Computing the output response of LTI Systems By

Computing the output response of LTI Systems. • By breaking or decomposing and representing the input signal to the LTI system into terms of a linear combination of a set of basic signals. • Using the superposition property of LTI system to compute the output of the system in terms of its response to these basic signals. 1

General Signal Representations By Basic Signal • The basic signal - in particular the unit impulse can be used to decompose and represent the general form of any signal. • Linear combination of delayed impulses can represent these general signals. 2

Response of LTI System to General Input Signal General Output LTI SYSTEM Input Response Signal Delayed Impulse Signal 1 Delayed Impulse LTI SYSTEM Response to Impulse signal N 3

Representation of Discrete-time Signals in Terms of Impulses. Discrete-time signals are sequences of individual impulses. x[n] -4 -1 -3 -2 2 3 0 1 -1 -4 -3 -2 -4 n 2 3 0 1 2 -1 -3 -2 4 0 1 3 4 n 4

Discrete-time signals are sequences of individual scaled unit impulses. x[n] -4 -1 -3 -2 2 3 0 1 -1 -4 -3 -2 -4 2 -1 -4 4 n 3 0 1 2 -1 -3 -2 n 2 3 0 1 -3 -2 4 0 1 4 n 3 5

Shifted Scaled Impulses Generally: - The arbitrary sequence is represented by a linear combination of shifted unit impulses d[n-k] , where the weights in this linear combination are x[k]. The above equation is called the sifting property of discrete-time unit impulse. 6

As Example consider unit step signal x[n]=u[n]: - Generally: - The unit step sequence is represented by a linear combination of shifted unit impulses d[n-k] , where the weights in this linear combination are ones from k=0 right up to k= This is identically similar to the expression we have derived in our previous lecture a few weeks back when we dealt with unit step. 7

The Discrete-time Unit Impulse Responses and the Convolution Sum Representation • To determine the output response of an LTI system to an arbitrary input signal x[n], we make use of the sifting property for input signal and the superposition and timeinvariant properties of LTI system. 8

Convolution Sum Representation • The response of a linear system to x[n] will be the superposition of the scaled responses of the system to each of these shifted impulses. • From the time-invariant property, the response of LTI system to the time-shifted unit impulses are simply time-shifted responses of one another. 9

Unit Impulse Response h[n] d[n] 0 ho[n] LTI System n=0 d[n-k] k hk[n] LTI System n=k 10

Response to scaled unit impulse input x[n]d[n-k] x[-k]. d[n+k] LTI System -k x[0]. d[n] 0 x[-k]. h-k[n] n=-k x[0]. ho[n] LTI System n=0 x[+k]. d[n-k] k LTI System x[+k]. hk[n] n=+k 11

Output y[n] of LTI System Thus, if we know the response of a linear system to the set of shifted unit impulses, we can construct the response y[n] to an arbitrary input signal x[n]. 12

h-1[n] x[n] h 0[n] 0 h 1[n] 13

x[-1]h-1[n] x[-1]d[n+1] 0 0 x[0]h 0[n] x[0]d[n] 0 x[1]d[n-1] 0 0 0 x[n] 0 x[1]h 1[n] y[n] 0 14

• In general, the response hk[n] need not be related to each other for different values of k. • If the linear system is also time-invariant system, then these responses hk[n] to time shifted unit impulse are all time-shifted versions of each other. • I. e. hk[n]=h 0[n-k]. • For notational convenience we drop the subscript on h 0[n] =h[n]. • h[n] is defined as the unit impluse (sample) response 15

Convolution sum or Superposition sum. 16

Convolution sum or Superposition sum. x[k] h[k] 17

Convolution sum or Superposition sum. x[k] h[k] 18

Convolution sum or Superposition sum. x[k] h[k] 19

Convolution sum or Superposition sum. x[k] h[k] 20

y[n]=x[0]h[n-0]+x[1]h[n-1]= 0. 5 h[n]+2 h[n-1] 1 1 1 x x h[n] x 0. 5 x 1 x[n] 0 0. 5 h[n] 2 2 h[n-1] 2. 5 2 y[n] 0. 5 21

Modified Example 2. 3 22

Modified Example 2. 5 23

Modified Example 2. 5 24