1. 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.

2. 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

3. 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.

4. 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.

5. 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.

6. 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

7. 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.

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

9. 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.

10. -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)

11. 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 .

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

13. 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