FourierSeriesSummary(FromSalivahananetal,2002)A.ppt

Fourier Series Summary (From Salivahanan et al, 2002) A periodic continuous signal f(t), -?? t ?, with the fundamental frequency w0 (i.e. f(t) = f(t + 2?/w0)) can be represented as a linear combination of sinusoidal functions with periods mw0, where m?N+. represented by sine and cosine: represented by complex exponential: (1) ;am and bm can be resolved as represented by complex exponential: (2) cm and c-m are complex conjugate pairs;The integral can also be performed within [0, 2?/w0] More concise relation between {am, bm} and cm: In (1) and (2), f(t) and cm (-?? t ?, m?Z) form a transform pair, where (2) is referred to as a forward transform, and (1) is an inverse transform. Fourier series specifies Fourier transform in situation of periodic signals.; Frequency domain: we say that the periodic signal f(t) is transformed to the frequency domain specified by mw0 (m?Z) by Fourier series. cm is the frequency component of f(t) with respect to the frequency mw0. The frequency domain (or spectrum) or a periodic continuous signal is discrete. In contrast, the domain which the signal is defined is referred to as the “time domain” or “space domain.” The frequency components cm is a complex number. Hence, the transform domain can be divided into two parts: magnitude and phase.; Example: f(t) 5 …… 0 2? 4? 6? 8? …… w0t;when m=0, c-m is cm’s complex conjugate, so magnitude in the transform domain (in this case, the magnitude is an even function that is symmetric to the y-axis); phase in the transform domain: (it is an odd function) The frequency domain of a Fourier transform contains both the magnitude and phase spectra. High frequency (w=mw0 is large): corresponds to