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"Notes on the Design of Optimal FIR Filters" Appendix C

Module by: John Treichler. E-mail the author

Using a Chebyshev Polynomial to Estimate

We desire that the oscillatory portion of the polynomial shown in Figure 1 in the module titled "Filter Sizing" correspond to the stopband region of the filter response and the xMxM portion to correspond to the transition from the stopband to the passband. This is achieved by employing a change of variables from frequency ff to the polynomial argument xx:

x = ( x 0 + 1 2 ) c o s ( 2 π f f s ) + ( x 0 - 1 2 ) . x = ( x 0 + 1 2 ) c o s ( 2 π f f s ) + ( x 0 - 1 2 ) .
(1)

While many different types of variable changes could be employed, this one matches the boundary conditions (an obvious requirement) but happens to employ the cosine function, a member of the same family used to define the Chebyshev polynomials.

With this change of variables we see that the transition band ΔfΔf is defined by the difference between x=1x=1 and x=xpx=xp. Using the closed, but nonintuitive form of the K-th order Chebyshev polynomial, valid for |x|>1|x|>1, we have that

P K ( x ) = c o s h ( K · c o s h - 1 ( x ) ) P K ( x ) = c o s h ( K · c o s h - 1 ( x ) )
(2)

To synthesize the desired impulse response using this windowing technique we multiply the resulting window function by the sampled sinc function. In this case, however, we desire that the cutoff frequency be as low as possible, limiting at zero Hz. The associated sinc function equals unity for all non-zero coefficients of the impulse response. Since the final impulse response is the point-by-point product of the window and the sampled sinc function, in this case the window itself is the resulting impulse response. It suffices then to examine the properties of the N-th order Chebyshev polynomial to see how the N-point optimal filter will behave.

To find the relationship between the required filter order NN and the attainable transition band ΔfΔf, we first determine the proper value of KK and then evaluate Equation 2 at the known combinations of xx and PK(x)PK(x). To select KK we note that all but one of the ripples in the polynomial's response are used in the stopband and these are split evenly between the positive and negative frequencies. Thus a filter and window of order NN implies a Chebyshev polynomial of order

K = N - 1 2 K = N - 1 2
(3)

With this resolved we observe from Figure 1 in the module titled "Filter Sizing" that

P N - 1 2 ( 1 ) = 1 P N - 1 2 ( 1 ) = 1
(4)
P N - 1 2 ( x p ) = 1 - δ 1 δ 2 P N - 1 2 ( x p ) = 1 - δ 1 δ 2
(5)
P N - 1 2 ( x 0 ) = 1 + δ 1 δ 2 P N - 1 2 ( x 0 ) = 1 + δ 1 δ 2
(6)

These equations are manipulated to yield an expression for xpxp. Equation 1 is then used to obtain values for fstfst, corresponding to x=1x=1, and fcfc, corresponding to x=xpx=xp. Their difference, defined earlier to be the transition band ΔfΔf, is then given by

Δ f = f s π ( N - 1 ) [ c o s h - 1 ( 1 + δ 1 δ 2 ) - { ( c o s h - 1 ( 1 + δ 1 δ 2 ) ) 2 - ( c o s h - 1 ( 1 - δ 1 δ 2 ) ) 2 } 1 2 ] . Δ f = f s π ( N - 1 ) [ c o s h - 1 ( 1 + δ 1 δ 2 ) - { ( c o s h - 1 ( 1 + δ 1 δ 2 ) ) 2 - ( c o s h - 1 ( 1 - δ 1 δ 2 ) ) 2 } 1 2 ] .
(7)

Under suitable conditions this equation can be simplified considerably. For example, in the limits of small δ1δ1 and large NN, Equation 7 reduces to Equation 4 in the module titled "Filter Sizing".

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