DISCRETE-TIME FOURIER SERIES (DTFS)

9.1 DISCRETE-TIME FOURIER SERIES (DTFS)

A continuous-time sinusoid cos ωt is a periodic signal regardless of the value of ω. Such is not the case for the discrete-time sinusoid cos Ωn (or exponential e j Ω n ). A sinusoid cos Ωn is periodic only if Ω/2π is a rational number. This can be proved by observing that if this sinusoid is N 0 periodic, then

This is possible only if Here, both m and N 0 are integers. Hence, Ω/2π = m/N 0 is a rational number. Thus, a sinusoid cos Ωn (or exponential e j Ω n ) is periodic

only if

When this condition ( Ω/2π a rational number) is satisfied, the period N 0 of the sinusoid cos Ωn is given by [ Eq. (9.1a) ]

To compute N 0 , we must choose the smallest value of m that will make m(2 π/Ω) an integer. For example, if Ω = 4π/17, then the smallest value of m that will make m(2 π/Ω) = m (17/2) an integer is 2. Therefore

However, a sinusoid cos(0.8n) is not a periodic signal because 0.8/2 π is not a rational number.

9.1-1 Periodic Signal Representation by Discrete-time Fourier Series

A continuous-time periodic signal of period T 0 can be represented as a trigonometric Fourier series consisting of a sinusoid of the fundamental frequency ω 0 =2 π/T 0 , and all its harmonics. The exponential form of the Fourier series consists of exponentials e j0t ,

e ±j ω0t ,e ±j2 ω0t ,e ±j3?0t ....

A discrete-time periodic signal can be represented by a discrete-time Fourier series using a parallel development. Recall that a periodic

signal x[n] with period N 0 is characterized by the fact that

The smallest value of N 0 for which this equation holds is the fundamental period. The fundamental frequency is Ω 0 ... 2 π/N 0 rad/sample. An N 0 -periodic signal x[n] can be represented by a discrete-time Fourier series made up of sinusoids of fundamental frequency Ω 0 2 π/N 0 and its harmonics. As in the continuous-time case, we may use a trigonometric or an exponential form of the Fourier series. Because of its compactness and ease of mathematical manipulations, the exponential form is preferable to the trigonometric. For this reason we shall bypass the trigonometric form and go directly to the exponential form of the discrete-time Fourier series.

The exponential Fourier series consists of the exponentials e j0n ,e ±j Ω0n ,e ±j2 Ω0n , ..., e ±jnO0n , ..., and so on. There would be an infinite number of harmonics, except for the property proved in Section 5.5-1 , that discrete-time exponentials whose frequencies are separated by 2 π (or integer multiples of 2π) are identical because

The consequence of this result is that the rth harmonic is identical to the (r + N 0 )th harmonic. To demonstrate this, let g n denote the nth harmonic e jn ω n0n . Then The consequence of this result is that the rth harmonic is identical to the (r + N 0 )th harmonic. To demonstrate this, let g n denote the nth harmonic e jn ω n0n . Then

Thus, the first harmonic is identical to the (N 0 + 1)st harmonic, the second harmonic is identical to the (N 0 + 2)nd harmonic, and so on. In other words, there are only N 0 independent harmonics, and their frequencies range over an interval 2 π (because the harmonics are separated by Ω 0 =2 π/N 0 ). This means, unlike the continuous-time counterpart, that the discrete-time Fourier series has only a finite number (N 0 ) of terms. This result is consistent with our observation in Section 5.5-1 that all discrete-time signals are bandlimited to a band from −π to π. Because the harmonics are separated by Ω 0 =2 π/N 0 , there can only be N 0 harmonics in this band. We also saw that this band can be taken from 0 to 2 π or any other contiguous band of width 2π. This means we may choose the N 0 independent harmonics e jr Ω0n over 0 ≤r≤N 0 − 1, or over −1 ≤ r ≤ N 0 − 2, or over 1 ≤ r ≤ N 0 , or over any other suitable choice for that matter. Every one of these sets will have the same harmonics, although in different order.

Let us consider the first choice, which corresponds to exponentials e jrO0n for r = 0,1,2,..., N 0 − 1. The Fourier series for an N 0 -periodic

signal x[n] consists of only these N 0 harmonics, and can be expressed as

To compute coefficients in the Fourier series (9.4) , we multiply both sides of (9.4) by e −jm Ω0n and sum over n from n = 0 to (N 0 − 1).

The right-hand sum, after interchanging the order of summation, results in

The inner sum, according to Eq. (8.28) in Section 8.5 , is zero for all values of r ≠ m. It is nonzero with a value N 0 only when r = m. This fact means the outside sum has only one term D m N 0 (corresponding to r = m). Therefore, the right-hand side of Eq. (9.5) is equal to D m N 0 , and

and

We now have a discrete-time Fourier series (DTFS) representation of an N 0 -periodic signal x[n] as

where

Observe that DTFS equations (9.8) and (9.9) are identical (within a scaling constant) to the DFT equations (8.22b) and (8.22a) . [ † ] Therefore, we can use the efficient FFT algorithm to compute the DTFS coefficients.

9.1-2 Fourier Spectra of a Periodic Signal x[n]

The Fourier series consists of N 0 components

The frequencies of these components are 0, Ω 0 , 2O 0 , ..., (N 0 − 1)Ω 0 where Ω 0 =2 π/N 0 . The amount of the rth harmonic is . We can plot this amount (the Fourier coefficient) as a function of index r or frequency Ω. Such a plot, called the Fourier spectrum of

x[n], gives us, at a glance, the graphical picture of the amounts of various harmonics of x[n]. In general, the Fourier coefficients are complex, and they can be represented in the polar form as

The plot of | | versus Ω is called the amplitude spectrum and that of ∟ versus Ω is called the angle (or phase) spectrum. These two plots together are the frequency spectra of x[n]. Knowing these spectra, we can reconstruct or synthesize x[n] according to Eq. (9.8) . Therefore, the Fourier (or frequency) spectra, which are an alternative way of describing a periodic signal x[n], are in every way equivalent (in terms of the information) to the plot of x[n] as a function of n. The Fourier spectra of a signal constitute the frequency- domain description of x[n], in contrast to the time-domain description, where x[n] is specified as a function of index n (representing time).

The results are very similar to the representation of a continuous-time periodic signal by an exponential Fourier series except that, generally, the continuous-time signal spectrum bandwidth is infinite and consists of an infinite number of exponential components

(harmonics). The spectrum of the discrete-time periodic signal, in contrast, is bandlimited and has at most N 0 components.