2.4 Complex form of the fourier series

The trigonometric form of the Fourier Series, shown in [link] can be converted into a more convenient form by doing the following substitutions:

After some straight-forward rearranging, we obtain

Keeping in mind that $a_n$ and $b_n$ are only defined for positive values of $n$ , lets sum over the negative integers in the second summation:

Next, let's assume that $a_n$ and $b_n$ are defined for both positive and negative $n$ . In this case, we find that $a_n=a_{-n}$ and $b_n=-b_{-n}$ , since

and

Using this fact, we can rewrite [link] as

If we define The notation “ $\equiv$ " is often used instead of the “ $=$ " sign when defining a new variable. $c_0\equiv a_0$ , and

then we can rewrite [link] as

which is called the complex form of the Fourier Series. Note that since $a_{-n}=a_n$ and $b_{-n}=-b_n$ , we have

This means that

and

Next, we must find formulas for finding the $c_n$ given $x\left(t\right)$ . We first look at a property of complex exponentials:

To see this, we note that

It's easy to see that $k{\Omega }_0$ also has period $T$ , hence the integral is over $k$ periods of $cos\left(k{\Omega }_{0}t\right)$ and $sin\left(k{\Omega }_{0}t\right)$ . Therefore, if $k\ne 0$ , then

otherwise

We use [link] to derive an equation for $c_n$ as follows. Consider the integral

Substituting the complex form of the Fourier Series of $x\left(t\right)$ in [link] , (using $k$ as the index of summation) we obtain

Rearranging the order of integration and summation, combining the exponents, and using [link] gives

Using this result, we find that

Example 2.1 Let's now find the complex form of the Fourier Series for the signal in Example [link] . The integral to be evaluated is

Integrating by parts yields

[link] shows the magnitude of the coefficients, $\left|c_n\right|$ . Note that the complex Fourier Series coefficients have even symmetry as was mentioned earlier.

Can the basic formula for computing the $c_n$ in [link] be simplified when $x\left(t\right)$ has either even, odd, or half-wave symmetry? The answer is yes. We simply use the fact that

and solve for $a_n$ and $b_n$ using the formulae given above for even, odd, or half-wave symmetric signals. This avoids having to integrate complex quantities. This can also be seen by noting that (setting $t_0=T/2$ in [link] ):

Alternately, if $x\left(t\right)$ has half-wave symmetry, we can use [link] , [link] , and [link] to get

Unlike the trigonometric form, we cannot simplify this further if $x\left(t\right)$ is even or odd symmetric since $e^{-jn{\Omega }_{0}t}$ has neither even nor odd symmetry.

Example 2.2 In this example we will look at the effect of adjusting the period of a pulse train signal. Consider the signal depicted in [link] .

The Fourier Series coefficients for this signal are given by

[link] shows the magnitude of $|c_n|$ , the amplitude spectrum , for $T=1$ and $\tau =1/2$ as well as the Fourier Series for the signal based on the first 30 coefficients

Similar plots are shown in Figures [link] , and [link] , for $T=4$ , and $T=8$ , respectively.

This example illustrates several important points about the Fourier Series: As the period $T$ increases, ${\Omega }_0$ decreases in magnitude (this is obvious since ${\Omega }_0=2\pi /T$ ). Therefore, as the period increases, successive Fourier Series coefficients represent more closely spaced frequencies. The frequencies corresponding to each $n$ are given by the following table:

This table establishes a relation between $n$ and the frequency variable $\Omega$ . In particular, if $T=1$ , we have ${\Omega }_0=2\pi$ and

If $T=T$ , then ${\Omega }_0=\pi /2$ and

and if $T=8$ , we have ${\Omega }_0=\pi /4$ and

Note that in all three cases, the first zero coefficient corresponds to the value of $n$ for which $\Omega =4\pi$ . Also, as $T$ gets bigger, the $c_n$ appear to resemble more closely spaced samples of a continuous function of frequency (since the $n\Omega$ are more closely spaced). Can you determine what this function is?

As we shall see, by letting the period $T$ get large (infinitely large), we will derive the Fourier Transform in the next chapter.

References