Fourier Transforms of Common Signals

Hello! Welcome to your eighth lesson in the course.

In our last lesson on Parseval's theorem, we saw how the total energy of a signal is conserved between the time and frequency domains. We used the Fourier transform pair of a rectangular pulse and a sinc function as a key example to demonstrate how the theorem can simplify complex integrals. However, we took that transform pair as a given.

Today, we will fill in that gap and build up our foundational toolkit. The goal of this lesson is to derive the Fourier transforms of common signals, including rectangular pulses, sinc functions, and Gaussian functions.

These three functions are not just arbitrary examples; they are fundamental building blocks in signal processing:

The rectangular pulse models ideal switching behavior and gating. Its transform is crucial for understanding the bandwidth requirements of digital signals.
The sinc function appears as the transform of the rectangular pulse and, as we'll see, is the impulse response of an ideal low-pass filter.
The Gaussian function is central to probability theory (e.g., thermal noise in circuits) and has the unique property that its Fourier transform is also a Gaussian, making it a cornerstone of time-frequency analysis.

By deriving these transforms, you'll gain a much deeper intuition for the relationship between a signal's shape in time and its spectral characteristics.

1. The Rectangular Pulse and the Sinc Function

Let's begin with the rectangular pulse. We'll define a general rectangular pulse, $x (t)$ , of height 1, centered at the origin, with a total width of $W$ . This can be written as:

x (t) = rect (\frac{t}{W}) = {10 ∣ t ∣ < W /2 ∣ t ∣ > W /2

Now, let's derive its Fourier transform. The following video provides a clear, step-by-step derivation.

Note on Notation: The video uses a pulse of width $2 T$ (from $- T$ to $T$ ), so its width is $W = 2 T$ . It also uses the unnormalized sinc function, defined as $sinc (x) = \frac{s i n ( x )}{x}$ . We will adopt this definition for consistency in this lesson.

Fourier Transform Duality Rect and Sinc Functions

This video, from the channel 'Iain Explains Signals, Systems, and Digital Comms', will walk you through the derivation of the Fourier transform for a rectangular pulse. Pay close attention to how the integration limits are set by the pulse width and how Euler's formula is used to obtain the sine function.

Please watch the segment from the beginning until 03:08. This part calculates the Fourier transform of a rectangular pulse, showing it results in a sinc function.

As the video demonstrates, the Fourier transform of a rectangular pulse of width $W = 2 T$ is:

X (j ω) = \int_{- T}^{T} 1 \cdot e^{- j ω t} d t = \frac{2 sin ( ω T )}{ω} = 2 T \frac{sin ( ω T )}{ω T} = 2 T sinc (ω T)

Or, using our general width $W$ :

F {rect (\frac{t}{W})} = W sinc (\frac{ω W}{2})

This result is profoundly important. It shows that a signal that is perfectly limited in time (the rectangular pulse is zero outside its width) has a spectrum that extends to infinity in frequency (the sinc function decays but never reaches zero). This is a manifestation of the time-frequency uncertainty principle: you cannot simultaneously confine a signal in both time and frequency. To create perfectly sharp edges in time, you need an infinite range of frequencies.

Fourier Transform of a Rectangular Pulse.

Caption: A rectangular pulse in the time domain (left) and its corresponding sinc function spectrum in the frequency domain (right). The main lobe of the sinc function is centered at \omega=0, and its width is inversely proportional to the pulse's duration. The side lobes extend infinitely.

2. The Sinc Function and the Duality Property

Now, let's reverse the problem. What is the Fourier transform of a sinc function in the time domain? We could attempt a direct integration, but that is mathematically complex. A much more elegant path is to use the duality property of the Fourier transform, which we touched upon in previous lessons.

The duality property states that if $x (t) \Leftrightarrow X (j ω)$ is a Fourier transform pair, then there is a related pair:

X (j t) \Leftrightarrow 2 π x (- ω)

In simpler terms, if a function shape in the time domain produces a certain shape in the frequency domain, then putting the frequency-domain shape into the time domain will produce the original time-domain shape in the frequency domain (with a scaling and time-reversal).

The same video we just watched provides an excellent explanation of this. It first calculates the inverse Fourier transform of a rectangular pulse in the frequency domain and then formalizes the concept with the duality property.

Fourier Transform Duality Rect and Sinc Functions

Continuing with the same video, this part will show how to find the transform of a sinc function. It first does this by finding the inverse transform of a rectangular frequency pulse and then explains the general duality principle that makes this result intuitive.

Please watch the rest of the video, from 03:08 to the end. The first part demonstrates that a rectangular spectrum corresponds to a sinc function in time. The second part explains the formal duality property.

From the video and the duality principle, we arrive at our second fundamental transform pair. If we start with a sinc function in the time domain, say $x (t) = W sinc (W t /2)$ , its Fourier transform is a rectangular pulse in the frequency domain:

F {W sinc (\frac{W t}{2})} = 2 π rect (\frac{- ω}{2 W}) = 2 π rect (\frac{ω}{2 W})

(The last step holds because the rect function is even).

This rect/sinc pair is the mathematical foundation for ideal filtering. An ideal low-pass filter should pass all frequencies up to a cutoff $ω_{c}$ and block all frequencies above it. Its frequency response is therefore a rectangular pulse. The inverse Fourier transform of this rectangular pulse is a sinc function, meaning the impulse response of an ideal low-pass filter is a sinc function.

For the mathematically curious: Direct derivation of the sinc transform

Your preference for mathematical depth is well-noted. While duality is the elegant way, deriving the sinc function's transform directly is possible, though it requires more advanced techniques. The main difficulty is that $sinc (t)$ is not absolutely integrable ( $\int_{- \infty}^{\infty} ∣ sinc (t) ∣ d t = \infty$ ), so the Fourier integral doesn't converge in the usual sense.

One clever method, explored in the StackExchange resource LINK, avoids direct integration by using an integral representation of the sinc function itself:
$sinc (t) = \frac{s i n ( t )}{t} = \frac{1}{2} \int_{- 1}^{1} e^{j ν t} d ν$
Plugging this into the Fourier transform definition:
$F {sinc (t)} = \int_{- \infty}^{\infty} (\frac{1}{2} \int_{- 1}^{1} e^{j ν t} d ν) e^{- j ω t} d t$
By swapping the order of integration (a step that requires careful justification), we get:
$= \frac{1}{2} \int_{- 1}^{1} (\int_{- \infty}^{\infty} e^{- j (ω - ν) t} d t) d ν$
The inner integral is the Fourier transform of the constant 1, which is $2 π δ (ω - ν)$ .
$= \frac{1}{2} \int_{- 1}^{1} 2 π δ (ω - ν) d ν = π \int_{- 1}^{1} δ (ω - ν) d ν$
This integral is 1 if $ω$ is between -1 and 1, and 0 otherwise. This is precisely a rectangular pulse of height $π$ . This confirms the result from duality.

3. The Gaussian Function

Our final fundamental signal is the Gaussian function. It is defined as:
$x (t) = e^{- a t^{2}}, for a > 0$
The parameter $a$ controls the width of the bell curve: a larger $a$ corresponds to a narrower pulse.

The remarkable property of the Gaussian is that its Fourier transform is also a Gaussian. The derivation involves a classic mathematical trick called "completing the square" inside the integral's exponent.

Fourier Transform of a Gaussian

This video from the channel 'Mostly Math' provides a concise and clear derivation of the Fourier transform of a Gaussian function. It uses the method of completing the square and a known integral identity.

Watch the video from the beginning up to 04:21. The video will state the transform definition and then proceed with the derivation, arriving at the final Gaussian form in the frequency domain.

The derivation shows that:

F {e^{- a t^{2}}} = \frac{π}{a} e^{- ω^{2} / (4 a)}

Again, observe the time-frequency relationship. If we make the time-domain Gaussian narrow (large $a$ ), the denominator $a$ in the time-domain exponent is large. In the frequency domain, this large $a$ appears in the denominator of the exponent's denominator ( $- ω^{2} / (4 a)$ ), making the frequency-domain Gaussian wide. Conversely, a wide Gaussian in time (small $a$ ) produces a narrow Gaussian in frequency.

The Gaussian pulse is the unique function that optimally minimizes the time-frequency uncertainty, meaning it is the most simultaneously localized function in both domains.

Test your understanding!

Using the results from this lesson and the convolution theorem from a previous lesson, find the Fourier transform of the signal $y (t) = rect (t) \cdot e^{- π t^{2}}$ . (Here, $rect (t)$ is 1 for $∣ t ∣ < 1/2$ ).

Hint: Multiplication in the time domain is convolution in the frequency domain.

Show answer

Let $x_{1} (t) = rect (t)$ and $x_{2} (t) = e^{- π t^{2}}$ . Then $y (t) = x_{1} (t) \cdot x_{2} (t)$ .

The convolution theorem states that $F {x_{1} (t) \cdot x_{2} (t)} = \frac{1}{2 π} [X_{1} (j ω) * X_{2} (j ω)]$ .

First, let's find the individual transforms:

Transform of $x_{1} (t)$ : From our derivation for a rectangular pulse with width $W = 1$ , we have:
$X_{1} (j ω) = F {rect (t)} = 1 \cdot sinc (\frac{ω \cdot 1}{2}) = sinc (\frac{ω}{2})$
Transform of $x_{2} (t)$ : This is a Gaussian with $a = π$ . Using our formula:
$X_{2} (j ω) = F {e^{- π t^{2}}} = \frac{π}{π} e^{- ω^{2} / (4 π)} = e^{- ω^{2} / (4 π)}$
Apply the Convolution Theorem: The Fourier transform of $y (t)$ is the convolution of these two results, scaled by $1/ (2 π)$ :
$Y (j ω) = \frac{1}{2 π} [X_{1} (j ω) * X_{2} (j ω)] = \frac{1}{2 π} [sinc (\frac{ω}{2}) * e^{- ω^{2} / (4 π)}]$
The full convolution integral is $Y (j ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} sinc (\frac{λ}{2}) e^{- (ω - λ)^{2} / (4 π)} d λ$ . While evaluating this integral is complex, expressing the transform as the convolution of the individual transforms is the key insight. This shows how we can find the spectrum of a complex windowed signal by convolving the spectrum of the window (the sinc) with the spectrum of the signal (the Gaussian).

Conclusion

In this lesson, we have systematically derived the Fourier transforms for three of the most important signals in engineering and science.

Key Takeaways:

Rectangular Pulse $\Leftrightarrow$ Sinc Function: A rectangular pulse of width $W$ in time transforms into a sinc function in frequency whose main lobe width is inversely proportional to $W$ .
$rect (\frac{t}{W}) \Leftrightarrow W sinc (\frac{ω W}{2})$
Duality: The symmetry of the Fourier transform means that a sinc function in time transforms into a rectangular pulse in frequency. This pair is fundamental to ideal filter theory.
Gaussian $\Leftrightarrow$ Gaussian: A Gaussian function in time transforms into another Gaussian function in frequency. A narrow Gaussian in time corresponds to a wide Gaussian in frequency, and vice versa.
$e^{- a t^{2}} \Leftrightarrow \frac{π}{a} e^{- ω^{2} / (4 a)}$
Time-Frequency Uncertainty: All three pairs illustrate a fundamental trade-off: a signal cannot be arbitrarily "short" in both the time and frequency domains simultaneously.

Preview of the Next Lesson:
We have now thoroughly explored the Fourier transform. However, it has a significant limitation: the Fourier integral does not converge for functions that don't decay to zero, such as $e^{a t}$ for $a > 0$ . To analyze a broader class of signals and systems, particularly in the context of stability and control systems, we need a more powerful tool. In our next lesson, we will begin a new module by tracing the historical development of this tool: the Laplace transform. We'll see how it grew from the work of mathematicians like Euler and Lagrange and was forged into a practical engineering method by Oliver Heaviside.

Recommended resources

1. The Rectangular Pulse and the Sinc Function

2. The Sinc Function and the Duality Property

3. The Gaussian Function

Conclusion