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Type 1. odd sampling

Samples of the frequency response Equation 29 from FIR Digital Filters for the filter where N is odd, L = N , and M = ( N - 1 ) / 2 , and where there is a frequency sample at ω = 0 is given as

A k = n = 0 M - 1 2 h ( n ) cos ( 2 π ( M - n ) k / N ) + h ( M ) .

Using the amplitude function A ( ω ) , defined in Equation 28 from FIR Digital Filters , of the form [link] and the IDFT [link] gives for the impulse response

h ( n ) = 1 N k = 0 N - 1 e - j 2 π M k / N A k e j 2 π n k / N

or

h ( n ) = 1 N k = 0 N - 1 A k e j 2 π ( n - M ) k / N .

Because h ( n ) is real, A k = A N - k and [link] becomes

h ( n ) = 1 N [ A 0 + k = 1 M - 1 2 A k cos ( 2 π ( n - M ) k / N ) ] .

Only M + 1 of the h ( n ) need be calculated because of the symmetries in Equation 27 from FIR Digital Filters .

This formula calculates the impulse response values h ( n ) from the desired frequency samples A k and requires M 2 operations rather than N 2 . An interesting observation is that not only are [link] and [link] a pair of analysis and design formulas, they are also a transform pair. Indeed, they are of thesame form as a discrete cosine transform (DCT).

Type 2. odd sampling

A similar development applied to the cases for even N from Equation 36 from FIR Digital Filters gives the amplitude frequency response samples as

A k = n = 0 N / 2 - 1 2 h ( n ) cos ( 2 π ( M - n ) k / N )

with the design formula of

h ( n ) = 1 N [ A 0 + k = 1 N / 2 - 1 2 A k cos ( 2 π ( n - M ) k / N ) ]

which is of the same form as [link] , except that the upper limit on the summation recognizes N as even and A N / 2 equals zero.

Even sampling

The schemes just described use frequency samples at

ω = 2 π k / N , k = 0 , 1 , 2 , . . . , N - 1

which are N equally-spaced samples starting at ω = 0 . Another possible pattern for frequency sampling that allows designformulas has no sample at ω = 0 , but uses N equally-spacedsamples located at

ω = ( 2 k + 1 ) π / N , k = 0 , 1 , 2 , . . . , N - 1

This form of frequency sampling is more difficult to relate to the DFT than the sampling of [link] , but it can be done by stretching A k and taking a 2N-length DFT [link] .

Type 1. even sampling

The two cases for odd and even lengths and the two for samples at zero and not at zero frequency give a total of fourcases for the frequency-sampling design method applied to linear- phase FIR filters of Types 1 and 2, as defined in the section Linear-Phase FIR Filters . For the case of an odd length and no zero sample, the analysisand design formulas are derived in a way analogous to [link] and [link] to give

A k = n = 0 M - 1 2 h ( n ) cos ( 2 π ( M - n ) ( k + 1 / 2 ) / N ) + h ( M )

The design formula becomes

h ( n ) = 1 N [ k = 0 M - 1 2 A k cos ( 2 π ( n - M ) ( k + 1 / 2 ) / N ) + A M cos π ( n - M ) ]

Type 2. even sampling

The fourth case, for an even length and no zero frequency sample, gives the analysis formula

A k = n = 0 N / 2 - 1 2 h ( n ) cos ( 2 π ( M - n ) ( k + 1 / 2 ) / N )

and the design formula

h ( n ) = 1 N [ k = 0 N / 2 - 1 2 A k cos ( 2 π ( n - M ) ( k + 1 / 2 ) / N ) ]

These formulas in [link] , [link] , [link] , and [link] allow a very straightforward design of the four frequency-sampling cases. Theyand their analysis companions in [link] , [link] , [link] , and [link] also are the four forms of discrete cosine and inverse-cosine transforms. Matlabprograms which implement these four designs are given in the appendix.

Type 3. odd sampling

The design of even-symmetric linear-phase FIR filters of Types 1 and 2 in the section Linear-Phase FIR Filters have been developed here. A similar development for the odd-symmetric filters, Types 3 and 4, can easily be performed with theresults closely related to the discrete sine transform. The Type 3 analysis and design results using the frequency sampling scheme of [link] are

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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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