Fourier Transform for Periodic Signals
This is a continuation from the previous tutorial - representation of aperiodic signals with continuous-time Fourier transform.
In the previous tutorial, we introduced the Fourier transform representation and gave several examples.
While our attention in that tutorial was focused on aperiodic signals, we can also develop Fourier transform representations for periodic signals, thus allowing us to consider both periodic and aperiodic signals within a unified context.
In fact, as we will see, we can construct the Fourier transform of a periodic signal directly from its Fourier series representation. The resulting transform consists of a train of impulses in the frequency domain, with the areas of the impulses proportional to the Fourier series coefficients. This will tum out to be a very useful representation.
To suggest the general result, let us consider a signal \(x(t)\) with Fourier transform \(X(j\omega)\) that is a single impulse of area \(2\pi\) at \(\omega=\omega_0\); that is,
\[\tag{4.21}X(j\omega)=2\pi\delta(\omega-\omega_0)\]
To determine the signal \(x(t)\) for which this is the Fourier transform, we can apply the inverse transform relation, eq. (4.8) [refer to the representation of aperiodic signals with continuous-time Fourier transform tutorial], to obtain
\[\begin{align}x(t)&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}2\pi\delta(\omega-\omega_0)e^{j\omega{t}}\text{d}\omega\\&=e^{j\omega_0t}\end{align}\]
More generally, if \(X(j\omega)\) is of the form of a linear combination of impulses equally spaced in frequency, that is,
\[\tag{4.22}X(j\omega)=\sum_{k=-\infty}^{+\infty}2\pi{a_k}\delta(\omega-k\omega_0)\]
then the application of eq. (4.8) [refer to the representation of aperiodic signals with continuous-time Fourier transform tutorial] yields
\[\tag{4.23}x(t)=\sum_{k=-\infty}^{+\infty}a_ke^{jk\omega_0t}\]
We see that eq. (4.23) corresponds exactly to the Fourier series representation of a periodic signal, as specified by eq. (3.38) [refer to the Fourier series representation of continuous-time periodic signals tutorial].
Thus, the Fourier transform of a periodic signal with Fourier series coefficients \(\{a_k\}\) can be interpreted as a train of impulses occurring at the harmonically related frequencies and for which the area of the impulse at the \(k\)th harmonic frequency \(k\omega_0\) is \(2\pi\) times the \(k\)th Fourier series coefficient \(a_k\).
Example 4.6
Consider again the square wave illustrated in Figure 4.1 [refer to the representation of aperiodic signals with continuous-time Fourier transform tutorial]. The Fourier series coefficients for this signal are
\[a_k=\frac{\sin{k\omega_0T_1}}{\pi{k}}\]
and the Fourier transform of the signal is
\[X(j\omega)=\sum_{k=-\infty}^{+\infty}\frac{2\sin{k\omega_0T_1}}{k}\delta(\omega-k\omega_0)\]
which is sketched in Figure 4.12 for \(T=4T_1\).
In comparison with Figure 3.7(a) [refer to the Fourier series representation of continuous-time periodic signals tutorial], the only differences are a proportionality factor of \(2\pi\) and the use of impulses rather than a bar graph.
Example 4.7
Let
\[x(t)=\sin\omega_0t\]
The Fourier series coefficients for this signal are
\[\begin{align}a_1&=\frac{1}{2j}\\a_{-1}&=-\frac{1}{2j}\\a_k=0, &\qquad{k}\ne1\text{ or }-1\end{align}\]
Thus, the Fourier transform is as shown in Figure 4.13(a). Similarly, for
\[x(t)=\cos\omega_0t\]
the Fourier series coefficients are
\[\begin{align}a_1&=a_{-1}=\frac{1}{2}\\a_k=0,&\qquad{k}\ne1\text{ or }-1\end{align}\]
The Fourier transform of this signal is depicted in Figure 4.13(b).
These two transforms will be of considerable importance when we analyze sinusoidal modulation systems in later tutorials.
Example 4.8
A signal that we will find extremely useful in our analysis of sampling systems in later tutorials is the impulse train
\[x(t)=\sum_{k=-\infty}^{+\infty}\delta(t-kT)\]
which is periodic with period \(T\), as indicated in Figure 4.14(a).
The Fourier series coefficients for this signal were computed in Example 3.8 [refer to the properties of continuous-time Fourier series tutorial] and are given by
\[a_k=\frac{1}{T}\int_{-T/2}^{+T/2}\delta(t)e^{-jk\omega_0t}\text{d}t=\frac{1}{T}\]
That is, every Fourier coefficient of the periodic impulse train has the same value, \(1/T\). Substituting this value for \(a_k\) in eq. (4.22) yields
\[X(j\omega)=\frac{2\pi}{T}\sum_{k=-\infty}^{+\infty}\delta\left(\omega-\frac{2\pi{k}}{T}\right)\]
Thus, the Fourier transform of a periodic impulse train in the time domain with period \(T\) is a periodic impulse train in the frequency domain with period \(2\pi/T\), as sketched in Figure 4.14(b).
Here again, we see an illustration of the inverse relationship between the time and the frequency domains. As the spacing between the impulses in the time domain (i.e., the period) gets longer, the spacing between the impulses in the frequency domain (namely, the fundamental frequency) gets smaller.
The next tutorial discusses about linear lightwave propagation in an optical fiber.