Design of FIR Filters using windowing

The basic theory used in designing the FIR filter by windowing assumes

Ideal low pass filter representation which has an instantaneous passband, transition band and a flat stop band

The impulse response of the filter results in a

non casual filter with an infinite response → The sinc funtion

To remove the infinite length of the filters implse repsonse

* Truncating the ideal impulse response to make it finite.
* Add delay for causality

To truncate the ideal impulse response, what needs to be done

Multiply the filters impulse response with a window of finite length L ( L = M +1 ) M is the order

In ensuring the windowed response is causal,

* add a delay (L-1)/2
* Prior to windowing, modify the ideal response to include linear phase factor

Factors affecting the frequency response of the filter windowed

Peak sidelobe

* affects :- pass band & stop bands ripple
* depends :- on the type of the window

Mainlobe widh

* affects :- width of the transition band
* depends :- window length

The effects of the convolution process or the filter design specifications.

* Oscillations due to windowing
* Transition region due to main lobe width
* Non zero side lobe amplitude

Comparison of commonly used windows

* Tapering the window smoothly to zero reduces the side lobe amplitude and the peak approximation error
* Increasing the order M decreases the main lobe width
* choosing a smoother window can result in a larger main lobe width
* All windows are symmetric leading to linear phase filters or zero phase if centered at 0

the cut off frequency is positioned in filters at

3dB cut-off - half the power attenation

Using fixed windows in Filter design implies

* that the stop band attenuation is independent of the window length - fixed
* the passband ripple = stopband ripple and independent of window shape → depends on the shape of the window
* width of transition bands depends on the length and shape of the window

Filter Design using Fixed windows involve the steps which are

* Check the design specification (Ω_p,Ω_s,A_p,A_s )
* Determine the cut off frequency of the ideal low pass prototype Ω_c= (Ω_p+Ω_s)/2
* Using window specs. Choose a window function that provides the smallest stopband attenuation greater than A_s.
* Determine required filter order (M = L-1) for the selected window that will give me the desired transition bandwidth
* Determine the impulse response of the ideal low pass filter with cut off frequency Ω_c
* Compute the impulse response h\[n\] = h_d\[n\]w\[n\] using the chosen window
* Check if the filter satisfies the design specifications