The discrete sinc function is defined by sinc% n
N &
= sin(nπ/N)
(nπ/N) , sinc(0) = 1 (3.18)
The signal sinc(n/N) equals zero at n = kN, k = ±1, ±2, . . .. At n = 0, sinc(0) = 0/0 and cannot be evaluated in the limit since n can take on only integer values. We therefore define sinc(0) = 1. The envelope of the sinc function shows a mainlobe and gradually decaying sidelobes. The definition of sinc(n/N) also implies that sinc(n) = δ[n].
3.4.5
Discrete Exponentials
Discrete exponentials are often described using a rational base. For example, the signal x[n] = 2nu[n] shows exponential growth while y[n] = (0.5)nu[n] is a decaying exponential. The signal f [n] = (−0.5)nu[n] shows values that alternate in sign. The exponential x[n] = αnu[n], where α = rejθ is complex, may be described using the various formulations of a complex number as
x[n] = αnu[n] =(rejθ)nu[n] = rnejnθu[n] = rn[cos(nθ) + j sin(nθ)]u[n] (3.19) This complex-valued signal requires two separate plots (the real and imaginary parts, for example) for a graphical description. If 0 < r < 1, x[n] describes a signal whose real and imaginary parts are exponentially decaying cosines and sines. If r = 1, the real and imaginary parts are pure cosines and sines with a peak value of unity. If r > 1, we obtain exponentially growing sinusoids.
3.5
Discrete-Time Harmonics and Sinusoids
If we sample an analog sinusoid x(t) = cos(2πf0t) at intervals of ts corresponding to a sampling rate of S = 1/ts samples/s (or S Hz), we obtain the sampled sinusoid
x[n] = cos(2πf nts+ θ) = cos(2πnSf + θ) = cos(2πnF + θ) (3.20) The quantities f and ω = 2πf describe analog frequencies. The normalized frequency F = f/S is called the digital frequency and has units of cycles/sample. The frequency Ω = 2πF is the digital radian frequency with units of radians/sample. The various analog and digital frequencies are compared in Figure 3.1. Note that the analog frequency f = S (or ω = 2πS) corresponds to the digital frequency F = 1 (or Ω = 2π).
REVIEW PANEL 3.13
The Digital Frequency Is the Analog Frequency Normalized by Sampling Rate S F (cycles/sample) = f (cycles/sec)
S(samples/sec) Ω(radians/sample) =
ω(radians/sec) S(samples/sec) = 2πF
3.5.1
Discrete-Time Sinusoids Are Not Always Periodic in Time
An analog sinusoid x(t) = cos(2πft + θ) has two remarkable properties. It is unique for every frequency. And it is periodic in time for every choice of the frequency f. Its sampled version, however, is a beast of a different kind.
50 Chapter 3 Discrete Signals ω = 2π f Ω = 2π F S 0.5 S − −0.5S −2πS −πS πS 2πS F = f/S ( ) F f (Hz) −1 −0.5 0.5 1 −2π −π π 2π S 0 0 0 0 Analog frequency Digital frequency Analog frequency Digital frequency Connection between analog and digital frequencies
Figure 3.1 Comparison of analog and digital frequencies
Are all discrete-time sinusoids and harmonics periodic in time? Not always! To understand this idea, suppose x[n] is periodic with period N such that x[n] = x[n + N]. This leads to
cos(2πnF + θ) = cos[2π(n + N)F + θ] = cos(2πnF + θ + 2πNF ) (3.21) The two sides are equal provided NF equals an integer k. In other words, F must be a rational fraction (ratio of integers) of the form k/N. What we are really saying is that a DT sinusoid is not always periodic but only if its digital frequency is a ratio of integers or a rational fraction. The period N equals the denominator of k/N , provided common factors have been canceled from its numerator and denominator. The significance of k is that it takes k full periods of the analog sinusoid to yield one full period of the sampled sinusoid. The common period of a combination of periodic DT sinusoids equals the least common multiple (LCM) of their individual periods. If F is not a rational fraction, there is no periodicity, and the DT sinusoid is classified as nonperiodic or almost periodic. Examples of periodic and nonperiodic DT sinusoids appear in Figure 3.2. Even though a DT sinusoid may not always be periodic, it will always have a periodic envelope.
0 4 8 12 16 20 24 28 −1 −0.5 0 0.5 1 DT Index n Amplitude
(a) cos(0.125πn) is periodic. Period N=16 Envelope is periodic 0 4 8 12 16 20 24 28 −1 −0.5 0 0.5 1 DT Index n Amplitude
(b) cos(0.5n) is not periodic. Check peaks or zeros. Envelope
is periodic
Figure 3.2 Discrete-time sinusoids are not always periodic
REVIEW PANEL 3.14
A DT Harmonic cos(2πnF0+ θ) or ej(2πnF0+θ) Is Not Always Periodic in Time It is periodic only if its digital frequency F0= k/N can be expressed as a ratio of integers. Its period equals N if common factors have been canceled in k/N.
3.5 Discrete-Time Harmonics and Sinusoids 51
EXAMPLE 3.7 (Discrete-Time Harmonics and Periodicity)
(a) Is x[n] = cos(2πF n) periodic if F = 0.32? If F =√3? If periodic, what is its period N? If F = 0.32, x[n] is periodic because F = 0.32 = 32
100 =258 = Nk. The period is N = 25.
If F =√3, x[n] is not periodic because F is irrational and cannot be expressed as a ratio of integers. (b) What is the period of the harmonic signal x[n] = ej0.2nπ+ e−j0.3nπ?
The digital frequencies in x[n] are F1= 0.1 =101 = Nk11 and F2= 0.15 = 203 = Nk22. Their periods are N1= 10 and N2= 20.
The common period is thus N = LCM(N1, N2) = LCM(10, 20) = 20.
(c) The signal x(t) = 2 cos(40πt) + sin(60πt) is sampled at 75 Hz. What is the common period of the sampled signal x[n], and how many full periods of x(t) does it take to obtain one period of x[n]? The frequencies in x(t) are f1 = 20 Hz and f2 = 30 Hz. The digital frequencies of the individual components are F1= 2075 = 154 = Nk11 and F2= 3075 = 25 = Nk22. Their periods are N1= 15 and N2= 5. The common period is thus N = LCM(N1, N2) = LCM(15, 5) = 15.
The fundamental frequency of x(t) is f0= GCD(20, 30) = 10 Hz. One period of x(t) is T = f10 = 0.1 s. Since N = 15 corresponds to a duration of Nts= NS = 0.2 s, it takes two full periods of x(t) to obtain one period of x[n]. We also get the same result by computing GCD(k1, k2) = GCD(4, 2) = 2.
3.5.2
Discrete-Time Sinusoids Are Always Periodic in Frequency
Unlike analog sinusoids, discrete-time sinusoids and harmonics are always periodic in frequency. If we start with the sinusoid x[n] = cos(2πnF + θ) and add an integer m to F , we get
cos[2πn(F + m) + θ] = cos(2πnF + θ + 2πnm) = cos(2πnF + θ) = x[n]
This result says that discrete-time (DT) sinusoids at the frequencies F ± m are identical. Put another way, a DT sinusoid is periodic in frequency (has a periodic spectrum) with period F = 1. The range −0.5 ≤ F ≤ 0.5 defines the principal period or principal range. A DT sinusoid with a frequency outside this range can always be expressed as a DT sinusoid whose frequency is in the principal period by subtracting out an integer. A DT sinusoid can be uniquely identified only if its frequency falls in the principal period.
To summarize, a discrete-time sinusoid is periodic in time only if its digital frequency F is a rational fraction, but it is always periodic in frequency with period F = 1.
REVIEW PANEL 3.15
A DT Harmonic cos(2πnF0+ θ) Is Always Periodic in Frequency but Not Always Unique Its frequency period is unity (harmonics at F0and F0± M are identical for integer M).
It is unique only if F0lies in the principal period −0.5 < F0≤ 0.5.
52 Chapter 3 Discrete Signals