ELEG 305: Digital Signal Processing
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1 ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 5 Outline Review of Previous Lecture Lecture Objectives 3 Motivation and Preliminaries IIR Filter Design by Approximation of Derivatives IIR Filter Design by the Bilinear ransformation K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 5
2 Review of Previous Lecture Review of Previous Lecture Ideal Filters Have infinite, noncausal impulse responses; not physically realizable Practical Filters Have pass and stop band ripple & a nonzero transition band Windowing Method for FIR Filter design An ideal filter impulse response h d (n) is truncated by a window w(n) h(n) h d (n)w(n) H(ω) H d (ω) W (ω) Windows radeoff between main lobe width and side lobe attenuation; Numerous windows exist, e.g., Rectangular, Hanning, Hamming, Blackman, etc. (See able 0. for a fuller listing) FIR Filter Design with Matlab B fir(n,wn) returns the linear phase, order N lowpass FIR filter with normalized cutoff frequency 0<Wn<; Can also be used for highpass, band pass/stop filter design, and with multiple windows K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 Lecture Objectives Lecture Objectives Objective Develop IIR filter design methods based on mapping continuous filters to the discrete domain Reading Chapters 0 (0.3); Next lecture, continuous and discrete IIR filter design (Chapter 0.3); Matlab filter design K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
3 Motivation and Preliminaries Objective: Design digital IIR filters using known analog filter results Analog filter design is mature and well understood Recall: Analog systems are defined by H a (s) B(s) M A(s) k0 β ks k N k0 α ks h(t)e st dt k [Laplace rans.] Or, noting that s is the differentiation operator, N k0 α k d k y(t) dt k M k0 β k d k x(t) dt k [Differential Equation] Approach: Map analog filters (s plane) to digital filters (z plane) Frequency Mapping: s domain jω axis z domain unit circle Stability Mapping: s domain LHP inside z domain unit circle K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 Motivation and Preliminaries Aside: Recall linear phase in FIR filters requires h(n) ±h(m n) which implies z (M ) H(z ) ±H(z) hus roots of H(z ) and H(z) are identical roots are in reciprocal (complex conj.) pairs Note: he z (M ) H(z ) ±H(z) equality can be shown to be a necessary condition for linear phase Observation: If z (M ) H(z ) ±H(z) in the IIR case poles and zeroes are in reciprocal (complex conj.) pairs, i.e., there must be poles outside the unit circle Result: Causal IIR filters can not have linear phase Zero configuration for linear phase FIR filter K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
4 IIR Filter Design by Approximation of Derivatives IIR Filter Design by Approximation of Derivatives Approach: Since s it is the differentiation operator, define a discreet time approximation Utilize the backward difference as a derivative approximation dy(t) dt tn y(n ) y(n ) Continuous time differentiator Discrete time backward difference difference differentiator y(n) y(n ) Which has the system function H(z) z. Equating the operations s z or H(z) H a (s) s z K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 IIR Filter Design by Approximation of Derivatives Question: What s plane z plane mapping does this perform? s z z s Restrict s to the frequency axis, i.e., s σ + jω, with σ 0 z ( ) jω + jω jω + jω + Ω + j Ω + Ω Check limiting points and lim z [DC to DC] Ω 0 jω lim z 0 [HF to??] Ω jω K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
5 o see general mapping, examine the point z (with s jω) IIR Filter Design by Approximation of Derivatives z jω ( jω ) ( jω ) + jω ( jω ) z + jω jω z induced s plane to jω z plane mapping Observations: Mapping yields stable filters LHP of the s plane inside the z plane unit circle Image inside the unit circle is in right half No high pass filters K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 Example Convert the analog bandpass filter H a (s) IIR Filter Design by Approximation of Derivatives (s + 0.) + 9 into a digital filter using backward difference derivative approximation. Utilizing s z H(z) in the above ( z + 0.) + 9 /( ) (+0. ) z z Given, this reduces a simple system H(z) + a z + a z K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
6 Objective: Design a discrete time IIR filter by sampling the impulse response of a continuous filter hus if h a (t) is a continuous time impulse response, set h(n) h a (n ), n 0,,... Consider the sampling induced frequency domain relation H a (s) h a (t)e st dt [Laplace rans.] 0 h a (n )e sn n0 h(n)e sn H(z) ze s n0 Result: his defines the the s plane to z plane mapping z e s K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 5 Question: What s plane z plane mapping does this perform? Utilize z re jω and s σ + jω representations in the mapping z e s re jω e σ e jω r e σ and ω Ω z-plane z es ""Unit circle jq 3n s-plane --~~~ a z e s induced s plane Observations: to z plane mapping Mapping introduces aliasing (i.e., s and (s + jkπ/ ) same z) Zeros and poles don t follow the same mapping Consider the pole mapping for arbitrary H a (s) N c k H a (s) s p k k Note H a (s) has N poles at p, p,... p N K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / n n 3n
7 Since H a (s) is in partial fraction form, the impulse response is h a (t) h(n) H(z) N c k e p k t, t 0 k N c k e p k n [after sampling] k ( N ) h(n)z n c k e p k n z n n0 n0 k N c k (e p k z ) n k n0 N c k e p k z k [z transform] Result: H(z) has N poles at e p, e p,... e p N, i.e., the s domain poles p, p,... p N are mapped to the z domain according to z e s K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 Note: Zero mapping is filter dependent show by example Example Convert the analog bandpass filter H a (s) s + 0. (s + 0.) + 9 into a digital filter using impulse invariance. Note that H a (s) has the following zero and poles zero: s 0. and partial fraction expansion H(s) poles: p k 0. ± j3 / s ( j) + / s ( 0. 3j) Mapping the poles according to z e s yields p 0.+j3 e 0. e j3 and p 0. j3 e 0. e j3 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
8 Q) hus the poles in the z domain are p k e 0. e ±j3. Mapping H(s) / s ( j) + / s ( 0. 3j) to the z domain (partial fractions pole mapping) yields H(z) / e 0. e j3 z + / e 0. e j3 z (e 0. cos 3 )z (e 0. cos 3 )z + e 0. z System Zero Poles Comments H(s) s 0. p k 0. ± j3 Poles in LHP stable H(z) z e 0. cos 3 p k e 0. e ±j3 Poles in unit circle stable Result: Zero mapping is not simply z e s. his and aliasing effects limit the use of impulse invariance K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / lq "0 0 --~ "0 :: t: bo cu -0 ~ -30 -~ "0 '-' "0 ;:l -0.-: c:: co ~ j\, 0.5, \./' ,/ -,-' / Frequency Normalized frequency Frequency response of H(s) Frequency response of H(z) for 0. and 0.5 Note: should be chosen small to avoid aliasing; Method only appropriate for lowpass and a bandpassed filters. Summary of Impulse Invariance IIR Filter Design Express H(s) in partial fraction form Map poles to the z plane via z e s 3 Express H(z) in partial fraction form, converting to other representations as appropriate (e.g., difference equations) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
9 IIR Filter Design by the Bilinear ransformation IIR Filter Design by the Bilinear ransformation Objective: Design a discrete time IIR filter by approximating the integral (rather than the derivative, as before) Suppose H(s) b s + a ẏ(t) + ay(t) bx(t) where ẏ dy(t) dt. Also note that t t y(t) ẏ(τ)dτ ẏ(τ)dτ + y(t 0 ) t 0 Let t n and t 0 (n ). hen (for small) t ẏ(τ)dτ (ẏ(t 0) + ẏ(t)) t 0 (ẏ(n) + ẏ(n )) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 IIR Filter Design by the Bilinear ransformation t ẏ(τ)dτ area under ẏ(t) between t 0 and t t 0 (ẏ(n) + ẏ(n )) average of range start & end points width Result: Approximately equal for small hus y(t) t t 0 ẏ(τ)dτ + y(t 0 ) y(n) (ẏ(n) + ẏ(n )) + y(n ) ( ) We need an expression for ẏ(n) in ( ). Recall H(s) Substitute ( ) into ( ) ẏ(t) + ay(t) bx(t) ẏ(n) ay(n) + bx(n) ( ) b s+a gives K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
10 IIR Filter Design by the Bilinear ransformation y(n) (ẏ(n) + ẏ(n )) + y(n ) ([ ay(n) + bx(n)] + [ ay(n ) + bx(n )]) + y(n ) Rearranging, and then taking the z transform, gives ( + a ) ( y(n) a ) y(n ) b (x(n) + x(n )) Y (z) ( + a ( a ) ) z X(z) b ( + z ) H(z) Y (z) X(z) b ( + z ) + a ( a )z he denominator needs to be rearranged to lend insight K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 he denominator is + a ( a )z + IIR Filter Design by the Bilinear ransformation [ a z a ] z a z ( a )z + a ( + z ) z + a ( + z ) hus H(z) is expressed as H(z) b ( + z ) ( z ) + a ( + z ) b ( z +z ) + a H(s) s ( z +z ) Result: he mapping s ( z +z ) defines the bilinear transformation K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
11 IIR Filter Design by the Bilinear ransformation Rearranging s ( z +z ), with s σ + jω, yields z + s s + σ + jω σ jω ( ) Observations: From s σ + jω and ( ) σ < 0 z < (s domain LHP maps to the inside of the z domain unit circle stability preserved) σ 0 z (s plane frequency axis maps to the z domain unit circle frequency to frequency mapping) Also, σ 0 s jω and z. hus s jω ( ) z + z ( ) e jω + e ( ) jω Ω e jω/ e jω/ j e jω/ + e jω/ tan (ω/) ω arctan (Ω /) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 5 IIR Filter Design by the Bilinear ransformation Result: Cont. to Disc. Freq. mapping: ω arctan (Ω /) Disc. to Cont. Freq. mapping: Ω tan (ω/) his is a nonlinear frequency warping [Cont. Freq.] < Ω < π ω π [Disc. Freq.] (J) tan- Q J[ L L.-----f------'------'-----~Q J[ Bilinear ransformation Ω ω mapping Observation: K. E. Barner (Univ. ofno Delaware) frequency ELEG 305: aliasing Digital Signal is introduced Processing by the warping Fall 008 / 5
12 IIR Filter Design by the Bilinear ransformation Bilinear ransformation Design Procedure Summary Make filter specifications in the digital domain Map specifications to the continuous domain 3 Use analog design techniques (look filter up) 4 Use bilinear mapping to obtain the digital filter Example Designed a single pole lowpass digital filter with a 3-dB bandwidth of 0.π, using the bilinear transformation of the analog filter H a (s) Ω c s + Ω c where Ω c is the 3-dB bandwidth of the analog filter. First, map ω c 0.π to the analog domain Ω c tan ω c/ tan 0.π 0.65/ K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5 hus for Ω c 0.65/ H a (s) IIR Filter Design by the Bilinear ransformation Ω c H a (s) 0.65/ s + Ω c s / Next, apply the bilinear transformation ( ) z ( ) to ( ) s + z H(z) 0.65/ ( ) z / +z 0.45( + z ) 0.509z Observations: is divided out of the final H(z) expression (it is a scale term, the value of which is irrelevant) H(0) and H(0.π) /, thus the 3-dB point is correct K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
13 Lecture Summary Lecture Summary IIR Filter Design Map well understood analog filters to the discrete domain (s to z plane mapping) Approximation of Derivatives Method for IIR Filter design z s z s Maps s LHP to right half of z unit circle; No highpass filters Impulse Invariance Method for IIR Filter design z e s ; Maps s LHP to inside of z unit circle; small to avoid aliasing; poles & zeroes have different mappings Bilinear ransformation Method for IIR Filter design s ( ) z + z z + s s Ω tan (ω/) ω arctan (Ω /) Maps s LHP to inside of z unit circle; No aliasing K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 5
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