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