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In this chapter you will learn: Frequency response, system functions, relationship between magnitude and phase, all-pass systems, minimum-phase systems, linear systems with generalized linear phase.
Digital Signal Processing, Fall 2006 Lecture 5: System analysis Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, V, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction System System analysis System structures MM6 MM5 Filter MM4 z-transform MM3 DFT/FFT Filter structures MM9,MM10 MM7 Filter design MM8 Digital Signal Processing, V, Zheng-Hua Tan, 2006 System analysis Three domains Time domain: impulse response, convolution sum y[n] = x[n] * h[n] = ∞ ∑ x[k ]h[n − k ] k = −∞ Frequency domain: frequency response Y ( e jω ) = X ( e jω ) H ( e jω ) z-transform: system function Y ( z) = X ( z)H ( z) LTI system is completed characterized by … Digital Signal Processing, V, Zheng-Hua Tan, 2006 Part I: Frequency response Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase Digital Signal Processing, V, Zheng-Hua Tan, 2006 Frequency response Relationship btw Fourier transforms of input and output Y ( e jω ) = X ( e jω ) H ( e jω ) In polar form Magnitude Ỉ magnitude response, gain, distortion | Y (e jω ) |=| X (e jω ) | ⋅ | H (e jω ) | Phase Ỉ phase response, phase shift, distortion ∠Y (e jω ) = ∠X (e jω ) + ∠H (e jω ) Digital Signal Processing, V, Zheng-Hua Tan, 2006 Ideal lowpass filter – an example Frequency response | ω |< ωc , ⎧ 1, H ( e jω ) = ⎨ ⎩0, ωc 0.9 − 0.9 z −1 − 0.9 z −1 0.9 z −1 H i ( z) = = − − 0.5 z −1 − 0.5 z −1 − 0.5 z −1 H ( z) = So, | z |> 0.5 hi [n] = (0.5) n u[n] − 0.9(0.5) n −1 u[n − 1] 16 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Part III: Magnitude and phase Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase 17 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Relationship btw magnitude and phase In particular, for systems with rational system functions, there is constraint btw magnitude and phase H (e jω ) =| H (e jω ) | e j∠H ( e jω ) Consider the square of the magnitude | H (e jω ) |2 = H (e jω ) H * (e jω ) = H ( z ) H * (1 / z * ) | z = e jω M M H ( z) = ( ∏ (1 − cm z −1 ) b0 m =1 ) a0 N H * (1 / z * ) = ( ∏ (1 − d k z −1 ) ∏ (1 − cm* z ) b0 m =1 ) a0 N ∏ (1 − d k * z ) k =1 k =1 M b C ( z ) = H ( z ) H (1 / z ) = ( ) a0 * * ∏ (1 − cm z −1 )(1 − cm z ) * m =1 N ∏ (1 − d k z −1 )(1 − d k * z) k =1 18 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example P271, Example 5.11 19 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example 20 Digital Signal Processing, V, Zheng-Hua Tan, 2006 10 Part VI: All-pass systems Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase 21 Digital Signal Processing, V, Zheng-Hua Tan, 2006 All-pass systems Consider the following stable system function z −1 − a * − az −1 H ap ( z ) = e − jω − a * − ae − jω * jω − jω − a e =e − ae − jω H ap (e jω ) = | H ap (e jω ) |= all-pass system: for which the frequency response magnitude is a constant General form z −1 − d k M c ( z −1 − ek )( z −1 − ek ) ∏ −1 * k =1 − d z k =1 (1 − e z −1 )(1 − e z −1 ) k k k Mr * H ap ( z ) = A ∏ 22 Digital Signal Processing, V, Zheng-Hua Tan, 2006 11 An example P275 Example 5.13, Firstorder all-pass system 23 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example Second-order all-pass system 24 Digital Signal Processing, V, Zheng-Hua Tan, 2006 12 Part V: Minimum-phase systems Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase 25 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Minimum-phase systems Magnitude does not uniquely characterize the system 26 Stable and causal Ỉ poles inside unit circle, no restriction on zeros Zeros are also inside unit circle Ỉ inverse system is also stable and causal (in many situations, we need inverse systems!) Ỉ such systems are called minimum-phase systems (explanation to follow): are stable and causal and have stable and causal inverses Digital Signal Processing, V, Zheng-Hua Tan, 2006 13 Minimum-phase and all-pass decomposition Any rational system function can be expressed as: H ( z ) = H ( z ) H ap ( z ) Suppose H(z) has one zero outside the unit circle at z = / c * , | c |< H ( z ) = H ( z )( z −1 − c * ) = H ( z )(1 − cz −1 ) z −1 − c * − cz −1 minimum-phase all-pass 27 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Frequency response compensation When the distortion system is not minimum-phase system: H d ( z ) = H d ( z ) H ap ( z ) H c ( z) = G ( z ) = H d ( z ) H c ( z ) = H ap ( z ) H d ( z ) Frequency response magnitude is compensated Phase response is the phase of the all-pass 28 Digital Signal Processing, V, Zheng-Hua Tan, 2006 14 Properties of minimum-phase systems From minimum-phase and all-pass decomposition H ( z ) = H ( z ) H ap ( z ) arg[ H (e jω )] = arg[ H (e jω )] + arg[ H ap (e jω )] From previous figures, the continuous-phase curve of an all-pass system is negative for ≤ ω ≤ π So change from minimum-phase to nonminimumphase (+all-pass phase) always decreases the continuous phase or increases the negative of the phase (called the phase-lag function) Minimumphase is more precisely called minimum phase-lag system 29 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Part VI: Linear-phase systems Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase 30 Digital Signal Processing, V, Zheng-Hua Tan, 2006 15 Design a system with non-zero phase System design sometimes desires Constant frequency response magnitude Zero phase, when not possible 31 accept phase distortion, in particular linear phase since it only introduce time shift Nonlinear phase will change the shape of the input signal though having constant magnitude response Digital Signal Processing, V, Zheng-Hua Tan, 2006 Ideal delay H id (e jω ) = e − jωα , | ω |< π | H id (e jω ) |= ∠H id (e jω ) = −ωα , | ω |< π grd [ H id (e jω )] = α hid [n] = when α = nd sin π (n − α ) π (n − α ) hid [n] = δ [n − nd ] 32 Ideal lowpass with linear phase hlp [n] = sin ω c (n − nd ) π ( n − nd ) Digital Signal Processing, V, Zheng-Hua Tan, 2006 16 Generalized linear phase Linear phase filters H (e jω ) =| H (e jω ) | e − jωα Generalized linear phase filters H (e jω ) = A(e jω )e − jωα + jβ A(e jω ) is a real function of ω , α and β are real constants 33 Digital Signal Processing, V, Zheng-Hua Tan, 2006 Summary Frequency response System functions Relationship between magnitude and phase All-pass systems Minimum-phase systems Linear systems with generalized linear phase 34 Digital Signal Processing, V, Zheng-Hua Tan, 2006 17 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction System System analysis System structure MM6 MM5 Filter MM4 z-transform MM3 35 DFT/FFT Filter structures MM9,MM10 MM7 Filter design MM8 Digital Signal Processing, V, Zheng-Hua Tan, 2006 18 ... 18 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example P271, Example 5. 11 19 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example 20 Digital Signal Processing, V, Zheng-Hua Tan, ... dω Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example of group delay Figure 5. 1, 5. 2, 5. 3 10 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example of group delay 11 Digital Signal. .. Digital Signal Processing, V, Zheng-Hua Tan, 2006 11 An example P2 75 Example 5. 13, Firstorder all-pass system 23 Digital Signal Processing, V, Zheng-Hua Tan, 2006 An example Second-order all-pass