The Fourier Transform

4.2 The Fourier Transform for Periodic Signals

Imron Rosyadi

The Fourier Transform

4.2 The Fourier Transform for Periodic Signals

Welcome to the fascinating world of Signals and Systems!

Today, we’ll extend our understanding of the Fourier Transform (FT) to periodic signals.

This allows us to analyze both periodic and aperiodic signals within a unified framework.

We’ll discover how the Fourier Transform of a periodic signal reveals itself as a train of impulses in the frequency domain.

These impulses are directly related to the signal’s Fourier Series coefficients.

Unifying Periodic and Aperiodic Signals

Traditionally, Fourier Series describes periodic signals, while Fourier Transform handles aperiodic signals. Today, we bridge this gap.

Key Idea

The Fourier Transform of a periodic signal is a train of impulses in the frequency domain. The areas of these impulses are proportional to the signal’s Fourier Series coefficients.

This approach provides a powerful, unified tool for signal analysis across ECE disciplines.

Explain that prior to this, students learned Fourier Series for periodic signals and Fourier Transform for aperiodic signals. Emphasize that this section unites these concepts, providing a more general framework. Highlight the conceptual leap: continuous spectrum for aperiodic signals vs. discrete spectrum (impulses) for periodic signals.

Building Intuition: A Single Impulse in Frequency

Let’s start with a simple Fourier Transform: a single impulse in the frequency domain. \[ X(j \omega)=2 \pi \delta\left(\omega-\omega_{0}\right) \] What time-domain signal \(x(t)\) does this represent? We use the inverse Fourier Transform: \[ x(t) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} X(j \omega) e^{j \omega t} d \omega \] Substituting \(X(j\omega)\): \[ x(t) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} 2 \pi \delta\left(\omega-\omega_{0}\right) e^{j \omega t} d \omega \] \[ x(t) = e^{j \omega_{0} t} \] This shows that a single impulse in the frequency domain corresponds to a complex exponential in the time domain.

Walk through the derivation step-by-step. Explain the role of the \(2\pi\) factor in the definition of the impulse. Emphasize that the sifting property of the Dirac delta function is crucial here. This is the foundational block for understanding the FT of periodic signals.

Generalizing to a Train of Impulses

If we have a linear combination of impulses in the frequency domain: \[ X(j \omega)=\sum_{k=-\infty}^{+\infty} 2 \pi a_{k} \delta\left(\omega-k \omega_{0}\right) \] Applying the inverse Fourier Transform, as before: \[ x(t) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} \left( \sum_{k=-\infty}^{+\infty} 2 \pi a_{k} \delta\left(\omega-k \omega_{0}\right) \right) e^{j \omega t} d \omega \] \[ x(t) = \sum_{k=-\infty}^{+\infty} a_{k} \left( \frac{1}{2 \pi} \int_{-\infty}^{+\infty} 2 \pi \delta\left(\omega-k \omega_{0}\right) e^{j \omega t} d \omega \right) \] \[ x(t) = \sum_{k=-\infty}^{+\infty} a_{k} e^{j k \omega_{0} t} \]

The Core Relationship

This is precisely the Fourier Series representation of a periodic signal! The Fourier Transform of a periodic signal with Fourier Series coefficients \(\{a_k\}\) is a train of impulses. The impulse at the \(k\)-th harmonic frequency \(k\omega_0\) has an area of \(2\pi a_k\). \[ \mathcal{F} \left\{ \sum_{k=-\infty}^{+\infty} a_{k} e^{j k \omega_{0} t} \right\} = \sum_{k=-\infty}^{+\infty} 2 \pi a_{k} \delta\left(\omega-k \omega_{0}\right) \]

Highlight the direct connection between the Fourier Series coefficients (\(a_k\)) and the impulse areas in the Fourier Transform. This is the main theoretical result of this section. Explain that the factor \(2\pi\) arises from the definition of the inverse Fourier Transform and the Fourier Series.

Example 4.6: The Periodic Square Wave

Consider a symmetric periodic square wave.

Its Fourier Series coefficients are given by: \[ a_{k}=\frac{\sin k \omega_{0} T_{1}}{\pi k} \] Therefore, its Fourier Transform is: \[ X(j \omega)=\sum_{k=-\infty}^{+\infty} \frac{2 \sin k \omega_{0} T_{1}}{k} \delta\left(\omega-k \omega_{0}\right) \]

Example 4.6: The Periodic Square Wave (cont.)

The transform consists of impulses located at integer multiples of the fundamental frequency \(\omega_0\).

The amplitude of each impulse is proportional to the corresponding Fourier Series coefficient.

Notice the sinc function envelope, characteristic of rectangular pulses.

This illustrates how the discrete spectrum of a periodic signal maps to impulses in the frequency domain.

Figure 4.12: Fourier transform of a symmetric periodic square wave (\(T=4T_1\)).

Point out the sinc function envelope and its relation to the rectangular pulse in the time domain. Explain that \(T_1\) is the pulse width and \(T\) is the period. Discuss how the spacing of the impulses is determined by \(\omega_0 = 2\pi/T\). Compare this to the bar graph of Fourier Series coefficients, noting the \(2\pi\) scaling and the use of impulses.

Interactive Demo: Square Wave Spectrum

Use the sliders to change the square wave’s properties:

• Period (T): How does it affect the spacing of the impulses?
• Pulse Width (T1): How does it affect the envelope of the spectrum?

viewof T_val = Inputs.range([1, 10], {label: "Period (T)", step: 0.1, value: 4})
viewof T1_val = Inputs.range([0.1, T_val/2 - 0.1], {label: "Pulse Width (T1)", step: 0.05, value: 1})

Guide students to observe:

1. Effect of T: As T increases, \(\omega_0\) decreases, and the impulses get closer together.
2. Effect of T1: As T1 changes, the sinc envelope stretches or compresses, affecting the relative magnitudes of the impulses. When \(T_1\) approaches \(T/2\), the square wave becomes a pulse train with 50% duty cycle, and even harmonics vanish. When \(T_1\) is very small, the spectrum broadens. This interactive element reinforces the inverse relationship between time and frequency domains.

Example 4.7: Sinusoidal Signals

The Fourier Transform provides a concise representation for simple sinusoids.

Sine Wave: \(x(t)=\sin \omega_{0} t\)

Recall its Fourier Series coefficients: \[ a_{1} = \frac{1}{2 j} \] \[ a_{-1} = -\frac{1}{2 j} \] \[ a_{k}=0, \quad k \neq 1 \text{ or } -1 \] The Fourier Transform consists of two impulses at \(\pm \omega_0\): \[ X(j \omega) = 2\pi \left( \frac{1}{2j} \delta(\omega - \omega_0) - \frac{1}{2j} \delta(\omega + \omega_0) \right) \] \[ X(j \omega) = \frac{\pi}{j} \delta(\omega - \omega_0) - \frac{\pi}{j} \delta(\omega + \omega_0) \] \[ X(j \omega) = -j\pi \delta(\omega - \omega_0) + j\pi \delta(\omega + \omega_0) \]

Cosine Wave: \(x(t)=\cos \omega_{0} t\)

Its Fourier Series coefficients are: \[ a_{1}=a_{-1}=\frac{1}{2} \] \[ a_{k}=0, \quad k \neq 1 \text{ or } -1 \] Its Fourier Transform also has two impulses at \(\pm \omega_0\): \[ X(j \omega) = 2\pi \left( \frac{1}{2} \delta(\omega - \omega_0) + \frac{1}{2} \delta(\omega + \omega_0) \right) \] \[ X(j \omega) = \pi \delta(\omega - \omega_0) + \pi \delta(\omega + \omega_0) \]

Emphasize the difference in phase (imaginary vs. real impulses) between sine and cosine, corresponding to their odd and even symmetry. Discuss the importance of these basic transforms as building blocks for more complex signals, especially in modulation systems.

Visualizing Sinusoidal Spectra

These transforms are fundamental in understanding modulation systems.

They clearly show that sinusoids are “pure” frequencies, represented by single points in the spectrum.

The impulses indicate the presence of specific frequency components.

Figure 4.13: Fourier transforms of (a) \(\sin \omega_{0} t\) and (b) \(\cos \omega_{0} t\).

Directly relate the visual representation to the derived impulse equations. Explain why sine has impulses pointing in opposite directions (due to imaginary coefficients) and cosine has impulses pointing in the same direction (due to real coefficients), although typically only magnitude spectra are plotted.

Interactive Demo: Sine and Cosine Spectra

Use the slider to change \(\omega_0\):

• Fundamental Frequency (\(\omega_0\)):

How does this affect the position of the impulses in the spectrum? Note that for real signals, if there is a positive frequency component, there must be a corresponding negative frequency component.

viewof omega0_val = Inputs.range([1, 10], {label: "ω₀ (rad/s)", step: 0.5, value: 2})

Point out that changing \(\omega_0\) simply shifts the impulses along the frequency axis. Reinforce that the magnitude of the impulses remains constant for a pure sinusoid. Discuss the physical meaning: a sinusoid contains energy only at its specific frequency.

Example 4.8: The Impulse Train

A signal of immense importance in sampling systems is the impulse train. \[ x(t)=\sum_{k=-\infty}^{+\infty} \delta(t-k T) \] This signal is periodic with period \(T\).

Its Fourier Series coefficients were previously found to be: \[ a_{k}=\frac{1}{T} \] This means every harmonic component has the same amplitude! Substituting this into our Fourier Transform definition for periodic signals: \[ X(j \omega)=\sum_{k=-\infty}^{+\infty} 2 \pi \left(\frac{1}{T}\right) \delta\left(\omega-k \omega_{0}\right) \] \[ X(j \omega)=\frac{2 \pi}{T} \sum_{k=-\infty}^{+\infty} \delta\left(\omega-\frac{2 \pi k}{T}\right) \] The Fourier Transform of an impulse train in the time domain is also an impulse train in the frequency domain!

(a) Periodic impulse train in time domain. (b) Its Fourier transform in frequency domain.

Inverse Relationship

As the spacing between impulses in the time domain (\(T\)) gets longer, the spacing between impulses in the frequency domain (\(2\pi/T\)) gets smaller. This is a profound illustration of the inverse relationship between time and frequency domains.

Highlight that this is a unique signal whose FT has the same form as itself. Emphasize its critical role in sampling theory (Chapter 7), where it models the sampling process. Discuss the inverse relationship: a widely spaced time-domain signal has a narrowly spaced frequency-domain signal, and vice-versa.

Interactive Demo: Impulse Train Spectrum

Use the slider to change the period T:

• Period (T):

See how changing the time-domain period directly impacts the frequency-domain spacing.

This phenomenon is central to understanding aliasing and the Nyquist-Shannon sampling theorem.

viewof T_impulse_val = Inputs.range([0.5, 5], {label: "Period (T) of Time-Domain Impulse Train", step: 0.1, value: 1})

Encourage students to actively manipulate the T slider and voice their observations. This direct visual feedback is crucial for grasping the inverse relationship. Connect this to the concept of sampling rate: a faster sampling rate (smaller T) means a wider frequency spectrum (larger \(\omega_F\)), which is good for avoiding aliasing.

Summary and Key Takeaways

We’ve explored the powerful connection between Fourier Series and Fourier Transform for periodic signals.

• Unified View: Both periodic and aperiodic signals can be analyzed using the Fourier Transform.
• Impulse Representation: The Fourier Transform of a periodic signal is a train of impulses in the frequency domain.
• Coefficient Link: The area of each impulse is \(2\pi\) times the corresponding Fourier Series coefficient \(a_k\).
• Inverse Relationship: A wider spacing in one domain implies a narrower spacing in the other (e.g., impulse train).

Why is this important?

This unified framework is foundational for:

• Sampling Theory: Understanding how continuous signals are converted to discrete samples.
• Modulation: Analyzing how information is carried on carrier waves in communication systems.
• Filter Design: Designing systems that selectively process frequency components.

Summarize the main points concisely. Reiterate the practical applications listed in the callout, emphasizing that these theoretical tools are directly applied in real-world ECE systems. Encourage questions and further discussion.