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**Digital filters: Design of FIR filters**

احسان احمد عرساڻي Lecture 23-24

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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 Hd(ω) Determine the unit (sample) pulse reponse hd(n)=F-1{Hd(ω)} hd(n) is generally infinite in length Truncate hd(n) to a finite length M

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

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**Truncating hd(n) Take only M terms Remove all others**

N=0 to N=M-1 Remove all others Multiplying hd(n) with a rectangular window

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**Determine H(ω) Take Fourier transform of h(n) Therefore, compute:**

Hd(ω) and W(ω) Hd(ω) depends on the required response hd(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 Hd(ω) is required Hd(n) happens to be non-causal having infinite duration

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**The impulse response hd(n)**

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Windowing the hd(n)

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The truncated hd(n)

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

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**Frequency of oscilation increases with M**

Magnitude of oscillation doesn’t increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli- cation of the rectangular window with Hd(ω)

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**Frequency of oscilation increases with M**

Magnitude of oscillation doesn’t increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli-cation of the rectangular window with Hd(ω)

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Other windows

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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 filter’s response with Rectangular window**

M=61

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**FIR filter’s response with Hamming window**

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**FIR filter’s response with Blackman window**

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**FIR filter’s response with Kaiser window**

M=61

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Using the FIR filter

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**Blackman’s filter output**

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