Skip to content

Fourier Transform Pairs

Copy this iframe to your website:

1
<iframe src="https://dmccreary.github.io/signal-processing/sims/fourier-transform-pairs/main.html" width="100%" height="672px" scrolling="no"></iframe>

Run MicroSim in Fullscreen

Description

A Fourier transform pair links a signal \(x(t)\) to its spectrum

\[X(f) = \int_{-\infty}^{\infty} x(t)\,e^{-j2\pi ft}\,dt, \qquad x(t) = \int_{-\infty}^{\infty} X(f)\,e^{j2\pi ft}\,df .\]

The drop-down list picks one of five pairs: rectangle ↔ sinc, with \(\operatorname{sinc}(u) = \sin(\pi u)/(\pi u)\); Gaussian ↔ Gaussian; one-sided exponential \(e^{-at}u(t)\) ↔ \(1/(a + j2\pi f)\), whose power \(|X(f)|^2\) has the Lorentzian shape; cosine ↔ two impulses; and impulse train ↔ impulse train. The top plot shows \(x(t)\), the middle plot \(|X(f)|\) and the bottom plot the phase \(\angle X(f)\). Both axes run from −4.25 to 4.25 (seconds and hertz), so equal widths look equal.

  • Time scale T (0.25 to 2 s) replaces \(x(t)\) with \(x(t/T)\), which transforms to \(T\,X(fT)\): a shorter signal has a wider, lower spectrum.
  • Shift t₀ (−2 to 2 s) delays the signal: \(x(t - t_0) \leftrightarrow e^{-j2\pi f t_0}X(f)\). The magnitude stays the same and the phase gains a ramp of \(-360°\,t_0\) per hertz.
  • Show phase hides or shows the phase plot.

Maroon curves are the analytic transforms. Green dots are a numerical estimate: the sketch samples \(x(t)\) every \(\Delta t = 1/128\) s over a 128 s window, takes a 16384-point FFT and multiplies by \(\Delta t\). The bottom read-out gives the largest difference between the two. Impulses are arrows whose height is their area; for the two periodic pairs the window holds exactly 64 periods (\(\Delta t = T/256\)) and each green dot is an FFT bin times its width \(\Delta f\). Gold bars mark a width in each domain, and the first read-out line compares them. The phase is drawn only where \(|X|\) exceeds 2% of \(X(0)\).

This MicroSim is used in Chapter 6: Fourier Analysis Fundamentals.

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Identify the transform of a rectangular pulse, a Gaussian, a one-sided exponential, a cosine and an impulse train
  2. Predict how the height and width of \(|X(f)|\) change when a signal is compressed or stretched in time, using \(x(t/T) \leftrightarrow T\,X(fT)\)
  3. Calculate the phase that a time shift adds at a given frequency, using \(x(t - t_0) \leftrightarrow e^{-j2\pi f t_0}X(f)\)
  4. Explain why the Gaussian reaches the smallest time-bandwidth product, \(\sigma_t\sigma_f = 1/(4\pi)\)
  5. Read a transform pair backwards, identifying \(x(t)\) from a given \(X(f)\) (the inverse Fourier transform)

Target Audience

Undergraduate students in a signals and systems or introductory signal processing course (sophomore or junior engineering, physics or computer science). Plan 25 minutes for the activities below.

Prerequisites

  • Complex exponentials and Euler's formula
  • The unit step \(u(t)\) and the unit impulse \(\delta(t)\)
  • The definition of the Fourier transform (Chapter 6)

Activities

Activity 1: Compress in time, expand in frequency (6 min). Choose Rectangle ↔ sinc with \(T = 1.00\) s and \(t_0 = 0\). The first zero of the sinc is at 1.00 Hz and \(X(0) = 1.00\), the area of the pulse. Set \(T = 0.50\) s: the pulse is half as wide, \(X(0)\) drops to 0.50 and the first zero moves out to 2.00 Hz. Set \(T = 2.00\) s: \(X(0) = 2.00\) and the first zero is at 0.50 Hz. The product of the width and the first-zero frequency stays 1. With \(T = 1.00\) s, look at the phase plot: 0° in the main lobe and 180° in the lobes between 1 and 2 Hz, 3 and 4 Hz, where the sinc is negative.

Activity 2: A delay is a phase ramp (6 min). Keep the rectangle with \(T = 1.00\) s and set \(t_0 = 0.50\) s. The green magnitude dots do not move at all, but the phase becomes a ramp of −180° per hertz: −45° at 0.25 Hz and −90° at 0.5 Hz. Set \(t_0 = -1.00\) s (an advance): the ramp is now +360° per hertz and the phase at 0.25 Hz is +90°.

Activity 3: The Gaussian time-bandwidth product (6 min). Choose Gaussian ↔ Gaussian with \(T = 1.00\) s. The two plots have the same shape because \(e^{-\pi t^2} \leftrightarrow e^{-\pi f^2}\), and the measured rms widths are \(\sigma_t = 0.282\) s and \(\sigma_f = 0.282\) Hz. Set \(T = 0.50\) s (\(\sigma_t = 0.141\) s, \(\sigma_f = 0.564\) Hz) and then \(T = 2.00\) s (\(\sigma_t = 0.564\) s, \(\sigma_f = 0.141\) Hz). The product stays \(\sigma_t\sigma_f = 0.0796 = 1/(4\pi)\). Now choose the exponential with \(T = 1.00\) s: \(|X|\) falls to \(X(0)/\sqrt{2}\) at \(a/(2\pi) = 0.159\) Hz, where the phase is about −45°.

Activity 4: Impulses and periodic signals (7 min). Choose Cosine ↔ two impulses with \(T = 0.50\) s: impulses of area ½ sit at ±2.00 Hz. Set \(t_0 = 0.10\) s: the areas do not change and the phases become −72° at +2 Hz and +72° at −2 Hz. Choose Impulse train ↔ impulse train with \(T = 2.00\) s: the spectrum is a train of impulses every 0.50 Hz, each of area 0.50. Set \(T = 1.00\) s and \(t_0 = 0.25\) s: the phases are −90° at 1 Hz, 180° at 2 Hz and +90° at 3 Hz. Finally set \(t_0 = 1.00\) s, one whole period: every phase returns to 0° and the read-out reports that nothing changed.

Assessment

  1. A rectangular pulse lasts 2 ms. At what frequency is the first zero of its spectrum, and what is \(X(0)\)? Answer: at \(1/T = 500\) Hz; \(X(0) = T = 0.002\) s (the area of the pulse).
  2. A signal is compressed in time by a factor of 3, from \(T = 1.5\) s to \(T = 0.5\) s. How do the height and the width of its spectrum change? Check with the rectangle. Answer: \(X(0)\) falls from 1.50 to 0.50 (one third) and the first zero moves from 0.667 Hz to 2.00 Hz (three times wider).
  3. Which signal has the transform \(X(f) = 0.5\,\operatorname{sinc}(0.5f)\,e^{-j\pi f}\)? Answer: \(e^{-j\pi f} = e^{-j2\pi f(0.5)}\), so it is a rectangular pulse of width 0.5 s delayed by 0.5 s, running from 0.25 s to 0.75 s. Setting \(T = 0.50\) s and \(t_0 = 0.50\) s in the sim gives a phase ramp of −180° per hertz.
  4. A signal is delayed by 0.25 s. What happens to its magnitude spectrum, and what phase is added at 2 Hz? Answer: the magnitude is unchanged; the added phase is \(-360° \times 2 \times 0.25 = -180°\).
  5. A Gaussian pulse has an rms duration \(\sigma_t = 1\) ms. What is its rms bandwidth \(\sigma_f\), and can any pulse with \(\sigma_t = 1\) ms have \(\sigma_f = 50\) Hz? Answer: \(\sigma_f = 1/(4\pi \times 0.001) = 79.6\) Hz. No: \(\sigma_t\sigma_f\) would be 0.05, below the minimum \(1/(4\pi) = 0.0796\).

Common Misconceptions

"Delaying a signal changes its spectrum." Only the phase changes. Drag the shift slider with any pair selected: the green magnitude dots stay exactly where they are, while the phase plot tilts into a steeper or shallower ramp.

"A narrower pulse has a narrower spectrum." The opposite is true. Lower \(T\) for the rectangle or the Gaussian: the pulse narrows while the spectrum spreads out and drops in height, because the product of the two widths is fixed.

References

  1. Fourier transform - Wikipedia - definition, inverse transform, scaling and shift properties, and a table of transform pairs
  2. Sinc function - Wikipedia - the normalized sinc and its role as the transform of a rectangular pulse
  3. Gaussian function - Wikipedia - the Gaussian and why its Fourier transform is another Gaussian
  4. Dirac comb - Wikipedia - the impulse train and its transform, an impulse train with reciprocal spacing
  5. Cauchy distribution - Wikipedia - the Lorentzian shape of the power spectrum of a decaying exponential
  6. Uncertainty principle - Wikipedia - the time-bandwidth limit that the Gaussian attains
  7. The Scientist and Engineer's Guide to Digital Signal Processing - Steven W. Smith - free online textbook with chapters on Fourier transform pairs and properties