Fourier Analysis
Joseph Fourier showed in 1822 that any periodic function can be decomposed into a sum of sines and cosines. The modern Fast Fourier Transform (FFT) computes this decomposition in O(N log N) time — enabling real-time audio equalisation, image compression and wireless communications.
Fourier Series
Build square, triangle and sawtooth waves by adding harmonics one at a time. Watch the Gibbs phenomenon (9% overshoot at discontinuities) emerge as you add terms.
FFT Spectrum Analyser
Live spectrum of microphone input or synthesised waveforms. Cooley-Tukey radix-2 FFT with Hann windowing — compare magnitude spectrum vs power spectral density.
Discrete Fourier Transform & Cooley-Tukey FFT
DFT: X[k] = Σ_{n=0}^{N-1} x[n] · e^{−j2πkn/N} O(N²)
FFT: Cooley-Tukey radix-2 butterfly → O(N log₂ N)
Nyquist: f_sample ≥ 2 · f_max to avoid aliasing
Parseval: Σ|x[n]|² = (1/N) Σ|X[k]|² (energy conserved)
Digital Filters
A digital filter shapes the frequency content of a sampled signal. FIR (Finite Impulse Response) filters have linear phase and guaranteed stability; IIR (Infinite Impulse Response) filters achieve steeper roll-off with fewer coefficients at the cost of potential instability and phase distortion.
Digital Filter Designer
Design low-pass, high-pass, band-pass and notch filters. Choose FIR (windowed sinc, Parks-McClellan) or IIR (Butterworth, Chebyshev). View impulse, step response and Bode plot.
RC Filter (Analogue)
First-order RC low-pass and high-pass filters. Time constant τ=RC, −3 dB at f_c=1/(2πRC). Bode magnitude and phase, unit-step response — bridge to digital filter theory.
Modulation & Wireless
Modulation encodes information onto a carrier wave so it can travel efficiently through a channel. AM and FM are analogue standards; OFDM is the backbone of 4G/5G, WiFi and digital TV — splitting the spectrum into hundreds of narrow orthogonal sub-carriers to defeat multipath fading.
AM & FM Modulation
Amplitude and frequency modulation side by side. Adjust modulation index m (AM) or deviation Δf (FM) and compare bandwidth, Carson's rule, and demodulation SNR.
OFDM (Orthogonal FDM)
N sub-carriers spaced at Δf=1/T — the orthogonality condition. Visualise cyclic prefix, guard intervals, channel equalisation and the PAPR problem that plagues 5G.
Circuits & Wave Phenomena
RLC Resonance
Series and parallel RLC circuits — resonant frequency ω₀=1/√(LC), quality factor Q=ω₀L/R. Tune through resonance and watch impedance dip, current peak, phase flip.
Doppler Effect
Moving source and observer — f_obs = f_s(v±v_obs)/(v∓v_src). Visualise wavefront compression ahead and expansion behind, Mach cone, and the relativistic Doppler shift.
Why does OFDM beat single-carrier modulation in multipath channels? In a multipath channel (reflections from walls, hills) each path arrives with a different delay, smearing a single-carrier symbol into adjacent symbols (ISI). OFDM splits the bandwidth into narrow sub-carriers where each sub-carrier's symbol duration is much longer than the channel delay spread — ISI affects only the cyclic prefix, not the payload, and a single complex multiplication per sub-carrier corrects the whole channel.
Learning Paths
Foundations Track
- Fourier Series
- FFT Spectrum Analyser
- RC Filter (Analogue)
- Digital Filter Designer
Communications Track
- AM & FM Modulation
- OFDM
- RLC Resonance
- Doppler Effect