Signals and Systems

5.2 The Fourier Transform for Periodic Signals

Imron Rosyadi

DTFT of a Complex Exponential

Consider the fundamental periodic signal:

\[ x[n]=e^{j \omega_{0} n} \quad \text{(Eq. 5.17)} \]

• In continuous time, \(e^{j\omega_0 t}\) has a Fourier transform that is an impulse at \(\omega = \omega_0\).
• For discrete-time, the DTFT must be periodic with period \(2\pi\).
• This implies impulses at \(\omega_0, \omega_0 \pm 2\pi, \omega_0 \pm 4\pi\), and so on.

The DTFT of \(x[n]=e^{j \omega_{0} n}\) is an impulse train:

\[ X\left(e^{j \omega}\right)=\sum_{l=-\infty}^{+\infty} 2 \pi \delta\left(\omega-\omega_{0}-2 \pi l\right) \quad \text{(Eq. 5.18)} \]

Figure 5.8: Fourier transform of \(x[n]=e^{j \omega_{0} n}\).

We start with the simplest periodic discrete-time signal: a complex exponential \(e^{j\omega_0 n}\). Recall that in continuous time, the Fourier transform of such a signal is a single impulse at \(\omega_0\). However, discrete-time Fourier transforms are inherently periodic in frequency with a period of \(2\pi\). Therefore, for discrete-time, this single impulse “repeats” every \(2\pi\). This means the DTFT of \(e^{j\omega_0 n}\) is an infinite train of impulses, spaced \(2\pi\) apart, with each impulse having an area of \(2\pi\). This is mathematically represented by Equation 5.18 and visually depicted in Figure 5.8.

Verifying the Inverse DTFT

To confirm the validity of Eq. 5.18, we apply the inverse DTFT (synthesis equation):

\[ \frac{1}{2 \pi} \int_{2 \pi} X\left(e^{j \omega}\right) e^{j \omega n} d \omega=\frac{1}{2 \pi} \int_{2 \pi} \sum_{l=-\infty}^{+\infty} 2 \pi \delta\left(\omega-\omega_{0}-2 \pi l\right) e^{j \omega n} d \omega \]

• The integral is taken over any interval of length \(2\pi\).
• Within any \(2\pi\) interval, there is exactly one impulse from the sum.
• Let’s assume the interval includes the impulse at \(\omega_0 + 2\pi r\) for some integer \(r\).

Using the sifting property of the impulse function:

\[ \frac{1}{2 \pi} \int_{2 \pi} X\left(e^{j \omega}\right) e^{j \omega n} d \omega=e^{j\left(\omega_{0}+2 \pi r\right) n} \]

Since \(e^{j 2\pi r n} = (e^{j 2\pi})^{rn} = 1^{rn} = 1\) for integer \(n\) and \(r\):

\[ e^{j\left(\omega_{0}+2 \pi r\right) n} = e^{j\omega_0 n} e^{j 2\pi r n} = e^{j\omega_0 n} \cdot 1 = e^{j\omega_0 n} \]

This correctly returns our original signal \(x[n]=e^{j \omega_{0} n}\).

To build confidence in this representation, we can perform the inverse DTFT. The key here is that the integration in the synthesis equation is over a finite \(2\pi\) interval. Because the impulse train is periodic with \(2\pi\), any \(2\pi\) interval will contain exactly one impulse. When we apply the sifting property of the impulse, the integral picks out the value of \(e^{j\omega n}\) at the location of that impulse. Due to the periodicity of discrete-time complex exponentials, \(e^{j 2\pi r n}\) simplifies to 1 for any integer \(n\) and \(r\). Thus, the inverse transform correctly yields the original complex exponential, confirming our DTFT representation.

DTFT of a General Periodic Signal

A general periodic sequence \(x[n]\) with period \(N\) can be represented by its Discrete-Time Fourier Series (DTFS):

\[ x[n]=\sum_{k=\langle N\rangle} a_{k} e^{j k(2 \pi / N) n} \quad \text{(Eq. 5.19)} \]

Since the DTFT is a linear transform, we can apply the result for a single complex exponential to each term in the sum:

\[ X\left(e^{j \omega}\right)=\sum_{k=-\infty}^{+\infty} 2 \pi a_{k} \delta\left(\omega-\frac{2 \pi k}{N}\right) \quad \text{(Eq. 5.20)} \]

• The DTFT of a periodic signal is an impulse train.
• The impulses are located at multiples of the fundamental frequency \(2\pi/N\).
• The area (strength) of each impulse is \(2\pi\) times the corresponding DTFS coefficient \(a_k\).

Now, let’s generalize this to any periodic discrete-time signal. We know that any periodic signal can be expressed as a sum of harmonically related complex exponentials using its DTFS, as shown in Equation 5.19. Since the DTFT is a linear operation, the DTFT of a sum is the sum of the DTFTs. Therefore, the DTFT of a general periodic signal will be a sum of impulse trains, where each train corresponds to one of the complex exponential components in the Fourier series. This leads to Equation 5.20, which states that the DTFT of a periodic signal is an impulse train located at the harmonics of the fundamental frequency, with each impulse scaled by \(2\pi\) times its corresponding Fourier series coefficient \(a_k\).

Visualizing the General Periodic DTFT

Let \(x[n]\) be a periodic signal with DTFS coefficients \(a_k\). Its DTFT \(X(e^{j\omega})\) is formed by summing the transforms of its individual Fourier series components.

Consider \(x[n] = a_{0}+a_{1} e^{j(2 \pi / N) n} + \ldots + a_{N-1} e^{j(N-1)(2 \pi / N) n}\).

(a) Transform of \(a_0\) (DC component)

(b) Transform of \(a_1 e^{j (2\pi/N) n}\)

(c) Transform of \(a_{N-1} e^{j (N-1)(2\pi/N) n}\)

(d) Combined DTFT \(X(e^{j\omega})\)

These figures illustrate how the DTFT of a periodic signal is constructed. Each term in the Fourier series, like \(a_0\) (a DC component), or \(a_1 e^{j (2\pi/N) n}\), contributes its own impulse train to the total DTFT. For instance, \(a_0\) corresponds to impulses at \(\omega = 0, \pm 2\pi, \ldots\), scaled by \(2\pi a_0\). Similarly, \(a_1 e^{j (2\pi/N) n}\) contributes impulses at \(\omega = 2\pi/N, 2\pi/N \pm 2\pi, \ldots\), scaled by \(2\pi a_1\). When all these individual impulse trains are summed up, we get the complete DTFT of the periodic signal, which is itself an impulse train with specific locations and scaled amplitudes, as shown in the final figure. The periodicity of the \(a_k\) coefficients ensures that the combined impulse train is also periodic.

Example 5.5: Cosine Signal

Consider the periodic signal:

\[ x[n]=\cos \omega_{0} n=\frac{1}{2} e^{j \omega_{0} n}+\frac{1}{2} e^{-j \omega_{0} n}, \quad \text { with } \quad \omega_{0}=\frac{2 \pi}{5} \quad \text{(Eq. 5.22)} \]

Using the DTFT of complex exponentials (Eq. 5.18):

\[ X\left(e^{j \omega}\right)=\sum_{l=-\infty}^{+\infty} \pi \delta\left(\omega-\omega_{0}-2 \pi l\right)+\sum_{l=-\infty}^{+\infty} \pi \delta\left(\omega+\omega_{0}-2 \pi l\right) \quad \text{(Eq. 5.23)} \]

For the principal interval \(-\pi \leq \omega < \pi\):

\[ X\left(e^{j \omega}\right)=\pi \delta\left(\omega-\omega_{0}\right)+\pi \delta\left(\omega+\omega_{0}\right) \quad \text{(Eq. 5.24)} \]

And \(X(e^{j\omega})\) repeats periodically with a period of \(2\pi\).

Figure 5.10: Discrete-time Fourier transform of \(x[n]=\cos \omega_{0} n\).

Let’s apply this to a concrete example: a discrete-time cosine signal. We can express \(\cos(\omega_0 n)\) as a sum of two complex exponentials. Each exponential has a coefficient of \(1/2\). Applying the DTFT rule for complex exponentials, we get two impulse trains. The first train is centered at \(\omega_0\) and its \(2\pi\) multiples, scaled by \(\pi\). The second train is centered at \(-\omega_0\) and its \(2\pi\) multiples, also scaled by \(\pi\). When we focus on a single \(2\pi\) interval, say from \(-\pi\) to \(\pi\), we typically see just two impulses at \(\omega_0\) and \(-\omega_0\), each with an area of \(\pi\). This pattern then repeats across the entire frequency axis, as shown in Figure 5.10.

Interactive Plot: Cosine DTFT Spectrum

Explore the DTFT of a cosine signal \(x[n] = \cos(\omega_0 n)\) for various fundamental frequencies.

viewof omega0_cos = Inputs.range([0.1, 3.14 - 0.1], {value: 3.14/4, step: 0.05, label: "Fundamental Frequency $\omega_0$ (rad)"});

Here, you can interactively change the fundamental frequency \(\omega_0\) of the cosine signal. Observe how the positions of the impulses in the frequency domain change accordingly. Each impulse represents a specific frequency component present in the cosine signal. Notice that as \(\omega_0\) approaches \(\pi\), the impulses at \(\omega_0\) and \(-\omega_0\) move towards the high-frequency end of the baseband \([-\pi, \pi]\), and they also merge with the impulses from the adjacent periodic copies. This interactive visualization helps solidify the concept of impulse trains representing periodic signals and the periodicity of the DTFT.

Example 5.6: Periodic Impulse Train

Consider the discrete-time periodic impulse train:

\[ x[n]=\sum_{k=-\infty}^{+\infty} \delta[n-k N] \quad \text{(Eq. 5.25)} \]

Figure 5.11(a): Discrete-time periodic impulse train.

First, find its Fourier series coefficients \(a_k\):

\[ a_{k}=\frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-j k(2 \pi / N) n} \]

Example 5.6: Periodic Impulse Train

Choosing the interval \(0 \leq n \leq N-1\), only \(\delta[0]\) is non-zero, so:

\[ a_{k}=\frac{1}{N} \cdot 1 \cdot e^{-j k(2 \pi / N) \cdot 0} = \frac{1}{N} \quad \text{(Eq. 5.26)} \]

Now, use Eq. 5.20 to find the DTFT:

\[ X\left(e^{j \omega}\right)=\frac{2 \pi}{N} \sum_{k=-\infty}^{+\infty} \delta\left(\omega-\frac{2 \pi k}{N}\right) \quad \text{(Eq. 5.27)} \]

Figure 5.11(b): Its Fourier transform.

Our final example for periodic signals is the discrete-time periodic impulse train. This signal consists of impulses repeating every \(N\) samples. To find its DTFT, we first calculate its Fourier series coefficients. Due to the nature of the impulse train, within any period of \(N\) samples, only one impulse is present (e.g., at \(n=0\) if we choose the period \(0\) to \(N-1\)). This simplifies the calculation significantly, yielding \(a_k = 1/N\) for all \(k\). Plugging these coefficients into Equation 5.20, we find that the DTFT of a periodic impulse train in time is also a periodic impulse train in frequency, with impulses located at integer multiples of \(2\pi/N\) and scaled by \(2\pi/N\). This demonstrates a beautiful duality: an impulse train in one domain transforms to an impulse train in the other.

Summary & ECE Applications

Key Takeaways for Periodic Signals:

• The DTFT of a periodic signal is an impulse train in the frequency domain.
• Each impulse corresponds to a harmonic component of the signal.
• The locations of impulses are at multiples of the fundamental frequency (\(k \cdot 2\pi/N\)).
• The strength (area) of each impulse is \(2\pi\) times its corresponding Discrete-Time Fourier Series (DTFS) coefficient (\(2\pi a_k\)).
• This directly links the DTFS and DTFT for periodic signals.

Summary & ECE Applications

Applications in ECE:

• Digital Communications: Understanding carrier signals, modulation schemes (e.g., QAM, PSK), and their spectral properties.
• Sampling Theory: The DTFT of a sampled continuous-time signal (which is periodic in its spectrum) explains aliasing and reconstruction.
• Filter Design: Analyzing how digital filters respond to periodic inputs, crucial for designing frequency-selective filters.
• Power Spectral Density (PSD): While the DTFT of a periodic signal has infinite energy, the concept of impulses relates to power distribution in frequency.

To recap this section, the key concept is that periodic discrete-time signals have discrete, line spectra in the frequency domain, represented by impulse trains. Each impulse directly corresponds to a harmonic component of the signal, and its strength is proportional to the Fourier series coefficient. This relationship is fundamental and bridges the concepts of Fourier series and Fourier transforms. In ECE, this understanding is vital for many areas. For instance, in digital communications, understanding the spectrum of periodic carrier signals helps in designing efficient modulation and demodulation schemes. In sampling theory, the periodic nature of the sampled signal’s spectrum is key to avoiding aliasing. And in filter design, knowing how a filter interacts with specific frequency components of a periodic input is paramount.