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

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 $\omega =0$ 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).

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

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.

Even sampling

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] .

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

The design formula becomes

Type 2. even sampling

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.

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