1. MMSE Optimal Design: The Least Squares method
Prof. Siripong Potisuk

2. MMSE Approximation Measurable feature of the approximation error
An averaging integral of the squared error over the time period of interest
Can be used to prove convergence of the continuous-time Fourier series
The mean-square error, denoted by EN, is the average power in the error signal and defined as

3. Windowing Method Revisited
Uses a truncated Fourier series expansion of the desired amplitude response, Hd(f).
Produces a filter that is optimal in the sense that it minimizes the objective function below:
where Hr(f) is the actual amplitude response, which
is real and has even or odd symmetry.

4. Least Squares Method
Uses frequency samples of desired frequency response
The filter coefficients obtained by minimizing a discrete version of the weighted objective function defined as
{F0,F1,. . . ,FP} is a set of P+1 uniformly-spaced distinct frequencies with F0= 0, FP= Fs, and F0 < F1 <. . . < FP such that F1 = i Fs/P
w(i) > 0 is a weighting function

6. Special case: N = M+1 = P Known as the frequency sampling method
The causal impulse response is the IDFT of equally-spaced samples of desired magnitude sequence.
The actual magnitude response will approximate or interpolate (i.e., passes through the N frequency samples of) the desired magnitude sequence.
It can be shown that

7. MATLAB Implementation
Function B = firfs(N, Hk)
MATLAB user-defined function for FIR filter
design using the frequency sampling method
(text, pp.290)
Input Arguments:
N = number of filter taps (must be an odd number)
= M+1 where M is a filter order (even number for Type 1)
Hk = sampled desired frequency response for k = 0,1,…,M/2
Output:
B = FIR filter coefficients

8. Example 7.12 Design a type 1 linear-phase FIR filter with
coefficients satisfying lowpass response characteristics with
7 taps and a cutoff frequency of c = 0.4 radians using the
frequency sampling method.

9. Example 7.13 Design a type 1 linear-phase FIR filter with
coefficients satisfying bandpass response characteristics with
25 taps and lower and upper cutoff frequencies of 1000 Hz and
3000 Hz, respectively, using the frequency sampling method.
Plot both the impulse response and the frequency response.