Hossein Sameti Department of Computer Engineering Sharif University of Technology FIR IIR Achieving a linear phase is always possible Difficult to control the linear-phase property. Almost no particular technique is available. Can be unstable Filter order: less Always stable Filter order: higher Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 2 We focus on IIR filters with a rational transfer function: P( z ) H ( z) Q( z ) P and Q are polynomials in z. N P( z ) a(n) z n0 n M Q( z ) b(n) z n n0 Filter Design: To determine the values of a(n) and b(n) such that specs given to us are met. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 3 IIR Filter Design Optimization techniques Pole-zero placement Impulse Invariance Bilinear Transformation Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4 1. 2. 3. 4. A set of specs for the digital (discrete-time) filter is given. We transform the specs from the D.T. to C.T. (zs) Design a C.T. IIR filter : H a (s ) sz H a (s) H ( z ) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5 The art of CT IIR filter design is highly advanced. Many CT IIR methods have relatively closed-form design formulas. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 6 1) Causal/stable analog filter should be transformed to a causal stable DT filter. H a ( s) H ( z ) Causal and stable Causal and stable Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 7 2) jΏ axis in the s-plane (CT) needs to be transformed to the unit circle in the z-domain. * Needed to translate the specs from discrete to analog domain 3) Rational transfer function in the s-domain should be transformed into a rational transfer function in the zdomain. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 8 Proposal: H d ( z ) H a ( s) s 2 ( 1 z 1 ) T: arbitrary parameter T 1 z 1 • Does this transformation satisfy the desirable properties that we just discussed? • Does a rational analog filter lead to a rational digital filter? 2 1 z 1 s ( ) 1 T 1 z A rational analog filter translated into a rational digital filter. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 9 • Is jΏ axis in the s-plane (CT) transformed to the unit circle in the z-domain? Let: z e j s : 0 : 0 : 0 : 0 2 j tan T 2 2 1 e j s ( ) T 1 e j j s j 2 j 2 j 2 2e (e e ) j j T j e 2 (e 2 e 2 ) 2 tan T 2 0 Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011 10 • Does a causal and stable analog filter lead to a causal and stable digital filter? • We need to show that LHP in the s-domain is mapped into inside the unit circle in the z-domain. 2 1 z 1 s ( ) T 1 z 1 j 2 z 1 2 re 1 s ( ) s ( j ) T z 1 T re 1 2 r 2 1 2r sin r 1 0 s ( j ) 2 2 T 1 r 2r cos 1 r 2r cos r 1 0 11 H () 1 0.89 0.178 0.2 0.3 • Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 12 1. 2. 3. 4. A set of specs for the digital (discrete-time) filter is given. We transform the specs from the D.T. to C.T. (zs) Design a C.T. IIR filter : H a (s ) sz H a (s) H ( z ) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 13 T 1 2 tan T 2 2 tan 2 p 0.2 p 2 tan 0.2 0.6498 2 0.3 s 0.3 s 2 tan 1.02 2 14 H a () 1 0.89 0.178 0.6498 1.02 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 15 Butterworth filter: H c ( s) a0 a N s N a N 1s N 1 ... a1s a0 2 H c () Monotonic in stop-band and pass-band 1 1 c 2N Cut-off frequency Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 16 1 2 H c () 1 c 2N s j H c ( s ) H c ( s ) H c ( ) H c ( s) H c ( s) Poles of 2 1 s 1 j c H c () 2 : 2N 1 2N sk (1) ( j c ) c e j ( 2 k N 1) 2N k 0,1, ..., 2 N 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 17 Poles of 𝐻 𝑠 𝐻 −𝑠 = |𝐻 Ω |2 : 1 2N sk (1) ( j c ) c e j ( 2 k N 1) 2N k 0,1, ..., 2 N 1 • • • • • • 2N poles equally spaced in angle on a circle of radius c in the s-plane Poles are symmetric with respect to the imaginary axis No poles on the imaginary axis Two poles on the real axis only if N is odd The poles are spaced by 𝜋/N radians To construct a stable and causal H(s), we choose all the poles on the LHP Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 18 Chebyshev: ripple in either pass-band or stop-band Elliptic: ripple in both pass-band or stop-band See Appendix B in the textbook for related formulae. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 19 H a () 1 0.89 0.178 0.6498 1.02 - Which one would you choose? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 20 H a () 1 • Butterworth filter of order N: 2 H a () 1 1 c 0.178 2N 2 0.6498 1 0.6498 1 c 0.6498 1.02 H a () 0.6498 0.89 H a ( ) 0.89 2N (0.89) 2 H a () 1.02 0.178 H a ( ) 2 1.02 1 1.02 1 c 2N (0.178) 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 21 1 2 H a () 1 c H a () 0.6498 0.89 2N N 5.305 N 6 2N 12 N 6 angle H a () 1.02 0.178 c 0.766 360 30 12 s s1 , s 2 , s 3 , s 4 , s 5 , s 6 22 K H a (s) ( s s1 )( s s 2 )(s s3 )(s s4 )( s s5 )( s s6 ) H a ( 0) 1 1 2 1 z s ( ) 1 T 1 z K 0.20238 T 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 23 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 25 H () 1 0.7 0.2 0.2 0.5 • Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 26 T 1 2 tan T 2 2 tan 2 p 0.2 p 2 tan 0.2 0.6498 2 s 0.5 2 tan 0.5 2 s 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 27 H a () 1 0.7 0.2 0.6498 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 28 • Butterworth filter of order N: 1 2 H a () 1 c 2N H a () 0.6498 0.7 H a ( ) 2 0.6498 1 0.6498 1 c 2N (0.7) 2 H a () 2 0.2 H a ( ) 2 2 1 2 1 c 2N (0.2) 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 29 1 2 H a () 1 c 2N H a () 0.6498 0.7 H a () 2 0.2 N 1.3957, c 0.6465 N 2 N 2 c 0.6437 2N 4 angle 360 90 4 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 30 s1 0.4549 j 0.4549 s2 0.4549 j 0.4549 0.64372 H a ( s) ( s s1 )( s s 2 ) 2 1 z 1 s ( ) 1 T 1 z ... Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 31 Frequency Transformation of Low-pass IIR Filters Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 32 1. 2. 3. 4. 5. 6. Start with spec. in D.T. For HPF. Translate the filter specs from D.T. to C.T. specs of a HPF in C.T. (using bilinear transform.) Translate specs of C.T. HPF to C.T. LPF Design the LPF (Butterworth) Transform C.T. LPF C.T. HPF Transform C.T. HPF D.T. HPF (using bilinear transform.) C.T. LPF Ha(s1) C/C C/D C.T. HPF Ha’(s2) D.T. HPF Hd(z) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 33 LPF H a ( s1 ) HPF 1 H a' (s2 ) Proposal: H ( s2 ) H a ( s1 ) k a' s1 s2 2 k: positive constant Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 34 Is jΏ1 axis in the s1-plane (CT) transformed to the jΏ2 axis in the s2-plane (CT)? j1 j 2 k s1 j1 s2 k s2 j 2 j 2 1 k s2 j 1 k s2 j 1 k 2 0, 2 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 35 • Does a causal and stable analog LPF filter lead to a causal and stable analog HPF? • Does LHP in the s1-domain (CT) map into the LHP in the s2-domain (CT)? • It is easy to prove the above statement. • It is also easy to show that a rational transfer function is mapped into another rational transfer function. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 36 k 2 1 ha (t ) real H a ' (2 ) H a (1 ) 1 k 2 Frequency response is symmetric. H a (1 ) H a (1 ) H a ' ( 2 ) H a (1 ) H a ' (2 ) H a (1 ) 1 k 2 k 1 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 37 H a' (2 ) H a (1 ) 2 1 k 2 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 38 *Assumption: prototype lowpass filter has band edge frequency Type of Transformation Lowpass Highpass Bandpass Bandstop Transformation s p ' p s s p ' p s s 2 l u s p s(u l ) s p s(u l ) s 2 l u p Band edge frequencies of the new filter ' p ' p u , l u , l Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 39 1. 2. 3. 4. 5. 6. Start with spec. in D.T. For HPF. Translate the filter specs from D.T. to C.T. specs of a HPF in C.T. (bilinear transformation) Translate specs of C.T. HPF to C.T. LPF Design the LPF (Butterworth) Transform C.T. LPF C.T. HPF Transform C.T. HPF D.T. HPF (bilinear transformation) C.T. LPF Ha(s1) C/C C/D C.T. HPF Ha’(s2) D.T. HPF Hd(z) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 40 1. 2. Start with spec. in D.T. For HPF. Translate the filter specs from D.T. HPF to D.T. LPF (using the transformation discussed shortly) 3. 4. Design the LPF : (a) translate the DT LPF specs CT LPF specs; (b) Design CT LPF;(c) Transform CT LPF to DT LPF. Transform D.T. LPF D.T. HPF C.T. LPF Ha(s) C/D D/D D.T.LPF H(z1) D.T. HPF Hd(z2) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 41 Proposal: H d ( z2 ) H ( z1) z z2 1 1z2 1 : real It can be shown that this transformation has the 3 properties that we usually investigate for transformations: 1- A rational transfer function is transformed to a rational transfer function. 2- Unit circle in one domain is mapped into the unit circle in the other domain. 3- Inside of the unit circle in one domain is mapped to the inside of the unit circle in the other domain. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 42 Proposal: H d ( z2 ) H ( z1) z z2 1 1z2 1 : real Proof of the second property: Unit circle in one domain is mapped into the unit circle in the other domain. z2 e j 2 j 2 z2 e z1 1z2 1e j 2 e j 2 (1e j 2 ) z1 z1 1 (1e j 2 ) e j 2 1e j 2 1e j 2 z1 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 43 Proposal: H d ( z2 ) H ( z1) z z2 1 e j1 j 2 e 1e j2 1z2 1 : real 1 f (2 , ) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 44 1. 2. Start with spec. in D.T. For HPF. Translate the filter specs from D.T. HPF to D.T. LPF (using the transformation discussed earlier) 3. 4. Design the LPF : (a) translate the DT LPF specs CT LPF specs; (b) Design CT LPF;(c) Transform CT LPF to DT LPF. Transform D.T. LPF D.T. HPF C.T. LPF Ha(s) C/D D/D D.T.LPF H(z1) D.T. HPF Hd(z2) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 45 Transform 𝐻𝑙𝑝 𝑍 to 𝐻 𝓏 𝐻𝑙𝑝 𝑍 is a lowpass system function 𝐻 𝓏 is a new system function that can have either lowpass, highpass, bandpass, or bandstop characteristics Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 46 Transformations from a lowpass digital filter prototype of cutoff frequency 𝜃𝑝 to highpass, bandpass, and bandstop filters Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011 47 Suppose we have designed a filter that has met the following specs: We have designed a Chebyshev lowpass filter with the following system function: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 48 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 49 To transfer this filter to a highpass filter with passband edge frequency of p 0.6 : This results in the following high-pass filter: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 50 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 51 IIR filters generally have lower order compared to FIR filters, however, linear-phase cannot be guaranteed. The most popular technique is the transformation technique, although other methods such as pole-zero placement also exist. Using transformation techniques, a low-pass prototype filter can be transformed into HP, BP and BS filters. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 52