<Dd> H 2 π (ω) = H ^ (z) z = e j ω = H ^ (e j ω). (\ displaystyle H_ (2 \ pi) (\ omega) = \ left. (\ widehat (H)) (z) \, \ right _ (z = e ^ (j \ omega)) = (\ widehat (H)) (e ^ (j \ omega)).) </Dd> <P> An FIR filter is designed by finding the coefficients and filter order that meet certain specifications, which can be in the time domain (e.g. a matched filter) and / or the frequency domain (most common). Matched filters perform a cross-correlation between the input signal and a known pulse shape . The FIR convolution is a cross-correlation between the input signal and a time - reversed copy of the impulse response . Therefore, the matched filter's impulse response is "designed" by sampling the known pulse - shape and using those samples in reverse order as the coefficients of the filter . </P> <P> When a particular frequency response is desired, several different design methods are common: </P> <Ol> <Li> Window design method </Li> <Li> Frequency Sampling method </Li> <Li> Weighted least squares design </Li> <Li> Parks - McClellan method (also known as the Equiripple, Optimal, or Minimax method). The Remez exchange algorithm is commonly used to find an optimal equiripple set of coefficients . Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order N. The algorithm then finds the set of (N + 1) (\ displaystyle \ scriptstyle (N \, + \, 1)) coefficients that minimize the maximum deviation from the ideal . Intuitively, this finds the filter that is as close as you can get to the desired response given that you can use only (N + 1) (\ displaystyle \ scriptstyle (N \, + \, 1)) coefficients . This method is particularly easy in practice since at least one text includes a program that takes the desired filter and N, and returns the optimum coefficients . </Li> <Li> Equiripple FIR filters can be designed using the FFT algorithms as well . The algorithm is iterative in nature . You simply compute the DFT of an initial filter design that you have using the FFT algorithm (if you don't have an initial estimate you can start with h (n) = delta (n)). In the Fourier domain or FFT domain you correct the frequency response according to your desired specs and compute the inverse FFT . In time - domain you retain only N of the coefficients (force the other coefficients to zero). Compute the FFT once again . Correct the frequency response according to specs . </Li> </Ol>

Finite impulse response of the system is used to find