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Lecture Digital signal processing - Lecture 7 presents the following content: Filter design, IIR filter design, analog filter design, IIR filter design by impulse invariance, IIR filter design by bilinear transformation.
Digital Signal Processing, Fall 2006 Lecture 7: Filter Design Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part I: Filter design Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Filter design process Filter, in broader sense, covers any system Three design steps Problem Performance constraints Specifications Magnitude response Phase response (frequency domain) Complexity System function Approximations realization IIR or FIR Subtype Structure Solution Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Specifications – an example Specifications for a discrete-time lowpass filter − 0.01 ≤| H (e jω ) |≤ + 0.01, ≤ ω ≤ ω p | H (e jω ) |≤ 0.001, ω ≥ ω s δ = 0.01 δ = 0.001 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Specifications of frequency response Typical lowpass filter specifications in terms of tolerable Passband distortion, as smallest as possible Stopband attenuation, as greatest as possible Width of transition band: as narrowest as possible Improving one often worsens others Ỉ a tradeoff Increasing filter order improves all Digital Signal Processing, VII, Zheng-Hua Tan, 2006 DT filter for CT signals Discrete-time filter for the processing of continuoustime signals Bandlimited input signal High enough sampling frequency Then, specifications conversion is straightforward ⎧ H (e jΩT ), | Ω |< π / T H eff ( jΩ) = ⎨ | Ω |> π / T ⎩0, H (e jω ) = H eff ( j ω T ), | ω |< π ω = ΩT Fig 7.1 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Specifications – an example Specifications for a continuous-time lowpass filter − 0.01 ≤| H eff ( jΩ) |≤ + 0.01, ≤ Ω ≤ 2π (2000) | H eff ( jΩ) |≤ 0.001, Ω ≥ 2π (3000) Fig 7.2(a)(b) δ = 0.01 δ = 0.001 Ω p = 2π (2000) Ω s = 2π (3000) Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Design a filter Design goal: find system function to make frequency response meet the specifications (tolerances) Infinite impulse response filter Poles insider unit circle due to causality and stability Rational function approximation Finite impulse response filter Linear phase is often required Polynomial approximation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 E.g IIR filter design M For rational system function H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 find the system coefficients such that the corresponding frequency response H ( e j ω ) = H ( z ) | z = e jω provides a good approximation to a desired response H (e jω ) ≈ H desired (e jω ) 10 H(z) •Rational system function •Stable •causal Digital Signal Processing, VII, Zheng-Hua Tan, 2006 FIR or IIR Either FIR or IIR is often dependent on the phase requirements Only FIR filter can be at the same time stable, causal and GLP Design principle 11 If H(z) is stable and GLP, any non-trivial pole p inside the unit circle corresponds a pole 1/p outside the unit circle, so that H(z) cannot have a causal impulse response (as ROC is a ring including unit circle) If GLP is essential Ỉ FIR If not Ỉ IIR preferable (can meet specifications with lower complexity) Digital Signal Processing, VII, Zheng-Hua Tan, 2006 FIR and IIR FIR IIR 12 Rational system function Poles + zeros Stable/unstable Hard to control phase Low order (4-20) Designed on the basis of analog filter Polynomial system function Zeros Stable Easy to get linear phase High order (202000) Unrelated to analog filter Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part II: IIR filter design Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation 13 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Design IIR filter based on analog filter The mapping is direct ⎧ H (e jΩT ), | Ω |< π / T H eff ( jΩ) = ⎨ | Ω |> π / T ⎩0, H (e jω ) = H eff ( j ω T ), | ω |< π Advanced analog filter design techniques Ỉ Designing DT filter by transforming prototype CT filter: 14 Transform (map) DT specifications to analog Design analog filter Inverse-transform analog filter to DT Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Transformation method Transform (map) DT specifications to analog ω = ΩT Design analog filter H c ( s ) or hc (t ) Inverse-transform to DT H ( z ) or h[n] The imaginary axis of the s-plane ặ the unit circle of the z-plane • Poles in the left half of the s-plane Ỉ poles inside the unit circle in the zplane (stable) 15 s = σ + jΩ ∞ H ( s) = ∫ h(t )e − st dt −∞ ∞ H ( jΩ) = ∫ h(t )e − jΩt dt −∞ z = re − jω ∞ X ( e jω ) = ∑ x[n]e − jωn n = −∞ X ( z) = ∞ ∑ x[n]z − n n = −∞ Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part III: Analog filter design 16 Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Analog filter design Butterworth Chebyshev I Chebyshev II Ellipical 17 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Butterworth lowpass filters The magnitude response Maximally flat in the passband Monotonic in both passband and stopband The squared magnitude response | H c ( jΩ ) | = 1 + (Ω / Ω c ) N FigB.1.2 18 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Poles in s-plane 19 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Chebyshev filters Chebyshev II filters: equiripple in passband, flat in the stopband Chebyshev II filters: equiripple in stopband, flat in the passband 20 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 10 Elliptic filters 21 Equiripple both in stopband and in the passband Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part IV: Design by impulse invariance 22 Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 11 Filter design by impulse invariance Impulse invariance: a method for obtaining a DT jω system whose H (e ) is determined by the H c ( jΩ) of a CT system h[n] = Td hc (nTd ) Td - ' design' sampling interval 23 In DT filter design, the specifications are provided in the discrete-time, so Td has no role Td is included for discussion though Td also has nothing to with C/D and D/C conversion in Fig 7.1 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Relationship btw frequency responses Impulse response h[n] = Td hc (nTd ) Frequency response 2π H (e ) = ∑ H c ( j + j k) Td Td k = −∞ jω ω ∞ sampling: x[n] = xc (nT ) X ( e jω ) = T ∞ ∑ k = −∞ Xc( j ω T −j 2πk ) T if the CT filter is bandlimited H c ( jΩ) = 0, | Ω |≥ π / Td then H ( e jω ) = H c ( j ω Td ), | ω |≤ π This is also the way to get CT filter specifications from H (e jω ) by applying the relation Ω = ω / Td 24 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 12 Aliasing in the impulse invariance design h[n] = Td hc (nTd ) H ( e jω ) = ∞ ∑ k = −∞ 25 Hc ( j ω Td +j 2π k) Td Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Relationship btw system functions The transform from CT to DT is easy to carry out as a transformation on the system function Rational system function, after partial fraction expansion N Ak k =1 s − s k H c (s) = ∑ h[n] = Td hc (nTd ) ⎧N s t ⎪∑ Ak e k , t ≥ hc (t ) = ⎨ k =1 ⎪0, t < ⎩ N = ∑ Td Ak e sk nTd u[n] k =1 N = ∑ Td Ak (e sk Td )n u[n] k =1 N Td Ak s k Td −1 z k =1 − e H ( z) = ∑ 26 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 13 Impulse invariance with a Butterworth filter Specifications 0.89125 ≤| H (e jω ) |≤ 1, ≤| ω |≤ 0.2π | H (e jω ) |≤ 0.17783, 0.3π ≤| ω |≤ π Since the sampling interval Td cancels in the impulse invariance procedure, we choose Td=1, so ω = Ω Magnitude function for a CT Butterworth filter 0.89125 ≤| H c ( jΩ) |≤ 1, ≤| Ω |≤ 0.2π | H c ( jΩ) |≤ 0.17783, 0.3π ≤| Ω |≤ π Due to the monotonic function of Butterworth filter | H c ( j 0.2π ) |≥ 0.89125 | H c ( j 0.3π ) |≤ 0.17783 27 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Impulse invariance with a Butterworth filter Squared magnitude function of a Butterworth filter | H c ( jΩ ) | = 1 + (Ω / Ω c ) N (1) | H c ( j 0.2π ) |≥ 0.89125 | H c ( j 0.3π ) |≤ 0.17783 (2) 1+ ( 0.2π N ) =( )2 Ωc 0.89125 1+ ( 0.3π N ) =( )2 Ωc 0.17783 (4) N =6 (5) Ω c = 0.7032 (3) N = 5.8858 Ω c = 0.70474 H c ( s) H c (− s) = = 28 1 + ( s / jΩ c ) N 1 + ( s / j 0.7032)12 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 14 Impulse invariance with a Butterworth filter 12 poles for the squared magnitude function The system function has the three pole pairs in the left half of the s-plane 29 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Impulse invariance with a Butterworth filter H c (s) = H ( z) = + 0.12093 ( s + 0.3640 s + 0.4945)( s + 0.9945s + 0.4945)( s + 1.3585s + 0.4945) 0.2871 − 0.4466 z −1 (1 − 1.2971z −1 + 0.6949 z − ) − 2.1428 + 1.1455 z −1 1.8557 − 0.6303 z −1 + (1 − 1.0691z −1 + 0.3699 z − ) (1 − 0.9972 z −1 + 0.2570 z − ) 30 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 15 Impulse invariance with a Butterworth filter 31 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part V: Design by bilinear transformation 32 Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 16 Bilinear transformation By using impulse invariance, the relation between CT and DT frequency is linear (except for aliasing), thus the shape of the frequency response is preserved But only proper for bandlimited filters, problem for e.g highpass Bilinear transformation between s and z−1 −1 s= H ( z) = H c [ Inverse z= 33 1− z ( ) Td + z −1 1− z ( )] Td + z −1 + (Td / 2) s − (Td / 2) s Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Bilinear transformation Given s = σ + jΩ + (Td / 2) s + σTd / + jΩTd / = z= − (Td / 2) s − σTd / − jΩTd / if σ < 0, | z |< for any Ω if σ > 0, | z |> for any Ω 34 s = jΩ if z= + jΩTd / − jΩTd / so, | z |= 1, for any s on the jΩ - axis i.e the jΩ - axis maps onto the unit circle Digital Signal Processing, VII, Zheng-Hua Tan, 2006 17 Bilinear transformation – frequency relationship Consider frequency s= − z −1 ( ) Td + z −1 jΩ = Ω= s= + jΩTd = e jsω == jΩ − jΩTd − e − jω 2e − jω / ( j sin ω / 2) 2j ( ) [ − jω / ]= tan(ω / 2) = − jω Td + e Td 2e Td (cos ω / 2) tan(ω / 2) Td ω = arctan(ΩTd / 2) 35 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Bilinear transformation The bilinear transformation maps the entire jΩ -axis in the s-plane to one revolution of the unit circle in the z-plane Compare with Ω = ω / Td 36 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 18 Bilinear transformation of a Butterworth filter Specifications 0.89125 ≤| H (e jω ) |≤ 1, ≤| ω |≤ 0.2π | H (e jω ) |≤ 0.17783, 0.3π ≤| ω |≤ π Magnitude function for a CT Butterworth filter 0.89125 ≤| H c ( jΩ) |≤ 1, ≤| Ω |≤ | H c ( jΩ) |≤ 0.17783, 0.2π tan( ) Td 0.3π tan( ) ≤| Ω |≤ ∞ Td Due to the monotonic function of Butterworth filter Choose Td = | H c ( j tan(0.1π )) |≥ 0.89125 | H c ( j tan(0.15π )) |≤ 0.17783 37 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Bilinear transformation of a Butterworth filter Squared magnitude function of a Butterworth filter | H c ( jΩ ) | = 1 + (Ω / Ω c ) N | H c ( j tan(0.1π )) |≥ 0.89125 | H c ( j tan(0.15π )) |≤ 0.17783 N = 5.305 N =6 Ω c = 0.766 38 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 19 Bilinear transformation of a Butterworth filter 39 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Summary 40 Filter design IIR filter design Analog filter design IIR filter design by impulse invariance IIR filter design by bilinear transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 20 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 41 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 21 ... transformation Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Analog filter design Butterworth Chebyshev I Chebyshev II Ellipical 17 Digital Signal Processing, VII, Zheng-Hua Tan, 2006... jΩ ) | = 1 + (Ω / Ω c ) N FigB.1.2 18 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Poles in s-plane 19 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Chebyshev filters Chebyshev... passband 20 Digital Signal Processing, VII, Zheng-Hua Tan, 2006 10 Elliptic filters 21 Equiripple both in stopband and in the passband Digital Signal Processing, VII, Zheng-Hua Tan, 2006 Part