discrete fourier series pdf

The discrete Fourier series (DFS): For infinitely long but periodic signals ⇒ basis for the discrete Fourier transform The discrete Fourier transform (DFT): For general finite length signals ⇒ Used in practice with signals from experiments Underlying these three concepts is the decomposition of signals into

1 Fourier series (review/summary) We consider functions in L2[0; 2 ] (with weight w(x) = 1) which have a Fourier series = X ckeikx; k=1 Z 2 1 ikx ck = f(x)e dx: 2 0 The basis functions 2 k = eikx are orthogonal in the inner product hf; gi = R f(x)g(x) dx: 0

The Discrete Fourier Fourier analysis is a family of mathematical sinusoids The discrete Fourier transform digitized signals This is the real first DFT a ofversion four chapters of the transform that uses real numbers c to mplex represent a more advanced technique that uses complex chapter we look at the mathematics and DFT

The DFS is an alternative representation of a periodic sequence x with period N Such a sequence can be represented as a sum of N complex exponentials with frequencies k 2 where = 0; 1; : : : ; N 1: 1 1 x[n] X = X[k]ejk2 n N N k=0 (5 1) for all times n where X[k] 2 C is the DFS coe cient corresponding to the complex expo-nential sequence fejk2

One reason the spectrum is useful is that any periodic signal can be written as a sum of cosines Fourier\'s theorem says that any x(t) that is periodic i e x(t + T0) = x(t) can be written as x(t) = Xkej2 kF0t k=1 which is a special case of the spectrum for periodic signals: fk = kF0 and ak = Xk and

7 1 The DFT The Discrete Fourier Transform (DFT) is the equivalent of the continuous Fourier Transform for signals known only at instants separated by sample times (i e a finite sequence of data) Let be the continuous signal which is the source of the data Let samples be denoted The Fourier Transform of the original signal would be