离散时间信号处理(英文第三版)第八章Discrete-Time Signal Proce

离散时间信号处理(英文第三版)第八章,作者: oppenheim,奥本海姆,pdf文档,文字可复制,Discrete-Time Signal Processing,3rd edition,chapter8 The Discrete Fourier Transform _3rd_edition

8

The Discrete Fourier Transform

8.0 INTRODUCTION

In Chapters 2 and 3, we discussed the representation of sequences and LTI systems in terms of the discre te-time Fourier and z-transforms, respectively. For finite-duration se­ quences, there is an alternative discrete-time Fourier represen tation, referred to as the discrete Fourier transform (DFT). The DFT is itself a sequence rather than a function of a continuous variable, and it corresponds to samples, equally spaced in frequency, of the DTFT of the signal. In addition to its theoretical importance as a Fourier repre­ sentation of sequences, the DFT plays a central role in the implementation of a variety of digital signal-processing algorithms. This is because efficient algorithms exist for the computation of the DFT. These algorithms will be discussed in detail in Chapter 9. The application of the DFT to spectrum analysis will be described in Chapter 10. Although several points of view can be taken toward the derivation and inter­ pretation of the DFT representation of a finite-duration sequence, we have chosen to base our presentation on the relationship between periodic sequences and finite-length sequences. We begin by considering the Fourier series representation of periodic se­ quences. A lthough this representation is important in its own right, we are most often interested in the application of Fourier series results to the representation of finite­ length sequences. We acco mplish this by constructing a periodic sequence for which each period is identical to the finite-length sequence. The Fourier series representation of the periodic sequence then corresponds to the DFT of the finite-length sequence. Thus, our approach is to define the Fourier series representation for periodic sequences and to study the properties of such representations. Then, we repeat essentially the same derivati ons, assuming that the sequence to be represented is a finite-length sequence.623

离散时间信号处理(英文第三版)第八章,作者: oppenheim,奥本海姆,pdf文档,文字可复制,Discrete-Time Signal Processing,3rd edition,chapter8 The Discrete Fourier Transform _3rd_edition

624

Chapter 8

The Discrete Fourier Transform

This approach to the DFT emphasizes the fundamental inherent periodicity of the DFT representation and ensures that this periodicity is not overlooked in applications of the DFT.

8.1REPRESENTATION OF PERIODIC SEQUENCES: THE DISCRETE FOURIER SERIESConsider a sequence x[nJ that is periodic1 with period N, so that i[n] x[n+ rN] for any integer values of nand r. As with continuous-time periodic signals, such a sequence can be represented by a Fourier series corresponding to a sum of harmonically related complex exponential sequences, i.e., complex exponentials with frequencies that are integer multiples of the fundamental frequency (2rr/ N) associated with the periodic sequence x[n]. These periodic complex exponentials are of the formek[n]= eJ (2rr/ N)kn= ek[n

+ r N],

(8.1)

where k is any integer, and the Fourier series representation then has the form 2x[n]

=~LX[k]eJ(2rr/N)kn.k

(8.2)

The Fourier series representation of a continuous-time period

ic signal gener­ ally requires infinitely many harmonically related complex exponentials, whereas the Fourier series for any discrete-time signal with period N requires only N harmoni­ cally related complex exponentials. To see this, note that the harmonically related com­ plex exponentials ek[n] in Eq. (8.1) are identical for values of k separated by N; i.e., eo[n]= eNln], edn]= eN+dn], and, in general,ekHN[n]= eJ(2rr:/N)(k+ eN )n eJ(2rr/N)kll e j21r:en e j (2rr/N)kn ek[n],

(8.3)

where f is any integer. Consequently, the set of N periodic complex exponentials eo[n], el[n], ..., eN-l[n] defines all the distinct periodic complex exponentials with frequen­ cies that are integer multiples of (2rr/ N). Thus, the Fourier series representation of a periodic sequence x[n] need contain only N of these complex exponentials. For nota­ 1; hence, Eq. (8.2) has the tional convenience, we choose k in the range of 0 to N formx[n]

=~

N-I

L X[k]e J(2rr j N)kn.k=O

(8.4)

However, choosing k to range over any full period of X[k] would be equally valid. To obtain the sequence of Fourier series coefficients X[k] from the periodic se­ quence x[n], we exploit the orthogonality of the set of complex exponential sequences.1 Henceforth. we will use the tilde C) to denote periodic sequences whenever it is important to clearly distinguish between periodic and aperiodic sequences. 2The multiplicative constant 1/ N is included in Eq. (8.2) for convenience. It could also be absorbed into the definition of X[k J.

离散时间信号处理(英文第三版)第八章,作者: oppenheim,奥本海姆,pdf文档,文字可复制,Discrete-Time Signal Processing,3rd edition,chapter8 The Discrete Fourier Transform _3rd_edition

Section 8.1

Representation of Periodic Sequences: The Discrete Fourier Series

625n

n

After mUltiplying both sides of Eq. (8.4) by e- J(2n/N)rn and summing from= N -1, we obtain N-l N-l 1 N-l i[n]e- J (2n/N)rn= N[k]e J (2n/N Hk-r)n.

0 to(8.5)

L

L LXk=O

n=O

n=O

After interchanging the order of summation on the right-hand side, Eq. (8.5) becomes

The following identity expresses the orthogonality of the complex exponentials: 1 Ne j (2n/N)(k-r)n

I: I:n=O n=O

i[n]e- j (2n/Nlrn

=

I:k=O

X[k]

[~r

I:n=O

e J(2n I N)(k-r)n] .

(8.6)

={I,

k

= mN,

m an integer,

(8.7)

0,

otherwise.

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