1 signals & systems spring 2009 week 3 instructor: mariam shafqat uet taxila

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Signals & Systems Spring 2009

Week 3

Instructor: Mariam Shafqat UET Taxila

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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

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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.

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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.

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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

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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

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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.

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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.

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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]}

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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

][][][][][ 321 4321 nxnxnxnxny

][][][][][ 321 4321 nnnnnh

},,,{]}[{ 4321

nh

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Impulse Response Example

Example - The impulse response of the discrete-time accumulator

is obtained by setting x[n] = [n] resulting in

nxny

][][

][][][ nnhn

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Time-Domain Characterization of LTI Time-Domain Characterization of LTI Discrete-Time SystemDiscrete-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

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Time-Domain Characterization of LTI Time-Domain Characterization of LTI Discrete-Time SystemDiscrete-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]

][.][][.][.][ 57502151250 nnnnnx

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Time-Domain Characterization of LTI Time-Domain Characterization of LTI Discrete-Time SystemDiscrete-Time System

Since the system is time-invariant

input output

][][ 11 nhn][][ 22 nhn][][ 55 nhn

][][ 22 nhn

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Time-Domain Characterization of LTI Time-Domain Characterization of LTI Discrete-Time SystemDiscrete-Time System

Likewise, as the system is linear

Hence because of the linearity property we get

][.][. 57505750 nhn

input output

][.][. 250250 nhn

][][ 22 nhn][.][. 151151 nhn

][.][.][ 151250 nhnhny][.][ 57502 nhnh

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Time-Domain Characterization of LTI Time-Domain Characterization of LTI Discrete-Time SystemDiscrete-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

kknkxnx ][][][

][][ knkx ][][ knhkx input output

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Discrete Time LTI Systems: THE CONVOLUTION SUM

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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.

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Graphically

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Mathematically

This is called sifting property .

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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] .

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

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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.

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Contd….

Suppose that the system is linear and define hk[n] as a response of impulse[n-k]; then

For superposition:

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Cntd… Now suppose that the system is LTI; and define

the unit sample response hk[n] as.

For TI

For LTI systems:

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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.

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Convolution sum representation of LTI system

Mathematically

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Graphically

Sum up all the responses for all K’s

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Contd….

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

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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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Continuous time systems: THE CONVOLUTION INTEGRAL

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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.

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

impulses.

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Ideally

Impulse has unit area:

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Sifting property of impulse

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Response of LTI system

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Convolution Integral

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Operation of convolution

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Example

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Properties of LTI systems

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

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The end