91
hSqrt  Nyq t  
ECE 6560
 4    f Sym  t  cos  1     f Sym  t   sin   1     f Sym  t 
f Sym  

2




1
4

f
t

f
t









Sym
Sym


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
Other Windows
•
Attempting to control the spectral side lobes by only applying other
windows to the weights of the prototype (sinc) filter results in significant
increase in the ISI levels at the receiver output.
•
This is illustrated in Figure 4.7, which illustrates the effect on spectral
side-lobes and ISI levels as a result of applying windows to the
prototype impulse response.
•
The increase in ISI is traced to the shift of the filter’s 3-dB point away
from the nominal band edge. The requirement for zero ISI at the output
of the matched filter requires that the shaping and matched filters each
exhibit 3-dB attenuation at the filter band edge, half the symbol rate.
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.
45
Other Windows
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.
46
Matlab code example
• Nyq_2aaTest.m
– Special f. harris functions
• NyquistTestv0.m
– Compare the number of symbols used (increased filter length)
• NyquistTestv1.m
– Validate Dr. Bazuin’s filter routines nyquistfilt.m as compared to
firrcos.m
• NyquistTestv2.m
– Validate f. harris nyq_4 filter routines (nyq_fharris) as compared to
firrcos.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.
47
Matlab code example (cont)
• NyquistTestv3.m
– Observing firrcos.m
• NyquistTestv4.m
– Testing nyquistfilt.m and nyquistfilt_even.m
• NyquistTest.m
– Testing nyquistfilt.m and sqnyquistfilt2.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.
48
Improved Nyquist Filter
• Software Defined Radio (SDR) Forum '05 Papers,
November 14-18, 2005 - Hyatt Regency - Orange County,
California
• An Improved Square-Root Nyquist Shaping Filter
– harris f., Chris Dick, S. Seshagir, Karl Moerder; San Diego State
University, Xilinx, Broadband Innovations
– http://groups.winnforum.org/d/do/2658
– This paper appears to be the original source for section 4.3.2.
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.
49
Recreating the Textbook
• See ISI-Sq_Nyq_2_Test.m
– Note: hh length is set by NN (odd), firrcos must be odd length
• Nyq_2aaTest.m compares nyq_fharris.m to new harris!
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.
50
MATLAB Comm Toolbox Demo
• See RCosTestDemo.m
– Raised cosine example
Direct FIR filtering
– Square Root Raised Cosine Transmit and Receive
Direct FIR filtering
– Polyphase Square Root Raised Cosine Transmit and Receive
MATLAB Toolbox Implementation
– Cost analysis using MATLAB Toolbox
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.
51