2.1 Trigonometric form of the fourier series

A major goal of this book is to develop tools which will enable us to study the frequency content of signals. An important first step is the Fourier Series . The Fourier Series enables us to completely characterize the frequency content of a periodic signal There are periodic signals for which a Fourier series doesn't exist, conditions for existence of the Fourier series are given below. . A periodic signal $x\left(t\right)$ can be expressed in terms of the Fourier Series, which is given by:

where

is the fundamental frequency of the periodic signal. Examination of [link] suggests that periodic signals can be represented as a sum of suitably scaled cosine and sine waveforms at frequencies of ${\Omega }_0,2{\Omega }_0,3{\Omega }_0,...$ . The cosine and sine terms at frequency $n{\Omega }_0$ are called the $n^{th}$ harmonics . Evidently, periodic signals contain only the fundamental frequency and its harmonics. A periodic signal cannot contain a frequency that is not an integer multiple of its fundamental frequency.

In order to find the Fourier Series, we must compute the Fourier Series coefficients . These are given by

From our discussion of even and odd symmetric signals, it is clear that if $x\left(t\right)$ is even, then $x\left(t\right)sin\left(n{\Omega }_{0}t\right)$ must be odd and so $b_n=0$ . Also if, $x\left(t\right)$ has odd symmetry, then $x\left(t\right)cos\left(n{\Omega }_{0}t\right)$ also has odd symmetry and hence $a_n=0$ (see exercise [link] ). Moreover, if a signal is even, since $x\left(t\right)cos\left(n{\Omega }_{0}t\right)$ is also even, if we use the fact that for any even symmetric periodic signal $v\left(t\right)$ ,

then setting $t_0=-T/2$ in [link] gives,

This can sometimes lead to a savings in the number of integrals that must be computed. Similarly, if $x\left(t\right)$ has odd symmetry, we have

Example 2.1 Consider the signal in [link] . This signal has even symmetry, hence all of the $b_n=0$ . We compute $a_0$ using,

which we recognize as the area of one period, divided by the period. Hence, $a_0=\tau /T$ . Next, using [link] we get

Note how the limits of integration only go from $-\tau /2$ to $\tau /2$ since $x\left(t\right)$ is zero everywhere else. Evaluating this integral leads to

[link] shows the first few Fourier Series coefficients for $\tau =1/2$ and $T=1$ . If we attempt to reconstruct $x\left(t\right)$ based on only a limited number (say, $N$ ) of Fourier Series coefficients, we have

Figures [link] and [link] show $\widehat{x}\left(t\right)$ for $N=10$ , and $N=50$ , respectively. The ringing characteristic is known as Gibb's phenomenon and disappears only as $N$ approaches $\infty$ .

The following example looks at the Fourier series of an odd-symmetric signal, a sawtooth signal.

Example 2.2 Now let's compute the Fourier series for the signal in [link] . The signal is odd-symmetric, so all of the $a_n$ are zero. The period is $T=3/2$ , hence ${\Omega }_0=4\pi /3$ . Using [link] , the $b_n$ coefficients are found by computing the following integral,

After integrating by parts, we get

These are plotted in [link] and approximations of $x\left(t\right)$ using $N=10$ and $N=50$ coefficients are shown in Figures [link] and [link] , respectively.

References