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DIGITAL FILTERS: DESIGN OF FIR FILTERS Lecture احسان احمد عرساڻي

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Introduction to FIR filters These have linear phase No feedback Output is function of the present and past inputs only These are also called all-zero and non-recursive filters These do not have any poles

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Applications Where: highly linear phase response is required Need to avoid complicated design

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FIR Filter Design Methods Windows Frequency-sampling

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FIR Filter Design: Windows Method Start from the desired frequency response H d ( ω ) Determine the unit (sample) pulse reponse h d (n)=F -1 {H d ( ω )} h d (n) is generally infinite in length Truncate h d (n) to a finite length M

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Truncating h d (n) Take only M terms N=0 to N=M-1 Remove all others

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Truncating h d (n) Take only M terms N=0 to N=M-1 Remove all others Multiplying h d (n) with a rectangular window

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Determine H( ω ) Take Fourier transform of h(n) Therefore, compute: H d ( ω ) and W( ω ) H d ( ω ) depends on the required response h d (n)

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Computing W( ω ) W( ω )=F{w(n)} w(n) is a rectangular pulse

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Example A low-pass linear phase FIR filter with the frequency response H d ( ω ) is required H d (n) happens to be non-causal having infinite duration

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The impulse response h d (n)

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Windowing the h d (n)

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The truncated h d (n)

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Example A low-pass linear phase FIR filter with the frequency response H d ( ω ) is required

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Frequency of oscilation increases with M Magnitude of oscillation doesnt increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli- cation of the rectangular window with H d ( ω )

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Frequency of oscilation increases with M Magnitude of oscillation doesnt increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli-cation of the rectangular window with H d ( ω )

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Other windows

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Spectrum of Kaiser window (Cycles per sample)

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Spectrum of Hanning window

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Spectrum of Hamming Window (Cycles per sample)

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Spectrum of Blackman Window (Cycles per sample)

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Spectrum of Tukey Window (Cycles per sample)

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Windows characteristics

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The FIR filters response with Rectangular window M=61

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FIR filters response with Hamming window M=61

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FIR filters response with Blackman window M=61

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FIR filters response with Kaiser window M=61

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Using the FIR filter

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Blackmans filter output

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