# Find the compact trigonometric Fourier series for the periodic signal.pdf

A

Find the compact trigonometric Fourier series for the periodic signal x(t) and sketch the amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra, sketch the exponential Fourier spectra. By inspection of spectra in part b), write the exponential Fourier series for x(t) Solution ECE 3640 Lecture 4 – Fourier series: expansions of periodic functions. Objective: To build upon the ideas from the previous lecture to learn about Fourier series, which are series representations of periodic functions. Periodic signals and representations From the last lecture we learned how functions can be represented as a series of other functions: f(t) = Xn k=1 ckik(t). We discussed how certain classes of things can be built using certain kinds of basis functions. In this lecture we will consider specifically functions that are periodic, and basic functions which are trigonometric. Then the series is said to be a Fourier series. A signal f(t) is said to be periodic with period T0 if f(t) = f(t + T0) for all t. Diagram on board. Note that this must be an everlasting signal. Also note that, if we know one period of the signal we can find the rest of it by periodic extension. The integral over a single period of the function is denoted by Z T0 f(t)dt. When integrating over one period of a periodic function, it does not matter when we start. Usually it is convenient to start at the beginning of a period. The building block functions that can be used to build up periodic functions are themselves periodic: we will use the set of sinusoids. If the period of f(t) is T0, let 0 = 2/T0. The frequency 0 is said to be the fundamental frequency; the fundamental frequency is related to the period of the function. Furthermore, let F0 = 1/T0. We will represent the function f(t) using the set of sinusoids i0(t) = cos(0t) = 1 i1(t) = cos(0t + 1) i2(t) = cos(20t + 2) . . . Then, f(t) = C0 + X n=1 Cn cos(n0t + n) The frequency n0 is said to be the nth harmonic of 0. Note that for each basis function associated with f(t) there are actually two parameters: the amplitude Cn and the phase n. It will often turn out to be more useful to represent the function using both sines and cosines. Note that we can write Cn cos(n0t + n) = Cn cos(n) cos(n0t) Cn sin(n)sin(n0t). ECE 3640: Lecture 4 – Fourier series: expansions of periodic functions. 2 Now let an = Cn cos n bn = Cn sin n Then Cn cos(n0t + n) = an cos(n0t) + bn sin(n0t) Then the series representation can be f(t) = C0 + X n=1 Cn cos(n0t + n) = a0 + X n=1 an cos(n0t) + bn sin(n0t) The first of these is the compact trigonometric Fourier series. The second is the trigonometric Fourier series.. To go from one to the other use C0 = a0 Cn = p a 2 n + b 2 n n = tan1 (bn/an). To complete the representation we must be able to compute the coefficients. But this is the same sort of thing we did before. If we can show that the set of functions {cos(n0t),sin(n0t)} is in fact an orthogonal set, then we can use the same.

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