Editing Finite impulse response (section)

=== Least mean square error (MSE) method ===
'''Goal:'''
:To design FIR filter in the MSE sense, we minimize the mean square error between the filter we obtained and the desired filter.
::<math>\text{MSE}=f_s^{-1} \int_{-f_s/2}^{f_s/2} |H(f)-H_d(f)|^2\,df </math> , where <math>f_s\,</math> is sampling frequency, <math>H(f)\,</math> is the spectrum of the filter we obtained, and  <math>H_d(f)\,</math> is the spectrum of the desired filter.

'''Method:'''
:Given an ''N''-point FIR filter <math> h[n] </math>, and <math>r[n] = h[n+k], k = \frac{(N-1)}{2}</math>.
:Step 1: Suppose <math> h[n] </math>even symmetric. Then, the discrete time Fourier transform of <math>r[n]</math> is defined as 
::<math>R(F) = e^{j2 \pi Fk}H(F) = \sum_{n=0}^k s[n] \cos (2 \pi nF) </math> 
:Step 2: Calculate mean square error.
::<math>\text{MSE}=\int_{-1/2}^{1/2} |R(F)-H_d(F)|^2\,dF </math>
::Therefore,
::<math>\text{MSE}= \int_{-1/2}^{1/2} \sum_{n=0}^k s[n] \cos (2 \pi nF) \sum_{\tau=0}^k s[\tau] \cos (2 \pi \tau F)\,dF -2\int_{-1/2}^{1/2} \sum_{n=0}^k s[n] \cos (2 \pi nF) H_d\,dF + \int_{-1/2}^{1/2} H_d(F)^2\,dF</math> 
:Step 3: Minimize the mean square error by doing partial derivative of MSE with respect to <math>s[n]</math>
::<math> \frac{\partial \text{MSE}}{\partial s[n]} = 2\sum_{\tau=0}^k s[\tau]  \int_{-1/2}^{1/2} \cos (2 \pi nF) \cos (2 \pi \tau F) \,dF - 2\int_{-1/2}^{1/2} H_d(F)^2 \cos (2 \pi nF)\,dF = 0</math>
::After organization, we have
::<math> s[0] = \int_{-1/2}^{1/2} H_d(F)\, dF </math>
::<math> s[n] = \int_{-1/2}^{1/2} \cos (2 \pi nF) H_d(F) \,dF, \ \text{ for } n \ne 0</math> 
:Step 4: Change <math> s[n] </math> back to the presentation of <math> h[n] </math>
::<math>h[k] = s[0], h[k+n] = s[n]/2, h[k-n]=s[n]/2, \; for \; n=1,2,3, \ldots, k, \text{ where } k = (N-1)/2 </math> and <math> h[n] =0 \text{ for } n < 0 \text{ and } n \ge N</math>

In addition, we can treat the importance of passband and stopband differently according to our needs by adding a weighted function, <math>W(f)</math>
Then, the MSE error becomes
:<math> \text{MSE} =\int_{-1/2}^{1/2} W(F)|R(F)-H_d(F)|^2\,dF </math>