CHAP. 1 1

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SIGNALS AND SYSTEMS

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form x ( t ) = x , ( t ) +ix2(t) where x,( t ) and x2( t ) are real signals and j = Note that in Eq. (I.l)t represents either a continuous or a discrete variable.

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D. Deterministic and Random Signals:

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Deterministic signals are those signals whose values are completely specified for any given time. Thus, a deterministic signal can be modeled by a known function of time I . Random signals are those signals that take random values at any given time and must be characterized statistically. Random signals will not be discussed in this text.

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E. Even and Odd Signals:

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A signal x ( t ) or x [ n ] is referred to as an even signal if

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x(-t) =x(r) x[-n] =x[n]

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A signal x ( t ) or x [ n ] is referred to as an odd signal if

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x(-t) x[-n]

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-x(t) -x[n]

Examples of even and odd signals are shown in Fig. 1-2.

Fig. 1-2 Examples of even signals (a and 6 ) and odd signals ( c and d l .

SlGNALS AND SYSTEMS

[CHAP. 1

Any signal x ( t ) or x [ n ] can be expressed as a sum of two signals, one of which is even and one of which is odd. That is,

where

x e ( t )= $ { x ( t )+ x ( - t ) ] x e [ n ]= i { x [ n ]+ x [ - n ] ) x , ( t ) = $ { x ( t )- x ( - t ) ) x,[n]

even part of x ( t ) even part of x [ n ] odd part of x ( t ) odd part of x [ n ]

(1.5)

$ { x [ n ]- x [ - n ] )

( 1.6 )

Note that the product of two even signals or of two odd signals is an even signal and that the product of an even signal and an odd signal is an odd signal (Prob. 1.7).

F. Periodic and Nonperiodic Signals:

A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive nonzero value of T for which

+ T )= x ( t )

(1.7)

An example of such a signal is given in Fig. 1-3(a). From Eq. (1.7) or Fig. 1-3(a) it follows that for all t and any integer m. The fundamental period T, of x ( t ) is the smallest positive value of T for which Eq. (1.7) holds. Note that this definition does not work for a constant

Fig. 1-3 Examples of periodic signals.

CHAP. 1 1

SIGNALS AND SYSTEMS

signal x ( t ) (known as a dc signal). For a constant signal x ( t ) the fundamental period is undefined since x ( t ) is periodic for any choice of T (and so there is no smallest positive value). Any continuous-time signal which is not periodic is called a nonperiodic (or aperiodic ) signal. Periodic discrete-time signals are defined analogously. A sequence (discrete-time signal) x[n] is periodic with period N if there is a positive integer N for which x[n

+N] =x[n]

all n

(1.9)

An example of such a sequence is given in Fig. 1-3(b). From Eq. (1.9) and Fig. 1-3(b) it follows that

for all n and any integer m. The fundamental period No of x[n] is the smallest positive integer N for which Eq. (1.9) holds. Any sequence which is not periodic is called a nonperiodic (or aperiodic sequence. Note that a sequence obtained by uniform sampling of a periodic continuous-time signal may not be periodic (Probs. 1.12 and 1.13). Note also that the sum of two continuous-time periodic signals may not be periodic but that the sum of two periodic sequences is always periodic (Probs. 1.14 and 1.l5).

G. Energy and Power Signals:

Consider v(t) to be the voltage across a resistor R producing a current d t ) . The instantaneous power p( t ) per ohm is defined as