The Fourier Series for Continuous-Time Periodic Signals The Fourier Series of a periodic analogue signal x(t) is given by (1) Where F0 is the fundamental frequency of the signal, k = 0,1,…,N-1 and ck are called the Fourier Coefficients which may be calculated as (2) Where Tp = 1/F0 is the period of the signal.

All periodic signals that satisfy the Dirichlet conditions, can be represented in a Fourier Series Representation. In general, the Fourier Coefficients ck are complex valued. If the periodic signal is real, ck and c-k are complex conjugates. As a result, if

Other forms of Fourier Series Representation As we have just mentioned The above equation can be re-written as since and This is called the Cosine Fourier Series.

Other forms of Fourier Series Representation Yet another form for the Fourier Series can be obtained by expanding the cosine Fourier series as Consequently, we may rewrite the above equation in the form This is called the Trigonometric form of the FS, where a0 = ck, ak = 2|ck|cosk and bk = 2|ck|sink.

Dirichlet Conditions The Dirichlet conditions guarantee that the Fourier series will converge to x(t). These conditions are listed below: The signal x(t) has a finite number of discontinuities in any period. The signal x(t) contains a finite number of maxima and minima during any period. The signal x(t) is absolutely integrable in any period, i.e. All periodic signals of practical interest satisfy these conditions.

Power Density Spectrum of Periodic Signals A periodic signal has infinite energy and a finite average power, which is given as If we take the complex conjugate of (1) and substitute for x*(t), we obtain

Therefore, we have established the relation Which is called Parseval’s relation for power signals. This relation states that the total average power in the periodic signal is simply the sum of the average powers in all the harmonics. If we plot the |ck| as a function of the frequencies, the diagram we obtain shows how the power of the periodic signal is distributed among the various frequency components. This diagram is called the Power Density Spectrum of the periodic signal x(t). A typical PSD is shown in the next slide.

|ck|2 F -2F0 -F0 F0 2F0 Power density spectrum of a continuous time periodic signal

Example1: Determine the Fourier Series and the Power Density Spectrum of the rectangular pulse train signal illustrated in the following figure. x(t) A Tp -/2 /2 Tp Solution: and where k = 1, 2, ….. Figure (a), (b) and (c) illustrate the Fourier coefficients when Tp is fixed and the pulse width  is allowed to vary.

-60 -40 -20 20 40 60 -0.05 0.05 0.1 0.15 0.2  = 0.2Tp Fig.(a) -60 -40 -20 20 40 60 -0.04 -0.02 0.02 0.04 0.06 0.08 0.1  = 0.1Tp Fig. (b)

From these three figures we observe that the -60 -40 -20 20 40 60 -0.02 -0.01 0.01 0.02 0.03 0.04 0.05  = 0.05Tp Fig. (c) From these three figures we observe that the effect of decreasing  while keeping Tp fixed is to spread out the signal power over the frequency range. The Spacing between the adjacent lines is independent of the value of the width .

The following figures demonstrate the effect of varying Tp when  is fixed.

The figures on the previous slide (i.e. slide 13) show that the spacing between adjacent spectral lines decreases as Tp increases. In the limit as Tp  , the Fourier coefficients ck approach zero. This behaviour is consistant with the fact that as Tp   and  remains fixed, the resulting signal is no longer a power signal. Indeed it becomes an energy signal and its average power is zero. The Power Density Spectrum for the rectangular pulse train is