FIR - Recapitulation

Nomenclature With a₀ = 1 in the linear constant coefficient difference equation,

a₀ y(n) + a₁ y(n-1) + ... + a_N y(n-N) = b₀ x(n) + b₁ x(n-1) + ... + b_M x(n-M),        a₀ ≠ 0

we have,

That shows an IIR filter if at least one of a₁ goes through a_N is nonzero, and all the roots of the denominator are not cut exactly by the roots of the numerator. Usually, there are M finite zeros and N finite poles. There is no limitation that M could be less than or greater than or similar to N. In each and every case, especially digital filters formed from analog designs, M ≤ N. Systems of that type are known as N^th order systems. That is the type with IIR filter design.

When M > N, the order of the process is no longer unambiguous. In that case, H(z) can be taken to be an Nth order system in master slave with an FIR filter of order (M - N).

When N = 0, as in the case of an FIR filter, related to our convention the order is 0. However, it is more meaningful in that case to focus on M and call the filter an FIR filter of M stages or (M+1) coefficients.

Example The system H(z) = (1 - z^-8 )/(1 - z^-1 ) is an FIR filter. Why (verify)?

An FIR filter then has only single the "b" coefficients and all the "a" coefficients (except a0 which equals 1) are zero. A question is the three-term going average filter y(n) = (1/3) x(n) + (1/3) x(n-1) + (1/3)x(n-2). Usually the difference equation of an FIR filter may be defined

There are (M + 1) coefficients; some take only M coefficients. That relation defines a nonrecursive implementation. Its impulse response h(n) is build up of the coefficients {b_r} = {b₀, b₁, ..., b_M}

Equivalently, the finite length impulse response may also be described in the form of a weighted sum of d functions.

The difference equation (1) is also similar to a direct convolution of the input and the impulse response:

where we have written br as b(r), i.e., the subscript in br is written as an index in b(r). The transfer method H(z) of the FIR filter may be calculated either from the difference equation or from the impulse response h(n):

The transfer method has M nontrivial zeros and an Mth order (trivial) pole at z = 0.  That is taken as an all-zero system.

We may calculate the frequency response H (e^jw ) or H(w) of the FIR filter either from H(z) as 

H (e^jw ) = H ( z)|_z = ejw

or, from the impulse response, h(n), as the discrete-time Fourier transform (DTFT) of h(n):

The inverse DTFT of H(ω) is of course the impulse response, shown as

The simple design problem is to calculate the impulse response h(n), or, the coefficients br, for r = 0 to M, needed to achieve a desired H(w). These coefficients are of course the type of constants that seems in the numerator of the transfer method H(z). The several transformations needed in IIR filter design may not be used here since they normally yield IIR functions, i.e., with both denominator and numerator coefficients.

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