# lect1 - DISCRETE TIME SIGNALS A discrete time signal x(n is...

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DISCRETE TIME SIGNALS A discrete time signal, x(n), is a sequence of numbers, real or complex. In general the signal can be bi-infinite; . . . . . . . . . . , x(-2), x(-1), x(0), x(1), x(2), . . . . . . . Real signals can be represented graphically, as in the example below. Several standard operations on signals are often used. Among them are: Time Shifting The signal x(n-k) is a shifted version of x(n), shifted to the right by k units. For example, with x(n) as above, x(n-2) is shown below. n 2 1 0 -1 -2 -3 3 x(n) n 4 3 2 1 0 -1 5 x(n-2)

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Time Reversal The sequence x(-n) is a time reversed version of x(n), as shown below for the example above. Two often used special sequences are the Impulse sequence, ) ( n d , and the Unit Step sequence, ) ( n U , shown below. DISCRETE TIME SYSTEMS A discrete time system is one, which takes as input a discrete time signal, and generates a discrete time signal as output. n 2 1 0 -1 -2 -3 3 x(-n) δ (n) 0 n 1 U(n) 0 n 1
A simple example might be a square law system in which ) ( ) ( 2 n x n y = . The systems of most interest to us in this course are the Linear, Time invariant systems. Let L denote the operator mapping the input of the system to its output, i.e. )} ( { ) ( n x L n y = . The system is Linear if, for any complex numbers, 1 a and 2 a , and any inputs, ) ( 1 n x and ) ( 2 n x , we have } ) ( { )} ( { } ) ( ) ( { 2 2 1 1 2 2 1 1 n x L n x L n x n x L a a a a + = + . (1.1) In other words, if ) ( 1 n y is the output due to ) ( 1 n x , and ) ( 2 n y is the output due to ) ( 2 n x , then the input ) ( ) ( 2 2 1 1 n x n x a a + results in the output ) ( ) ( 2 2 1 1 n y n y a a + . It easily follows from this that for a Linear system, arbitrary linear combinations of inputs give rise to the same linear combinations of the respective outputs, as shown in the diagram below. x(n) y(n) S ) ( n x i i a ) ( n y i i a L

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The system is called Time Invariant (or Shift Invariant) if, for any input, ) ( n x , and corresponding output, ) ( n y , an arbitrary time shift of the input, say ) ( k n x - , yields the correspondingly time shifted output ) ( k n y - , as depicted below. A system that is both Linear and Time Invariant is called a Linear, Time Invariant (LTI) system. IMPULSE RESPONSE The Impulse Response of an LTI system is the response of the system to the input ) ( n d , and is usually denoted by ) ( n h . The diagram below illustrates the situation. We will see next that the impulse response completely determines the behavior of an LTI system. ) ( k n x - ) ( k n y - L ) ( n d ) ( n h L
THE CONVOLUTION SUM FOR LTI SYSTEMS Using nothing more than the linearity and time invariance properties of an LTI system, we can easily show that the output is the convolution of the input and the impulse response. Indeed, an arbitrary input sequence,

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lect1 - DISCRETE TIME SIGNALS A discrete time signal x(n is...

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