FIR Filter Design Procedure

1.   select whether H_R(ω) or H_I(ω) is to be set similar to zero. Typically,

• H_d(ω) = H_R(ω) + j 0 for filtering, and
• H_d(ω) = 0 + j H_I(ω) for differentiators, integrators and Hilbert transformers

2.   illustrate H_d(ω) into Fourier series h_d(n). That is the needed impulse response.

3.   Decide on the size N of the impulse response duration. Change the sequence h_d(n) to N samples {h_t(n), n = - (N-1)/2 to (N-1)/2}. Even numbers of N result in delays of half-sample periods; even value or odd values of N avoid that problem.

4.   Apply window method {ω(n), n = - (N-1)/2 to (N-1)/2}

5.   Search the transfer method H(z) = z-(N-1)/2 H_t(z) and the frequency response H(ω). If not satisfactory the value of N can have to be increased or a different window method can be tried.

Phase delay and group delay If we take a signal that has of various frequency components (such as a modulated signal or a speech waveform) the phase delay of the filter is the value of time delay each frequency element of the signal suffers in taking through the filter. Mathematically, the phase delay Τp is provided by secant

The group delay on the different hand is the average time delay the computes signal suffers at each frequency. The group delay τg is provided by the slope (tangent) at ω

where Θ(ω) = ∠H (e^jw ) of the filter.

A nonlinear phase characteristic can cause phase distortion, which is not desirable in several applications, for example, data transmission, music, video and biomedicine.

A filter is called to have a linear phase response if its phase response computes one of the given relationships:

where k and β are constants. If a filter computes equation (A) its set delay and phase delay are the similar constant k. It may be provided that for condition (A) to be calculated the impulse response of the filter has to be positive symmetry (aka even symmetry or just symmetry). The phase response in that case is simply a function of the filter size N:

If equation (B) is satisfied the filter will have a constant group delay only.  In this case, the impulse response h(n) has negative symmetry (aka odd symmetry or antisymmetry):

h(n) = - h(N-1-n)

k = (N-1)/2

β = Π /2

Analog filter background of phase and group delay Phase delay "at a provided frequency" is the slope of the secant line from dc to the specific frequency and is a kind of overall average delay parameter. Phase delay is calculated over the frequency range showing the major part of the input signal spectrum.

The group delay at a provided frequency shows the slope of the tangent line at the particular frequency and shows a local or narrow size of range (neighborhood of F₁ in the figure) delay parameter.

A type of significance adding both phase delay and group delay is that of a narrow band modulated signal. When a narrow band calculated signal is gone through a filter, the carrier is delayed by a time similar to the phase delay, while the envelope (or intelligence) is delayed by a time approximately same to the group delay. Since the intelligence (calculating signal) shows the desired data contained in such signals, strong emphasis on good set delay characteristics is often prepared in filters designed for processing calculated waveforms.

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