Design of FIR Filters using frequency sampling

Benefits of the FIR filter

* Simple to understand
* Easy to use
* Provides practical useful filters

The drawbacks of sing windowing

* Filters have approximately equal size ripples in each band
* Band edges and maximum ripple size cannot be specified precisely due to frequency convolution of window and signal spectrum
* Analytical expression required for desired frequency response
* Analytical expression for ideal impulse response that corresponds to a desired F.R isn’t always easy to get
* Filters are suboptimal and no optimal criterion is satisfied.

Design steps for frequency sampling

* In the frequency domain, sample the desired frequency response at L points(2kπ/L) . L = length of the filter and M+1 = L
* Inverse DFT of the length L to obtain impulse response. Impulse response will be periodic.
* multiply with the window function of length L, to keep only one period

Effect of transition band sharpness

* Transition band sharpness affects passband and stopband ripple.
* A sharp transition creates a discontinuity and leads to Gibbs phenomenon
* A smooth transition band (linear or raised cosine) eliminates the gibbs phenomenon
* penalty paid for ripple elimination is a wider transition band

Effect of window used

* rectangular window will lead to substantial ripple
* a smoother window results in less ripple but wider transition band
* the frequency response of the resulting filter does not include the samples of the the d F.R near the transition band
* matlab uses hamming window by default