1 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.

2 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.

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

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

5 Discrete-time signals are sequences of individual scaled unit impulses n x[n] n n n

6 Shifted Scaled Impulses Generally:- The arbitrary sequence is represented by a linear combination of shifted unit impulses  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.

7 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  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.

8 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 time- invariant properties of LTI system.

9 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.

10 LTI System  n] h o [n] LTI System  n-k] h k [n] n=k n  Unit Impulse Response h[n] k 0

11 LTI System x[0].  n] x[0].h o [n] LTI System x[-k].  n+k] x[-k].h -k [n] n=-k n  Response to scaled unit impulse input x[n]  n-k] -k 0 LTI System x[+k].  n-k] x[+k].h k [n] k n 

12 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].

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

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

15 In general, the response h k [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 h k [n] to time shifted unit impulse are all time-shifted versions of each other. I.e. h k [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

16 Convolution sum or Superposition sum.

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

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

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

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

21 xxx x h[n] 2h[n-1] y[n] h[n] x[n] 0 y[n]=x[0]h[n-0]+x[1]h[n-1]= 0.5h[n]+2h[n-1] 11

22 Modified Example 2.3

23 Modified Example 2.5

24 Modified Example 2.5