ELEN 5346/4304 DSP and Filter Design Fall 20081 Lecture 10: LTI IIR design: Analog to Digital and spectral transformations Instructor: Dr. Gleb V. Tcheslavski Contact:
[email protected] Office Hours: Room 2030 Class web site: http://ee.lamar.edu/gleb/dsp/in dex.htm Let’s go! Army will make a human out of you. ELEN 5346/4304 DSP and Filter Design Fall 2008 2 Preliminary considerations Techniques for analog filter design are well developed, so it is quite natural to start designing a digital filter from an analog prototype. An analog LTI system is stable if all poles of its transfer function H(s) lie in the left half of the s-plane. Therefore: 1) The jO axis of the s-plane should map into the unit circle in the z-plane. 2) The left half of the s-plane should map into the inside of the unit circle in the z-plane to convert a stable analog filter to a stable digital filter. A linear phase filter must have a transfer function satisfying: 1 ( ) ( ) N H z z H z ÷ ÷ = ± (10.2.1) However, in this case, filter would have a reciprocal pole outside the UC and, therefore, would be unstable. A causal and stable IIR filter cannot have linear phase. As a result, when designing IIR filters, only the magnitude response is specified. ELEN 5346/4304 DSP and Filter Design Fall 2008 3 Approximation of derivatives If the analog filter is specified by the LCC Differential Equation: 0 0 ( ) ( ) k k N M k k k k k k d y t d x t dt dt o | = = = ¿ ¿ (10.3.1) the following approximation for the derivative can be used: ( ) 1 ( ) ( ) n n t nT y nT y nT T y y dy t dt T T ÷ = ÷ ÷ ÷ = = Where T is the sampling period. The system function of an analog differentiator: The system function of a digital differentiator: ( ) H s s = 1 1 ( ) z H z T ÷ ÷ = (10.3.2) (10.3.3) (10.3.4) ELEN 5346/4304 DSP and Filter Design Fall 2008 4 Approximation of derivatives Therefore, the frequency-domain equivalent for (10.3.2) is: 1 1 z s T ÷ ÷ = Similarly, the k th derivative can be expressed as: (10.4.1) 1 1 k k z s T ÷ | | ÷ = | \ . The system transfer function for the digital IIR filter can be approximated as: 1 1 ( ) ( ) z a s T H z H s ÷ ÷ = = 1 1 z sT = ÷ Equivalently: (10.4.2) (10.4.3) (10.4.4) ELEN 5346/4304 DSP and Filter Design Fall 2008 5 Approximation of derivatives However, it can be shown that this method is suitable for a quite limited class of filters due to its mapping property: There is an attempt to overcome this limitation by using an alternative mapping: 1 ( ) 1 L nT kT nT kT k k t nT y y dy t dt T T o + ÷ = = ÷ = ¿ Therefore: ( ) 1 1 L k k k k s z z T ¸ ÷ = = ÷ ¿ By the proper selection of coefficients ¸ k , it is possible to map the jO axis into the unit circle. However, this selection is a difficult problem in general. (10.5.1) (10.5.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 6 Impulse invariance A digital IIR filter can be obtained by sampling the impulse response of the analog prototype ( ) n g h nT ÷ which may lead to aliasing in the frequency domain ¬ chose small T. (10.6.1) Assuming that the analog filter having N distinct poles is specified by: 1 ( ) N i a i i A H s s o = = ÷ ¿ poles coefficients in the partial fraction expansion (10.6.2) Periodical sampling will lead to: 1 1 ( ) ( ) i N t l i L i h t Ae u t o ÷ = ÷ = ¿ 1 ( ) i N Tn n l i n i g h nT Ae u o = ÷ = ¿ (10.6.3) (10.6.4) ELEN 5346/4304 DSP and Filter Design Fall 2008 7 Impulse invariance The resulting digital filter will be: ( ) 0 0 1 1 1 0 ( ) i i N nT n n n i n N n i i n n i T h z e H z A e z A z o o · · ÷ · ÷ = = ÷ = = = | | = = = | \ . ¿ ¿ ¿ ¿ ¿ (10.7.1) The inner sum converges since o i < 0 ( ) 1 1 0 1 1 i i n T T n e z e z o o · ÷ ÷ = = ÷ ¿ Therefore: 1 1 ( ) 1 i N i T i A H z e z o ÷ = = ÷ ¿ Digital poles at: 1 i i i T p p BIBO e o = ¬ < ¬ (10.7.2) (10.7.3) (10.7.4) Not the best method due to aliasing. Also, impulse response can be infinite… ELEN 5346/4304 DSP and Filter Design Fall 2008 8 Bilinear transform The techniques described so far, have severe limitations since they are appropriate only for LPFs and some BPFs. The bilinear transform does not have such limitations. The bilinear transform from the s-plane to the z-plane is derived via the trapezoidal numerical integration of the differential equation describing the analog prototype. For the given step size T, the BT is given by: 1 1 2 1 1 z s T z ÷ ÷ | | ÷ = | + \ . This transform is a one-to-one mapping; that is, it maps a single point in the s- plane to a unique point in the z-plane, and vice versa. The digital transfer function: 1 1 2 1 1 ( ) ( ) z a s T z G z H s ÷ ÷ | | ÷ = | | + \ . = (10.8.1) (10.8.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 9 Bilinear transform ( ) ( ) 0 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 s j sT T T T j j jx z sT T T T jx j j o o o o c o c o | | = + O < O + + + O + + ÷ + = = = = O + ÷ ÷ ÷ + O ÷ ÷ We can derive from (10.8.1) that (10.9.1) In this situation: 1 z BIBO < ¬ Similarly, when o > 0, |z| > 1; therefore, the mapping is correct. ELEN 5346/4304 DSP and Filter Design Fall 2008 10 Bilinear transform j z e e = When : ( ) ( ) ( ) 2 2 2 1 1 2 2 2 2 1 2 1 2 1 1 2 sin 2 2 2 tan 2 2cos 2 j j j j j j j j e e e z e s T z T e T e e e j j j T T e e e e e e e e e e e ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ( ÷ | | ÷ ÷ ¸ ¸ = = = | + + \ . ( + ¸ ¸ = = = O Therefore: tan 2 2 T e O = Analog freq. Digital freq. (10.10.1) (10.10.2) (10.10.3) s s p p e e O = O We can notice that therefore, a “frequency warping” (10.10.3) is needed. ELEN 5346/4304 DSP and Filter Design Fall 2008 11 Bilinear transform The BT destroys the phase response of an analog filter but preserves linear magnitude. analog digital Therefore, to design a digital filter meeting the given specs for the magnitude response, we need to: 1) prewarp the critical bandage frequencies (e p and e s ) to find their analog equivalents (O p and O s ) using (10.10.3); 2) design the analog prototype using the prewarped critical frequencies; 3) apply the BT to obtain the desired digital transfer function. Note: o p,a = o p,d , o s,a = o s,d. (10.11.1) BT is very good for LPF, HPF, BPF, BSF… – piecewise constant magnitude filters. BT maps the point s = · into the point z = -1. ELEN 5346/4304 DSP and Filter Design Fall 2008 12 Bilinear transform – zero/pole conversion Assuming that the analog zeros and poles are known: i.e. the analog transfer function is in the form: Applying the BT: ( ) ( ) ( ) 1 1 M m m a N k k s H s k s o µ = = ÷ = ÷ [ [ ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 2 1 2 2 1 2 2 1 2 M M m m N M m m m a N N N M k k k k k T z T T H z k z T T T z T o o o µ µ µ ÷ ÷ = ÷ = ÷ ÷ = = | | + ÷ ÷ | ÷ \ . = · + | | + ÷ ÷ | ÷ \ . [ [ [ [ Constant: new gain factor N-M zeros at z = -1 (from zeros at ·) New poles New zeros (10.12.1) (10.12.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 13 Bilinear transform – zero/pole conversion Therefore: ( ) ( ) 1 1 2 2 M m m d a N N M k k T k k T T o µ = ÷ = ÷ = ÷ [ [ Digital gain: Digital poles: Digital zeros: Need to add N-M zeros at z = -1: 2 2 m m m T z T o o + = ÷ 2 2 k k k T p T µ µ + = ÷ (10.13.1) (10.13.2) (10.13.3) ELEN 5346/4304 DSP and Filter Design Fall 2008 14 “Theoretical” Example ( ) c c H s s O = + O Example 10.1: Design a single-pole digital LPF with a 3-dB bandwidth of 0.2t, by use of the BT applied to the analog filter with the 3-dB bandwidth : c O Using the frequency warping: ( ) 2 2 0.65 tan tan 0.1 2 c c T T T e t | | O = = = | \ . 0.65 0.65 ( ) 0.65 0.65 T H s s T sT = = + + The analog filter: Applying BT: ( ) 1 1 1 1 1 1 1 0.245 1 0.65 0.325 ( ) 1 1 0.325 0.325 1 0.509 2 0.65 1 1 z H z z z z z z z ÷ ÷ ÷ ÷ ÷ ÷ ÷ · + = = = ÷ ÷ + + ÷ + + + Which leads to: ( ) 0.245 1 ( ) 1 0.509 j j j e H e e e e e ÷ ÷ · + = ÷ (10.14.1) (10.14.2) (10.14.3) (10.14.4) (10.14.5) ELEN 5346/4304 DSP and Filter Design Fall 2008 15 “Practical” Example Example 10.2: Design a digital LPF starting from the elliptic prototype. The specs for the digital filter are: 1000 ; 1500 ; 10 2 2 1 0.3 ; 50 p s p st s p s f Hz f Hz f kHz dB dB e e t t o o = = = = = ÷ = ÷ = ÷ 4 1 10 s T f s ÷ = = Note: The critical frequencies normalized for T = 1: 2 0.2 ; 2 0.3 ; 1.5 p st s p s s s p f f f f e e t t e t t e = = = = = 1. Frequency pre-warping (for T = 1): 2 tan 0.6498 ; 1.01905 ; 1.568 2 s p s p rad s rad s T e O O = ¬ O = O = ¬ = O ELEN 5346/4304 DSP and Filter Design Fall 2008 16 “Practical” Example 2. Table lookup (software): For the specs (actually, slightly higher than given ones): 1 0.28 ; 50.1 ; 5; 1.566 p s s p dB dB N o o ÷ = ÷ = ÷ = O O = we find the poles and zeros for O p = 1 rad/s: 1,2 3,4 5 1,2 3,4 0.09699 1.0300 0.33390 0.7177 0.49519 1.6170 2.4377 j j j j µ µ µ o o = ÷ ± = ÷ ± = ÷ = ± = ± -3 -2 -1 0 1 2 3 -2 -1 0 1 2 Real Part I m a g i n a r y P a r t Slightly different from Matlab Stable – all poles are in the left half of the plane. ELEN 5346/4304 DSP and Filter Design Fall 2008 17 “Practical” Example 3. De-normalize the analog design: Multiply zeros and poles by O p : 1,2 1,2 3,4 3,4 5 5 1,2 1,2 3,4 3,4 0.0630 0.6694; 0.2170 0.4664; 0.3216 1.0508; 1.5842 p p p p p j j j j µ µ µ µ µ µ o o o o = · O = ÷ ± = · O = ÷ ± = · O = ÷ = · O = ± = · O = ± 4. BT transform of poles and zeros: 2 2 k k k T p T µ µ + = ÷ 2 2 m m m T z T o o + = ÷ 1,2 3,4 5 1,2 3,4 0.7542 0.5692 0.7278 0.3635 0.7228 0.5673 0.8235 0.2289 0.9734 p j p j p z j z j = ± = ± = = ± = ± -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Part I m a g i n a r y P a r t Since N-M = 1, we must add one zero: 5 1 z = ÷ ELEN 5346/4304 DSP and Filter Design Fall 2008 18 “Practical” Example 5. Verification Need to check whether specs are satisfied. Yes: we are done! No: start over (perhaps, a higher order of the prototype will help) In general (especially, for more complicated filters), multiple iterations may be needed to satisfy specifications (meet or exceed them). ELEN 5346/4304 DSP and Filter Design Fall 2008 19 Frequency (spectral) transformation It is often needed to modify the characteristics of existing digital filter to meet new specs without starting the design from scratch. Frequency transformation can convert digital LPF to either another LPF, or HPF, or BPF, or BSF. The transformation involves replacing the variable z by a rational function F(ž), while satisfying the following conditions to preserve BIBO: 1 1 z z > > = ¬ = < < Note: F(z) is an allpass filter: 1 * 1 1 1 1 1 ( ) ( ) 1 L l l l z z F z z F z z z o o ÷ ÷ ÷ ÷ ÷ = ÷ = ¬ = ¬ = ± ÷ [ 1 l o < Here to ensure that a stable filter is transferred to another stable filter. (10.19.1) (10.19.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 20 Frequency (spectral) transformation 1. LPF to LPF: ( ) 1 1 1 1 1 1 z F z z z z z o o o o ÷ ÷ ÷ ÷ ÷ ÷ = = = ÷ ÷ 1 j j j e e e e e e o o ÷ ÷ ÷ ÷ ¬ = ÷ Where e is the “old frequency” and is the “new frequency”. e (10.20.1) (10.20.2) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 1 j j j j j j j j j j j j j j j j j j j j e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e o o o ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ = ÷ ¬ = = ÷ ÷ sin 2 sin 2 e e o e e ÷ | | | \ . = + | | | \ . We can relate ANY frequency component of the new LPF to the frequency component of the prototype LPF. (10.20.3) (10.20.4) ELEN 5346/4304 DSP and Filter Design Fall 2008 21 Frequency (spectral) transformation sin 2 sin 2 c c c c e e o e e ÷ | | | \ . = + | | | \ . Usually: From (10.20.2): ( )( ) 2 2 2 1 2 1 1 1 j j j j j j j j j e e e e e e e e e e e e e e e e e e o o o o o o o o ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ + + + + + = = = + + + 2 1 2 1 sin tan 2 1 cos o e e o o e ÷ | | ( ÷ ¸ ¸ | = ÷ | ( + + ¸ ¸ \ . Clearly: s s p p e e e e = (10.21.1) (10.21.2) (10.21.3) (10.21.4) ELEN 5346/4304 DSP and Filter Design Fall 2008 22 Frequency (spectral) transformation Example of mapping: -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 e/t e " / t o = 0 o = 0.6 o = -0.9 e e c -e c e c e c e ÷ Note: the gain in unchanged ELEN 5346/4304 DSP and Filter Design Fall 2008 23 Frequency (spectral) transformation 2. LPF to HPF: ( ) 1 1 1 1 1 1 1 1 z F z z z z z o o o o ÷ ÷ ÷ ÷ ÷ ÷ + ÷ + + = = ÷ = + 1 j j j e e e e e e o o ÷ ÷ ÷ + ¬ = ÷ + (10.23.1) (10.23.2) j j j e e e e e e o o ÷ ÷ ÷ + = ÷ ÷ cos 2 co 2 1 s j j j j e e e e e e e e e e o e e ÷ ÷ ÷ ÷ ÷ ÷ = ÷ | | | \ . = ÷ + | | | + \ . 2 1 2 1 sin tan 2 1 cos o e e o o e ÷ | | ( ÷ ¸ ¸ | = ÷ | ( ÷ ÷ + ¸ ¸ \ . (10.23.3) (10.23.4) (10.23.5) ELEN 5346/4304 DSP and Filter Design Fall 2008 24 Frequency (spectral) transformation Example of mapping: -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 e/t e " / t o = 0 o = 0.6 o = -0.9 e e c -e c e c e c e ÷ ELEN 5346/4304 DSP and Filter Design Fall 2008 25 Frequency (spectral) transformation 3. LPF to BPF (filter order will be doubled): 2 1 1 2 1 1 2 1 1 1 2 1 1 1 z z z z z o| | | | | o| | | ÷ ÷ ÷ ÷ ÷ ÷ ÷ + + + = ÷ ÷ ÷ + + + 2 1 2 1 cos 2 cos 2 c c c c e e o e e + | | | \ . = ÷ | | | \ . 2 1 cotan tan 2 2 c c c e e e | ÷ | | | | = | | \ . \ . (10.25.1) (10.25.2) (10.25.3) Here and are new upper and lower cutoff frequencies. 2 c e 1 c e ELEN 5346/4304 DSP and Filter Design Fall 2008 26 Frequency (spectral) transformation A special case: bandwidth preserving 2 1 c c c e e e ÷ = Therefore: 2 1 1 1 1 1 1 1 1 z z z z z z z o o o o ÷ ÷ ÷ ÷ ÷ ÷ ÷ | | ÷ ÷ = ÷ = ÷ | ÷ + ÷ \ . 1 | = and: (10.26.1) (10.26.2) (10.26.3) ELEN 5346/4304 DSP and Filter Design Fall 2008 27 4. LPF to BSF (filter order will be doubled): 2 1 1 2 1 1 2 1 1 1 2 1 1 1 z z z z z o | | | | o | | ÷ ÷ ÷ ÷ ÷ ÷ ÷ + + + = ÷ ÷ ÷ + + + 2 1 2 1 cos 2 cos 2 c c c c e e o e e + | | | \ . = ÷ | | | \ . Frequency (spectral) transformation 2 1 tan tan 2 2 c c c e e e | ÷ | | | | = | | \ . \ . (10.27.1) (10.27.2) (10.27.3) ELEN 5346/4304 DSP and Filter Design Fall 2008 28 Spectral transformation: Example Example 10.3: Design a BW preserving BPF from the LP prototype: 1 1 1 1 ( ) 1 0.9 0.9 z z G z z z ÷ ÷ + + = = ÷ ÷ -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Part I m a g i n a r y P a r t ( ) 0 1 2 ( ) 20 26 1 0.9 j z e G z dB = = = = ~ ÷ 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 Fractional frequency | G ( z ) | DC gain: (10.28.1) (10.28.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 29 Spectral transformation: Example 1 1 1 1 1 1 j j j j z e z z e e z e e e e e o o o o ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ | | | | ÷ ÷ = ÷ ¬ = ÷ | | ÷ ÷ \ . \ . Since the filter is BW preserving, | = 1 For e = 0: 0 0 0 1 1 j j j e e e e e e o o ÷ ÷ ÷ | | ÷ = ÷ | ÷ \ . Assuming - this is where we map DC. ( ) 0 0 0 0 0 0 1 cos j j j j j e e e e e e e e e e o o o o o e ÷ ÷ ÷ ÷ ¬ ÷ = ÷ ¬ ÷ = ÷ ¬ = 0 0.4 e t = 0.309016994 o ¬ = 1 1 1 1 1 ( ) 1 z z F z z z o o ÷ ÷ ÷ ÷ ÷ | | ÷ = = ÷ | ÷ \ . 1 1 1 1 ( ) ( ) ( ) 1 z z z F z z z z o o o o ÷ ÷ ÷ ÷ ÷ = = ÷ ÷ ÷ In a general case, both o and | can be found from (10.27.2) and (10.27.2). ELEN 5346/4304 DSP and Filter Design Fall 2008 30 Spectral transformation: Example Method 1: 1 1 1 ( ) ( ) 1 0.9 LP BP z G z G z z ÷ ÷ + = ÷ ÷ ( )( ) ( )( ) 1 1 1 1 2 1 1 1 1 1 1 2 1 1 2 1 2 1 1 1 1 ( ) 1 1 0.9 0.9 1 0.9 1 1 1 1.9 0.9 BP z z z z z z z G z z z z z z z z z z z o o o o o o o o o o o ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ | | ÷ + | ÷ + ÷ ÷ ÷ \ . = = | | ÷ ÷ ÷ ÷ + ÷ | ÷ \ . ÷ = ÷ + Method 2: SFG replacement… ELEN 5346/4304 DSP and Filter Design Fall 2008 31 Spectral transformation: Example Method 3 (preferred): Zeros: 1 2 2 , 2 ( ) ( ) 1 1 1 1 1 k k z z z F z z z z z z z o o o o ÷ = ¬ ÷ = ÷ ¬ ÷ + = ÷ + ÷ ¬ = = ± ¬ Poles: 2 2 2 1 2 2 , 2 1 , ( ) ( ) 1 1.9 0.9 0 0.2936 0 1.9 1.9 4 0 .902 .9 2 1 k k p p p F p p p p p p p p p j o o o o o o o ÷ = ¬ 0.9 = ÷ ¬ 0.9÷0.9 = ÷ + ÷ ¬ ÷ + = ± ÷ · ¬ = ± = ¬ Therefore: { } 2 2 1 2 1 2 1 1 ( ) 1 1 1.9 0.9 1 0.5871 0.9 BP z z G z k k z z z z o ÷ ÷ ÷ ÷ ÷ ÷ + + = = = = ÷ + ÷ + Gain factor The new gain can be determined experimentally…