Signals and Systems - Electrical Engineering

8.2 Discrete-Time Signals 471

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(a) (b)

(c) (d)

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− 1

−0.5

0

0.5

1

− 1

−0.5

0

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− 1

−0.5

0.5

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− 1

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n

0 10203040

n

x^1

[n

]

x^3

[n

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x^4

[n

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x^2

[n

]

0 10203040

n

0 10203040

n

FIGURE 8.6
Periodic signalsxi[n],(a)i=1,(b)i=2,(c)i= 3 , and(d)i= 4 , given in Example 8.14.

three times that ofx 2 [n]. When plotting these signals using MATLAB, the first two resemble analog

sinusoids but not the other two. See Figure 8.6. n

Remarks

n The discrete-time sine and cosine signals, as in the continuous-time case, are out of phaseπ/ 2 radians.
n The discrete frequencyωis given in radians since n, the sample index, does not have units. This can also
be seen when we sample a sinusoid using a sampling period Tsso that

cos( 0 t)|t=nTs=cos( 0 Tsn)=cos(ω 0 n)

where we definedω 0 = 0 Ts, and since 0 has rad/sec as units and Tshas seconds as units, thenω 0 has
radians as units.
n The frequencyof analog sinusoids can vary from 0 (dc frequency) to∞. Discrete frequenciesωas
radian frequencies can only vary from 0 toπ. Negative frequencies are needed in the analysis of real-
valued signals; thus−∞<  <∞and−π < ω≤π. A discrete-time cosine of frequency 0 is constant
for all n, and a discrete-time cosine of frequencyπvaries from− 1 to 1 from sample to sample, giving the
largest variation possible for the discrete-time signal.