In this chapter include all of the following: Sinusoidal response of LTI systems, response of LTI systems in the frequency domain, distortion of signals passing through LTI systems, ideal and practical filters, frequency response for rational system functions, dependency of frequency response on poles and zeros,...
Nguyễn Công Phương SIGNAL PROCESSING Transform Analysis of LTI Systems Contents I Introduction II Discrete – Time Signals and Systems III The z – Transform IV Fourier Representation of Signals V Transform Analysis of LTI Systems VI Sampling of Continuous – Time Signals VII.The Discrete Fourier Transform VIII.Structures for Discrete – Time Systems IX Design of FIR Filters X Design of IIR Filters XI Random Signal Processing sites.google.com/site/ncpdhbkhn Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal and Practical Filters Frequency Response for Rational System Functions Dependency of Frequency Response on Poles and Zeros Design of Simple Filters by Pole – Zero Placement Relationship between Magnitude and Phase Responses Allpass Systems Invertibility and Minimum – Phase Systems Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn Sinusoidal Response of LTI Systems (1) x[ n ] = z → y[n ] = H ( z ) z , all n; H ( z ) = ∑ h[ k ]z ∞ n n −k k =−∞ z = e jω → x[n] = e jωn → y[n] = H (e jω )e jωn , all n jω H (e ) = H ( z ) z =e jω = jω jω H (e ) = H (e ) e x[n ] = Ae j ( ω n +φ ) j∠H ( e jω ) ∞ − jωk h [ k ] e ∑ k =−∞ = H R (e jω ) + jH I (e jω ) jω → y[n] = A H (e ) e j [ωn +φ +∠H ( e jω )] The response of a stable LTI system to a complex exponential sequence is a complex exponential sequence with the same frequency, only the amplitude and phase are changed by the system sites.google.com/site/ncpdhbkhn Sinusoidal Response of LTI Systems (2) Ax jφx jωn Ax − jφx − jωn x[n ] = Ax cos(ωn + φx ) = e e + e e 2 x[n ] = Ae j ( ω n +φ ) jω → y[n] = A H (e ) e j [ωn +φ +∠H ( e jω )] Ax jφx jωn Ax jφx j [ ωn +∠H ( e jω )] jω x1[n ] = e e → y1[n ] = H (e ) e e 2 Ax − jφx − jωn Ax − jφ x j [ − ωn +∠H ( e − jω )] − jω x2 [n] = e e → y2 [ n ] = H (e ) e e 2 Ax Ax j [ω n +φ x +∠H ( e jω )] j [ − ωn −φx +∠H ( e − jω )] jω − jω → y[ n ] = H (e ) e + H (e ) e 2 sites.google.com/site/ncpdhbkhn Sinusoidal Response of LTI Systems (3) Ax Ax j[ ωn +φ x +∠H ( e jω )] j[ − ωn −φ x +∠H ( e − jω )] jω − jω y[ n ] = H (e ) e + H (e ) e 2 H (e − jω ) = H (e jω ) ; ∠H (e− jω ) = −∠H (e− jω ) Ax Ax j [ ωn +φx +∠H ( e jω )] − j [ ωn +φx +∠H ( e jω )] jω jω → y[ n ] = H (e ) e + H (e ) e 2 Ax j [ ωn +φx +∠H ( e jω )] − j [ ωn +φ x +∠H ( e jω )] jω = H (e ) {e +e } = Ax H (e jω ) cos ωn + φx + ∠H (e jω ) x[ n ] = Ax cos(ωn + φx ) → y[n ] = Ay cos(ωn + φ y ) Ay = Ax H ( e jω ) ( magnitude response/gain ) φ y = ∠H (e jω ) + φx (phase response) sites.google.com/site/ncpdhbkhn Ex Sinusoidal Response of LTI Systems (4) Find the frequency response of the system y[n] = ay[n – 1] + bx[n], –1 < a < x[n] = e jωn → y[n] = H (e jω )e jωn y[n − 1] = H (e jω )e jω ( n −1) → H ( e jω )e jωn = aH ( e jω )e jω ( n −1) + be jωn b y[ n] = ay[ n − 1] + bx[ n] → H (e jω ) = − ae − jω e − jω = cos ω − j sin ω → H (e jω ) = b − cos ω a sin ω − jb − 2a cos ω + a − 2a cos ω + a b jω H (e ) = − a cos ω + a → ∠H (e jω ) = atan −a sin ω − a cos ω sites.google.com/site/ncpdhbkhn Sinusoidal Response of LTI Systems (5) Ex Find the frequency response of the system y[n] = ay[n – 1] + bx[n], b H (e jω ) = − 2a cos ω + a ∠H (e jω ) = atan ; –1 < a < −a sin ω − a cos ω Magnitude 1.5 0.5 -3 -2 -1 omega (rad) -3 -2 -1 omega (rad) 0.6 0.4 Phase 0.2 -0.2 -0.4 sites.google.com/site/ncpdhbkhn Sinusoidal Response of LTI Systems (6) x[n] = e jωn → y[n ] = H (e jω )e jωn , all n x[n] = e jω n n n k =0 k =0 u[n] → y[n ] = ∑ h[k ]x[n − k ] = ∑ h[k ]e jω ( n −k ) ∞ ∞ − jω k jω n − jωk jωn = ∑ h[k ]e e − h [ k ] e ∑ e k =0 k =n +1 jω = H (e )e jω n y steady state [ n ] ∞ − jωk jωn − ∑ h[k ]e e k = n +1 ytransient response [ n ] ytr [n] ≤ ∞ ∑ k = n +1 ∞ h[k ] ≤ ∑ h[k ] < ∞ k =0 lim y[n] = H (e jω )e jωn = yss [n] n →∞ sites.google.com/site/ncpdhbkhn Ex Sinusoidal Response of LTI Systems (7) Given a system h[n] = 0.7nu[n] and an input x[n] = cos(0.03πn)u[n], find the output? y[ n] = h[ n]* x[n ] → Y ( z ) = H ( z ) X ( z ) H ( z) = , − 0.7 z −1 z > 0.7 − cos(0.03π ) z −1 X ( z) = , −1 −2 − cos(0.03π ) z + z z >1 − cos(0.03π ) z −1 → Y ( z) = (1 − 0.7 z −1 )[1 − cos(0.03π ) z −1 + z −2 ] − cos(0.03π )z −1 + z −2 = ∆ = cos2 α − = − sin α −1 → ( z )1,2 cos α ± −4 sin a = = cos α ± j sin α = e ± jα = e ± j 0.03π → − cos(0.03π ) z −1 + z −2 = ( z −1 − e j 0.03π )( z −1 − e − j 0.03π ) = (1 − e j 0.03π z −1 )(1 − e − j 0.03π z −1 ) − cos(0.03π ) z −1 → Y ( z) = (1 − 0.7 z −1 )(1 − e j 0.03π z −1 )(1 − e − j 0.03π z −1 ) sites.google.com/site/ncpdhbkhn 10 Ex Systems with Rational System Functions (4) s − 3s + 2.3077 0.9806∠2.3007 0.9806∠ − 2.3007 H (s) = = + + s + 8s + 29 s + 52 s+4 s + − j3 s + + j3 → h(t ) = 2.3077e −4 t + (0.9806∠2.3007)e −( 2− j 3) t + (0.9806∠ − 2.3007)e− ( 2+ j 3)t = 2.3077e −4 t u(t ) + 0.9806e j 2.3007e −( 2− j 3)t u(t ) + 0.9806e− j 2.3007e −( + j 3) t u(t ) = 2.3077e −4 t u( t ) + 0.9806e−2t [ e( 3t +2.3007 ) + e− ( 3t + 2.3007) ]u(t ) = 2.3077e −4 t u( t ) + 0.9806e−2t [ cos(3t + 2.3007)]u( t ) = [ 2.3077e −4 t + 1.9612e −2 t cos(3t + 2.3007)]u(t ) sites.google.com/site/ncpdhbkhn 99 Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal and Practical Filters Frequency Response for Rational System Functions Dependency of Frequency Response on Poles and Zeros Design of Simple Filters by Pole – Zero Placement Relationship between Magnitude and Phase Responses Allpass Systems Invertibility and Minimum – Phase Systems Transform Analysis of Continuous – Time Systems a) b) c) d) e) f) System Function and Frequency Response The Laplace Transform Systems with Rational System Functions Frequency Response from Pole – Zero Location Minimum – Phase and Allpass Systems Ideal Filters sites.google.com/site/ncpdhbkhn 100 Frequency Response from Pole – Zero Location (1) M H (s) = M ∑b s k ∑a s k k =0 N k =0 k b = 0× a0 k ∏ (s − z ) k =1 N ∏ (s − p ) H ( jΩ ) = H ( s ) s = jΩ = k =0 N M k k ∑ a ( jΩ ) k =0 k k =1 M ∑ b ( jΩ ) k k b = 0× a0 k jΩ − z k = Qk e jΘk ; ∏ ( jΩ − z ) k =1 N k ∏ ( jΩ − p ) k =1 k jΩ − pk = Rk e jΦ k M b0 → H ( jΩ ) = a0 ∏ Q ( jΩ) M N k =1 exp j ∠(b0 / a0 ) + ∑ Θk ( jΩ) − ∑ Φ k ( jΩ) N k =1 k =1 ∏ Rk ( jΩ) k k =1 sites.google.com/site/ncpdhbkhn 101 Frequency Response from Pole – Zero Location (2) M b H ( jΩ ) = a0 ∏ Q ( jΩ) M N exp j ∠(b0 / a0 ) + ∑ Θk ( jΩ) − ∑ Rk ( jΩ) k =1 k =1 ∏ Rk ( jΩ) k =1 N k k =1 b0 Product of zero vectors to s = jΩ H ( j Ω ) = × → a0 Product of pole vectors to s = jΩ ∠H ( jΩ ) = ∠(b / a ) + (Sum of zero angles to s = jΩ) − 0 − (Sum of pole angles to s = jΩ ) sites.google.com/site/ncpdhbkhn 102 Frequency Response from Pole – Zero Location (3) H ( s) = G s+a H ( jΩ ) = L P G H ( jΩ ) = jΩ + a jΩ × s = −a φ σ G G G = = jΩ + a Ω + a PL ∠H ( jΩ) = ∠G − ∠( jΩ + a ) Ω = ∠G − atan a = −φ sites.google.com/site/ncpdhbkhn 103 Frequency Response from Pole – Zero Location (4) Ω2n H ( s) = s + 2ζΩn s + Ωn2 If (2ζΩ n )2 − 4Ω n2 < ↔ −1< ζ < then p1,2 = −ζΩ n ± jΩ n − ζ = −α ± jβ → h(t ) = Ωn ( ) e − ζΩn sin Ω n − ζ t u(t ), < ζ < 1− ζ sites.google.com/site/ncpdhbkhn 104 Frequency Response from Pole – Zero Location (5) Ω2n Ω2n Ω2n H (s) = = = s + 2ζΩn s + Ωn ( s − p1 )( s − p2 ) ( s + α − jβ )( s + α + jβ ) p1,2 = −ζΩn ± jΩ n − ζ = −α ± j β p1 = p2 = α + β 2 s − p1 = ( −ζΩ n )2 + (Ω n − ζ )2 = Ω n2 H ( jΩ ) = Ω Ω = s − p1 s − p2 PL × P2 L n n ∠H ( jΩ) = ∠Ω − ∠( s − p1 ) − ∠( s − p2 ) n = −φ1 − φ2 P1 β × φ1 −α φ1 P2 × −β −Ω n sites.google.com/site/ncpdhbkhn jΩ σ s − p2 105 Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal and Practical Filters Frequency Response for Rational System Functions Dependency of Frequency Response on Poles and Zeros Design of Simple Filters by Pole – Zero Placement Relationship between Magnitude and Phase Responses Allpass Systems Invertibility and Minimum – Phase Systems Transform Analysis of Continuous – Time Systems a) b) c) d) e) f) System Function and Frequency Response The Laplace Transform Systems with Rational System Functions Frequency Response from Pole – Zero Location Minimum – Phase and Allpass Systems Ideal Filters sites.google.com/site/ncpdhbkhn 106 Minimum – Phase and Allpass Systems (1) ∠H ( jΩ) = α H ( jΩ) = H ( jΩ) H * ( jΩ) H * ( jΩ) = H ( − jΩ) jΩ L = jΩ α H ( s ) s = jΩ = H ( jΩ) × P1 → H ( jΩ ) = H ( s ) H ( − s ) s = jΩ Z1 σ Hmin H ∠H ( jΩ) = β ∠H(jΩ) jΩ L = jΩ β -1 × -2 P1 -3 -3 -2 -1 Ω (rad) sites.google.com/site/ncpdhbkhn σ Z1 107 Minimum – Phase and Allpass Systems (2) H (s) = s − zk s − pk s − pk = Rk ∠Φ k , Rk = Qk , jΩ ∠H allpass ( jΩ) = α L = jΩ Rk s − zk = Qk ∠Θk Φ k + Θk = π , for all Ω s − zk Rk H ( jΩ ) = = = (allpass system) s − pk Qk × Pk s = −a α Qk Θk Φk Zk s=a ∠H ( jΩ) = ∠( s − zk ) − ∠( s − pk ) = α = Θk − Φ k Ω = π − 2Φ k = π − 2atan a H ( jΩ) = 1∠α sites.google.com/site/ncpdhbkhn 108 σ Minimum – Phase and Allpass Systems (3) ( s − zk )( s − zk* ) H (s) = ( s − pk )( s − pk* ) Φk s − pk = Rk ∠Φ k , s − zk = Qk ∠Θk s − pk* = Rk*∠Φ *k , s − zk* = Qk*∠Θ*k Rk = Qk , Rk* = Qk* H ( jΩ ) = jΩ b × P k β L = jΩ Rk Zk Qk α −a Rk* s − zk s − zk* s − pk s − pk* Qk Qk* = = (allpass system) * Rk Rk Θk × Φ * k * k P Qk* a Θ*k −b Z k* ∠H ( jΩ) = ∠( s − zk ) + ∠( s − zk* ) − ∠( s − pk ) − ∠( s − pk* ) = Θk + Θ*k − Φ k − Φ *k sites.google.com/site/ncpdhbkhn 109 σ Minimum – Phase and Allpass Systems (4) An Nth order allpass system has the following properties: jΩ Φk b × P k The zeros and poles are symmetric with respect to the jΩ axis, that is, if the poles are pk, then the zeros are − pk* Therefore, the system function is: Zk Qk α −a The phase response is monotonically Pk* decreasing from 2πN to zero as Ω increases from –∞ to ∞ sites.google.com/site/ncpdhbkhn β L = jΩ Rk ( s + p1* ) ( s − p N* ) H (s) = ( s − p1 ) ( s − p*N ) Θk × Rk* Φ * k Qk* a Θ*k −b Z k* 110 σ Minimum – Phase and Allpass Systems (5) The process of decomposing a nonminimum – phase into a product of a minimum – phase and an allpass system function H ( s) = H ( s) H allpass ( s) For each zero in the right – half plane, include a pole and a zero at its mirror position in the left half plane Assign the left – half plane zeros and the original poles to Hmin(s) Assign the right – half plane zeros and the left – half plane poles introduced in step to Hallpass(s) sites.google.com/site/ncpdhbkhn 111 Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal and Practical Filters Frequency Response for Rational System Functions Dependency of Frequency Response on Poles and Zeros Design of Simple Filters by Pole – Zero Placement Relationship between Magnitude and Phase Responses Allpass Systems Invertibility and Minimum – Phase Systems Transform Analysis of Continuous – Time Systems a) b) c) d) e) f) System Function and Frequency Response The Laplace Transform Systems with Rational System Functions Frequency Response from Pole – Zero Location Minimum – Phase and Allpass Systems Ideal Filters sites.google.com/site/ncpdhbkhn 112 Ideal Filters y (t ) = Gx(t − td ) H ( jΩ) = Ge − jΩtd e− jΩtd , H lowpass ( jΩ) = 0, Ω ≤ Ωc otherwise Ω c sin Ωc (t − td ) hlowpass (t ) = πΩc (t − td ) sites.google.com/site/ncpdhbkhn 113 ... x[n] y[n] -1 -2 -3 -4 20 40 60 80 Time index (n) sites.google.com/site/ncpdhbkhn 100 120 140 160 14 Transform Analysis of LTI Systems Sinusoidal Response of LTI Systems Response of LTI Systems. .. (e jω ) dB 20 Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal... sites.google.com/site/ncpdhbkhn 23 Transform Analysis of LTI Systems 10 11 Sinusoidal Response of LTI Systems Response of LTI Systems in the Frequency Domain Distortion of Signals Passing through LTI Systems Ideal