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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 $\omega =0$ is given as
Using the amplitude function $A\left(\omega \right)$ , defined in Equation 28 from FIR Digital Filters , of the form [link] and the IDFT [link] gives for the impulse response
or
Because $h\left(n\right)$ is real, ${A}_{k}={A}_{N-k}$ and [link] becomes
Only $M+1$ of the $h\left(n\right)$ need be calculated because of the symmetries in Equation 27 from FIR Digital Filters .
This formula calculates the impulse response values $h\left(n\right)$ 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).
A similar development applied to the cases for even $N$ from Equation 36 from FIR Digital Filters gives the amplitude frequency response samples as
with the design formula of
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.
The schemes just described use frequency samples at
which are $N$ equally-spaced samples starting at $\omega =0$ . Another possible pattern for frequency sampling that allows designformulas has no sample at $\omega =0$ , but uses $N$ equally-spacedsamples located at
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] .
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
The design formula becomes
The fourth case, for an even length and no zero frequency sample, gives the analysis formula
and the design formula
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.
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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