Convergence of Fourier series It is known that a periodic signal x(t) has a Fourier series representation if it satisfies the following Dirichlet conditions: x(t) is absolutely integrable over any period, that is, x(t) has a finite number of maxima and minima within any finite interval of t. x(t) has a finite number of discontinuities within any finite interval of t, and each of these discontinuities is finite. 1Signals and systems analysis د. عامر الخيري

Convergence of Fourier series Dirichlet conditions are sufficient but not necessary. we can have examples of Fourier series for functions that violate some of the Dirichlet conditions. 2Signals and systems analysis د. عامر الخيري

Fourier series properties Linearity: Let F denote the transformation from f(t) to the Fourier coefficients. F is a linear transformation. If F f(t) = c n and F g(t) = d n. Then and 3Signals and systems analysis د. عامر الخيري

Fourier series properties Time Shifting: Shifting in time equals a phase shift of Fourier coefficients. Remark: The magnitudes of the Fourier series coefficients are not changed, only their phases. 4Signals and systems analysis د. عامر الخيري

Fourier series properties Frequency shifting: The spectrum of a signal, x(t), can be shifted by multiplying the signal by a complex exponential,, where k 0 is an integer and ω 0 is the fundamental frequency. 5Signals and systems analysis د. عامر الخيري

Fourier series properties Time scaling: Time scaling applied on a periodic signal changes the fundamental frequency of the signal. For example, f(  t) has fundamental frequency  0 and fundamental period  –1 T. The Fourier series coefficients do not change: 6Signals and systems analysis د. عامر الخيري

Fourier series properties Time reversal: Time reversal leads to a “sequence reversal” of the corresponding sequence of Fourier series coefficients: Interesting consequences are that For f(t) even, the sequence of coefficients is also even (c -n = c n ). For f(t) odd, the sequence of coefficients is also odd (c -n = -c n ). 7Signals and systems analysis د. عامر الخيري

Fourier series properties Multiplication of two signals: Suppose that x(t) and y(t) are both periodic with period T. 8Signals and systems analysis د. عامر الخيري

Fourier series properties Differentiation in Fourier domain: The derivative of is. Therefore, if the FS spectrum for a time-domain function f(t) is c n, then the FS spectrum for its derivative is jnω 0 c n, where ω 0 is the fundamental frequency. Note that the FS coefficient with k = 0 is zero, as the dc component is lost in differentiating a signal. In general, 9Signals and systems analysis د. عامر الخيري

Fourier series properties Integration in Fourier domain: 10Signals and systems analysis د. عامر الخيري

Fourier series properties Conjugation & conjugate symmetry: Taking the conjugate of a periodic signal has the effect of conjugation and index reversal on the spectral coefficients: Interesting consequences are that: 11Signals and systems analysis د. عامر الخيري

Fourier series properties For x(t) real, and the sequence of coefficients is conjugate symmetric; that is, This implies (magnitude is even), (phase is odd). 12Signals and systems analysis د. عامر الخيري

Fourier series properties For x(t) real and even, the sequence of coefficients is also real and even ( ). For x(t) real and odd, the sequence of coefficients is imaginary and odd ( purely imaginary ). 13Signals and systems analysis د. عامر الخيري

Gibbs phenomenon An interesting observation can be made when one looks at the graph of a truncated Fourier series of a square wave. The graph over one period looks like: 14Signals and systems analysis د. عامر الخيري

Gibbs phenomenon We can see that there are ripples of a certain amplitude in the approximation, especially close to the discontinuities in the signal. The surprising thing is that the peak amplitude of these ripples does not diminish when we add more terms in the truncated Fourier series. For example, for N=19 in the Figure shown in the next slide; the approximation gets closer to a square wave, but we can still see rather large, but narrow, ripples around the discontinuities. This is called the Gibbs phenomenon 15Signals and systems analysis د. عامر الخيري

Gibbs phenomenon 16Signals and systems analysis د. عامر الخيري