Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir E-mail: demir@disi.unitn.it Web page: http://rslab.disi.unitn.it
Linear Time-Invariant (LTI) Systems Impulse Response [ n] Discrete Time System hn [ ] yn [ ] xn [ k] k 0 hn [ ] [ n k] k 0 hn [ ] un [ ] yn [ ] xl [ ] 1 2 3 1 n 0 hn [ ] 0 n 0 hn [ ] un [ ] 2 n l yn [ ] axn 1 [ ] axn 2 [ 1] axn 3 [ 2] hn [ ] a[ n] a[ n1] a[ n2]
Linear Time-Invariant (LTI) Systems [ n] h[ n] [ nk] h[ nk] x[ k] [ n k] x[ k] h[ nk] k k xn [ ] yn [ ] Convolution sum: yn [ ] xkhn [ ] [ k] k 3
Linear Time-Invariant Systems LTI Systems: Convolution
Example n xn [ ] (0.2) un [ ] n hn [ ] (0.6) un [ ] yn [ ] xn [ ] hn [ ]? y n u n n1 n1 [ ] 2.5(0.6 0.2 ) [ ] 5
Convolution Sum-Example The output of an LTI system can be obtained as the superposition of responses to individual samples of the input. This approach is shown to estimate y[n] in the case of x[n] and h[n] given in the following: y[n]?
Convolution Sum-Example-cont
Convolution Sum-Example-cont
Example y[n]? 9
Example-cont
Example-cont
LTI Systems-Convolution The output sequence y[n] can be also obtained by multiplying the input sequence (expressed as a function of k) by the sequence whose values are h[n k], <k< for any fixed value of n, and then summing all the values of the products x[k]h[n k], with k a counting index in the summation process. Therefore, the operation of convolving two sequences involves doing the computation specified by k y [ n] x[ k] h[ n k] for each value of n, thus generating the complete output sequence y[n], < n<.
LTI Systems-Convolution The process of computing the convolution between x[n] and h[n] involves the following steps: 1. Express x[n] and h[n] as a function of k and obtain x[k] and h[k] 2. Obtain h[-k] from h[k]. 3. Shift h[-k] by n 0 to the right if n is positive to obtain h[n 0 k] (shift h[- k] by n 0 to the left if n is negative). 4. Multiply x [k] by h[n 0 -k] to obtain the product sequence. 5. Sum all the values of the product sequence to obtain the value of the output at time n= n 0. The steps from 3 to 5 should be repeated for each values of n 0 (i.e., for any fixed value of n). 13
LTI Systems-Convolution k y [ n] x[ k] h[ n k] 14
LTI Systems-Convolution 15
LTI Systems-Convolution For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n]. 16
LTI Systems-Convolution 17
LTI Systems-Convolution yn [ ] xn [ ]* hn [ ] xkhn [ ] [ k] k 0123456 x[k] k 0123456 h[k] k 0123456 h[0k] k
LTI Systems-Convolution How to compute y[n]? 0123456 0123456 0123456 x[k] h[0k] h[1k] k compute y[0] k compute y[1] k For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n].
Example xn [ ] 1 2 3 4 5 hn [ ] 1 2 1 yn [ ]?
Example hn [ ] 1 1 1 1 1 1 xn [ ] 1 0.5 0.25 0.125 yn [ ]?
Convolution Features commutative: x [ n] x [ n] x [ n] x[ n] 1 2 2 1 associative: x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] 1 2 3 1 2 3 distributive: x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] c [ n] 1 2 2 1 2 3 1 2 1 3 x1[ n m] x2[ nk] c2[ nmk] x1[ n] [ n] x1[ n] if x [ n] has m samples and x [ n] k samples, 1 2 cn [ ] x[ n] x[ n] will have mk-1 samples. 1 2 x [ n] [ n] x [ n] 1 1
LTI Systems-Parallel Connection of Systems h 1 [n] x[n] y[n] x[n] h 1 [n]+h 2 [n] y[n] h 2 [n]
LTI Systems-Cascade Connection of Systems x[n] h 1 [n] h 2 [n] y[n] x[n] h 1 [n]*h 2 [n] y[n]
LTI Systems-Inverse Systems x[n] h[n] y[n] h'[n] x[n] hn [ ] h'[ n] [ n]
Example-Inverse Systems yn [ ] xn [ k] k0 hn [ ] [ nk] un [ ] k0 h'[ n]? h'[ n] [ nk] [ n] k0 k0 h'[ nk] [ n] h'[ n] h'[ n1] h'[ n2]... [ n] h'[0] 1 h'[1] h'[0] 0 h'[1] 1 h'[2] 0, h'[ n] 0, n1 h'[ n] [ n] [ n1]
Stability of Systems All LTI systems are BIBO stable if and only if h[n] is absolutely summable: This is due to the fact that: k hk [ ] yn [ ] hkxn [ ] [ k] B, k y[ n] h[ k] x[ nk] h[ k] x[ nk] k if xn [ ] is bounded ( xn [ ] B),so that yn [ ] B hk [ ], hk [ ] x k k y k x
Stability of Systems-Example n hn [ ] un [ ] is stable? k hk [ ] k0 a k if 1 the system is stable if 1 the system is NOT stable
Stability of Systems-Example hn [ ] [ k]? 1 n 0 = 0 n<0 un [ ] n k The system is NOT stable
Stability of Systems-Example M 2 1 hn [ ] [ nk]? M M 1 1 2 km 1 M n M = M1M2 1 0 otherwise 1 1 2 The system is stable
Causality of Systems A LTI system is causal if an only if hn [ ] 0 for n 0
Causality of Systems yn [ ] xn [ n0] Is hn [ ] causal? hn [ ] [ nn] 0 Since h[n] includes only one sample at n=n 0, h[n] is causal. It is stable because it is absolutely summable. yn [ ] xn [ 1] xn [ ] Is hn [ ] causal? hn [ ] [ n1] [ n] Since h[n] includes sample at n=-1, h[n] is not causal, however it is stable because it is absolutely summable.
Causality of Systems N 1 1 yn [ ] xn [ k] N k0 Is hn [ ] causal? 1 N 1 1 0 n N 1 hn [ ] [ n k] N N k 0 0 n 0, n N 1 Since h[n] does not include samples when at n<0, h[n] is causal. It is stable because it is absolutely summable.
Causality of Systems yn [ ] xk [ ] n k Is hn [ ] causal? n 1 n 0 hn [ ] [ k] k 0 n 0 hn [ ] un [ ] Since h[n] does not include samples when at n<0, h[n] is causal. It is NOT stable because it is NOT absolutely summable.
FIR and IIR Systems Finite Impulse Response (FIR) Systems: Systems with only a finite length of nonzero values in h[n] are called FIR systems. Examples: Ideal delay, moving average filter FIR systems are always STABLE Infinite Impulse Response (IIR) Systems: Systems with infinite length of nonzero values in h[n] are called IIR systems. Examples:Accumulator, filters IR systems can be STABLE or UNSTABLE
FIR and IIR Systems FIR Systems: IIR Systems: hn [ ] [ n n] 0 hn [ ] un [ ] hn [ ] [ n1] [ n] n hn [ ] aun [ ] N 1 1 hn [ ] [ n k] N k0