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CHAP. 11
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SIGNALS AND SYSTEMS
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Then the unit step function u(t) can be expressed as
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( t )=
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(1.31)
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Note that the unit step function u(t) is discontinuous at t = 0; therefore, the derivative of u(t) as shown in Eq. (1.30)is not the derivative of a function in the ordinary sense and should be considered a generalized derivative in the sense of a generalized function. From Eq. (1.31) see that u(t) is undefined at t = 0 and we
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C. Complex Exponential Signals: The complex exponential signal
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Fig. 1-7 ( a ) Exponentially increasing sinusoidal signal; ( b )exponentially decreasing sinusoidal signal.
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[CHAP. 1
is an important example of a complex signal. Using Euler's formula, this signal can be defined as ~ ( t = eiUo'= cos o,t )
+jsin w0t
(1.33)
Thus, x ( t ) is a complex signal whose real part is cos mot and imaginary part is sin o o t . An important property of the complex exponential signal x ( t ) in Eq. (1.32) is that it is periodic. The fundamental period To of x ( t ) is given by (Prob. 1.9)
Note that x ( t ) is periodic for any value of o,.
General Complex Exponential Signals:
Let s = a + jw be a complex number. We define x ( t ) as ~ ( t = eS' = e("+~")'= e"'(cos o t )
+j sin wt )
( 1-35)
Then signal x ( t ) in Eq. (1.35) is known as a general complex exponential signal whose real part eu'cos o t and imaginary part eu'sin wt are exponentially increasing (a > 0) or decreasing ( a < 0) sinusoidal signals (Fig. 1-7).
Real Exponential Signals:
Note that if s = a (a real number), then Eq. (1.35) reduces to a real exponential signal x(t)
= em'
(1.36)
Fig. 1-8 Continuous-time real exponential signals. ( a ) a > 0; ( b )a < 0.
CHAP. 1 1
SIGNALS AND SYSTEMS
As illustrated in Fig. 1-8, if a > 0, then x(f ) is a growing exponential; and if a < 0, then x ( t ) is a decaying exponential.
Sinusoidal Signals:
A continuous-time sinusoidal signal can be expressed as
where A is the amplitude (real), w , is the radian frequency in radians per second, and 8 is the phase angle in radians. The sinusoidal signal x ( t ) is shown in Fig. 1-9, and it is periodic with fundamental period
The reciprocal of the fundamental period To is called the fundamental frequency fo:
fo=70 .
h ertz (Hz)
From Eqs. (1.38) and (1.39) we have
which is called the fundamental angular frequency. Using Euler's formula, the sinusoidal signal in Eq. (1.37) can be expressed as
where "Re" denotes "real part of." We also use the notation "Im" to denote "imaginary part of." Then
Fig. 1-9 Continuous-time sinusoidal signal.
SIGNALS AND SYSTEMS
[CHAP. 1
1.4 BASIC DISCRETE-TIME SIGNALS A. The Unit Step Sequence: T h e unit step sequence u[n] is defined as
which is shown in Fig. 1-10(a). Note that the value of u[n] at n = 0 is defined [unlike the continuous-time step function u(f) at t = 01 and equals unity. Similarly, the shifted unit step sequence ii[n - k ] is defined as
which is shown in Fig. 1-lO(b).
Fig. 1-10 ( a ) Unit step sequence; (b) shifted unit step sequence.
B. The Unit Impulse Sequence:
T h e unit impulse (or unit sample) sequence 6[n] is defined as
which is shown in Fig. 1 - l l ( a ) . Similarly, the shifted unit impulse (or sample) sequence 6[n - k ] is defined as
which is shown in Fig. 1-1l(b).
Fig. 1-11 ( a ) Unit impulse (sample) sequence; (6) shifted unit impulse sequence.
CHAP. 11
SIGNALS AND SYSTEMS
Unlike the continuous-time unit impulse function S(f), S [ n ] is defined without mathematical complication or difficulty. From definitions (1.45) and (1.46) it is readily seen that
which are the discrete-time counterparts of Eqs. (1.25) and (1.26), respectively. From definitions (1.43) to (1.46), 6 [ n ] and u [ n ] are related by (1.49) ( 1SO) which are the discrete-time counterparts of Eqs. (1.30) and (1.31), respectively. Using definition (1.46), any sequence x [ n ] can be expressed as
which corresponds to Eq. (1.27) in the continuous-time signal case. C. Complex Exponential Sequences: The complex exponential sequence is of the form x[n]= e ~ n ~ " Again, using Euler's formula, x [ n ] can be expressed as
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