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(1) the larger the magnitude of ?0 , the higher is the rate of oscillation in the signal; and (2) ej?t is
0
periodic for any value of ?0 . In this section we describe the discrete-time versions of both of these properties, and as we will see, there are definite differences between each of these and its continuous-time counterpart.
The fact that the first of these properties is different in discrete time is a direct consequence of another extremely important distinction between discrete-time and continuous-time complex exponentials. Specifically, consider the discrete-time complex exponential with frequency ?0+2π: (1.51)
From eq. (1.51), we see that the exponential at frequency ?0+2π is the same as that at frequency???0. Thus, we have a very different situation from the continuous-time case, in which the signals ? ? j?t
eare all distinct for distinct values of ??0. In discrete time, these signals are not distinct, as the signal with frequency ??0 is identical to the signals with frequencies ??0 ± 2π, ??0 ± 4π, and so on.
0
Therefore, in considering discrete-time complex exponentials, we need only consider a frequency interval of length 2π in which to choose ??0. Although, according to eq. (1.51), any interval of length 2π will do, on most occasions we will use the interval 0 ≤ ??0 ≤ 2π or the interval －π ≤ ??0 ＜ π.
Because of the periodicity implied by eq. (1.51), the signal ej?n does not have a continually increasing rate of oscillation as ??0 is increased in magnitude. Rather, as illustrated in Figure 1.27, as
0
we increase ??0 form 0, we obtain signals that oscillate more and more rapidly until we reach ??0 = π. As we continue to increase ??0 , we decrease the rate of oscillation until we reach ??0 =2π, which produces the same constant sequence as ??0 = 0. Therefore, the low-frequency (that is, slowly varying) discrete-time exponentials have values of ??0 near 0, 2π, and any other even multiple of π, while the high frequencies (corresponding to rapid variations) are located near ??0 = ±π and other odd multiples of π. Note in particular that for ??0 = π or any other odd multiple of π,
1.27(e)].
The second property we wish to consider concerns the periodicity of the discrete-time complex exponential. In order for the signal ej?n to be periodic with period N ＞ 0, we must have
?0
（1.52）
So that this signal oscillates rapidly, changing sign at each point in time [as illustrated in Figure
(1.53)
or equivalently,
(1.54)
For eq.(1.54) to hold, ??0 N must be a multiple of 2π. That is, there must be an integer m such that
???0N = 2??m (1.55)
or equivalently,
(1.56)
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According to eq.(1.56), the signal ej?n is periodic if ??0 /2π is a rational number and is not periodic
0
otherwise. These same observations also hold for discrete-time sinusoids. For example, the signals depicted in Figure 1.25(a) and (b) are periodic, while the signal in Figure 1.25(c) is not.
Using the calculations that we have just made, we can also determine the fundamental period and frequency of discrete-time complex exponentials, where we define the fundamental frequency of discrete-time periodic signal as we did in continuous time. That is, if x[n] is periodic with fundamental period N, its fundamental frequency is 2π/N. Consider, then, a periodic complex exponential x[n] = ej?n with ??0 ≠ 0. As we have just seen, ??0 must satisfy eq. (1.56) for some pair of integers m and N, with N ＞ 0. In Problem 1.35, it is shown that if ??0 ≠ 0 and if N and m have
0
no factors in common, then the fundamental period of x[n] is N. Using this fact together with eq.
j?n
(1.56), we find that the fundamental frequency of the periodic signal eis
0
2??0?， (1.57) NmNote that the fundamental period can also be written as
N?m(2?/?0)， (1.58)
These last two expressions again differ from their continuous-time counterparts. In Table 1.1,
we have summarized some of the differences between the continuous-time signal ej?t and the
0
discrete-time signal e. Note that as in the continuous-time case, the constant discrete-time signal resulting from setting ??0 =0 has a fundamental frequency of zero, and its fundamental period is undefined.
TABLE 1.1 Comparison of the signals e
0
j?0n
j?0t
and e
j?0n
.
0
ej?t ej?n
Distinct signals for distinct values of ??0 Identical signals for values of ?0 separated by multiples of 2? Period for any choice of ??0 Periodic only if ??0 = 2?m/N for some integers N > 0 and m. Fundamental frequency ??0 Fundamental frequency * ??0 / m Fundamental period Fundamental period*
??0 = 0: undefined ??0 = 0: undefined ??0 ≠???????? ?0 ??0 ≠????m(???? ??0)
*Assumes that m and N do not have any factors in common
To gain some additional insight into these properties, let us examine again the signals depicted in Figure 1.25. First, consider the sequence x[n] =cos (2πn/12), depicted in Figure 1.25(a), which we can think of as the set of samples of the continuous-time sinusoid x (t) =cos (2πt/12) at integer time points. In this case, x (t) is periodic with fundamental period 12 and x[n] is also periodic with fundamental period 12. That is, the values of x[n] repeat every 12 points, exactly in step with the fundamental period of x (t).
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In contrast, consider the signal x[n] =cos (8πn/31), depicted in Figure 1.25(b), which we can view as the set of samples of x (t) =cos (8πt/31) at integer points in time. In this case, x (t) is periodic with fundamental period 31/4. On the other hand, x[n] is periodic with fundamental period 31. The reason for this difference is that the discrete-time signal is defined only for integer values of the independent variable. Thus, there is no sample at time t = 31/4, when x (t) completes one period (starting from t = 0). Similarly, there is no sample at t = 2·31/4 or t = 3·31/4, when x(t) has completes two or three periods, but there is a sample at t=4·31/4=31, when x(t) has completes four periods. This can be seen in Figure 1.25(b), where the pattern of x[n] values does not repeat with each single cycle of positive and negative values. Rather, the pattern repeats after four such cycles, namely, every 31 point.
Similarly, the signal x[n] =cos (n/6) can be viewed as the set of samples of the signal x (t) = cos (t/6) at integer time points. In this case, the values of x(t) at integer sample points never repeat, as these sample points never span an interval that is an exact multiple of the period, 12π, of x(t). Thus, x[n] is not periodic, although the eye visually interpolates between the sample points, suggesting the envelope x (t), which is periodic. The use of the concept of sampling to gain insight into the periodicity of discrete-time sinusoidal sequences is explored further in Problem 1.36.
Example 1.6
Suppose that we wish to determine the fundamental period of the discrete-time signal
(1.59)
The first exponential on the right-hand side of eq. (1.59) has a fundamental period of 3. While this can be verified from eq. (1.58), there is a simpler way to obtain that answer. In particular, note that the angle (2π/3) n of the first term must be incremented by a multiple of 2π for the values of this exponential to begin repeating. we then immediately see that if n is incremented by 3, the angle will be incremented by a single multiple of 2π. With regard to the second term, we see that incrementing the angle (3π/4)n by 2 would require n to be incremented by 8/3, which is impossible, since n is restricted to being an integer. Similarly, incrementing the angle by 4π would require a noninteger increment of 16/3 to n. However, incrementing the angle by 6π requires an increment of 8 to n, and thus the fundamental period of the second term is 8.
Now, for the entire signal x[n] to repeat, each of the terms in eq. (1.59) must go through an integer number of its own fundamental period. The smallest increment of n that accomplishes this is 24. That is, over an interval of 24 points, the first term on the right-hand side of eq. (1.59) will have gone through eight of its fundamental periods, the second term through three of its fundamental periods, and the overall signal x[n] through exactly one of its fundamental periods.
As in continuous time, it is also of considerable value in discrete-time signal and system analysis to consider sets of harmonically related periodic exponentials——that is, periodic exponentials with a common period N. From eq. (1.56), we know that these are precisely the signals which are at frequencies which are multiples of 2π/N. That is,
（1.60）
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In the continuous-time case, all of the harmonically related complex exponentials ejk (2π/T) t, k = 0, ±1, ±2, …, are distinct. However, because of eq. (1.51), this is not the case in discrete time. Specifically,
(1.61)
This implies that there are only N distinct periodic exponentials in the set given in eq. (1.60). For example,
?0[n]?1, ?1[n]?ej2??n/N, ?2[n]?ej4??n/N,…, ?N?1[n]?ej2?(N?1)n/N (1.62)
are all distinct, and any other ??k [n] is identical to one of these (e.g., ??N [n] = ??0 [n] and ??-1 [n] = ??N -1 [n]).
1.4 THE UNIT IMPULSE AND UNIT STEP FUNCTIONS
In this section, we introduce several other basic signals__---specifically, the unit impulse and step functions in continuous and discrete time----that are also of considerable importance in signal and system analysis. In Chapter 2, we will see how we will see how we can use unit impulse signals as basic building blocks for the construction and representation of other signals. We begin with the discrete-time case.
1.4.1 The Discrete-Time Unit Unit impulse and Unit Step Sequences One of the simplest discrete-time signals is the unit impulse (or unit sample), which is defined as
?[n]???0,n?0 (1.63)
?1,n?0and which is shown in Figure 1.28. Throughout the book, we will refer to ??[n] interchangeably as the unit impulse or unit sample.
Figure 1.28 Discrete-time unit impulse (sample).
A second basic discrete-time signals is the discrete-time unit step, denoted by u[n] and defined by
?0,n?0u[n]?? (1.64)
?1,n?0The unit step sequence is shown in Figure 1.29.
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