Design of FIR Filters using windowing

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

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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

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2

The impulse response of the filter results in a

non casual filter with an infinite response → The sinc funtion

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3

To remove the infinite length of the filters implse repsonse

  • Truncating the ideal impulse response to make it finite.

  • Add delay for causality

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4

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

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5

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

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6

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

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7

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

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8

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

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9

the cut off frequency is positioned in filters at

3dB cut-off - half the power attenation

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10

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

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11

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

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