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Lecture 11: FIR Filter DesignsXILIANG LUO
2014/11
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WindowingDesired frequency response:
Fourier series for a periodic function with period 2pi
Convergence of the Fourier series
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Windowing
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Windowing
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WindowingRectangular window:
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Common Windows
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Common Windows
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Common Windows
Rectangular Window
M=50
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Common Windows
Hamming Window
M=50
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Common Windows
Blackman Window
M=50
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Comparisons
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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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Kaiser Window
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Optimal FIR FilterDesign Type-1 FIR filter:
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Optimal FIR Filter
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Optimal FIR FilterParks-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]
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Optimal FIR FilterParks-McClellan, MinMax criterion:
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Optimal FIR Filter
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Parks-McClellanAlternation theorem gives necessary and sufficient conditions on theerror for optimality in the Chebyshev or minimax sense!
Optimal FIR should satisfy:
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Parks-McClellan
2(L+2) unknowns are two alternation frequencies
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Parks-McClellanGiven set of the extremal frequencies, we can have:
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Parks-McClellanGiven set of the extremal frequencies, we can have:
Evaluate on other frequencies
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Parks-McClellan
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Flow Chart ofParks-McClellen
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0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
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