Signal processing - Discrete – Time signals and systems

Discrete – Time Signals Discrete – Time Systems Convolution Description of Linear Time – Invariant Systems Properties of Linear Time – Invariant Systems Analytic Evaluation of Convolution Numerical Computation of Convolution Real – Time Implementation of FIR Filters FIR Spatial Filters Systems Described by Linear Constant – Coefficient Difference Equations

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Nguyễn Công Phương SIGNAL PROCESSING Discrete – Time Signals and 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 2 Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 3 Discrete – Time Signals (1) • x[n]: a sequence of numbers defined for every value of the integer variable n • It is not defined for noninteger value of n • :the samples in the range from N1 to N2 • T (sampling period): the interval between two successive samples, measured in seconds (s) • Fs (sampling frequency, 1/T): the number of samples per unit of time, measured in hertz (Hz) sites.google.com/site/ncpdhbkhn 4 2 1 N Nx n{ [ ]} sites.google.com/site/ncpdhbkhn 5 Discrete – Time Signals (2) 0 0 1 0 3 , [ ] , n n x n n <  =  ≥ n –2 –1 0 1 2 x[n] 0 0 1 1/3 1/9 1 1 10 1 3 9 27 [ ] {... ...}x n ↑ = 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Function Table GraphSequence sites.google.com/site/ncpdhbkhn 6 Discrete – Time Signals (3) 2 [ ] x n E x n ∞ =−∞ = ∑Energy: Power: 21 2 1 lim [ ] L x L n L P x n L→∞ =−   =  +  ∑ sites.google.com/site/ncpdhbkhn 7 Discrete – Time Signals (4) 1 0 0 0 , [ ] , n n n δ ==  ≠ Unit sample sequence: -2 -1 0 1 2 3 40 0.2 0.4 0.6 0.8 1 n 1 0 , [ ] , n k n k n k δ =− =  ≠ 3nδ −[ ]nδ[ ] sites.google.com/site/ncpdhbkhn 8 Discrete – Time Signals (5) 1 0 0 0 , [ ] , n u n n ≥ =  < Unit step sequence: -2 -1 0 1 2 3 40 0.2 0.4 0.6 0.8 1 n sites.google.com/site/ncpdhbkhn 9 Discrete – Time Signals (6) 0 0[ ] cos( ),x n A n nω φ= + −∞ < < ∞Sinusoidal sequence: n sites.google.com/site/ncpdhbkhn 10 Discrete – Time Signals (7) n 0 1[ ] ,nx n Aa a= < <Exponential sequence: sites.google.com/site/ncpdhbkhn 11 Discrete – Time Signals (8) 0 1 2 3 4 5 6 7 8 9 10 -4 -3 -2 -1 0 1 2 3 4 5 n 1 0[ ] ,nx n Aa a= − < <Exponential sequence: sites.google.com/site/ncpdhbkhn 12 Discrete – Time Signals (9) 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n Periodic sequence: x n x n N= +[ ] [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 13 Discrete – Time Systems (1) • Causality • Stability • Linearity • Time – invariance sites.google.com/site/ncpdhbkhn 14 [ ] [ ] H x n y n֏ [ ] { [ ]}y n H x n= Discrete – time system x[n] y[n]H Discrete – Time Systems (2) • Definition: A system is called causal if the present value of the output does not depend on future values of the input • That is, y[n0] is determined by the values of x[n] for n ≤ n0, only • Ex. 1: y[n] = x[n] + 2x[n – 1] + x[n – 2] • Ex. 2: y[n] = x[n – 1] + x[n] + x[n + 1] sites.google.com/site/ncpdhbkhn 15 Discrete – Time Systems (3) • Definition: A system is called stable, in the Bounded-Input Bounded-Output (BIBO) sense, if every bounded input signal results in a bounded output signal • That is, |x[n]| ≤ Mx < ∞  |y[n]| ≤ My < ∞ • A signal x[n] is bounded if there exists a positive finite constant Mx such that |x[n]| ≤ Mx for all n • Ex. 3: y[n] = x[n] + 2x[n – 1] + x[n – 2] • Ex. 4: y[n] = 2nx[n] sites.google.com/site/ncpdhbkhn 16 Discrete – Time Systems (4) • Definition: A system is called linear if and only if for every real or complex constant a1, a2 and every input signal x1[n] and x2[n]: H{a1x1[n] + a2x2[n]} = a1H{x1[n]} + a2H{x2[n]} • A.k.a. the principle of superposition • Ex. 5: y[n] = 2x[n] • Ex. 6: y[n] = x2[n] sites.google.com/site/ncpdhbkhn 17 Discrete – Time Systems (5) • Definition: A system is called time – invariant or fixed if and only if: y[n] = H{x[n]}  y[n – n0] = H{x[n – n0]} for every input x[n] and every time shift n0 • That is, a time shift in the input results in a corresponding time shift in the output • Ex. 7: y[n] = 3x[n] • Ex. 8: y[n] = x[n]/n sites.google.com/site/ncpdhbkhn 18 sites.google.com/site/ncpdhbkhn 19 Discrete – Time Systems (6) +1[ ]x n 2[ ]x n 1 2[ ] [ ] [ ]y n x n x n= + 1[ ]x n 2[ ]x n 1 2[ ] [ ] [ ]y n x n x n= + Adder Summing node sites.google.com/site/ncpdhbkhn 20 Discrete – Time Systems (7) Multiplier Gain branch [ ]x n [ ] [ ]y n ax n=a [ ]x n [ ] [ ]y n ax n=a sites.google.com/site/ncpdhbkhn 21 Discrete – Time Systems (8) Unit delay Unit delay branch [ ]x n 1[ ] [ ]y n x n= −1z− [ ]x n 1[ ] [ ]y n x n= −1z− sites.google.com/site/ncpdhbkhn 22 Discrete – Time Systems (9) Splitter Pick-off node [ ]w n [ ]w n [ ]w n [ ]w n [ ]w n [ ]w n sites.google.com/site/ncpdhbkhn 23 Discrete – Time Systems (10) [ ]x n [ ]w n [ ]y n 1z− a bc Ex. 8 Find y[n]? 1w n x n cw n= + −[ ] [ ] [ ] 1y n aw n bw n= + −[ ] [ ] [ ] 1 1a x n cw n bw n= + − + −( [ ] [ ]) [ ] 1ax n ac b w n= + + −[ ] ( ) [ ] bx n cy n w n b ac + → = + [ ] [ ] [ ] 1 11 bx n cy nw n b ac − + − → − = + [ ] [ ] [ ] 1 1 1 1bx n cy ny n ax n ac b ax n bx n cy n b ac − + − → = + + = + − + − + [ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] Discrete – Time Systems (11) • Discrete – time system properties: – Causality: – Stability: – Linearity: – Time – invariance: • Practical realizability: – A finite amount of memory for the storage of signal samples and – A finite number of arithmetic operations for the computation of each output sample sites.google.com/site/ncpdhbkhn 24 0 00 0x n for n n y n for n n= ≤ → = ≤[ ] [ ] x x n M y n M≤ < ∞→ ≤ < ∞[ ] [ ] k k k kk k c x n c y n→∑ ∑[ ] [ ] 0 0x n n y n n− → −[ ] [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant (LTI) Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 25 Convolution Description of LTI Systems (1) • A great need for the evaluation of the performance of an LTI system • The response of a system can be used to evaluate its performance • This can be obtained from impulse responses by using the convolution sites.google.com/site/ncpdhbkhn 26 LTI system0 1 Unit impulse 0 1 Impulse response nδ [ ] h n[ ] sites.google.com/site/ncpdhbkhn 27 Convolution Description of LTI Systems (2) LTI system x[n] y[n] = ?H LTI system H LTI system H LTI system H y n→ [ ] Convolution sites.google.com/site/ncpdhbkhn 28 Convolution Description of LTI Systems (3) 0 1 2 0 1 2 0 1 2 0 1 2 0k x k n k x n n k = =  ≠ [ ], [ ] , 0x nδ[ ] [ ] 1 1x nδ −[ ] [ ] 2 2x nδ −[ ] [ ] 0 1 1 2 2x n x n x n x nδ δ δ→ = + − + −[ ] [ ] [ ] [ ] [ ] [ ] [ ] k x n x k n kδ ∞ =−∞ = −∑[ ] [ ] [ ] sites.google.com/site/ncpdhbkhn 29 Convolution Description of LTI Systems (4) LTI system0 1 0 1 nδ [ ] h n[ ] LTI systemk 1 k 1 n kδ −[ ] h n k−[ ] LTI systemk x[k] k x[k] x k n kδ −[ ] [ ] x k h n k−[ ] [ ] k y n x k h n k x n h n n ∞ =−∞ = − = −∞ < < ∞∑[ ] [ ] [ ] [ ]* [ ], (Convolution) sites.google.com/site/ncpdhbkhn 30 Convolution Description of LTI Systems (5) [ ] [ ] [ ] [ ]* [ ], k y n x k h n k x n h n n ∞ =−∞ = − = −∞ < < ∞∑ 1 1 1 0 1 1 1 2 1 , , , , [ ] ... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ... n k n k n k n k y x k h n k x k h n k x k h n k x k h n k =− =− =− = =− = =− = − = + − + − + + − + − 1 0 0 1 1 2 2 3... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ...x h x h x h x h= + − + − + − + − + 0 1 0 0 0 1 0 2 0 , , , , [ ] ... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ... n k n k n k n k y x k h n k x k h n k x k h n k x k h n k = =− = = = = = = = + − + − + + − + − 1 1 0 0 1 1 2 2... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ...x h x h x h x h= + − + + − + − + 1 1 1 0 1 1 1 2 1 , , , , [ ] ... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ... n k n k n k n k y x k h n k x k h n k x k h n k x k h n k = =− = = = = = = = + − + − + + − + − 1 2 0 1 1 0 2 1... [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ...x h x h x h x h= + − + + + − + sites.google.com/site/ncpdhbkhn 31 Convolution Description of LTI Systems (6)Ex. 9 1 2 3 4 5 1 2 1[ ] { }, [ ] { }.x n h n ↑ ↑ = = −Given Find x[n]*h[n]? 0 0 1 2 3 4 5 0 0 0 1 2 –1 0 0[ ]y = × 0 × 2+ × 2− 0= 0 0 1 2 3 4 5 0 0 0 1 2 –1 0 1[ ]y = × 1 × 4+ × 3− 2= 0 0 1 2 3 4 5 0 0 0 1 2 –1 0 2[ ]y = × 2 × 6+ × 4− 4= sites.google.com/site/ncpdhbkhn 32 Convolution Description of LTI Systems (7)Ex. 9 1 2 3 4 5 1 2 1[ ] { }, [ ] { }.x n h n ↑ ↑ = = −Given Find x[n]*h[n]? 0 0 5 4 3 2 1 0 0 0 –1 2 1 0 0[ ]y = × 2− × 2+ × 0 0= 0 0 5 4 3 2 1 0 0 0 –1 2 1 0 1[ ]y = × 3− × 4+ × 1 2= 0 0 5 4 3 2 1 0 0 0 –1 2 1 0 2[ ]y = × 4− × 6+ × 2 4= sites.google.com/site/ncpdhbkhn 33 Convolution Description of LTI Systems (8) [ ] [ ] [ ] [ ] [ ] k k y n x k h n k h k x n k ∞ ∞ =−∞ =−∞ = − = −∑ ∑ [ ] [ ]* [ ] [ ]* [ ]y n x n h n h n x n= = Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems a) Properties of Convolution b) Causality and Stability c) Convolution of Periodic Sequences d) Response to Simple Test Sequences 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 34 sites.google.com/site/ncpdhbkhn 35 Properties of Convolution (1) δ[n] [ ]x n [ ]x n δ[n – n0] [ ]x n 0[ ]x n n− h[n] [ ]x n [ ]y n x[n] [ ]h n [ ]y n h1[n] [ ]x n [ ]y n h2[n] h[n] = h1[n]*h2[n] [ ]x n [ ]y n [ ] [ ]* [ ]x n n x nδ= 0 0[ ] [ ]* [ ]x n n n n x nδ− = − [ ] [ ]* [ ] [ ]* [ ]y n x n h n h n x n= = 1 2 1 2( [ ]* [ ])* [ ] [ ]* ( [ ]* [ ])x n h n h n x n h n h n= sites.google.com/site/ncpdhbkhn 36 Properties of Convolution (2) h1[n] [ ]x n [ ]y n h2[n] h2[n] [ ]x n [ ]y n h1[n] h1[n] [ ]x n y n[ ] + h2[n] h[n] = h1[n]+h2[n] [ ]x n [ ]y n 1 2 2 1( [ ]* [ ])* [ ] ( [ ]* [ ])* [ ]x n h n h n x n h n h n= 1 2 1 2x n h n h n x n h n x n h n+ = +[ ]* ( [ ] [ ]) [ ]* [ ] [ ]* [ ] Causality and Stability • A linear time – invariant system with impulse response h[n] is causal if: • A linear time – invariant system with impulse response h[n] is stable, in the bounded – input bounded – output sense, if and only if the impulse response is absolutely summable, that is, if: sites.google.com/site/ncpdhbkhn 37 0 for 0h n n= <[ ] n h n ∞ =−∞ < ∞∑ [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems a) Properties of Convolution b) Causality and Stability c) Convolution of Periodic Sequences d) Response to Simple Test Sequences 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 38 sites.google.com/site/ncpdhbkhn 39 Convolution of Periodic Sequences k y n x k h n k ∞ =−∞ = −∑[ ] [ ] [ ] k y n N x k h n N k ∞ =−∞ → + = + −∑[ ] [ ] [ ] x n x n N= +[ ] [ ] x n k x n N k→ − = + −[ ] [ ] k y n N x k h n k ∞ =−∞ → + = −∑[ ] [ ] [ ] y n N y n→ + =[ ] [ ] 0u n[ ] 1 sites.google.com/site/ncpdhbkhn 40 Response to Simple Test Sequences (1) k s n h k u n k ∞ =−∞ = −∑[ ] [ ] [ ]The step response: 0 h n[ ] 0 0 0s u h=[ ] [ ] [ ] 1 0h= × [ ] 0h= [ ] 0 0k h k = =∑ [ ] 1 0 0 1 1s u h u h= +[ ] [ ] [ ] [ ] [ ] 0 1h h= +[ ] [ ] 1 0k h k = =∑ [ ] 2 0 0 1 1 2 2s u h u h u h= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 1 2h h h= + +[ ] [ ] [ ] 2 0k h k = =∑ [ ] 3 0 0 1 1 2 2 3 3s u h u h u h u h= + + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 1 2 3h h h h= + + +[ ] [ ] [ ] [ ] 3 0k h k = =∑ [ ] n k h k =−∞ = ∑ [ ] sites.google.com/site/ncpdhbkhn 41 Response to Simple Test Sequences (2) The step response: 0 0 0s u h=[ ] [ ] [ ] 1 0h= × [ ] 0h= [ ] 1 0 0 1 1s u h u h= +[ ] [ ] [ ] [ ] [ ] 0 1h h= +[ ] [ ] 2 0 0 1 1 2 2s u h u h u h= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 1 2h h h= + +[ ] [ ] [ ] 3 0 0 1 1 2 2 3 3s u h u h u h u h= + + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 1 2 3h h h h= + + +[ ] [ ] [ ] [ ] k s n h k u n k ∞ =−∞ = −∑[ ] [ ] [ ] n k h k =−∞ = ∑ [ ] 1 0 1s s h→ − =[ ] [ ] [ ] 2 1 2s s h→ − =[ ] [ ] [ ] 3 2 3s s h→ − =[ ] [ ] [ ] 1h n s n s n= − −[ ] [ ] [ ] sites.google.com/site/ncpdhbkhn 42 Response to Simple Test Sequences (3) x n n y n h nδ= → =[ ] [ ] [ ] [ ] n k x n u n y n s n h k =−∞ = → = = ∑[ ] [ ] [ ] [ ] [ ] n nx n a y n H a a= → =[ ] [ ] ( ) j n j n j nx n e y n H e eω ω ω= → =[ ] [ ] ( ) k k H a h k a ∞ − =−∞ = ∑( ) [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 43 sites.google.com/site/ncpdhbkhn 44 Analytic Evaluation of Convolution k h n k−[ ] 2n M− 1n M− x k[ ] 1N 2N k h n k−[ ] 2n M− 1n M− x k[ ] 1N 2N k h n k−[ ] 2n M− 1n M− x k[ ] 1N 2N k h n k−[ ] 2n M− 1n M− x k[ ] 1N 2N Partial overlap (left): 1 1 1 1 1 2for n M k N y n x k h n k N M n N M − = = − + ≤ ≤ +∑[ ] [ ] [ ] , Full overlap: 1 2 1 2 2 1for n M k n M y n x k h n k N M n N M − = − = − + < < +∑[ ] [ ] [ ] , Partial overlap (right): 2 2 2 1 2 2for N k n M y n x k h n k N M n N M = − = − + ≤ ≤ +∑[ ] [ ] [ ] , 1 1 1 1 2 1 1 2 n M N n N M n M N n N M − ≥ ≥ +  →  − ≤ ≤ +  2 1 1 2 1 2 2 1 n M N n N M n M N n N M − > > +  →  − < < +  1 2 1 2 2 2 2 2 n M N n M N n M N n M N − ≥ ≥ +  →  − ≤ ≤ +  1 2 1 2 1 1 2 2 h n n M M x n n N N y n n M N M N ∈ ∈ → ∈ + + [ ], [ , ]; [ ], [ , ] [ ], [ , ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 45 sites.google.com/site/ncpdhbkhn 46 Numerical Computation of Convolution (1) 0 x n[ ] 0 h n[ ] 1 0 0 1 0 2 0y h h h− = + +[ ] [ ] [ ] [ ] 0 0 0 1 0 2 0y h x h h= + +[ ] [ ] [ ] [ ] [ ] 1 0 1 1 0 2 0y h x h x h= + +[ ] [ ] [ ] [ ] [ ] [ ] 2 0 2 1 1 2 0y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 3 0 3 1 2 2 1y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 4 0 4 1 3 2 2y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 5 0 5 1 4 2 3y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 6 0 0 1 5 2 4y h h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] 7 0 0 1 0 2 5y h h h x= + +[ ] [ ] [ ] [ ] [ ] 8 0 0 1 0 2 0y h h h= + +[ ] [ ] [ ] [ ] No overlap Partial overlap Full overlap Partial overlap No overlap sites.google.com/site/ncpdhbkhn 47 Numerical Computation of Convolution (2) 1 0 0 1 0 2 0y h h h− = + +[ ] [ ] [ ] [ ] 0 0 0 1 0 2 0y h x h h= + +[ ] [ ] [ ] [ ] [ ] 1 0 1 1 0 2 0y h x h x h= + +[ ] [ ] [ ] [ ] [ ] [ ] 2 0 2 1 1 2 0y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 3 0 3 1 2 2 1y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 4 0 4 1 3 2 2y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 5 0 5 1 4 2 3y h x h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] [ ] 6 0 0 1 5 2 4y h h x h x= + +[ ] [ ] [ ] [ ] [ ] [ ] 7 0 0 1 0 2 5y h h h x= + +[ ] [ ] [ ] [ ] [ ] 8 0 0 1 0 2 0y h h h= + +[ ] [ ] [ ] [ ] No overlap Partial overlap Full overlap Partial overlap No overlap 0 0 0 0 1 1 0 0 2 2 1 0 0 3 3 2 1 1 4 4 3 2 2 5 5 4 3 6 0 5 4 7 0 0 5 y x y x x y x x x h y x x x h y x x x h y x x x y x x y x                           =                             [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 0 0 1 0 0 2 1 0 3 2 1 0 1 2 4 3 2 5 4 3 0 5 4 0 0 5 x x x x x x x x x h h h x x x x x x x x x                                     = + +                                     [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 48 sites.google.com/site/ncpdhbkhn 49 Real – Time Implementation of FIR Filter h[0] h[1] h[2] h[3] +  +  +  +  0 0 0 0x[0]1 x[0]2 1 x[0]3 2 1 x[0] Signal memory Coefficient memory In Out 0 0 1 0 2 0 3 0h h h h+ + +[ ] [ ] [ ] [ ]0 1 0 2 0 3 0x h h h+ + +[ ] [ ] [ ] [ ]1 0 2 0 3 0x h h+ +] [ ] [ ] [ ] [ ]2 1 3 0x x h+[ [ ] [ ]3 2 1 0x] [ ] [ ] Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 50 sites.google.com/site/ncpdhbkhn 51 FIR Spatial Filters 1 1 1 1 1 1 9 1 1 1 1 1 1 1 y m n x m n x m n x m n x m n x m n x m n x m n x m n x m n = − − + − + − + + − + + + + + − + + + + + [ , ] ( [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ]) 1 1 1 1 1 9k l x m k n l =− =− = − −∑∑ [ , ] 1 1 1 9 0 otherwise m nh m n  − ≤ ≤ =   , , [ , ] , 1 1 1 1k l y m n h k l x m k n l =− =− → = − −∑∑[ , ] [ , ] [ , ] K L k K l L m K n L k m K l n L y m n h k l x m k n l h k l x m k n l =− =− + + = − = − = − − = − − ∑ ∑ ∑ ∑ [ , ] [ , ] [ , ] [ , ] [ , ] H Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 52 sites.google.com/site/ncpdhbkhn 53 Systems Described by Linear Constant – Coefficient Difference Equations (1) 1 1[ ] [ ],nh n ba u n a= − < < 21 2 1 2 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ... [ ] ( [ ] [ ] ...) n n k k y n x k h n k h k x n k bx n bax n ba x n bx x a bx n bax n =−∞ =−∞ = − = − = + − + − + = + − + − + ∑ ∑ 1 1 2[ ] [ ] [ ] ...y n bx n bax n− = − + − + 1[ ] [ ] [ ]y n ay n bx n→ = − + [ ]x n y n[ ]+ z–1 b a sites.google.com/site/ncpdhbkhn 54 Systems Described by Linear Constant – Coefficient Difference Equations (2) 1[ ] [ ] [ ]y n ay n bx n= − + 0 1 0[ ] [ ] [ ]y ay bx= − + 21 0 1 1 0 1[ ] [ ] [ ] [ ] [ ] [ ]y ay bx a y bax bx= + = − + + 3 22 1 2 1 0 1 2[ ] [ ] [ ] [ ] [ ] [ ] [ ]y ay bx a y ba x bax bx= + = − + + + ⋮ 1 1 1 1 0 1 [ ] [ ] [ ] [ ] [ ] [ ] ... [ ] n n n y n ay n bx n a y ba x ba x bx n+ − = − + = = − + + + + [ ] [ ] nh n ba u n= 1 1 0 1 1 0[ ] [ ] [ ] [ ] [ ] [ ] ... [ ] [ ]ny n a y n h n x h n x h x n+→ = − + + − + + sites.google.com/site/ncpdhbkhn 55 Systems Described by Linear Constant – Coefficient Difference Equations (3) 1 1 0 1 1 0[ ] [ ] [ ] [ ] [ ] [ ] ... [ ] [ ]ny n a y n h n x h n x h x n+= − + + − + + 1If 0 for 0 then 1 0[ ] [ ] [ ],nzix n n y n a y n + = ≥ = − ≥ 0 If 1 0 then[ ] [ ] [ ] [ ] n zs k y y n h k x n k = − = = −∑ (Zero – input response) (Zero – state response) 1 0 1[ ] [ ] [ ] [ ] [ ] [ ] n n zi zs k zero input response zero state response y n a y n h k x n k y n y n+ = − − → = − + − = +∑  sites.google.com/site/ncpdhbkhn 56 Systems Described by Linear Constant – Coefficient Difference Equations (4) 1 1 0 1 1 0[ ] [ ] [ ] [ ] [ ] [ ] ... [ ] [ ]ny n a y n h n x h n x h x n+= − + + − + + [ ] [ ]x n u n= 1 1 1 0 11 1 0 1 [ ] [ ] [ ], nn k n n k ay n ba a y b a y n a + + + = − → = + − = + − ≥ − ∑ 1a < 1 0 1 [ ] lim [ ] ,ss n y n y n b n a→∞ → = = ≥ − (Steady – state response) 1 1 1 0 1 [ ] [ ] [ ] [ ], n n tr ss bay n y n y n a y n a + + = − = + − ≥ − (Transient response)0lim [ ]tr n y n →∞ =  1 1 1 11 1 1 1 1 1 [ ] [ ] [ ] [ ] [ ] [ ] [ ] zi ss tr zs n n n n y n y n y n y n b a b ay n b a y b a y a a a a + + + +− − = + + − = + + − − − − −    sites.google.com/site/ncpdhbkhn 57 Systems Described by Linear Constant – Coefficient Difference Equations (5)  1 1 1 11 1 1 1 1 1 [ ] [ ] [ ] [ ] [ ] [ ] [ ] zi ss tr zs n n n n y n y n y n y n b a b ay n b a y b a y a a a a + + + +− − = + + − = + + − − − − −    0 2 4 6 8 10 12 14 16 18 20 -8 -6 -4 -2 0 2 4 6 8 [ ]u n [ ]y n [ ]try n sites.google.com/site/ncpdhbkhn 58 Systems Described by Linear Constant – Coefficient Difference Equations (6)  1 1 1 11 1 1 1 1 1 [ ] [ ] [ ] [ ] [ ] [ ] [ ] zi ss tr zs n n n n y n y n y n y n b a b ay n b a y b a y a a a a + + + +− − = + + − = + + − − − − −    [ ]u n [ ]y n [ ]try n 0 2 4 6 8 10 12 14 16 18 20 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 sites.google.com/site/ncpdhbkhn 59 Systems Described by Linear Constant – Coefficient Difference Equations (7) 1 0 1[ ] [ ] [ ] [ ] n n k y n a y n h k x n k+ = = − + −∑ 0[ ] j n x n e ω = ( )0 0 01 1 0 0 1 1( )[ ] [ ] [ ] n n kj n k j n jn k n k k y n a y a e a y e aeω ω ω− −+ + = = → = − + = − +∑ ∑ 0 0 0 11 1 11 1 ( ) [ ] j nn j nn j a e a y e ae ω ω ω − ++ + − − = − + − 0 0 0 0 0 11 1 11 1 1 [ ] [ ] [ ] ( ) [ ] [ ] [ ] zs zs zi tr ss y n y n y n j nn j n j nn j j y n y n a e a y e e ae ae ω ω ω ω ω − ++ + − − − = − + + − −         sites.google.com/site/ncpdhbkhn 60 Systems Described by Linear Constant – Coefficient Difference Equations (8) 0 -1 -0.5 0 0.5 1 1.5 Input Output R e a l p a r t I m a g i n a r y p a r t 0 0 -1 -0.5 0 0.5 1 1.5 Systems Described by Linear Constant – Coefficient Difference Equations (9) • Linear constant – coefficient difference equation (LCCDE) • ak: feedback coefficients • bk: feedforward coefficients • If ak & bk are fixed, then the system is time – invariant • If they depend on n, then time – varying • N is the order of the system sites.google.com/site/ncpdhbkhn 61 1 1 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = = − − + −∑ ∑ Systems Described by Linear Constant – Coefficient Difference Equations (10) • Is the system linear time – invariant? • Find its impulse response in analytical form • Given an analytical input x[n], find its analytical output y[n] • Is the system stable (given ak & bk)? sites.google.com/site/ncpdhbkhn 62 1 1 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = = − − + −∑ ∑ Discrete – Time Signals and Systems 1. Discrete – Time Signals 2. Discrete – Time Systems 3. Convolution Description of Linear Time – Invariant Systems 4. Properties of Linear Time – Invariant Systems 5. Analytic Evaluation of Convolution 6. Numerical Computation of Convolution 7. Real – Time Implementation of FIR Filters 8. FIR Spatial Filters 9. Systems Described by Linear Constant – Coefficient Difference Equations 10. Continuous – Time LTI Systems sites.google.com/site/ncpdhbkhn 63 sites.google.com/site/ncpdhbkhn 64 Continuous – Time LTI Systems (1) [ ] [ ] [ ] [ ] [ ] k k y n x k h n k h k x n k ∞ ∞ =−∞ =−∞ = − = −∑ ∑ ( ) ( ) ( ) ( ) ( )y t x h t d h x t dτ τ τ τ τ τ ∞ ∞ −∞ −∞ = − = −∫ ∫ t0 t0 τ0 τ0 ( )x t ( )h t ( )h τ ( )h t τ− ( )x t τ− ( )x τ sites.google.com/site/ncpdhbkhn 65 Continuous – Time LTI Systems (2) t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− 0 1: ( ) 1; ( ) 0t h t x t< < = = ( ) * ( ) 0h t x t = 0 1: ( )* ( ) 0t h t x t< < = Ex. 1 Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 66 Continuous – Time LTI Systems (3)Ex. 1 t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 τ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− 1 2 : ( ) 1; ( ) 2t h t x t< < = = 11 1 ( ) * ( ) ( ) ( ) 1 2 2 2( 1)t t th t x t h t x d d t τ τ τ τ τ τ = = − = × = = −∫ ∫ 0 1: ( )* ( ) 0t h t x t< < = 1 2 : ( )* ( ) 2( 1)t h t x t t< < = − τ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− t 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 67 Continuous – Time LTI Systems (4) t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− 2 3: ( ) 1; ( ) 2t h t x t< < = = 1 2 11 1 ( ) * ( ) ( ) ( ) 1 2 2 2t t t tt t h t x t f t f d d τ τ τ τ τ τ = − − − = − = × = =∫ ∫ 0 1: ( )* ( ) 0t h t x t< < = 1 2 : ( )* ( ) 2( 1)t h t x t t< < = − 2 3: ( ) * ( ) 2t h t x t< < = Ex. 1 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− t1t − Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 68 Continuous – Time LTI Systems (5) t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 0 1: ( )* ( ) 0t h t x t< < = 1 2 : ( )* ( ) 2( 1)t h t x t t< < = − 2 3: ( ) * ( ) 2t h t x t< < = λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− 3 4 : ( ) 1; ( ) 2t h t x t< < = = 3 3 3 11 1 ( ) * ( ) ( ) ( ) 1 2 2 8 2 tt t h t x t h t x d d t τ τ τ τ τ τ = − − − = − = × = = −∫ ∫ 3 4 : ( )* ( ) 8 2t h t x t t< < = − Ex. 1 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− t1t − Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 69 Continuous – Time LTI Systems (6) t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 0 1: ( )* ( ) 0t h t x t< < = 1 2 : ( )* ( ) 2( 1)t h t x t t< < = − 2 3: ( ) * ( ) 2t h t x t< < = 3 4 : ( )* ( ) 8 2t h t x t t< < = − λ0 2 ( )x τ 1 2 3 4 1 ( )h t τ− 4 : ( ) 1; ( ) 0t h t x t> = = ( ) * ( ) 0h t x t = 4 : ( )* ( ) 0t h t x t> = Ex. 1 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 70 Continuous – Time LTI Systems (7) t0 1 1 ( )h t 2 3 4 t0 2 ( )x t 1 2 3 4 0 1: ( )* ( ) 0t h t x t< < = 1 2 : ( ) * ( ) 2( 1)t h t x t t< < = − 2 3: ( ) * ( ) 2t h t x t< < = 3 4 : ( )* ( ) 8 2t h t x t t< < = − 4 : ( )* ( ) 0t h t x t> = t0 2 1 2( )* ( )f t f t 1 2 3 4 Ex. 1 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 71 Continuous – Time LTI Systems (8)Ex. 2 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ t0 1 1 ( )w t 2 3 4 t0 2 2 te− 1 2 3 4 ( ) 00 0 2 : ( ) 2 1 2 2(1 )t tt t tt f t e d e eττ τ τ τ = − − − − = < < = × = = −∫ 2 2( ) 2 00 2 : ( ) 2 1 2 2( 1)t t tt f t e d e e eττ τ τ τ = − − − − = > = × = = −∫ t0 2 2e τ− 1 2 3 4 ( )w t t0 2 1 2 3 4 Method 1 Find the convolution of the two signals? sites.google.com/site/ncpdhbkhn 72 Continuous – Time LTI Systems (9)Ex. 2 Find the convolution of the two signals? 0 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t ty t h t x t h t x d h t dτ τ τ τ τ τ τ= = − = −∫ ∫ t0 1 1 ( )w t 2 3 4 t0 2 2 te− 1 2 3 4 00 0 2 : ( ) 1 2 2 2(1 )t t tt f t e d e eττ τ τ τ = − − − = < < = × = − = −∫ 2 22 2 : ( ) 1 2 2 2( 1)t t t tt t f t e d e e eττ τ τ τ = − − − = − − > = × = − = −∫ t0 2 2 te− 1 2 3 4 ( )w τ− t0 2 1 2 3 4 Method 2 sites.google.com/site/ncpdhbkhn 73 Continuous – Time LTI Systems (10) 1 0 0 0 , [ ] , n n n δ ==  ≠ 1 0 0 0 , ( ) , t t t δ ==  ≠ 1 2 2 0 otherwise / , / / ( ) , t tδ∆ ∆ − ∆ < < ∆ =   t0 ( )tδ∆ 2 −∆ 2 ∆ τ0 ( )x τ t τ= ( ) ( ) ( )y t x t dτ δ τ τ∞ ∆ −∞ = −∫ ( ) ( ) ( ) ( )x t x t tτ δ τ δ τ∆ ∆− ≈ − ( ) ( ) ( ) ( ) ( ) ( ) y t x t d x t t d x t τ δ τ τ δ τ τ ∞ ∆ −∞ ∞ ∆ −∞ → = − ≈ − = ∫ ∫ sites.google.com/site/ncpdhbkhn 74 Continuous – Time LTI Systems (11) ( ) ( ) ( ) ( )y t x t d x tτ δ τ τ∞ ∆ −∞ = − ≈∫ ( ) ( ) ( ) ( )t x t d x tδ τ δ τ τ∞ −∞ → − =∫ ( ) * ( ) ( ) ( ) ( )t x t x t d x tδ τ δ τ τ∞ −∞ = − =∫ 0 0 0( ) ( ) ( ) ( )x t t t x t t tδ δ− = − 0 0 0 0( ) ( ) ( ) ( ) ( )x t t t dt x t t t dt x tδ δ ∞ ∞ −∞ −∞ − = − =∫ ∫ t0 ( )A tδ∆A

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