2.1 Fir filter design by frequency sampling or interpolation

Since samples of the frequency response of an FIR filter can be calculated by taking the DFT of the impulse response $h\left(n\right)$ , one could propose a filter design method consisting of taking the inverse DFT of samples of adesired frequency response. This can indeed be done and is called frequency sampling design . The resulting filter has a frequency response that exactly interpolates the given samples, but there is no explicit controlof the behavior between the samples [link] , [link] .

Three methods for frequency sampling design are:

1. Take the inverse DFT (perhaps using the FFT) of equally spaced samples of thedesired frequency response. Care must be taken to use the correct phase response to obtain a real valued causal $h\left(n\right)$ with reasonable behavior between sample response. This method works for general nonlinear phasedesign as well as for linear phase.
2. Derive formulas for the inverse DFT which take the symmetries, phase, and causality into account. It is interesting to notice these analysis and designformulas turn out to be the discrete cosine and sine transforms and their inverses.
3. Solve the set of simultaneous linear equations that result from calculating the sampledfrequency response from the impulse response. This method allows unevenly spaced samples of the desired frequency response but the resulting equationsmay be ill conditioned.

Frequency sampling filter design by inverse dft

The most direct frequency sampling design method is to simply take the inverse DFT of equally spaced samples of the desired complex frequencyresponse $H_d\left({\omega }_k\right)$ . This is done by

where care must be taken to insure that the real and imaginary parts (or magnitude and phase) of $H_d\left({\omega }_k\right)$ satisfy the symmetry conditions that give a real, causal $h\left(n\right)$ . This method will allow a general complex $H\left(\omega \right)$ as well as a linear phase. In most cases, it is easier to specify proper and consistentsamples if it is the magnitude and phase that are set rather than the real and imaginary parts. For example, it is important that the desiredphase be consistent with the specified length being even or odd as is given in Equation 28 from FIR Digital Filters and Equation 24 from FIR Digital Filters .

Since the frequency sampling design method will always produce a filter with a frequency response that interpolates the specified samples, theresults of inappropriate phase specifications will show up as undesired behavior between the samples.

Frequency sampling filter design by formulas

When equally spaced samples of the desired frequency response are used, it is possible to derive formulas for the inverse DFT and, therefore,for the filter coefficients. This is because of the orthogonal basis function of the DFT. These formulas can incorporate the various constraints of a real $h\left(n\right)$ and/or linear phase and eliminate the problems of inconsistency in specifying $H\left({\omega }_k\right)$ .

To develop explicit formulas for frequency-sampling design of linear-phase FIR filters, a direct use of the inverse DFT is most straightforward. When $H\left(\omega \right)$ has linear phase, [link] may be simplified using the formulas for the four types of linear-phase FIR filters.