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## Chapter 7. Filter Design Techniques

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**Chapter 7. Filter Design Techniques**7.1 Introduction - Digital Filter Design 7.2 IIR Filter Design by Impulse Invariance 7.3 IIR Filter Design by Bilinear Transformation 7.4 FIR Filter Design by Windowing 7.5 Kaiser Window based FIR Filter Design 7.6 Approximation based Optimal FIR Filter Design BGL/SNU**1. Introduction -- Digital Filter Design**(1) Frequency selective filters : spectral shapers lowpass highpass bandpass bandstop (2)Filter Design Techniques IIR : - mapping from analog filters - impulse invariance FIR: - windowing - equiripple design BGL/SNU**(3)Filter Specification(LPF)**In some IIR filter design 1 2 3 1 2 3 BGL/SNU**2. IIR Filter Design by Impulse Invariance**(1) Design Concept - Utilize existing analog filter design technique - Convert analog impulse response into digital impulse response h[n] by taking samples - Take Then by fundamental sampling property, we get - If analog filter were bandlimited, Then BGL/SNU**(2) Aliasing Problem**- But the issue is that the above assumption is not likely, so aliasing is inevitable in reality aliasing • Therefore this design technique is useful only when • designing a narrowband sharp lowpass filters BGL/SNU**(3) Parameter Conversion**- Let the analog filter has the partial-fraction expansion - After sampling, = = h [ n ] T h (nT ) d c d • Therefore, • pole at in is mapped to Pole at BGL/SNU*** Impulse Invariance Filter Design Procedure**1) Given specification in domain. 2) Convert it into specification in domain 3) Design analog filter meeting the specification 4) Convert it into digital filter function H(z) by putting [ 5) Implement it in 2nd order cascade form] BGL/SNU**Design Example**1 2 * Choose Td=1 3 BGL/SNU**4*** You plot the pole locations in the z-plans! BGL/SNU**3. IIR Filter Design by Bilinear Transformation**(1) Design Concept - s-plane to z-plane conversion • any mapping than maps stable region is s-plane (left half plane) • to stable region in z-plane (inside u.c) ? or bilinear transform! * Td inserted for convention may put to any convenient value for practical use.*** IIR Filter Design Procedure**Given specification in digital domain Convert it into analog filter specification Design analog filter (Butterworth, Chebyshov, elliptic):H(s) Apply bilinear transform to get H(z) out of H(s) 1 2 3 4 3 2 4 1**3**1 W = 2 | H ( j ) | c + W W 2 N 1 ( / ) c Design Example (Butterworth Filter) Given specification 1 Specification Conversion 2 (Set Td=1) Butterworth filter design***Comparison of Butterworth, Chebyshev, elliptic filters**-Filter equations 1 Butterworth filter B Chebyshev filter (type I) C 1 Chebyshev polynomial***Comparison of Butterworth, Chebyshev, elliptic filters**(Cont’d) Chebyshev filter (type II) 1 Elliptic filter E 1 Jacobian elliptic function***Comparison of Butterworth, Chebyshev, elliptic filters:**Example -Given specification -Order Butterworth Filter : N=14. ( max flat) Chebyshev Filter : N=8. ( Cheby 1, Cheby 2) Elliptic Filter : N=6 ( equiripple) B C E**-Pole-zero plot (analog)**B C1 C2 E -Pole-zero plot (digital) B C1 C2 E (14) (8)**-Magnitude**-Group delay C1 B B E C1 E C2 C2**4. FIR Filter Design by Windowing**(1) Design Concept - Given a desired frequency response evaluate - Then, is the desired filter coefficients. However, is infinitely long, so not practical. Therefore, take a finite segment of , or such that the resulting frequency spectrum fall in the given specification. This process of getting out of is called Windowing**)**w ) ( j W ( e**(3) Design Point**w e ( ) : depends on the attenuation of the peak sidelobe of w j W ( e ). But M cannot improve this. (due to Gibb’s phenomena). Therefore, once a specific window is given, is fixed. e ( w ) w D w j : depends on the width of main lobe of W ( e ). This can be improved by increasing M.**(4) Commonly Used Windows**BGL/SNU**Frequency Spectrum of Windows**(a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman , (M=50) BGL/SNU**5. Kaiser Window based FIR Filter Design**(1) Design Concept ( I : 0th order modified Bessel function) 0 • targets at limited duration in time and energy concentration at • low frequency • - compromisable. (choose appropriate ) • Performance comparable to Hamming window ( when ) BGL/SNU**Frequency Spectrum of Kaiser Window**(a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6 BGL/SNU**(2) Determination of Filter Order ( Kaiser, 1974 )**① ② ③ BGL/SNU**- Design Example :**① ② ③ ④ (Note)**6. Approximation based Optimal FIR Filter Design**(1) Design Concept - Linear phase filters possess the property - More Generally, constant delay filters have the expression BGL/SNU**- Approximation error**BGL/SNU**- Approximation ( Chebyshev)**BGL/SNU**(2) Type I Lowpass Filter case**- desired : - approximation - weighting BGL/SNU**(3) Type II Lowpass Filter case**- desired (original) : - approximation BGL/SNU**- Weighting (modified)**- Desired (modified) - error function BGL/SNU**(4) Alternation Theorem**BGL/SNU**(5) Parks-McClellan Algorithm**① ② ③ ④ BGL/SNU**⑤**Remez Exchange Algorithm (1934) (multiple exchange) ① ② BGL/SNU**③**BGL/SNU**④**⑤ ⑥ BGL/SNU**Selection of new extrema**BGL/SNU**Initial guess of**(L+2) extremal frequencies Remez Exchange Algorithm Calculate the optimum On extremal set Interpolate through (L+1) Points to obtain Ae(ej) Best approximation Calculate error E() And find local maxima Where | E()|>= Unchanged Check whether the Extremal points changed Changed More than (L+2) extrema? Yes Retain (L+2) Largest extrema No BGL/SNU**Design Examples**① Kaiser, (1974) BGL/SNU