Continuous-Time Signals and Systems

Fourier Transform of Aperiodic Signals: Development and Examples

Imron Rosyadi

Fourier Transform of Aperiodic Signals: Development and Examples

Representing Aperiodic Signals: The Continuous-Time Fourier Transform

Objective:

To extend the concept of Fourier series, which is for periodic signals, to represent aperiodic signals in the frequency domain. This is crucial for analyzing a wide range of real-world signals.

4.1.1 Development of the Fourier Transform Representation of an Aperiodic Signal

We begin by considering a continuous-time periodic square wave. Its definition over one period is:

\[ x(t)= \begin{cases}1, & |t|<T_{1} \\ 0, & T_{1}<|t|<T / 2\end{cases} \]

This signal periodically repeats with period \(T\).

The Fourier series coefficients \(a_{k}\) for this square wave are given by:

\[ a_{k}=\frac{2 \sin \left(k \omega_{0} T_{1}\right)}{k \omega_{0} T} \quad \text{where } \omega_{0}=\frac{2 \pi}{T} \]

Recall that the Fourier series allows us to represent any periodic signal as a sum of complex exponentials. Each exponential has a specific frequency, which is a multiple of the fundamental frequency, and a corresponding complex amplitude, the Fourier series coefficient. For a square wave, these coefficients describe the amplitude and phase of each harmonic component.

\(a_k\) as Samples of an Envelope Function

The Fourier series coefficients \(a_k\) can be viewed as samples of a continuous envelope function.

Specifically, \(T a_{k}\) samples the function:

\[ T a_{k}=\left.\frac{2 \sin \omega T_{1}}{\omega}\right|_{\omega=k \omega_{0}} \]

As \(T\) increases (or \(\omega_0\) decreases), the samples become more closely spaced.

This continuous function, \(\frac{2 \sin \omega T_{1}}{\omega}\), represents the envelope of \(T a_k\) and is independent of \(T\).

Let’s visualize this as \(T\) changes.

Observe how the discrete spectral lines become denser as the period \(T\) increases.

Tip

This concept is key to bridging the gap between Fourier series (discrete spectrum) and Fourier transform (continuous spectrum).

The core idea here is to observe what happens to the Fourier series coefficients as the period of the signal becomes very large. As T increases, the fundamental frequency omega_0 decreases, meaning the harmonic frequencies k*omega_0 get closer together. The coefficients themselves, when multiplied by T, start to look like samples of an underlying continuous function. This continuous function is what we will define as the Fourier Transform.

Interactive Demo: Fourier Series Coefficients as Samples

Visualize the discrete Fourier series coefficients approaching a continuous envelope as the period \(T\) increases.

viewof T_val = Inputs.range([1, 20], {label: "Period (T)", step: 1, value: 4})
viewof T1_val = Inputs.range([0.1, 2], {label: "Pulse Width (T1)", step: 0.1, value: 0.5})

Here, students can manipulate T and T1 to see the effect. As T increases, omega0 decreases, and the red bars (representing T*a_k) get closer together, visually approximating the continuous blue curve (the envelope). This demonstrates how a discrete spectrum transitions to a continuous one. Observe that the shape of the envelope remains the same, only the sampling density changes.

Aperiodic Signal as a Limiting Case

We can think of an aperiodic signal \(x(t)\) as the limit of a periodic signal \(\tilde{x}(t)\) as its period \(T\) approaches infinity.

Original Aperiodic Signal \(x(t)\)

• Finite duration.
• \(x(t) = 0\) for \(|t| > T_1\).

Constructed Periodic Signal \(\tilde{x}(t)\)

• \(\tilde{x}(t)\) is formed by repeating \(x(t)\) with period \(T\).
• As \(T \to \infty\), \(\tilde{x}(t)\) becomes identical to \(x(t)\) over any finite interval.

graph LR
  A["Aperiodic Signal $$x(t)$$"] --> B{"Construct Periodic $$\tilde{x}(t)$$"};
  B --> C{"Increase Period $$T$$"};
  C --> D{"As $$T \to \infty$$"};
  D --> E["$$\tilde{x}(t) \to x(t)$$"];
  E --> F["F. Series becomes F. Transform of $$x(t)$$"];

This conceptual step is fundamental. We are essentially taking a finite segment of an aperiodic signal, making it periodic, and then letting the period grow infinitely large. As the period grows, the “copies” of the original signal in the periodic version move infinitely far apart, so for any finite observation window, the periodic signal looks exactly like the original aperiodic signal.

Deriving the Fourier Transform: Coefficients

Let’s examine the Fourier series representation of \(\tilde{x}(t)\):

\[ \tilde{x}(t)=\sum_{k=-\infty}^{+\infty} a_{k} e^{j k \omega_{0} t} \]

The Fourier coefficients \(a_k\) are:

\[ a_{k}=\frac{1}{T} \int_{-T / 2}^{T / 2} \tilde{x}(t) e^{-j k \omega_{0} t} d t \]

Since \(\tilde{x}(t)=x(t)\) for \(|t|<T/2\) and \(x(t)=0\) outside this interval (assuming \(T_1 < T/2\)), we can rewrite \(a_k\) as:

\[ a_{k}=\frac{1}{T} \int_{-T / 2}^{T / 2} x(t) e^{-j k \omega_{0} t} d t=\frac{1}{T} \int_{-\infty}^{+\infty} x(t) e^{-j k \omega_{0} t} d t \]

We start with the familiar Fourier series equations. The key simplification happens when we realize that the periodic signal \(\tilde{x}(t)\) is equal to the aperiodic signal \(x(t)\) within one period, and \(x(t)\) is zero outside a certain range. This allows us to change the integration limits from one period to infinity, effectively focusing on the aperiodic signal itself.

Defining the Fourier Transform \(X(j\omega)\)

From the previous slide, we have:

\[ a_k = \frac{1}{T} \int_{-\infty}^{+\infty} x(t) e^{-j k \omega_0 t} dt \]

We define the Fourier Transform \(X(j\omega)\) as the integral:

\[ X(j \omega)=\int_{-\infty}^{+\infty} x(t) e^{-j \omega t} d t \quad \text{(Analysis Equation)} \]

Using this definition, the Fourier coefficients \(a_k\) can be expressed as samples of \(X(j\omega)\):

\[ a_{k}=\frac{1}{T} X\left(j k \omega_{0}\right) \]

This is the formal definition of the Continuous-Time Fourier Transform (CTFT). It takes a time-domain signal \(x(t)\) and transforms it into a frequency-domain representation \(X(j\omega)\). Notice the similarity to the Fourier series coefficient formula, but now it’s an integral over continuous frequency \(\omega\).

Deriving the Fourier Transform: Synthesis Equation

Substitute \(a_k = \frac{1}{T} X(j k \omega_0)\) back into the Fourier series of \(\tilde{x}(t)\):

\[ \tilde{x}(t)=\sum_{k=-\infty}^{+\infty} \frac{1}{T} X\left(j k \omega_{0}\right) e^{j k \omega_{0} t} \]

Since \(\omega_0 = 2\pi/T\), we can write \(\frac{1}{T} = \frac{\omega_0}{2\pi}\). So,

\[ \tilde{x}(t)=\frac{1}{2 \pi} \sum_{k=-\infty}^{+\infty} X\left(j k \omega_{0}\right) e^{j k \omega_{0} t} \omega_{0} \]

We’re taking the expression for the Fourier series of the periodic signal \(\tilde{x}(t)\) and substituting our newly defined \(X(j\omega)\) into it. This sets the stage for the crucial limiting step.

The Limit: From Sum to Integral

As \(T \to \infty\), we have:

• \(\tilde{x}(t) \to x(t)\)
• \(\omega_0 \to 0\)
• The summation becomes an integral.

The summation: \[ \tilde{x}(t)=\frac{1}{2 \pi} \sum_{k=-\infty}^{+\infty} X\left(j k \omega_{0}\right) e^{j k \omega_{0} t} \omega_{0} \] Can be graphically interpreted as a Riemann sum. Each term is the area of a rectangle of height \(X(j k \omega_0)e^{j k \omega_0 t}\) and width \(\omega_0\).

The Limit: From Sum to Integral (cont.)

The summation resembles a Riemann sum in integral calculus:

• A Riemann sum approximates an integral by summing rectangles under a curve.
• Each term in the Fourier series can be seen as a rectangle:
  • Height: \(X(jk\omega_0) e^{jk\omega_0 t}\) — the value of the function (complex amplitude) at frequency \(k\omega_0\)
  • Width: \(\omega_0\) — the spacing between frequency samples

So the entire sum approximates the inverse Fourier transform integral:

\[ x(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} X(j\omega) e^{j\omega t} d\omega \]

The discrete version (your summation) is a Riemann sum approximation of this integral, where the continuous spectrum \(X(j\omega)\) is sampled at intervals of \(\omega_0\).

graph LR
    A["Summation $$\sum_{k=-\infty}^{+\infty} f(k \omega_0) \omega_0$$"] --> B{"As $$\omega_0 \to 0$$"};
    B --> C["Integral $$\int_{-\infty}^{+\infty} f(\omega) d\omega$$"];

As \(\omega_0 \to 0\), the sum converges to an integral.

This is the heart of the derivation. The discrete sum, with infinitesimally small frequency spacing \(\omega_0\), transforms into a continuous integral. This is analogous to how a Riemann sum approximates and then becomes a definite integral in calculus.

The Continuous-Time Fourier Transform Pair

Taking the limit, we arrive at the Fourier Transform Pair:

Synthesis Equation (Inverse Fourier Transform)

Reconstructs the time-domain signal \(x(t)\) from its frequency-domain representation \(X(j\omega)\):

\[ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} X(j \omega) e^{j \omega t} d \omega \]

Analysis Equation (Forward Fourier Transform)

Computes the frequency-domain representation \(X(j\omega)\) from the time-domain signal \(x(t)\):

\[ X(j \omega)=\int_{-\infty}^{+\infty} x(t) e^{-j \omega t} d t \]

Important

The function \(X(j\omega)\) is often referred to as the spectrum of \(x(t)\). It describes the signal’s content at different frequencies.

These two equations are the cornerstone of continuous-time Fourier analysis. The analysis equation tells us “what frequencies are present and how much of each,” while the synthesis equation tells us “how to put those frequencies back together to get the original signal.” Emphasize that \(X(j\omega)\) is a continuous function of frequency \(\omega\), unlike the discrete \(a_k\) coefficients for periodic signals.

Fourier Series Coefficients from Fourier Transform

We established a direct relationship between the Fourier series coefficients \(a_k\) of a periodic signal \(\tilde{x}(t)\) and the Fourier transform \(X(j\omega)\) of one period of that signal.

If \(\tilde{x}(t)\) is periodic with period \(T\) and Fourier coefficients \(a_k\), and \(x(t)\) is a finite-duration signal equal to \(\tilde{x}(t)\) over one period (and zero otherwise), then:

\[ a_{k}=\left.\frac{1}{T} X(j \omega)\right|_{\omega=k \omega_{0}} \]

where \(X(j\omega)\) is the Fourier transform of \(x(t)\).

Tip

This means the discrete spectrum of a periodic signal is proportional to samples of the continuous spectrum of a single period of that signal.

This equation is very useful in practice. If you know how to find the Fourier transform of a single pulse, you can immediately find the Fourier series coefficients for a periodic repetition of that pulse by simply sampling the Fourier transform at the harmonic frequencies and scaling by \(1/T\).

Convergence of Fourier Transforms

For the Fourier Transform to be a valid representation, certain conditions must be met.

Finite Energy Condition

If \(x(t)\) has finite energy (is square integrable):

\[ \int_{-\infty}^{+\infty}|x(t)|^{2} d t<\infty \]

Then \(X(j\omega)\) is finite, and the energy in the error between \(x(t)\) and its Fourier representation \(\hat{x}(t)\) is zero:

\[ \int_{-\infty}^{+\infty}|e(t)|^{2} d t=0 \]

Note

This means that even if \(x(t)\) and \(\hat{x}(t)\) differ at isolated points, their difference contains no energy.

These conditions are analogous to those for Fourier series. The finite energy condition is a very broad and useful criterion for convergence in an engineering context. It implies that the signal doesn’t “blow up” to infinity and has a well-behaved frequency spectrum.

Dirichlet Conditions for Pointwise Convergence

An alternative set of conditions, known as the Dirichlet conditions, ensures that \(\hat{x}(t)\) equals \(x(t)\) for all \(t\), except at discontinuities where it equals the average of the values on either side.

1. Absolutely Integrable: \[ \int_{-\infty}^{+\infty}|x(t)| d t<\infty \]
2. Finite Maxima and Minima: \(x(t)\) must have a finite number of maxima and minima within any finite interval.
3. Finite Discontinuities: \(x(t)\) must have a finite number of discontinuities within any finite interval, and each must be finite.

Tip

Most physically realizable signals satisfy these conditions. However, periodic signals are not absolutely integrable over an infinite interval, requiring a different approach (e.g., using impulse functions in the transform, which we’ll see later).

The Dirichlet conditions are stricter and guarantee pointwise convergence. They are easier to check for many common signals. It’s important to note that periodic signals, while having Fourier series, do not satisfy the absolute integrability condition over an infinite range, which is why their Fourier transforms require special handling (impulse functions).

Examples of Continuous-Time Fourier Transforms

Let’s explore some common signals and their Fourier Transforms. These examples build intuition about how signals in the time domain relate to their frequency domain representations.

Note

Pay attention to the shape of the time-domain signal and the corresponding shape of its spectrum. Look for symmetries and characteristics in both domains.

Example 4.1: Exponential Decay \(x(t) = e^{-at}u(t)\), \(a>0\)

Signal Definition

\[ x(t)=e^{-a t} u(t) \quad a>0 \]

Fourier Transform Derivation

\[ X(j \omega)=\int_{0}^{\infty} e^{-a t} e^{-j \omega t} d t = \frac{1}{a+j \omega} \]

Magnitude and Phase

\[ |X(j \omega)|=\frac{1}{\sqrt{a^{2}+\omega^{2}}}, \quad \angle X(j \omega)=-\tan ^{-1}\left(\frac{\omega}{a}\right) \]

This is a fundamental example. The signal starts at 1 and decays exponentially. It’s causal because of the unit step function \(u(t)\). The Fourier transform is complex-valued, so we typically look at its magnitude and phase.

Interactive Demo: Fourier Transform of \(e^{-at}u(t)\)

Explore how the parameter \(a\) affects both the time-domain signal and its frequency spectrum.

viewof a_val_exp = Inputs.range([0.1, 5], {label: "Decay Rate (a)", step: 0.1, value: 1})

As a increases, the time-domain signal decays faster (becomes narrower). In the frequency domain, the magnitude spectrum |X(jω)| becomes wider, indicating a broader distribution of frequencies. A faster decay in time means more high-frequency components are needed to represent the sharp transition.

Example 4.2: Double-Sided Exponential \(x(t) = e^{-a|t|}\), \(a>0\)

Signal Definition

\[ x(t)=e^{-a|t|}, \quad a>0 \]

This signal is symmetric around \(t=0\).

Fourier Transform Derivation

\[ \begin{aligned} X(j \omega) & =\int_{-\infty}^{+\infty} e^{-a|t|} e^{-j \omega t} d t \\ & = \int_{-\infty}^{0} e^{a t} e^{-j \omega t} d t + \int_{0}^{\infty} e^{-a t} e^{-j \omega t} d t \\ & = \frac{1}{a-j \omega} + \frac{1}{a+j \omega} \\ & = \frac{2a}{a^{2}+\omega^{2}} \end{aligned} \]

This signal is an even function of time. A key property of Fourier transforms is that if \(x(t)\) is real and even, then \(X(j\omega)\) is also real and even. Notice that the result here is purely real, unlike the previous example.

Interactive Demo: Fourier Transform of \(e^{-a|t|}\)

Observe the symmetry in both time and frequency domains for this real and even signal.

viewof a_val_abs = Inputs.range([0.1, 5], {label: "Decay Rate (a)", step: 0.1, value: 1})

Similar to the previous example, as a increases, the time-domain signal becomes narrower (more concentrated around \(t=0\)). Consequently, its frequency spectrum X(jω) becomes wider, indicating a broader range of frequencies. This inverse relationship between time-domain spread and frequency-domain spread is a fundamental concept.

Example 4.3: Unit Impulse \(x(t) = \delta(t)\)

Signal Definition

\[ x(t)=\delta(t) \]

The unit impulse is a signal that is zero everywhere except at \(t=0\), where its integral is one.

Fourier Transform Derivation

Using the sifting property of the impulse function:

\[ X(j \omega)=\int_{-\infty}^{+\infty} \delta(t) e^{-j \omega t} d t = e^{-j \omega (0)} = 1 \]

Note

The Fourier transform of a unit impulse is \(1\). This means the unit impulse contains equal contributions from all frequencies. It has a “flat” or “constant” spectrum.

This is a very important result. An impulse in time is perfectly localized in time, and to achieve this perfect localization, it must contain all possible frequencies with equal amplitude. This tells us that very short, sharp signals require a very wide range of frequencies to represent them.

Example 4.4: Rectangular Pulse

Signal Definition

\[ x(t)= \begin{cases}1, & |t|<T_{1} \\ 0, & |t|>T_{1}\end{cases} \]

This is a pulse of width \(2T_1\) and height 1, centered at \(t=0\).

Fourier Transform Derivation

\[ X(j \omega)=\int_{-T_{1}}^{T_{1}} e^{-j \omega t} d t = 2 \frac{\sin \omega T_{1}}{\omega} \]

This is another classic example. The Fourier transform of a rectangular pulse is a sinc function (or a scaled version of it). This function has a main lobe and then decaying side lobes. The width of the main lobe is inversely proportional to the width of the pulse in the time domain.

Interactive Demo: Fourier Transform of a Rectangular Pulse

Adjust the pulse width \(T_1\) and observe its effect on the frequency spectrum.

viewof T1_val_rect = Inputs.range([0.1, 5], {label: "Pulse Half-Width (T1)", step: 0.1, value: 1})

Notice the inverse relationship: a wider pulse in time (T1 increases) results in a narrower main lobe in the frequency spectrum. This means that a longer pulse has a more concentrated frequency content. This is a crucial concept for understanding bandwidth and signal duration. Also, observe the Gibbs phenomenon if you try to reconstruct this signal from a finite number of frequency components.

Example 4.5: Ideal Low-Pass Filter (Sinc Function in Time)

Transform Definition

\[ X(j \omega)= \begin{cases}1, & |\omega|<W \\ 0, & |\omega|>W\end{cases} \]

This is a rectangular pulse in the frequency domain. It represents an ideal low-pass filter, passing all frequencies below \(W\) and blocking those above.

Inverse Fourier Transform Derivation

\[ x(t)=\frac{1}{2 \pi} \int_{-W}^{W} e^{j \omega t} d \omega=\frac{\sin W t}{\pi t} \]

This is the dual of the previous example! Here, the frequency spectrum is a rectangle. When we take the inverse Fourier transform, the time-domain signal is a sinc function. This signal is non-causal (extends to negative infinity) and oscillates, which are characteristics of an ideal filter’s impulse response.

Interactive Demo: Inverse FT of a Frequency Pulse

Adjust the bandwidth \(W\) of the frequency-domain pulse and see its effect on the time-domain signal.

viewof W_val_sinc = Inputs.range([0.1, 10], {label: "Bandwidth (W)", step: 0.1, value: 3})

Here, a wider frequency bandwidth W results in a narrower main lobe in the time-domain signal. This again illustrates the inverse relationship. A signal with a broader frequency content can be more concentrated in time. This is fundamental to understanding concepts like time-bandwidth product.

The Sinc Function

Functions of the form \(\frac{\sin(\theta)}{\theta}\) appear frequently in Fourier analysis. A commonly used precise form for the sinc function is:

\[ \operatorname{sinc}(\theta)=\frac{\sin \pi \theta}{\pi \theta} \]

Examples Using Sinc

• Fourier Transform of Rectangular Pulse: \[ \frac{2 \sin \omega T_{1}}{\omega} = 2 T_{1} \operatorname{sinc}\left(\frac{\omega T_{1}}{\pi}\right) \]
• Inverse Fourier Transform of Rectangular Frequency Pulse: \[ \frac{\sin W t}{\pi t} = \frac{W}{\pi} \operatorname{sinc}\left(\frac{W t}{\pi}\right) \]

The Sinc Function

Plotting \(\operatorname{sinc}(\theta)\)

Let’s see how the \(\operatorname{sinc}\) function behaves.

The sinc function is normalized such that sinc(0) = 1 and its zero crossings occur at integer values (e.g., \(\theta = \pm 1, \pm 2, \dots\)). It’s important to be aware of this specific definition as sometimes sin(x)/x is also informally called sinc. The properties of the sinc function (main lobe, side lobes, decay) are critical for understanding signal reconstruction and filter design.

Time-Frequency Duality and Bandwidth

A fundamental concept in Fourier analysis is the inverse relationship between the spread of a signal in the time domain and its spread in the frequency domain.

Wide in Time, Narrow in Frequency

• A signal that is spread out (long duration) in the time domain tends to have a narrow (concentrated) spectrum in the frequency domain.
• Example: A long rectangular pulse in time has a narrow \(\operatorname{sinc}\) function in frequency.

Narrow in Time, Wide in Frequency

• A signal that is concentrated (short duration) in the time domain tends to have a wide (broad) spectrum in the frequency domain.
• Example: A short rectangular pulse in time has a wide \(\operatorname{sinc}\) function in frequency.

Important

This time-frequency duality implies that you cannot simultaneously localize a signal arbitrarily well in both time and frequency. This is often referred to as the uncertainty principle in signal processing.

This duality is a cornerstone of signal processing. It explains why a very short burst of sound (like a click) contains a wide range of frequencies, and conversely, why a pure tone (a single frequency) must extend infinitely in time. It has implications for filter design, communication system bandwidth, and quantum mechanics (Heisenberg’s Uncertainty Principle).

Interactive Demo: Time-Frequency Duality

Observe the inverse relationship between signal duration and spectral width.

viewof param_val = Inputs.range([0.1, 5], {label: "Pulse Parameter (T1 or W)", step: 0.1, value: 1})
viewof type_val = Inputs.select(["Time Pulse (Rect)", "Freq Pulse (Sinc)"], {label: "Signal Type", value: "Time Pulse (Rect)"})

Use the Signal Type dropdown to switch between a time-domain rectangular pulse and a frequency-domain rectangular pulse. Then, adjust Pulse Parameter (T1 or W). When T1 increases for the “Time Pulse”, the pulse gets wider, and its sinc spectrum gets narrower. When W increases for the “Freq Pulse”, the frequency pulse gets wider, and its sinc time function gets narrower. This visually reinforces the time-frequency duality.

Summary

Key Takeaways

• From Series to Transform: The Fourier Transform represents aperiodic signals as the limit of Fourier series as the period approaches infinity.
• Fourier Transform Pair: \[ X(j \omega)=\int_{-\infty}^{+\infty} x(t) e^{-j \omega t} d t \] \[ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} X(j \omega) e^{j \omega t} d \omega \]
• Convergence: Conditions like finite energy or Dirichlet conditions ensure a valid Fourier Transform representation.
• Examples: We explored the transforms of exponential decay, double-sided exponential, unit impulse, rectangular pulse, and the sinc function.
• Time-Frequency Duality: A fundamental principle showing an inverse relationship between a signal’s spread in time and its spread in frequency.

Note

The Fourier Transform is a powerful tool for analyzing signals and systems, providing insights into their frequency content and behavior.

This summary covers the main points discussed. Encourage students to remember the core definitions, the examples as building blocks, and especially the concept of time-frequency duality, which will be revisited many times in the course.