1. No category

ECE 6560
Quadrature Mirror Filters
&
Project Simulations
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
Filter Bank Processing
Observing Narrowband and Reforming Wideband
Significant filter restriction are required if the output is required to approximate the input!
Quadrature Mirror Filter Definition and Requirements
h0 m 
y0 m 
x0 m 
h0 m 
h1 m 
y1 m 
x1 m 
h1 m 
xn 
y n 
h2 m 
hM 1 m 
Analysis
ECE 6560
IFFT
y2 m 
x2 m 
y M 1 m  xM 1 m 
FFT
h2 m 
hM 1 m 
Synthesis
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.
2
Filter Bank Equations
• Analysis
  k    1






yk m    exp j 2 
h
r
x
m
r



  


M


0

 r 0
M 1
• Synthesis (substitute –t and L=M)
 

t m
xˆ s  M  t    ht r    exp  i  2 
  ym s  r 
M 

r 0
m 0 

 1
M 1
• Inverse mathematical processes
– IFFT with inverse FFT
– Broadband to narrowband and back to broadband
– Filters are critical for overall performance.
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
Perfect Reconstruction Applications
• Frequency domain filtering or equalization
• Time-Spectral Analysis with reconstruction
• Arbitrarily take signals apart and then reconstruct them
– Partial-Band Synthesis to one or more arbitrary bandwidths
(universal base station receiver)
– Partial-Band Analysis with frequency domain summation and fullband synthesis (universal base station transmitter)
– Applications: cellular telephone base stations, satellite relay
stations, etc.
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
Quadrature Mirror Filters
• Architecturally, the structure is mirrored across the central
axis in terms of the analysis and synthesis processing
• Perfect Reconstruction of the input signal is desired, but …
– Quasi-Perfect is usually accomplished
– Look for PR-QMF or Quasi-PR QMF
• The QMF operation is typically described in terms of two
filters
H 0 Z 
2
w0 n 
2
H 0 Z 
xˆ n 
xn 
H 1 Z 
ECE 6560
2
w1 n 
2
H 1 Z 
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.
5
2-Bank QMF
H 0 Z 
2
w0 n 
2
H 0 Z 
xˆ n 
xn 
H 1 Z 
2
w1 n 
2
H 1 Z 
• Perfect Reconstruction Conditions
1
Xˆ  z    H 0  z   H 0 z   H1 z   H1 z  X  z 
2
1
  H 0  z   H 0  z   H1  z   H1 z  X  z 
2
1
H1 z   H 0  
z
1
Xˆ  z    H 0  z   H 0 z   H 0  z   H 0  z  X  z 
2
1
  H 0  z   H 0  z   H 0  z   H 0  z  X  z 
2
1
Xˆ z    H 0 z   H 0  z   H 0  z   H 0  z  X  z 
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.
6
2-Bank QMF
• Perfect Reconstruction Conditions
1
Xˆ z    H 0 z   H 0  z   H 0  z   H 0  z  X  z 
2
H1  z   H 0  z 
T w   H 0 w   H 0 w     1
  1
H0  
2 2
Error Mag 1
Error Mag 1b
• Additional Conditions (optimization)
H 0 w  1,
for 0  w  wPassband 

2
H 0 w  0,
for     w  
Error Mag 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.
7
Project Assignment Discussion
1.
Find a paper describing the implementation of a filter that can be used in a
Quadrature Mirror Filter
•
•
2.
Implement the filter using MATLAB
•
•
3.
Look in IEEE Transactions on Circuits and Systems II: Analog and Digital
Signal Processing
Dr. Bazuin must approve of all papers/filters selected.
Provide a function that generates the filter coefficients based on a desired set
of input criteria (i.e. filtercoef=Name(inputparameters))
Verify that you have generated the correct filter coefficients by comparing
them to values or curves provided in the paper.
Use the QMF analysis-synthesis MATLAB script that will be developed
or provided in class to characterize the results of your filter.
•
•
ECE 6560
Post-analysis filter ripple, bandwidth, stopbands, etc.
Post-synthesis input to output error, error frequency response, etc.
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.
8
Project Test Code
•
Multi-band/bank operation – 64 in the code provided.
– Mx where M = 64 and  = 4, 8, 12, or 16 should be sufficient
– Sinc function interpolation of smaller filters up to Mx where necessary
(used in project code - qmf_NxM(weights,win_size))
•
Operate analysis bank to provide twice as many samples as a critically
sampled filter bank.
– Nyquist rate of output supports bandwidth better.
– Filter can be ½ of output sample rate
– Limit aliasing and avoid synthesis cancellation requirements
•
Operate Synthesis bank at ½ the critical sample synthesis rate.
– Nyquist rate of bandlimited input supported.
– No requirement for adjacent band cancellation or “unwrapping” an alias
term.
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.
9
Performance Comparison
• Execute the random sample test and determine your signalto-error ratio from the results.
– Insure that your filter can be loaded and used in chan_filt
– Can you perform better than the “4 weight” or 8-coefficient type 2
FIR filters provided ? (e.g. better than 53.5 dB)
• Can you generate the same filter values that are present in
the paper?
• Can you plot the same performance curves?
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