Fourier Series & Transform
Any signal can be built from sinusoids. Learn the Fourier series of periodic signals (square wave harmonics), the Fourier transform and its properties (linearity, time shift, scaling, convolution ↔ multiplication, Parseval), spectra of common signals, bandwidth, and frequency response of LTI systems.
Joseph Fourier showed that complex waveforms are sums of simple sine waves. Looking at a signal's frequency content (its spectrum) explains bandwidth, filtering, distortion, modulation and much more.
An analogy
The Fourier transform is like a prism splitting white light into colours: a complex signal is split into the pure tones (frequencies) that make it up.
Fourier series
Periodic signals contain only harmonics of f₀. A square wave has odd harmonics falling as 1/n; a triangle wave falls as 1/n². Sharp edges need many harmonics — hence wide bandwidth.
Fourier transform
Non-periodic signals have continuous spectra. Short pulses have wide spectra (time–bandwidth trade-off). A delay adds linear phase; modulation shifts the spectrum.
Frequency response
For LTI systems, Y(f) = X(f)H(f). Magnitude |H| shows gain versus frequency; phase ∠H shows delay. Bode plots use log scales.
Practical spectra
The DFT/FFT computes spectra of sampled data; windowing controls leakage; spectrograms show how spectra change over time (speech, radar, vibration analysis).
Formulas
- Fourier transform:
X(f) = ∫ x(t) e^(−j2πft) dt - Square wave series (±A):
x(t) = (4A/π) Σ_odd n (1/n) sin(2πn f₀ t) - Convolution theorem:
x * h ↔ X(f) H(f) - Parseval:
∫|x(t)|² dt = ∫|X(f)|² df - Rectangular pulse:
width τ ↔ τ sinc(fτ), first null at 1/τ
Worked example: Square wave harmonics
Given: 1 kHz square wave, ±1 V
Find: Amplitudes of 1st and 3rd harmonics.
- Fundamental = 4/π = 1.273 V at 1 kHz
- 3rd = 4/(3π) = 0.424 V at 3 kHz
Answer: 1.27 V and 0.42 V
Worked example: Pulse bandwidth
Given: 1 µs rectangular pulse
Find: First spectral null.
- f = 1/τ = 1 MHz
Answer: 1 MHz — shorter pulses need more bandwidth
Common mistakes
- Wrong: Thinking a square wave has only one frequency. — Right: It contains infinite odd harmonics.
- Wrong: Ignoring phase. — Right: Phase carries timing and shape information.
Applications
- Audio analysis: Equalisers and spectrum analysers.
- EMC: Harmonics of fast digital edges.
- Power quality: Harmonic distortion in mains.
Key takeaways
- Signals = sums of sinusoids.
- Square wave: odd harmonics 4A/(nπ).
- Convolution ↔ multiplication.
- Short in time = wide in frequency.
Going further
- Short-time Fourier transform: Spectrograms.
- Wavelets: Multi-resolution analysis.
- Uncertainty principle: Δt·Δf ≥ 1/(4π).