1 Signals & Systems Spring 2009 Week 3 Instructor: Mariam Shafqat UET Taxila

2 Today's lecture Linear Time Invariant Systems  Introduction  Discrete time LTI systems: Convolution Sum  Continuous time LTI systems: Convolution Integral  Properties of LTI systems  Quiz at the end of lecture

3 Linear Time Invariant Systems A system satisfying both the linearity and the time-invariance property LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design. Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades. They possess superposition theorem.

4 How superposition is applicable If we represent the input to an LTI system in terms of linear combination of a set of basic signals, we can then use superposition to compute the output of the system in terms of responses to these basic signals.

5 Representation of LTI systems Any linear time-invariant system (LTI) system, continuous-time or discrete-time, can be uniquely characterized by its  Impulse response: response of system to an impulse  Frequency response: response of system to a complex exponential e j 2  f for all possible frequencies f.  Transfer function: Laplace transform of impulse response Given one of the three, we can find other two provided that they exist

6 Significance of unit impulse Every signal whether large or small can be represented in terms of linear combination of delayed impulses. Here two properties apply:  Linearity  Time Invariance

7 Basic building Blocks For DT or CT case; there are two natural choices for these two basic building blocks  For DT: Shifted unit samples  For CT: Shifted unit impulses.

8 Convolution Sum This fact together with the property of superposition and time invariance, will allow us to develop a complete characterization of any LTI system in terms of responses to a unit impulse. This representation is called  Convolution sum in discrete time case  Convolution integral in continuous time case.

9 Impulse Response The response of a discrete-time system to a unit sample sequence {  [n]} is called the unit sample response or simply, the impulse response, and is denoted by {h[n]} The response of a discrete-time system to a unit step sequence {  [n]} is called the unit step response or simply, the step response, and is denoted by {s[n]}

10 Impulse Response Example The impulse response of the system is obtained by setting x[n] =  [n] resulting in The impulse response is thus a finite-length sequence of length 4 given by

11 Impulse Response Example Example - The impulse response of the discrete- time accumulator is obtained by setting x[n] =  [n] resulting in

12 Time-Domain Characterization of LTI Discrete-Time System Input-Output Relationship -A consequence of the linear, time-invariance property is that an LTI discrete-time system is completely characterized by its impulse response Knowing the impulse response one can compute the output of the system for any arbitrary input

13 Time-Domain Characterization of LTI Discrete-Time System Let h[n] denote the impulse response of a LTI discrete-time system We compute its output y[n] for the input: As the system is linear, we can compute its outputs for each member of the input separately and add the individual outputs to determine y[n]

14 Time-Domain Characterization of LTI Discrete-Time System Since the system is time-invariant inputoutput

15 Time-Domain Characterization of LTI Discrete-Time System Likewise, as the system is linear Hence because of the linearity property we get input output

16 Time-Domain Characterization of LTI Discrete-Time System Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences in the form The response of the LTI system to an input will be inputoutput

17 Discrete Time LTI Systems: THE CONVOLUTION SUM

18 The representation of discrete time signal in terms of impulses Simplest way is to visualize discrete time signal in terms of individual impulses. Here we use scaled unit impulse sequences. Where the scaling on each impulse equals the value of x[n] at the particular instant the unit impulse occurs.

19 Graphically

20 Mathematically This is called sifting property.

21 Sifting property This property corresponds to the representation of an arbitrary sequence as a linear combination of shifted unit impulses ; where the weights in the linear combination are x[k].

22 The discrete time unit impulse response and the convolution sum representation of LTI systems

23 Significance of sifting property Represent any input signal as a superposition of scaled version of a very simple set of elementary functions; namely; shifted unit impulses:  Each of which is non-zero at a single point in time specified by the corresponding value of K.  Moreover property of time invariance states that the response of a time invariant system to the time shifted unit impulses are simply time shifted version of one another.

24 Contd…. Suppose that the system is linear and define h k [n] as a response of impulse[n-k]; then For superposition:

25 Cntd… Now suppose that the system is LTI; and define the unit sample response h k [n] as. For TI For LTI systems:

26 Convolution sum/ Superposition sum The last equation is called superposition sum or the convolution sum. Operation on the right hand side is known as convolution of the sequence x[n] and h[n]. We will represent the operation of the convolution symbolically  y[n]=x[n]*h[n]  LTI system is completely characterized by its response to a single signal namely; its response to the unit impulse.

27 Convolution sum representation of LTI system Mathematically

28 Graphically Sum up all the responses for all K’s

29 Contd…. Develop the sequence y[n] generated by the convolution of the sequences x[n] and h[n] shown below

30 y(0)=x(0)h(0) y(1)=x(0)h(1) +x(1)h(0) y(2)=x(2)h(0) +x(1)h(1) h(1-k)=h[-(k-1)] h(2-k)=h[-(k-2)]

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32 Continuous time systems: THE CONVOLUTION INTEGRAL

33 What we had for discrete time signals Convolution sum was the sifting property of discrete time unit impulse – that is, the mathematical representation of a signal as a superposition of scaled and shifted unit impulse functions. For CT signals consider impulse as an idealization of a pulse that is too short. Rep CT signal as idealized pulses with vanishingly small duration impulses.

34 Rep of CT signal in terms of impulses App any signal x(t) as sum of shifted, scaled impulses.

35 Ideally Impulse has unit area:

36 Sifting property of impulse

37 Response of LTI system

38 Convolution Integral

39 Operation of convolution

40 Example

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42 Properties of LTI systems Commutative Distributive Associative With and without memory Invertibility Causality Stability The unit step response of an LTI system

43 The end