Lecture 11:
FIR Filter Designs
XILIANG LUO
2014/11
1
Windowing
Desired frequency response:
Fourier series for a periodic function with period 2pi
Convergence of the Fourier series
2
Windowing
3
Windowing
4
Windowing
Rectangular window:
5
Common Windows
6
Common Windows
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Common Windows
M=50
Rectangular Window
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Common Windows
M=50
Hamming Window
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Common Windows
M=50
Blackman Window
10
Comparisons
11
Kaiser Window
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Kaiser Window
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Kaiser Window
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Kaiser Window
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Kaiser Window
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Optimal FIR Filter
Design Type-1 FIR filter:
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Optimal FIR Filter
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Optimal FIR Filter
Parks-McClellan algorithm is based on the reformulating the filter
design problem as a problem in polynomial approximation.
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Optimal FIR Filter
Approx. Error:
only defined in interested subintervals of [0, pi]
20
Optimal FIR Filter
Parks-McClellan, MinMax criterion:
21
Optimal FIR Filter
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Parks-McClellan
Alternation theorem gives necessary and sufficient conditions on the
error for optimality in the Chebyshev or minimax sense!
Optimal FIR should satisfy:
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Parks-McClellan
2(L+2) unknowns
𝜔𝑝 , 𝜔𝑠 are two alternation frequencies
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Parks-McClellan
Given set of the extremal frequencies,
we can have:
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Parks-McClellan
Given set of the extremal frequencies,
we can have:
Evaluate on other frequencies
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Parks-McClellan
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Flow Chart of
Parks-McClellen
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1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
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