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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

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