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Explain frequency sampling method of designing FIR filter?
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The frequency sampling method is use to design recursive and non-recursive FIR filters for both standard frequency selective filters and with arbitrary frequency response. The main idea of the frequency sampling design method is that a desired frequency response can be approximated by sampling it at N evenly spaced points and then obtaining N-point filter response.

A continuous frequency response is then calculated as an interpolation of the sampled frequency response. The approximation error would then be exactly zero at the sampling frequencies and would be finite in frequencies between them. The smoother the frequency response being approximated, the smaller will be the error of interpolation between the sample points.

There are two distinct types of Non-Recursive Frequency Sampling method of FIR filter design, depending on where the initial frequency sample occurred. The type 1 designs have the initial point at $\omega=0$ , whereas the type 2 designs have the initial point at $f=\frac{1}{2 N}$ or $\omega=\frac{\pi}{N}$

Procedure for Type-1 Design:

1) Choose the desired frequency response $H_{d}(\omega)$

2) Sample $H_{d}(\omega)$ at $\mathrm{N}$ -points by taking $\omega=\omega_{k}=\frac{2 \pi k}{N}$ where $\mathrm{k}=0,1,2,3, \ldots . . .(\mathrm{N}-1),$ generate the sequence $\mathrm{H} (k)$. To obtain a good approximation of the desired frequency response, a sufficiently large number of the frequency samples should be taken. $H(k)=\left.H_{d}(\omega)\right|_{\omega=2 \pi k / N}$ for $\mathrm{k}=0,1, \ldots(\mathrm{N}-1)$

3) The N-point inverse DFT of the sequence $H(k)$ gives the impulse response of the filter $h(\text { n)}$. For practical realization of the filter, samples of impulse responsed should be real. This can happen if all the complex terms appear in conjugate pairs.

Desired filter coefficients

$h(n)=Inv D F T\{H(k)\}=\frac{1}{N} \sum_{k=0}^{N-1} H(k) e^{j 2 \pi k n / N}$

For linear phase filters, with positive symmetrical impulse response,

$h(n)=\frac{1}{N}\left\{h(0)+2 \sum_{k=1}^{U L} \operatorname{Re}\left[H(k) e^{j 2 \pi k n / N}\right]\right\},$

when

N is odd $U L=\frac{N-1}{2}$ and when N is even $U L=\frac{N}{2}-1$

4) Take z-transform of the impulse response h (n) to get the filter transfer function, H(z)

$H(z)=\sum_{n=0}^{N-1} h(n) \cdot z^{-n}$

Procedure for Type-2 Design:

(Same steps as above except step 2)

2) Sample $H_{d}(\omega)$ at N-points by taking $\omega=\omega_{k}=\frac{2 \pi}{2 N}(2 k+1)$ where $k=0,1,2,3, \ldots(N-1)$ generate the sequence H(z)

$H(k)=\left.H_{d}(\omega)\right|_{\omega=\pi k(2 k+1) / N}$ for $k=0,1, \ldots N-1$

Type 2 frequency samples give additional flexibility in the design method to specify the desired frequency response at a second possible set of frequencies.

Advantage

  • Unlike the window method, this technique can be used for any given magnitude response.
  • This method is useful for the design of non-prototype filters where the desired magnitude response can take any irregular shape.
  • Major advantage of Frequency sampling method lies in the efficient frequency sampling structure, which is obtained when most of the frequency samples are zero.

Disadvantage

One disadvantage with this method is that the frequency response obtained by interpolation is equal to the desired frequency response only at the sampled points. At the other points, there will be a finite error present.

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