ELEC2134 Chap.6 Fourier Transform Pairs
Fourier Transform Pairs
The Fourier transform extends frequency analysis from periodic signals with discrete lines to aperiodic signals with continuous spectra. This chapter follows the limiting argument from a pulse train to one isolated pulse, states the course transform convention, and develops high-value pairs for rectangles, sinc functions, decaying exponentials, impulses and cosines.
Area at zero frequency provides a fast numerical check.
The transform is generally complex. Magnitude tells how strongly a continuous frequency contributes, while phase records timing and waveform alignment. A time shift can leave the entire magnitude curve unchanged while adding a phase ramp, so magnitude alone cannot reconstruct a general signal.
For real signals, conjugate symmetry links negative frequency to positive frequency.
Known pairs should be read through their geometry. A wider rectangular pulse has a narrower sinc spectrum, and the central transform value equals pulse area. A faster decaying exponential spreads further across frequency.
A shifted impulse has constant magnitude because it excites every frequency equally, while a persistent cosine concentrates content into two delta lines. These shape cues are as valuable as the algebraic formulas.
What this chapter covers
- 01
Fourier series limit as period tends to infinity
- 02
Forward and inverse transform convention
- 03
Rectangle and sinc transform relationship
- 04
One-sided exponential transform
- 05
Impulse shifts and constant magnitude
- 06
Cosine transform as two frequency impulses
Worked example · free
Transform a decaying causal signal
- +2The unit step makes the lower integration limit zero: X(omega) = integral from zero to infinity of 2 exp[-(3 + j omega)t] dt.
- +1Evaluating the convergent integral gives X(omega) = 2/(3 + j omega).
- +2Magnitude is 2/sqrt(9 + omega squared), and phase is -atan(omega/3).
- +1X(0) = 2/3, equal to the signed area of x(t).
Key terms
- Fourier transform
- The integral mapping an aperiodic time signal to a continuous frequency spectrum.
- Inverse transform
- The integral that reconstructs a time signal from its continuous spectrum.
- Transform pair
- Two functions related by the forward and inverse Fourier transform operations.
- Delta impulse
- A distribution concentrated at one point with unit integral and a sifting property.
- Continuous spectrum
- Frequency content defined across a continuum rather than only at harmonic indices.
Fourier Transform Pairs FAQ
Why does an aperiodic signal have a continuous spectrum?
As a periodic repetition interval grows, the fundamental frequency shrinks and the harmonic lines become more densely spaced. In the infinite-period limit, the isolated event is no longer periodic and the discrete sum becomes an integral over continuous angular frequency. The transform describes the resulting spectral density.
What does X(0) tell me?
Setting omega to zero removes the complex exponential from the forward transform. Therefore X(0) equals the signed area under x(t), provided the integral exists. This gives a fast check for pulses and decaying signals, and it fixes the central height of many transform sketches.
Why does a cosine transform into impulses?
A cosine persists forever and its ordinary transform integral does not converge as a conventional function. It contains exactly two complex exponential frequencies, positive and negative omega0, so delta impulses represent concentrated spectral content at those locations. Their paired weights reconstruct the real cosine under the inverse transform.
How does pulse width affect the sinc spectrum?
Increasing time-domain width narrows the main spectral lobe and brings its zeros closer to zero frequency. Decreasing width spreads the spectrum. The central value still equals pulse area, so changing width also changes that height when amplitude is held fixed. Use both height and zero spacing when checking a sketch.
When should I integrate instead of using a pair?
Use a known pair whenever the signal matches one directly or after a simple property. Integrate when no suitable parent pair is available or when the question specifically asks for derivation. Even then, predict symmetry, zero-frequency value and convergence before calculating so the integral has independent checks.
Exam move
Learn transform pairs by their landmarks, not only their formulas. For a rectangle, connect time width to sinc-zero spacing and area to X(0). For an exponential, connect decay rate to spectral width and phase sign. Practise reading every pair in both directions. Use direct integration when no pair applies, but always finish with an area, symmetry or limiting-value check.
Create flashcards that show a time shape on one side and require both the transform formula and two qualitative features on the other. Mix forward and inverse prompts so recognition does not depend on which domain appears first.
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