Fir digital filters  (Page 5/7)

Examples of the four types of linear-phase FIR filters with the symmetries for odd and even length are shown in [link] . Note that for $N$ odd and $h\left(n\right)$ odd symmetric, $h\left(M\right)=0$ .

For the analysis or design of linear-phase FIR filters, it is necessary to know the characteristics of $A\left(\omega \right)$ . The most important characteristics are shown in [link] .

Examples of the amplitude function for odd and even length linear-phase filter $A\left(\omega \right)$ are shown in [link] .

These characteristics reveal several inherent features that are extremely important to filter design. For Types 3 and 4, $A\left(0\right)=0$ for any choice of filter coefficients $h\left(n\right)$ . This would not be desirable for a lowpass filter. Types 2 and 3 alwayshave $A\left(\pi \right)=0$ which is not desirable for a highpass filter. In addition to the linear-phase characteristic that represents atime shift, Types 3 and 4 give a constant 90-degree phase shift, desirable for a differentiator or Hilbert transformer. The firststep in the design of a linear-phase FIR filter is the choice of the type most compatible with the specifications.

It is possible to uses the formulas to express the frequency response of a general complex or non-linear phase FIR filter by taking theeven and odd parts of $h\left(n\right)$ and calculating a real and imaginary “amplitude" that would be added to give the actual frequency response.

Calculation of fir filter frequency response

As shown earlier, $L$ equally spaced samples of $H\left(\omega \right)$ are easily calculated for $L>N$ by appending $L-N$ zeros to $h\left(n\right)$ for a length-L DFT. This appears as

This direct method of calculation is a straightforward and flexible approach. Only the samples of $H\left(\omega \right)$ that are of interest need to be calculated. In fact, even nonuniform spacingof the frequency samples can be achieved by sampling the DTFT defined in [link] . The direct use of the DFT can be inefficient, and for linear-phase filters, it is $A\left(\omega \right)$ , not $H\left(\omega \right)$ , that is the most informative. In addition to the direct application of theDFT, special formulas are developed in Equation 5 from FIR Filter Design by Frequency Sampling or Interpolation for evaluating samples of $A\left(\omega \right)$ that exploit the fact that $h\left(n\right)$ is real and has certain symmetries. For long filters, even these formulasare too inefficient, so the DFT is used, but implemented by a Fast Fourier Transform (FFT) algorithm.