2.1 Fir filter design by frequency sampling or interpolation  (Page 2/3)

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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 $M=\left(N-1\right)/2$ , and where there is a frequency sample at $\omega =0$ is given as

${A}_{k}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{n=0}^{M-1}2h\left(n\right)cos\left(2\pi \left(M-n\right)k/N\right)+h\left(M\right).$

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

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\sum _{k=0}^{N-1}{e}^{-j2\pi Mk/N}{A}_{k}{e}^{j2\pi nk/N}$

or

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\sum _{k=0}^{N-1}{A}_{k}{e}^{j2\pi \left(n-M\right)k/N}.$

Because $h\left(n\right)$ is real, ${A}_{k}={A}_{N-k}$ and [link] becomes

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\left[{A}_{0}+\sum _{k=1}^{M-1}2{A}_{k}cos\left(2\pi \left(n-M\right)k/N\right)\right].$

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).

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

${A}_{k}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{n=0}^{N/2-1}2h\left(n\right)cos\left(2\pi \left(M-n\right)k/N\right)$

with the design formula of

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\left[{A}_{0}+\sum _{k=1}^{N/2-1}2{A}_{k}cos\left(2\pi \left(n-M\right)k/N\right)\right]$

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

$\omega \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}2\pi k/N,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}k=0,1,2,...,N-1$

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

$\omega \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left(2k+1\right)\pi /N,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}k=0,1,2,...,N-1$

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

${A}_{k}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{n=0}^{M-1}2h\left(n\right)cos\left(2\pi \left(M-n\right)\left(k+1/2\right)/N\right)+h\left(M\right)$

The design formula becomes

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\left[\sum _{k=0}^{M-1}2{A}_{k}cos\left(2\pi \left(n-M\right)\left(k+1/2\right)/N\right)+{A}_{M}cos\pi \left(n-M\right)\right]$

Type 2. even sampling

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

${A}_{k}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{n=0}^{N/2-1}2h\left(n\right)cos\left(2\pi \left(M-n\right)\left(k+1/2\right)/N\right)$

and the design formula

$h\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{N}\left[\sum _{k=0}^{N/2-1}2{A}_{k}cos\left(2\pi \left(n-M\right)\left(k+1/2\right)/N\right)\right]$

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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