CHAP. 11

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SIGNALS AND SYSTEMS

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Finally, an exponential sequence is defined by

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where a may be a real or complex number. Of particular interest is the exponential sequence that is formed when a = e ~ mwhere q, a real number. In this case, x(n) is a complex exponential , is

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As we will see in the next chapter, complex exponentials are useful in the Fourier decomposition of signals.

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1.2.3 Signal Duration

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Discrete-time signals may be conveniently classified in terms of their duration or extent. For example, a discretetime sequence is said to be a finite-length sequence if it is equal to zero for all values of n outside a finite interval [ N 1 , N2].Signals that are not finite in length, such as the unit step and the complex exponential, are said to be infinite-length sequences. Infinite-length sequences may further be classified as either being right-sided, left-sided, or two-sided. A right-sided sequence is any infinite-length sequence that is equal to zero for all values of n < no for some integer no. The unit step is an example of a right-sided sequence. Similarly, an infinite-length sequence x ( n ) is said to be lefr-sided if, for some integer no, x ( n ) = 0 for all n > no. An example of a left-sided sequence is

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which is a time-reversed and delayed unit step. An infinite-length signal that is neither right-sided nor left-sided, such as the complex exponential, is referred to as a two-sided sequence.

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1.2.4 Periodic and Aperiodic Sequences

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A discrete-time signal may always be classified as either being periodic or aperiodic. A signal x(n) is said to be periodic if, for some positive real integer N ,

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for all n. This is equivalent to saying that the sequence repeats itself every N samples. If a signal is periodic with period N , it is also periodic with period 2 N , period 3 N , and all other integer multiples of N. The fundamental period, which we will denote by N , is the smallest positive integer for which Eq. (I . I ) is satisfied. If Eq. (1. I ) is not satisfied for any integer N , x ( n ) is said to be an aperiodic signal.

EXAMPLE 1.2.1 The signals

and are not periodic, whereas the signal

xZ(n)= cos(n2) x3(n) = e ~ ~ ' ' l '

is periodic and has a fundamental period of N = 16.

If xl (n) is a sequence that is periodic with a period N1,and x2(n)is another sequence that is periodic with a period N2, the sum x(n)=x ~ ( n ) xdn)

will always be periodic and the fundamental period is

SIGNALS AND SYSTEMS

[CHAP. 1

where gcd(NI, N2) means the greatest common divisor of N1 and N2. The same is true for the product; that is,

will be periodic with a period N given by Eq. (1.2). However, the fundamental period may be smaller. Given any sequence x ( n ) , a periodic signal may always be formed by replicating x ( n ) as follows:

where N is a positive integer. In this case, y ( n ) will be periodic with period N.

1.2.5 Symmehic Sequences

A discrete-time signal will often possess some form of symmetry that may be exploited in solving problems. Two symmetries of interest are as follows:

Definition: A real-valued signal is said to be even if, for all n ,

x(n) = x(-n)

whereas a signal is said to be odd if, for all n ,

x(n) = -x(-n)

Any signal x ( n ) may be decomposed into a sum of its even part, x,(n), and its odd part, x,(n), as follows:

x(n> = x d n )

+ x,(n>

(1.3)

To find the even part of x ( n ) we form the sum

x,(n) = ( x ( n )

+x(-n))

whereas to find the odd part we take the difference

x,(n) = i ( x ( n ) - x ( - n ) )

For complex sequences the symmetries of interest are slightly different.

Definition: A complex signal is said to be conjugate symmetric3 if, for all n ,

x(n) = x*(-n)

and a signal is said to be conjugate antisymmetric if, for all n ,

x(n) = -x*(-n)

Any complex signal may always be decomposed into a sum of a conjugate symmetric signal and a conjugate antisymmeuic signal.