FIR%20Filter%20Design%20Using%20Windows

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FIR Filter Design Using Windows
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The FIR Filter Challenge

• We are given a filter specification
• Examples

p
p
0
0
low pass
high pass
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The FIR Filter Challenge

• We are given a filter specification
• Examples

p
p
0
0
band pass
band stop
Goal Find a filter which is a good
approximation of the specification
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Outline

• Goals
• Impulse Response truncation
• Using Windows
• Practical Filter Design

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Impulse Response Truncation -Reminder
Write the amplitude response
Choose filter type and order
Compute impulse response of ideal filter
Truncate the impulse response
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Example Band Pass Filter

• The desired impulse response
• An FIR filter of order N has the form

p
0
An approximation of Ad(q)
group delay -.5N
initial phase
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Choosing Filter Type and Order
IV III II I Type
Anti-symmetric Anti-symmetric symmetric symmetric Symmetry
Odd even odd even Order
.5p .5p 0 0 f

• Suppose we choose a type I filter
• The Ideal filter with phase is

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Computing the Impulse Response

• Recall that if
• The impulse response is
• Using linearity we get

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Truncating

• The impulse response of the filter is
• And we end up with an approximated filter

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Optimality

• The integral of the square error is
• Theorem
• For a given ideal frequency response and for a
• given order, the filter obtained by impulse
• response truncation has a minimum error
• among all causal FIR filters of the given order.

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The Gibbs Phenomenon

• Peaks near the transition band
• Both within and outside the pass band
• Peaks height 0.09, regardless of the filters
  order
• Become narrower as the order gets larger

N10
N40
N100
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Other Optimality Criteria -Reminder
1dp
Transition band
pass band ripple
ds
qs
qp
Pass band
Stop band
Given the filters order, our aim is to minimize
dp, ds and qs -qp
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A Deeper Look At Gibbs Phenomenon

• Rewriting the truncated impulse response
• where
• Therefore, by the modulation theorem
• If Wr was the delta function -gt an ideal filter

The DTFT of wr
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The DTFT of a Rectangular Window

• Suppose wr is a window with length N .
• The fourier transform of wr is
• The function
• Is called the Dirichlet kernel

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Properties of Dirichlet Kernel
D(0,N)N
Main lobe region between two zeros near the
origin
zeros at q2mp/N m is integer
Nearset zeros to the origin at q2p/N
Side lobes regions between adjacent zeros
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Understanding Gibbs

• Main lobe width determineDistance of peaks from
  transition point
• Side lobe level
• The ratio between main lobe and next lobes
  height
• Roughly Determines peak height
• 2/(3p), or about -13.5db in Dirichlet kernel

Can we do better?
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Outline

• Goals
• Impulse Response truncation
• Using Windows
• Practical Filter Design

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FIR filter Design Using Windows

• Define the ideal frequency response Hd (q)
• For each pair qp,qs take the midpoint
  0.5(qpqs)
• Obtain the ideal impulse response hd n as in
  the IRT method
• Compute coefficients

qs
qp
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Criteria for Choosing a Window

• Given the desired order of the FIR filter,
• we would like the kernel function to have
• A main lobe that is as narrow at possible
• Side lobes that are as low as possible

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Reducing Side Lobes by Squaring

• The side lobe level of Dirichlet kernel may be
  reduced by squaring it
• The corresponding window
• Window size is nearly doubled
• To maintain Filters order we need a smaller
  window

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Bartlett Window

• Defined as a convolution of rectangular windows
• Size of rectangular window 0.5(N-1)
• Normalized

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Kernel of Bartlett Window

• For an odd N
• The kernel function becomes

Bartlett
Dirichlet
Side lobe level -27db
Side lobe level -13.5db
Main lobe width 4p/N
Main lobe width 8p/(N1)
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The Window Design Challenge

• Choosing a window always involves a trade-off
  between
• the width of the main lobe and the level of the
  side lobes
• The rectangular window
• Narrowest possible main lobe of all windows of
  the same length
• But its side lobes are the highest
• We are ready to increase the main-lobe width in
  order to reduce the side lobe level
• As a consequence we increase transition band
  width in order to reduce the pass band ripple
• Our aim choose a window with a good trade-off

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Window Representation - Rectangular
Time Domain Plot
Frequency Magnitude
FIR Filter Magnitude
1.09
-13.5db
Ripple 1
4p/L
Side lobe level
Transition band width (L is the length of the
window)
Amplitude response of an FIR filter based on the
window near a discontinuity point
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Window Representation - Bartlett
Time Domain Plot
Frequency Magnitude
FIR Filter Magnitude
.95
-27db
1 - Ripple
8p/L
Side lobe level
Transition band width (L is the length of the
window)
Amplitude response of an FIR filter based on the
window near a discontinuity point
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Hann Window (aka Hanning)
Time Domain Plot
Frequency Magnitude
FIR Filter Magnitude
1.0063
.95
-32db
8p/L

• Constructed by combining three Dirichlet
  Kernels W(q)0.5D(q,N)0.25D(q-2p/(N-1),N)0
  .25D(q2p/(N-1),N)
• The resulting window sequence

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Hamming Window
Time Domain Plot
Frequency Magnitude
FIR Filter Magnitude
1.0022
.95
-43db
8p/L

• Constructed by combining three Dirichlet
  Kernels W(q)0.54D(q,N)0.23D(q-2p/(N-1),N)
  0.23D(q2p/(N-1),N)
• The resulting window sequence

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Blackman Window
Time Domain Plot
Frequency Magnitude
FIR Filter Magnitude
1.0002
.95
-57db
12p/L

• Constructed by combining five Dirichlet
  Kernels
• lower lobes at the cost of larger main lobe
  width

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Optimal Windows

• Previous windows were derived by intuition and
  educated guess
• Modern windows are based on optimality criteria
• Kaisers criterion
• Minimize the width of the main lobe kernel
• Constraints
• Window length should be fixed
• Energy in the side lobes level do not exceed a
  percentage of total energy
• Dolph criterion
• Similar to Kaisers, except
• Energy in side lobes do not exceed a given
  maximum value

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Windows Comparison
Band pass ripple, dp Side lobe level, db Main lobe width Window
0.09 -13.5 4p/L Rectangular
0.05 -27 8p/L Bartlett
0.0065 -32 8p/L Hann
0.0022 -43 8p/L Hamming
0.0002 -57 12p/L Blackman
Depends on specific parameters Depends on specific parameters Depends on specific parameters Kaiser
Depends on specific parameters Depends on specific parameters Depends on specific parameters Dolph
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Summary of Design by Windows

• FIR design by impulse response truncation is
  optimal on the average
• However, it is not optimal given constrained
  criteria
• Windowing allows to tradeoff transition band
  width and pass band ripple
• Specific window should be chosen according to
  desired constraints
• Recall that this is not the optimal method

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Outline

• Goals
• Impulse Response truncation
• Using Windows
• Practical Filter Design

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Designing a Filter

• We can use the function fir1
• theta-pi2pi/1023pi
• tf fir1(40,0.5,'low',rectwin(41))
• plot(theta,fftshift(abs(fft(tf,1024))))

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Demonstrating Gibbs Phenomenon

• tf1 fir1(40,0.5,'low',rectwin(41))
• tf2 fir1(100,0.5,'low',rectwin(101))
• plot(x,fftshift(abs(fft(tf1,1024))),'b','LineWidth
  ',2)
• hold on
• plot(x,fftshift(abs(fft(tf2,1024))),'g','LineWidth
  ',2)

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Using Windowing

• tf1 fir1(40,0.5,'low',hamming(41))
• plot(x,fftshift(abs(fft(tf1,1024))),'b','LineWidth
  ',2)
• tf2 fir1(40,0.5,'low',blackman(41))
• hold on
• plot(x,fftshift(abs(fft(tf2,1024))),'k','LineWidth
  ',2)

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Using Least Square Design

• lsffirls(40,0 0.5 0.5001 1,1 1 0 0)
• plot(x,fftshift(abs(fft(lsf,1024))),'b','LineWidth
  ',2)
• hold on
• lsf2firls(40,0 0.5 0.6 1,1 1 0 0)
• plot(x,fftshift(abs(fft(lsf2,1024))),'k','LineWidt
  h',2)

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Usign Remez Exchange

• ReFremez(40,0 0.5 0.6 1,1 1 0 0)
• plot(x,fftshift(abs(fft(ReF,1024))),'k','LineWidth
  ',2)

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Applying a Filter

• tfh fir1(80,0.1,'low',hamming(81))
• outx filter (tfh,y)
• soundsc(y,fs)
• soundsc(outx,fs)
• freqz(y)
• freqz(outx)

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Beyond Low-pass

• lsffirls(500,0 0.4 0.4001 0.6 0.6001 0.7 0.7001
  1,
• 0 0 5 5 10 10 0 0)
• outx filter(lsf,1,y)
• plot(x,fftshift(abs(fft(lsf,1024))),'b','LineWidth
  ',2)
• soundsc(outx,fs)
• plot(outx,fftshift(abs(fft(lsf,1024))),'b','LineWi
  dth',2)

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