Signal processing - Transform analysis of lti systems

nalysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 18 sites.google.com/site/ncpdhbkhn 19 Response to Aperiodic Inputs ( ) ( ) ( ) j j jY e H e X eω ω ω= ( ) ( ) ( ) ( ) ( ) ( ) j j j j j j Y e H e X e Y e H e X e ω ω ω ω ω ω  = →  ∠ = ∠ +∠ sites.google.com/site/ncpdhbkhn 20 Power Gain 2 10Gain in dB 10( ) log ( ) j j dB H e H eω ω= = ( ) ( ) ( ) ( ) ( ) ( )j j j j j j dB dB dB Y e H e X e Y e H e X eω ω ω ω ω ω= → = + Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 21 sites.google.com/site/ncpdhbkhn 22 Distortion of Signals Passing through LTI Systems (1) A system has distortionless response if the input signal x[n] and the output signal y[n] have the same “shape” 0[ ] [ ],dy n Gx n n G= − > ( ) ( )d j nj jY e Ge X eωω ω−→ = ( ) ( ) ( ) d j j nj j Y eH e Ge X e ω ωω ω −→ = = ( ) ( ) j j d H e G H e n ω ω ω  = →  ∠ = − sites.google.com/site/ncpdhbkhn 23 Distortion of Signals Passing through LTI Systems (2) [ ] ( ) cos ( )j jx xy n A H e n H e ω ωω φ = + +∠  ( ) ( ) cos j j x x H eA H e n ω ω φω ω ω   ∠ = + +      ( ) ( ) j phase delay H e ω τ ω ω ∠ = − ( ) ( )group delay d d ω τ ω ω Ψ = − [ ] [ ]cos [ ] ( ) [ ( )]cos{ [ ( )]}c j c gd c c pd cx n s n n y n H e s n n ωω τ ω ω τ ω= → ≈ − − Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 24 sites.google.com/site/ncpdhbkhn 25 Ideal and Practical Filters (1) ω ( )jlpH e ω 1 cωcω− pipi− 0 Lowpass Highpass Bandpass Bandstop ω 1 cω ( )jhpH e ω 0pi− cω− pi ω 1 lω uω ( )jbpH e ω 0l ω−uω− pipi− ω 1 lω uω ( )jbsH e ω 0lω−uω− pi− pi sites.google.com/site/ncpdhbkhn 26 Ideal and Practical Filters (2) -20 -15 -10 -5 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 n x [ n ] -20 -15 -10 -5 0 5 10 15 20 -0.5 0 0.5 1 n h [ n ] -20 -15 -10 -5 0 5 10 15 20 -2 0 2 4 6 n y [ n ] 0 , ( ) , dj n cj lp c e H e ω ω ω ω ω ω pi − < =  < ≤ sin ( ) [ ] ( ) c d lp d n nh n n n ω pi − → = − [ ]lp n h n ∞ =−∞ = ∞∑ The ideal lowpass filter is unstable and practically unrealizable ω ( )jlpH e ω 1 cωcω− pipi− 0 sites.google.com/site/ncpdhbkhn 27 Ideal and Practical Filters (3) ω 1 lω uω ( )jbpH e ω 0l ω−uω− pipi− ω 1 1lω 1uω ( )jbpH e ω 0 pi2uω2lω Transition – band Transition – band Passband Stopband Stopband sin ( ) [ ] ( ) c d lp d n nh n n n ω pi − = − 0 1 0 otherwise sin ( ) , ˆ ( )[ ] , c d dlp n n n M n nh n ω pi − ≤ ≤ − −=    0 , ( ) , dj n cj lp c e H e ω ω ω ω ω ω pi − < =  < ≤ Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 28 sites.google.com/site/ncpdhbkhn 29 Frequency Response for Rational System Functions (1) 1 1 1 1[ ] [ ] [ ] N M k k k k y n a y n b x n = = = − − + −∑ ∑ 1 1 1 ( ) ( ) ( ) M k k k N k k k b z B zH z A z a z − = − = → = = + ∑ ∑ 1 1 1 ( ) ( ) ( ) j M j k k j k N j kz e k k b e B zH e A z a e ω ω ω ω − = − = = → = = + ∑ ∑ 1 1 1 0 0 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) j M M j k k k k N N j k k k kz e z z z e b b p z p e ω ω ω − − = = − − = == − − = = − − ∏ ∏ ∏ ∏ sites.google.com/site/ncpdhbkhn 30 Frequency Response for Rational System Functions (2) 1 0 1 1 1 ( ) ( ) ( ) M j k j k N j k k z e H e b p e ω ω ω − = − = − = − ∏ ∏ → 1 1 0 1 1 1 1 ( ) M k j k N k k z z H e b p z ω − = − = − = − ∏ ∏ 0 1 1 1 1( ) ( ) ( ) M N j j j k k k k H e b z e p eω ω ω− − = = ∠ = ∠ + ∠ − − ∠ −∑ ∑ 1 1 1 1( ) [ ( )] [ ( )] M N j j group delay k k k k d d z e p e d d ω ωτ ω ω ω − − = = = ∠ − − ∠ −∑ ∑ sites.google.com/site/ncpdhbkhn 31 Frequency Response for Rational System Functions (3) 1( ) ( )j jC e eβ ωω α −= − 1 cos( ) sin( )jα ω β α ω β= − − + − 2 1 1*( ) ( ) ( ) ( )( )j j j jC C C e eβ ω β ωω ω ω α α− − += = − − 21 2 cos( )α α ω β= + − − → 1 sin( ) ( ) atan cos( ) C α ω βω α ω β −∠ = − − 2 21 2 ( ) cos( ) ( ) cos( ) gd d d ω α α ω β τ ω ω α α ω β Ψ − − = − = + − − sites.google.com/site/ncpdhbkhn 32 Frequency Response for Rational System Functions (4) ;k k j j k k k kz q e p r e θ φ = = 2 1 0 2 1 0 1 1 2 2 2 1 1 2 1 2 atan atan 1 1 atan atan 1 2 cos( ) ( ) cos( ) sin( ) sin( ) ( ) cos( ) cos( ) cos( ) cos( ( ) cos( ) M k k k j k N k k k k M N j k k k k k kk k k k N k k k k k gd k k k k q q H e b r r q rH e b q r r r q q r r ω ω ω θ ω φ ω θ ω φ ω θ ω φ ω φ ω τ ω ω φ = = = = = + − − = + − − − −→ ∠ = ∠ + − − − − − − − − = − + − − ∏ ∏ ∑ ∑ ∑ 2 1 1 2 ) cos( ) M k k k k kq q θ ω θ =          −  + − −   ∑ sites.google.com/site/ncpdhbkhn 33 Frequency Response for Rational System Functions (5)Ex. Draw the magnitude spectrum of 0 1 2 2 01 2 ( ) cos( ) bH z r z r zω − − = − + Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros a) Geometrical Evaluation of H(ejω) from Poles and Zeros b) Significance of Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 34 sites.google.com/site/ncpdhbkhn 35 Geometrical Evaluation of H(ejω) from Poles and Zeros (1) 1 0 1 1 1 ( ) ( ) ( ) M j k j k N j k k z e H e b p e ω ω ω − = − = − = − ∏ ∏ 1 0 1 ( ) ( ) ( ) M j k j N M k N j k k e z b e e p ω ω ω − = = − = − ∏ ∏ 0 1 Re Im O Z ω kZ kθ j k ke z ω − → −Z Z k−Z Z kΘkQ kj kQ e Θ= × kP kφ j k ke p ω − → −Z P kjkR e Φ = k R k−Z P kΦ kα kβ 1 0 1 0 1 1 ( ) ( ) ( ) exp ( ) ( ) ( ) M k j k N k k M N k k k k Q H e b R j b N M ω ω ω ω ω ω = = = = = ×    × ∠ + − + Θ − Φ      ∏ ∏ ∑ ∑ sites.google.com/site/ncpdhbkhn 36 Geometrical Evaluation of H(ejω) from Poles and Zeros (2) 1 0 0 1 1 1 ( ) ( ) exp ( ) ( ) ( ) ( ) M k M N j k k kN k k k k Q H e b j b N M R ω ω ω ω ω ω = = = =    = × ∠ + − + Θ − Φ      ∏ ∑ ∑ ∏ 1 0 1 0 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) M k j k N k k M N j k k k k Q H e b R H e b N M ω ω ω ω ω ω ω = = = =   =  →   ∠ = ∠ + − + Θ − Φ  ∏ ∏ ∑ ∑ 0 1 Re Im O Z ω kZ kθ k−Z Z kΘkQ× kP kφ kR k−Z P kΦ kα kβ Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros a) Geometrical Evaluation of H(ejω) from Poles and Zeros b) Significance of Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 37 sites.google.com/site/ncpdhbkhn 38 Significance of Poles and Zeros (1) 1 0 0 1 1 1 ( ) ( ) exp ( ) ( ) ( ) ( ) M k M N j k k kN k k k k Q H e b j b N M R ω ω ω ω ω ω = = = =    = × ∠ + − + Θ − Φ      ∏ ∑ ∑ ∏ ( )j j j k k K KH e e p R e ω ω ω = = − 0 1 Re Im O Z ω kZ kθ k−Z Z kΘkQ× kP kφ kR k−Z P kΦ kα kβ ( ) ( ) ( ) j k j k KH e R H e ω ω ω ω  = →  ∠ = −Φ ( ) ( )k k k kα φ ω pi φ pi+ − + Φ + − = k kω α→ −Φ = ( ) j kH e ω α→∠ = sites.google.com/site/ncpdhbkhn 39 Significance of Poles and Zeros (2) ( )j j j k k K KH e e p R e ω ω ω = = − 0 0.5 1 1.5 2 3 4 5 6 7 8 9 ω | H ( j ω ) | pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 0 0.5 1 1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 ω ∠ H ( j ω ) pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 sites.google.com/site/ncpdhbkhn 40 Significance of Poles and Zeros (3) j j j j j k k k k K KH e e p e p R e R e ω ω ω ω ω− = = − − * ( ) ( )( ) 0 0.5 1 1.5 1 2 3 4 5 6 7 8 9 ω | H ( j ω ) | pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 0 0.5 1 1.5 0 0.5 1 1.5 2 ω ∠ H ( j ω ) pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 sites.google.com/site/ncpdhbkhn 41 Significance of Poles and Zeros (4) 1 0 0 1 1 1 ( ) ( ) exp ( ) ( ) ( ) ( ) M k M N j k k kN k k k k Q H e b j b N M R ω ω ω ω ω ω = = = =    = × ∠ + − + Θ − Φ      ∏ ∑ ∑ ∏ ( ) ( ) j j j k kH e K e z KQ eω ω ω= − = 0 1 Re Im O Z ω kZ kθ k−Z Z kΘkQ× kP kφ kR k−Z P kΦ kα kβ ( ) ( ) ( ) j k j k H e KQ H e ω ω ω ω  = →  ∠ = −Θ ( ) ( )j kH e ω ω ω β→∠ = − + Θ = sites.google.com/site/ncpdhbkhn 42 Significance of Poles and Zeros (5) ( ) ( ) j j j k kH e K e z KQ eω ω ω= − = 0 0.5 1 1.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω | H ( j ω ) | zk = 0.8e jpi/6 zk = 0.9e jpi/6 zk = 0.1e jpi/6 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 ω ∠ H ( j ω ) zk = 0.8e jpi/6 zk = 0.9e jpi/6 zk = 0.1e jpi/6 sites.google.com/site/ncpdhbkhn 43 Significance of Poles and Zeros (6) j j j j j k k k kH e K e z e z KQ e Q eω ω ω ω ω−= − − =*( ) ( )( ) 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 1.2 1.4 ω | H ( j ω ) | pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 0 0.5 1 1.5 -3 -2 -1 0 1 2 3 ω ∠ H ( j ω ) pk = 0.8e jpi/6 pk = 0.9e jpi/6 pk = 0.1e jpi/6 sites.google.com/site/ncpdhbkhn 44 Significance of Poles and Zeros (7) {ak, bk} Time - Domain Frequency - Domain z - Domain jz e ω= j e zω = 1 1 1 1[ ] [ ] [ ] N M k k k k y n a y n b x n = = = − − + −∑ ∑ 1 1 1 ( ) M k k k N k k k b z H z a z − = − = = + ∑ ∑ 1 1 1 ( ) M j k k j k N j k k k b e H e a e ω ω ω − = − = = + ∑ ∑ Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement a) Discrete – Time Resonators b) Notch Filters c) Comb Filters d) Pole – Zero Pattern Rotation 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 45 Design of Simple Filters by Pole – Zero Placement • To suppress a frequency component at ω = ω0, we should place a zero at angle θ = ω0 on the unit circle • To enhance or amplify a frequency component at ω = ω0, we should place a pole at angle ϕ = ω0 close but inside the unit circle • Complex poles or zeros should appear in complex conjugate pairs to assure that the system has real coefficients • Poles or zeros at the origin do not influence the magnitude response because their distance from any point on the unit circle is unity. However, a pole (or zero) at the origin adds (or subtracts) a linear phase of ω rads to the phase response. We often introduce poles and zeros at z = 0 to assure that N = M sites.google.com/site/ncpdhbkhn 46 sites.google.com/site/ncpdhbkhn 47 Discrete – Time Resonators (1) 0 0 1 2 2 1 11 2 1 1 ( ) cos ( )( ) j j b bH z r z r z re z re zφ φφ − − − − −= =− + − − 2 0 0 2 1 2 1 [ ] ( cos ) [ ] [ ] [ ] sin[( ) ] [ ] [ ] sin n y n r y n r y n b x n nh n b r u n φ φ φ  = − − − +  → + =  -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 2 Real Part I m a g i n a r y P a r t sites.google.com/site/ncpdhbkhn 48 Discrete – Time Resonators (2) 0 0 1 2 2 1 11 2 1 1 ( ) cos ( )( ) j j b bH z r z r z re z re zφ φφ − − − − −= =− + − − 2 0 0 2 1 2 1 [ ] ( cos ) [ ] [ ] [ ] sin[( ) ] [ ] [ ] sin n y n r y n r y n b x n nh n b r u n φ φ φ  = − − − +  → + =  0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -100 -50 0 50 100 Normalized Frequency (×pi rad/sample) P h a s e ( d e g r e e s ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -10 -5 0 5 10 15 20 Normalized Frequency (×pi rad/sample) M a g n i t u d e ( d B ) cω 21 2 cos cos c r r ω φ+= A discrete – time resonator sites.google.com/site/ncpdhbkhn 49 Discrete – Time Resonators (3) 0 0 1 2 2 1 11 2 1 1 ( ) cos ( )( ) j j b bH z r z r z re z re zφ φφ − − − − −= =− + − − 2 0 0 2 1 2 1 [ ] ( cos ) [ ] [ ] [ ] sin[( ) ] [ ] [ ] sin n y n r y n r y n b x n nh n b r u n φ φ φ  = − − − +  → + =  0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 0 1 2 3 4 5 6 7 8 9 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) sites.google.com/site/ncpdhbkhn 50 Discrete – Time Resonators (4) 0 0 1 2 2 1 11 2 1 1 ( ) cos ( )( ) j j b bH z r z r z re z re zφ φφ − − − − −= =− + − − 2 0 0 2 1 2 1 [ ] ( cos ) [ ] [ ] [ ] sin[( ) ] [ ] [ ] sin n y n r y n r y n b x n nh n b r u n φ φ φ  = − − − +  → + =  0 1 If sin r b A φ =  = 1[ ] sin[( ) ] [ ]h n A n u nφ→ = + A discrete – time sinusoidal oscillator Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement a) Discrete – Time Resonators b) Notch Filters c) Comb Filters d) Pole – Zero Pattern Rotation 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 51 sites.google.com/site/ncpdhbkhn 52 Notch Filters (1) 1 2 2 0 1 2( ) [ cos ]H z b r z r zφ − −= − + -3 -2 -1 0 1 2 3 0.5 1 1.5 2 omega M a g n i t u d e -3 -2 -1 0 1 2 3 -1 -0.5 0 0.5 1 omega P h a s e ( r a d ) sites.google.com/site/ncpdhbkhn 53 Notch Filters (2) 1 2 0 1 2 2 1 2 1 2 cos ( ) cos z zH z b r z r z φ φ − − − − − + = − + -3 -2 -1 0 1 2 3 0.5 1 1.5 2 omega M a g n i t u d e -3 -2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 omega P h a s e ( r a d ) Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement a) Discrete – Time Resonators b) Notch Filters c) Comb Filters d) Pole – Zero Pattern Rotation 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 54 sites.google.com/site/ncpdhbkhn 55 Comb Filters (1) ( ) [ ] n n H z h n z ∞ − =−∞ = ∑ ( ) ( ) [ ]L nL n G z H z h n z ∞ − =−∞ → = = ∑ ( ) [ ] ( )j j Ln j L n G e h n e H eω ω ω ∞ =−∞ → = =∑ 0 2 0 otherwise [ / ], , , , ... [ ] , h n L n L L g n = ± ± → =   sites.google.com/site/ncpdhbkhn 56 Comb Filters (2) 11( )H z z−= − 1( ) LG z z−→ = − -3 -2 -1 0 1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 omega (rad) M a g n i t u d e H G -3 -2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 omega (rad) P h a s e ( r a d ) H G Ex. 1 sites.google.com/site/ncpdhbkhn 57 Comb Filters (3) 1 1[ ] [ ] [ ],y n ay n D x n a= − + − < < 0 1 1 [ ] [ ] ( )k D k h n a n kD H z az δ ∞ − = = − ↔ = − ∑ Ex. 2 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 7 Real Part I m a g i n a r y P a r t sites.google.com/site/ncpdhbkhn 58 Comb Filters (4) 1 1[ ] [ ] [ ],y n ay n D x n a= − + − < < 0 1 1 [ ] [ ] ( )k D k h n a n kD H z az δ ∞ − = = − ↔ = − ∑ Ex. 2 0 10 20 30 40 50 600 0.2 0.4 0.6 0.8 1 n (samples) A m p l i t u d e Impulse Response sites.google.com/site/ncpdhbkhn 59 Comb Filters (5) 1 1[ ] [ ] [ ],y n ay n D x n a= − + − < < 0 1 1 [ ] [ ] ( )k D k h n a n kD H z az δ ∞ − = = − ↔ = − ∑ Ex. 2 -3 -2 -1 0 1 2 3 0.8 1 1.2 1.4 1.6 1.8 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 omega (rad) M a g n i t u d e Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement a) Discrete – Time Resonators b) Notch Filters c) Comb Filters d) Pole – Zero Pattern Rotation 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 60 sites.google.com/site/ncpdhbkhn 61 Pole – Zero Pattern Rotation (1) 1 0 1 [ ] [ ] M k y n x n k M − = = −∑ 1 1 1 0 1 0 1 1 1 1 1 1 10 elsewhere , [ ] ( ) ( ) , M MM k M k n M z zh n H z zM M M z M z z − − − − − =  ≤ < − − = ↔ = = × = × − − ∑ -3 -2 -1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 omega (rad) M a g n i t u d e -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 9 Real Part I m a g i n a r y P a r t M – point moving average filter sites.google.com/site/ncpdhbkhn 62 Pole – Zero Pattern Rotation (2) 1 1 1 0 1 1 1 1 1 1 1 ( ) ( ) M MM k M k z zH z z M M z M z z − − − − − = − − = = × = × − − ∑ 1 0 0 1 1, , , ...,M kk Mz z W k M− = → = = − [ ] [ ] mk Mg k W h k= DTFT [ ] [ ] ( )m m j k j e h k H eω ω ω−←→ 1 0 1 ( ) ( ) M m k M k G z W z M − − − = → = ∑ 1 1 1 1 ( ) ( ) m M M m M W z M W z − − − − − = × − 1 1 1 0 1 1 1 1 1 ( ) M M k Mm kM k m z W z M W z M − − − − = ≠ − = × = − − ∏ sites.google.com/site/ncpdhbkhn 63 Pole – Zero Pattern Rotation (3) 1 1 1 0 1 1 1 1 1 ( ) ( ) M M k Mm kM k m zG z W z M W z M − − − − = ≠ − = × = − − ∏ 1 0 0 1 1, , , ...,M kk Mz z W k M− = → = = − 6 1010 6 0 3090 0 9511, . .M m W j= = → = + 10 1 1 1 10 1 0 3090 0 9511 ( ) ( . . ) zG z j z − − − → = × − + -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 9 Real Part I m a g i n a r y P a r t -3 -2 -1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 omega (rad) M a g n i t u d e Complex bandpass filter sites.google.com/site/ncpdhbkhn 64 Pole – Zero Pattern Rotation (4) 1 1 1 0 1 1 1 1 1 1 1 ( ) ( ) M MM k M k z zH z z M M z M z z − − − − − = − − = = × = × − − ∑ 1 0 0 1 1, , , ...,M kk Mz z W k M− = → = = − 2[ ] cos( / )f k m M kpi= DTFT 1 1 2 2 [ ] [ ] [ ]cos ( ) ( )m m j j m h k k H e H eω ω ω ωω − +←→ + 1 0 2 ( ) cos M k k mF z k z M pi− − =   → =     ∑ ( )1 0 1 2 M mk mk k M M k W W z − − − = = +∑ 1 1 1 1 1 21 1 1 1 1 1 2 1 2 1 1 1 ( ) cos ( )( ) M M M m m m m M M M M z z m z z M W z W z W z W z pi − − − − − − − − − −   − −  − −   = × + × = − − − − sites.google.com/site/ncpdhbkhn 65 Pole – Zero Pattern Rotation (6) -3 -2 -1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 omega (rad) M a g n i t u d e -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 9 Real Part I m a g i n a r y P a r t -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 9 Real Part I m a g i n a r y P a r t -3 -2 -1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 1 2 3 4 5 6 7 8 9 10 omega (rad) M a g n i t u d e -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 9 Real Part I m a g i n a r y P a r t Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 66 sites.google.com/site/ncpdhbkhn 67 Relationship between Magnitude and Phase Responses (1) ( ) ( ) j j z e H e H z ωω = = ( ) [ ] k k H z h k z ∞ − =−∞ = ∑ 2 * ( ) ( ) ( ) ( )j j j j z e R z H e H e H eω ω ω ω = = = * * *( ) [ ] [ ] j j j k k k k z e H e h k e h k z ω ω ω ∞ ∞ =−∞ =−∞ = = =∑ ∑ * *[ ]( )k k h k z ∞ =−∞   =     ∑ 1 * *[ ]( / ) k k h k z ∞ − =−∞   =     ∑ 1* *( / ) ( )H z V z= = V(z) is obtained by conjugating the coefficients of H(z) and replacing everywhere z–1 by z sites.google.com/site/ncpdhbkhn 68 Relationship between Magnitude and Phase Responses (2) 1 1 1 0 1 1 1 1 1 1 ( ) ( ) ( ) M M k k k k k N N k k k k k b z z z H z b a z p z − − = = − − = = − = = + − ∑ ∏ ∑ ∏ 1* *( ) ( ) ( / )R z H z H z= 1 0 1 1 1 1 * * * * ( ) ( / ) ( ) M k k N k k z z H z b p z = = − = − ∏ ∏ 1 2 1 0 1 1 1 1 1 1 * * ( )( ) ( ) ( )( ) M k k k N k k k z z z z R z b p z p z − = − = − − → = − − ∏ ∏ sites.google.com/site/ncpdhbkhn 69 Relationship between Magnitude and Phase Responses (3) 1 11 1 1 1 1 1( ) ( )( ), ,H z az bz a b− −= − − − < < − < < 1 2,z a z b= = 1* *( ) ( ) ( / )R z H z H z= 1( ) ( / )H z H z= Ex. [ ]1 11 1 1 1( )( ) ( )( )az bz az bz− − = − − − −  1 2 3 1 1, , / , /z a z b z a z b= = = = 1 1 1 1 2 1 3 4 1 1 1 1 1 1 1 1 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) H z az bz H z az bz H z az bz H z az bz − − − −  = − −  = − − →  = − −  = − − sites.google.com/site/ncpdhbkhn 70 Relationship between Magnitude and Phase Responses (4)Ex. 1 11 1 1 1 1 1( ) ( )( ), ,H z az bz a b− −= − − − < < − < < 1 1 1 1 1 2 3 41 1 1 1 1 1 1 1( ) ( )( ); ( ) ( )( ); ( ) ( )( ); ( ) ( )( )H z az bz H z az bz H z az bz H z az bz − − − − = − − = − − = − − = − − 1H 2H 3H 4H -3 -2 -1 0 1 2 3 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 omega (rad) M a g n i t u d e o f H 1 -3 -2 -1 0 1 2 3 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 omega (rad) P h a s e o f H 1 ( r a d ) -3 -2 -1 0 1 2 3 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 omega (rad) M a g n i t u d e o f H 1 -3 -2 -1 0 1 2 3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 omega (rad) P h a s e o f H 1 ( r a d ) -3 -2 -1 0 1 2 3 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 omega (rad) M a g n i t u d e o f H 1 -3 -2 -1 0 1 2 3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 omega (rad) P h a s e o f H 1 ( r a d ) -3 -2 -1 0 1 2 3 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 omega (rad) M a g n i t u d e o f H 1 -3 -2 -1 0 1 2 3 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 omega (rad) P h a s e o f H 1 ( r a d ) Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 71 sites.google.com/site/ncpdhbkhn 72 Allpass Systems (1) ( ) jH e Gω = ( ) k allpassH z Gz − = 1 1 1 1 1 1 1 * * ( ) k kk k k p z z pH z z p z p z − − − − − − = = − − 1 1 1 1 * ( ) N j k ap k k z pH z e p z β − − = − = − ∏ 0 1 Re Im O Z ω kZ k−Z Z ( )k ωΘ × kPkφ k−Z P ( )k ωΦ ( )kα ω 1 1 1 1 */ , / j kk kj k kk e p p pe p ω ω − − = = < − − Z P Z Z 2 2 1 2 1 1 1 2 ( ) sin( ) ( ) atan cos( ) ( ) cos( ) j k j k k k k k k k k k k H e rH e r r r r ω ω ω φω ω φ τ ω ω φ = −∠ = − − − − − = + − − sites.google.com/site/ncpdhbkhn 73 Allpass Systems (2) 1 6 6 1 0 7 1 0 7 / / . ( ) . j k j z eH z e z pi pi − − − − = − Ex. 1 -3 -2 -1 0 1 2 3 1 1 1 1 1 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 omega (rad) P h a s e ( r a d ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Part I m a g i n a r y P a r t sites.google.com/site/ncpdhbkhn 74 Allpass Systems (3) 1 1 0 7 1 0 7 . ( ) . k zH z z − − − = − Ex. 2 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Real Part I m a g i n a r y P a r t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) -3 -2 -1 0 1 2 3 1 1 1 1 1 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 omega (rad) P h a s e ( r a d ) sites.google.com/site/ncpdhbkhn 75 Allpass Systems (4) 1 1 0 7 1 0 7 . ( ) . k zH z z − − + = + Ex. 3 -1.5 -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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) -3 -2 -1 0 1 2 3 1 1 1 1 1 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 omega (rad) P h a s e ( r a d ) sites.google.com/site/ncpdhbkhn 76 Allpass Systems (5) 1 6 1 6 6 1 6 1 0 7 0 7 1 0 7 1 0 7 / / / / . . ( ) . . j j k j j z e z eH z e z e z pi pi pi pi − − − − − − − − = × − − Ex. 4 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Part I m a g i n a r y P a r t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) -3 -2 -1 0 1 2 3 1 1 1 1 1 omega (rad) M a g n i t u d e -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 omega (rad) P h a s e ( r a d ) sites.google.com/site/ncpdhbkhn 77 Allpass Systems (6) 1 1 1 1 * ( ) N j k ap k k z pH z e p z β − − = − = − ∏ 2 0,N β= = 1 2 2 22 1 1 2 1 2 1 2 1 2 1 2 1 1 1 * * * * ( ) ap a a z z a z a zH z z a z a z a z a z − − − − − − − + + + + → = = + + + + 1 1 2 2 1 2( );a p p a p p= − + = 1 1 1 1 1 1 * * * *... ( / ) ( ) ... ( ) N N NN ap N N a z a z A zH z z z a z a z A z − − − − + + + = = + + + sites.google.com/site/ncpdhbkhn 78 Allpass Systems (7) 1 1 1 ( ) D k D z aH z az a − − − = − − < < Ex. 5 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Part I m a g i n a r y P a r t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 20 22 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) - 3 -2 -1 0 1 2 3 1 1 1 1 1 omega (rad) M a g n i t u d e 0 5 10 15 20 25 30 35 40 45 50 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 n (samples) A m p l i t u d e Impulse Response Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems sites.google.com/site/ncpdhbkhn 79 sites.google.com/site/ncpdhbkhn 80 Invertibility and Minimum – Phase Systems (1) An LTI system H(z) with input x[n] and output y[n] is said to be invertible if we can uniquely determine x[n] from y[n] [ ]* [ ] [ ]invh n h n nδ= 1( ) ( )invH z H z = 1 1 1 ( ) ( ) ( ) M k k k N k k k b z B zH z A z a z − = − = = = + ∑ ∑ 1 ( ) ( ) ( ) ( ) inv A zH z H z B z → = = - A causal and stable system H(z) with a causal and stable invers Hinv(z) is known as a minimum – phase system - H(z) is minimum – phase if both its poles & zeros are inside the unit circle sites.google.com/site/ncpdhbkhn 81 Invertibility and Minimum – Phase Systems (2) 1 1 1 *( ) ( )( ),H z H z z a a−= − < 1 1 1 11 1 *( ) ( )( ) ( ) z aH z az az − − − − = − − 1( ) is minimum phaseH z 11( ) is minimum phaseaz−− 1 1 1( ) ( )( ) is minimum phaseminH z H z az −→ = − 1 1 is allpass1 * ( ) allpass z aH z az − − − = − ( ) ( ) ( ) min apH z H z H z→ = sites.google.com/site/ncpdhbkhn 82 Invertibility and Minimum – Phase Systems (3) 1 1 1 1 0 1 1 azH z a b bz − − + = > < < + ( ) , , Ex. 1 1 1 11 ( ) z aH z a bz − − − + = + 1 1 1 1 1 1 1 1 1 1 z a a z a bz a z − − − − − − − + + = × + + 1 1 1 1 1 1 1 1 1 1 ( ) ( )min apH z H z a z z a a bz a z − − − − − − − + + = × + +

 We can obtain the minimum – phase system by replacing each factor (1 + az–1) (|a| > 1), by a factor of the form a(1 + a–1z–1) sites.google.com/site/ncpdhbkhn 83 Invertibility and Minimum – Phase Systems (4) 1 11 1 1 1( ) ( )( ), ,minH z az bz a b − − = − − − < < Ex. 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) mix mix max H z a a z bz H z b az b z H z ab a z b z − − − − − − − − − − = − − = − − = − − -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 2 Real Par t I m a g i n a r y P a r t Hmin -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 2 Real Part I m a g i n a r y P a r t Hmix1 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 2 Real Par t I m a g i n a r y P a r t Hmix2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 2 Real Part I m a g i n a r y P a r t Hmax sites.google.com/site/ncpdhbkhn 84 Invertibility and Minimum – Phase Systems (5) 1 11 1 1 1( ) ( )( ), ,minH z az bz a b − − = − − − < < Ex. 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) mix mix max H z a a z bz H z b az b z H z ab a z b z − − − − − − − − − − = − − = − − = − − -3 -2 -1 0 1 2 3 0.5 1 1.5 2 Magnitude omega (rad) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Phase omega (rad) Hmin Hmix1 Hmix2 Hmax sites.google.com/site/ncpdhbkhn 85 Invertibility and Minimum – Phase Systems (6) 1 11 1 1 1( ) ( )( ), ,minH z az bz a b − − = − − − < < Ex. 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) mix mix max H z a a z bz H z b az b z H z ab a z b z − − − − − − − − − − = − − = − − = − − 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -4 -3 -2 -1 0 1 2 3 4 5 6 Normalized Frequency (×pi rad/sample) G r o u p d e l a y ( s a m p l e s ) Hmin Hmix1 Hmix2 Hmax sites.google.com/site/ncpdhbkhn 86 Invertibility and Minimum – Phase Systems (7) 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( )( ), , ( ) ( )( ) min max H z az bz a b H z ab a z b z − − − − − − = − − − < < = − − 2 1( ) ( / )max minH z z H z −→ = 1 11( ) ... N min NA z a z a z − − = + + + 1 1 1 * * * * ( ) ... ( / ) N N max N N minA z a a z z z A z − − − − → = + + + = sites.google.com/site/ncpdhbkhn 87 Invertibility and Minimum – Phase Systems (8) ( )distortionH z ( ) ( ) ( ) ( ) delay delay n n distortion equalizer equalizer distortion GzH z H z Gz H z H z − − = → = ( ) ( ) delayn equalizer min GzH z H z − = ( ) ( ) ( )distortion min allpassH z H z H z= ( ) ( ) delayn allpass equalizer distortion GH z H z H z − → = Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems a) System Function and Frequency Response b) The Laplace Transform c) Systems with Rational System Functions d) Frequency Response from Pole – Zero Location e) Minimum – Phase and Allpass Systems f) Ideal Filters sites.google.com/site/ncpdhbkhn 88 sites.google.com/site/ncpdhbkhn 89 System Function and Frequency Response all[ ] [ ] ( ) ,n j n j n j nx n r e y n H re r e nω ω ω= → = ( ) [ ] j n j n n H re h n r eω ω ∞ − − =−∞ = ∑ all( )( ) ,j t stx t e e tσ + Ω= = ( ) ( ) ( )y t h x t dτ τ τ ∞ −∞ = −∫ ( ) ( ) ( ) s t st sh e d e h e dτ ττ τ τ τ ∞ ∞ − − −∞ −∞ = =∫ ∫ all( ) ( ) ( ) ,st stx t e y t H s e t= → = ( ) ( ) sH s h e dττ τ ∞ − −∞ = ∫ sites.google.com/site/ncpdhbkhn 90 The Laplace Transform (1) ( ) ( ) stX s x t e dt ∞ − −∞ = ∫ ( ) ( ) t j tX j x t e e dtσσ ∞ − − Ω −∞ + Ω = ∫ sites.google.com/site/ncpdhbkhn 91 The Laplace Transform (2) Ex. 1 Find the Laplace transform of x(t) = e–atu(t), a > 0. ( ) ( ) stX s x t e dt ∞ − −∞ = ∫ 0 0 ( )at st a s te e dt e dt ∞ ∞ − − − + = =∫ ∫ 0 ( )s a te s a ∞ − + = − + 1 10 s a s a − = − = + + 0 0( ) Re{ }a te dt a s aσ σ ∞ − + → > −∫ 1 ROC( ) ( ) ( ) , : Re{ }atx t e u t X s s a s a − = ↔ = ≥ − + Im{ }s Re{ }s a− 0 Re{ }s a> − sites.google.com/site/ncpdhbkhn 92 The Laplace Transform (3) Ex. 2 Find the Laplace transform of x(t) = cos(Ω0t)u(t). ( ) ( ) stX s x t e dt ∞ − −∞ = ∫ 00 cos( ) stt e dt ∞ − = Ω∫ 0 0 0 1 2 ( ) j t j t st e e e dt ∞ Ω − Ω − = +∫ 0 0 0 1 2 ( ) ( ) [ ] s j t s j t st e e e dt ∞ − + Ω − − Ω − = +∫ 0 0 0 0 0 1 2 ( ) ( )s j s j t e e s j s j ∞ − + Ω − − Ω =   − = + + Ω − Ω  2 2 0 0 0 1 1 1 2 s s j s j s   = + = + Ω − Ω +Ω  Im{ }s Re{ }s 0 Re{ }s a> − 00 0 0 0cos( ) Re{ }st tt e dt e dt sσ σ ∞ ∞ − −Ω ≤ → >∫ ∫ 0 2 2 0 ROC 0( ) cos( ) ( ) ( ) , : Re{ }sx t t u t X s s s = Ω ↔ = > +Ω sites.google.com/site/ncpdhbkhn 93 The Laplace Transform (4) 1 1 2 2 1 1 2 2( ) ( ) ( ) ( ) ( ) ( )x t a x t a x t X s a X s a X s= + ↔ = + ( ) ( )sx t e X sττ −− ↔ ( ) * ( ) ( ) ( )h t x t H s X s↔ ( ) ( ) ( ) ( ) t X sy t x d Y s s τ τ −∞ = ↔ =∫ ( ) ( ) ( ) ( ) dx ty t Y s sX s dt = ↔ = Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems a) System Function and Frequency Response b) The Laplace Transform c) Systems with Rational System Functions d) Frequency Response from Pole – Zero Location e) Minimum – Phase and Allpass Systems f) Ideal Filters sites.google.com/site/ncpdhbkhn 94 sites.google.com/site/ncpdhbkhn 95 Systems with Rational System Functions (1) 0 0 ( ) ( )k kN M k kk k k k d y t d x t a b dt dt = = =∑ ∑ ( ) ( ) N N n d f t s F s dt ↔ 0 1 0 1( ... ) ( ) ( ... ) ( ) N M N Ma a s a s Y s b b s b s X s→ + + + = + + + 0 1 0 1 ...( ) ( ) ( ) ( ) ... ( ) M M N N b b s b sY s B sH s X s a a s a s A s + + + → = = = + + + 1 2 1 2 ( )( )...( ) ( )( )...( ) M M N N b s z s z s z a s p s p s p − − − = × − − − sites.google.com/site/ncpdhbkhn 96 Systems with Rational System Functions (2) 1 2 1 2 ( )( )...( ) ( ) ( )( )...( ) M M N N b s z s z s zH s a s p s p s p − − − = × − − − 1 2 1 2 ... N N KK K s p s p p p = + + + − − − 12 1 1 1 2 ( )( ) ( ) ( ) ... N N K s pK s p s p H s K s p p p −− → − = + + + − − 1 2 1 ( )( )...( ) ( ) M M N b s z s z s z a s p − − − → × − 12 1 1 22 ( )( ) ... ( )...( ) N NN K s pK s pK s p p ps p s p −− = + + + − −− − 1 2 1 ( )( )...( ) ( ) M M N b s z s z s z a s p − − − → × − 1 11 12 1 1 22 ( )( ) ... ( )...( ) N NN s p s ps p K s pK s pK s p p ps p s p = = = −− = + + + − −− − 1 2 1 1 ( )( )...( ) ( ) M M N b s z s z s zK a s p − − − → = × − 1 2( )...( )N s p s p s p = − − sites.google.com/site/ncpdhbkhn 97 Systems with Rational System Functions (3) 1 2 1 2 ( )( )...( ) ( ) ( )( )...( ) M M N N b s z s z s zH s a s p s p s p − − − = × − − − 1 2 1 1 ( )( )...( ) ( ) M M N b s z s z s zK a s p − − − = × − 1 2( )...( )N s p s p s p = − − 1 2 1 2 ( )( )...( ) ( )( )... ( ) M M k N k b s z s z s zK a s p s p s p − − − = × − − − ...( ) k N s p s p = − ( )at K Ke u t s a ↔ − 1 ( ) ( )k N p t k k h t K e u t = → =∑ 1 1 0 ( ) ( ) is stable Re{ } , all ( ) M k kM k kN N k k s z bH s p k a s p σ= = − = × ⇔ = < − ∏ ∏ sites.google.com/site/ncpdhbkhn 98 Systems with Rational System Functions (4) 2 3 2 3 2 8 29 52 ( ) s sH s s s s − + = + + + Ex. 3 2 1 2 38 29 52 0 4 2 3,,s s s s s j+ + + = → = − = − ± 2 31 23 2 4 2 3 2 3 4 2 3 2 3 ( ) ( )( )( ) Ks s K KH s s s j s j s s j s j − + = = + + + + − + + + + − + + 2 1 3 2 4( ) s sK s − + = + 2 4 4 3 4 2 2 3077 4 2 3 4 2 32 3 2 3 ( ) ( ) . ( )( )( )( ) s j js j s j =− − − − + = = − + − − + ++ − + + 2 2 3 2 4 2 3( ) ( ) s sK s s j − + = + + − 2 3 0 6538 0 7308 0 9806 2 3007 2 3 . . . . ( ) s j j s j =− + = − + = ∠ + + 2 3 3 2 4 2 3 2 3( )( ) ( ) s sK s s j s j − + = + + − + + 2 3 0 6538 0 7308 0 9806 2 3007. . . . s j j =− − = − − = ∠ − 2 3077 0 9806 2 3007 0 9806 2 3007 4 2 3 2 3 . . . . . ( )H s s s j s j ∠ ∠− = + + + + − + + sites.google.com/site/ncpdhbkhn 99 Systems with Rational System Functions (4)Ex. 4 2 3 2 32 3077 0 9806 2 3007 0 9806 2 3007( ) ( )( ) . ( . . ) ( . . )t j t j th t e e e− − − − +→ = + ∠ + ∠ − 4 2 3007 2 3 2 3007 2 32 3077 0 9806 0 9806. ( ) . ( ). ( ) . ( ) . ( )t j j t j j te u t e e u t e e u t− − − − − += + + 4 2 3 2 3007 3 2 30072 3077 0 9806 ( . ) ( . ). ( ) . [ ] ( )t t t te u t e e e u t− − + − += + + 2 3 2 3 2 8 29 52 ( ) s sH s s s s − + = + + + 2 3077 0 9806 2 3007 0 9806 2 3007 4 2 3 2 3 . . . . . s s j s j ∠ ∠− = + + + + − + + 4 22 3077 0 9806 2 3 2 3007. ( ) . [ cos( . )] ( )t te u t e t u t− −= + + 4 22 3077 1 9612 3 2 3007[ . . cos( . )] ( )t te e t u t− −= + + Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems a) System Function and Frequency Response b) The Laplace Transform c) Systems with Rational System Functions d) Frequency Response from Pole – Zero Location e) Minimum – Phase and Allpass Systems f) Ideal Filters sites.google.com/site/ncpdhbkhn 100 sites.google.com/site/ncpdhbkhn 101 Frequency Response from Pole – Zero Location (1) 0 0 1 0 0 1 ( ) ( ) ( ) M M k k k k k N N k k k k k b s s zbH s a a s s p = = = = − = = × − ∑ ∏ ∑ ∏ 0 0 1 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) M M k k k k k N Ns j k k k k k b j j zbH j H s a a j j p = = = Ω = = Ω Ω− Ω = = = × Ω Ω − ∑ ∏ ∑ ∏ ;k k j j k k k kj z Q e j p R eΘ ΦΩ − = Ω − = 0 1 0 0 1 10 1 M k M N k k kN k k k k Q jbH j j b a j j a R j = = = = Ω    → Ω = ∠ + Θ Ω − Φ Ω     Ω ∏ ∑ ∑ ∏ ( ) ( ) exp ( / ) ( ) ( ) ( ) sites.google.com/site/ncpdhbkhn 102 Frequency Response from Pole – Zero Location (2) 0 1 0 0 1 10 1 ( ) ( ) exp ( / ) ( ) ( ) ( ) M k M N k k kN k k k k Q jbH j j b a j R j a R j = = = = Ω   Ω = ∠ + Θ Ω − Ω     Ω ∏ ∑ ∑ ∏ 0 0 0 0 Product of zero vectors to Product of pole vectors to Sum of zero angles to Sum of pole angles to b s jH j a s j H j b a s j s j  = ΩΩ = × = Ω→  ∠ Ω = ∠ + = Ω − − = Ω ( ) ( ) ( / ) ( ) ( ) sites.google.com/site/ncpdhbkhn 103 Frequency Response from Pole – Zero Location (3) ( ) GH s s a = + ( ) GH j j aΩ = Ω + jΩ σ0s a= − × L P φ 2 2 ( ) G G G H j j a PLa Ω = = = Ω + Ω + ( ) ( ) atan H j G j a G a φ ∠ Ω =∠ −∠ Ω+ Ω = ∠ − = − sites.google.com/site/ncpdhbkhn 104 Frequency Response from Pole – Zero Location (4) 2 2 22 ( ) n n n H s s sζ Ω = + Ω +Ω 2 22 4 0 1 1If ( )n nζ ζΩ − Ω < ↔ − < < 2 1 2then 1, n np j jζ ζ α β= − Ω ± Ω − = − ± ( )22 1 0 11( ) sin ( ),nn nh t e t u tζ ζ ζζ − ΩΩ  → = Ω − < <  − sites.google.com/site/ncpdhbkhn 105 Frequency Response from Pole – Zero Location (5) 2 2 2 2 1 2 2 2 2 21( ) ( ) n n n p p α β ζ ζ = = + = − Ω + Ω − = Ω 2 2 2 2 2 1 22 ( ) ( )( ) ( )( ) n n n n n H s s s s p s p s j s jζ α β α β Ω Ω Ω = = = + Ω +Ω − − + − + + 2 1 2 1, n np j jζ ζ α β= − Ω ± Ω − = − ± 2 2 1 2 1 2 ( ) n nH j s p s p PL P L Ω ΩΩ = = − − × 2 1 2 1 2 ( ) ( ) ( ) n H j s p s p φ φ ∠ Ω =∠Ω −∠ − −∠ − = − − jΩ σ0 1P α− β× ×2P β− 1s p− n −Ω 2s p− 1φ 1φ Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems a) System Function and Frequency Response b) The Laplace Transform c) Systems with Rational System Functions d) Frequency Response from Pole – Zero Location e) Minimum – Phase and Allpass Systems f) Ideal Filters sites.google.com/site/ncpdhbkhn 106 sites.google.com/site/ncpdhbkhn 107 Minimum – Phase and Allpass Systems (1) 2 *( ) ( ) ( )H j H j H jΩ = Ω Ω * ( ) ( )H j H jΩ = − Ω ( ) ( ) s jH s H j= Ω = Ω 2 ( ) ( ) ( ) s jH j H s H s = Ω→ Ω = − jΩ σ01P × L j= Ω 1Z α ( ) minH j α∠ Ω = jΩ σ01P × L j= Ω 1Z β ( )H j β∠ Ω = -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Ω (rad) ∠ H ( j Ω ) Hmin H sites.google.com/site/ncpdhbkhn 108 Minimum – Phase and Allpass Systems (2) jΩ σ0kP × L j= Ω kZ α ( ) allpassH j α∠ Ω = kΦ kΘkR kQ s a= − s a= ( ) k k s zH s s p − = − for all, ,k k k kR Q pi= Φ +Θ = Ω ,k k k k k ks p R s z Q− = ∠Φ − = ∠Θ 1 allpass system( ) ( )k k k k s z RH j s p Q − Ω = = = − ( ) ( ) ( )k k k kH j s z s p α∠ Ω = ∠ − −∠ − = = Θ −Φ 2 2atank a pi pi Ω = − Φ = − 1( )H j αΩ = ∠ sites.google.com/site/ncpdhbkhn 109 Minimum – Phase and Allpass Systems (3) * * ( )( ) ( ) ( )( ) k k k k s z s zH s s p s p − − = − − * * ,k k k kR Q R Q= = ,k k k k k ks p R s z Q− = ∠Φ − = ∠Θ jΩ σ0 kP L j= Ω kZ α × kΦ kΘβ kQ a− a × * kP * kZ b b− * kΦ * kΘ * kQ kR * kR * * * * * * ,k k k k k ks p R s z Q− = ∠Φ − = ∠Θ 1 allpass system * * * * ( ) ( ) k k k k k k k k s z s z H j s p s p Q Q R R − − Ω = − − = = * * * * ( ) ( ) ( ) ( ) ( )k k k k k k k k H j s z s z s p s p∠ Ω = ∠ − +∠ − −∠ − −∠ − = Θ + Θ −Φ −Φ Minimum – Phase and Allpass Systems (4) An Nth order allpass system has the following properties: 1. The zeros and poles are symmetric with respect to the jΩ axis, that is, if the poles are pk, then the zeros are . Therefore, the system function is: 2. The phase response is monotonically decreasing from 2πN to zero as Ω increases from –∞ to ∞. sites.google.com/site/ncpdhbkhn 110 1 1 * * * ( )...( ) ( ) ( )...( ) N N s p s pH s s p s p + − = − − jΩ σ0 kP L j= Ω kZ α × kΦ kΘβ kQ a− a × * kP * kZ b b− * kΦ * kΘ * kQ kR * kR * kp− 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 1. For each zero in the right – half plane, include a pole and a zero at its mirror position in the left half plane. 2. Assign the left – half plane zeros and the original poles to Hmin(s). 3. Assign the right – half plane zeros and the left – half plane poles introduced in step 1 to Hallpass(s). sites.google.com/site/ncpdhbkhn 111 ( ) ( ) ( ) min allpassH s H s H s= Transform Analysis of LTI Systems 1. Sinusoidal Response of LTI Systems 2. Response of LTI Systems in the Frequency Domain 3. Distortion of Signals Passing through LTI Systems 4. Ideal and Practical Filters 5. Frequency Response for Rational System Functions 6. Dependency of Frequency Response on Poles and Zeros 7. Design of Simple Filters by Pole – Zero Placement 8. Relationship between Magnitude and Phase Responses 9. Allpass Systems 10. Invertibility and Minimum – Phase Systems 11. Transform Analysis of Continuous – Time Systems a) System Function and Frequency Response b) The Laplace Transform c) Systems with Rational System Functions d) Frequency Response from Pole – Zero Location e) Minimum – Phase and Allpass Systems f) Ideal Filters sites.google.com/site/ncpdhbkhn 112 sites.google.com/site/ncpdhbkhn 113 Ideal Filters ( ) ( )dy t Gx t t= − ( ) d j tH j Ge− ΩΩ = 0 otherwise , ( ) , dj t c lowpass e H j − Ω Ω ≤ ΩΩ =   sin ( ) ( ) ( ) c c d lowpass c d t th t t tpi Ω Ω − = Ω −