The Discrete Fourier Series

THE DISCRETE FOURIER SERIES 1. Lecture 8 - 43 minutes Convolution Proper-ly 1,(')~X (k) 1(n) "X,(k) Xz(k) 2-0 c. Dual Property X4(k)--L E FX,(f)Xk) 8 . x2(r) Jit Ti? ~tI 0 Illustration of the sequences involved in forming a periodic convolution. (N-1) m 0 X2 (2-m) t I~t,. It.,tiLLIti.. 0 ?II i m X1(m)~X2(2-m) O d. Correction: Note that the view-graph shown in the lecture illustrating periodic convolution does not correctly illustrate the sequence i1 (m) R2(2 - m). The copy above of this figure includes the correction 2. Comments In the past several lectures we considered the Fourier transform and z-transform as tools for the representation and analysis of discretetime signals and systems. In this lecture we introduce the Fourier series representation for periodic sequences. In the next lecture the Fourier series will be applied to developing a Fourier representation of finite length sequences, referred to as the Discrete Fourier Transform by utilizing the fact that finite length and periodic sequences are Because of the importance of this relationship in eventually interpreting properties of the Discrete Fourier Transform, this lecture begins with a discussion of the relationship betweeen closely related. finite length and periodic sequences. The remainder of the lecture concentrates on the Fourier series representation of periodic sequences. representing continuous-time signals through the Fourier series, the representation in general requires an infinite number of harmonically In contrast, in related complex exponentials (or sines and cosines). In the discrete-time case there are only a finite number (N) of distin­ guishable harmonically related complex exponentials with fundamental Thus there are only a finite number (N) of distinguishable Discrete Fourier Series coefficients. In developing the Fourier series representation it is convenient to interpret the Fourier series period N. coefficients as a periodic sequence with period N. 8.2 This interpretation is consistent with the property that there are only N distinguishable coefficients since the distinguishable values in a periodic sequence are represented by a single period. The lecture concludes with a discussion of some of the properties of Fourier series coefficients. A property which will play an important role in lecture 10 is the convolution property, and in particular the definition of periodic convolution. 3. Reading Text: 4. Sections 8.0 (page 514) through 8.3. Problems Problem 8.1 (a) The sequence x1 (n) of Figure P8.1-1 is periodic with period 4. X1 (k) Determine the Fourier series coefficients x1 (n) *0 1 '1 2 -l 0 e e e e 1 2 3 4 * * 5 Figure P8.1-1 Problem 8.2 In the lecture it was stated that for a real periodic sequence x(n) the Fourier series coefficients X(k) are conjugate symmetric, i.e. X(k) = X (-k). By using the Discrete Fourier Series analysis and synthesis pair, (eqs 8.11 and 8.12 in text) show that this symmetry property is true. Problem 8.3 Another important symmetry property of the Fourier series coefficients states that if x(n) is real and even, then X(k) is real and even. By using the Discrete Fourier Series analysis and synthesis pair show that the symmetry property is true. 8.3 Problem 8.4 In figure P8.4-1 is shown a real periodic signal x(n). Utilizing the properties discussed in section 8.2 and without explicitly evaluating the Fourier series coefficients, determine whether or not the following are true or not for the Fourier series coefficients X(k). (i) X(k) = X(k + 10) for all k (ii) X(k) = X(-k) for all k (iii) X(0) = 0 jk2'T ( (k) e v); is real for all k 5 x (n) Figure P8.4-1 Problem 8.5 In figure P8.5-1 are shown two periodic sequences both with period 6. Determine and sketch x3 (n) the result of a periodic convolution of these two sequences. x 1 (n) 900 000 -l 0 1 1 000 2 2 3 (n) 4 171 @000 p Figure P8.5-1 8.4 Problem 8.6 x(n) denotes a periodic sequence with period N and X(k) denotes its discrete Fourier series coefficients. periodic sequence with period N. The sequence X(k) is also a Determine, in terms of x(n), the discrete Fourier series coefficients of X(k). Problem 8.7* In Figure P8.7-1 are shown several periodic sequences x(n). sequences can be expressed in a Fourier Series as N-1 X(k) ej( 2 Tr/N) kn x(n) = These k=0 ?? (a) (b) (c) TT~T n n n Figure P8.7-1 For which sequences can the time origin be chosen such that all the X(k) are real? (a) For which sequences can the time origin be chosen such that all the X(k) (except X(0)) are imaginary? (b) Problem 8.8 If x(n) is a periodic sequence with period N, it is also periodic with period 2N. Let XW(k) denote the DFS coefficients of x(n) considered as a periodic sequence with period N and X2 (k) denote the DFS coefficients of x(n) considered as a periodic sequence with period 2N. X (k) is of course, periodic with period N and X 2 (k) is periodic with period 2N. Determine X2 (k) in terms of X1 (k). 8.5