Unit 3: Fourier series representation of periodic signals

The Fourier series expresses any well-behaved periodic signal as a weighted sum of harmonically related complex exponentials (or equivalently sines and cosines). Developed from Joseph Fourier's 1807 work on heat conduction, it converts a time-domain waveform into a discrete set of frequency-domain coefficients, providing the foundation for spectral analysis, filtering, and modulation theory.

• Periodic signal: a signal satisfying x(t) = x(t + T) for all t, where T > 0 is the fundamental period.
• Fundamental frequency: ω₀ = 2π/T rad/s, the smallest angular frequency present; f₀ = 1/T Hz.
• Harmonics: integer multiples kω₀ of the fundamental; k = 1 is the fundamental, k ≥ 2 are higher harmonics.
• Orthogonality basis: the set {e^{jkω₀t}} is orthogonal over any interval of length T, which is what allows unique coefficient extraction.
• Linearity of the model: the signal is a linear combination of basis functions, so superposition applies throughout.

II. Fourier Series Representation of Continuous-Time Periodic Signals

Synthesis and analysis of a periodic waveform from harmonic components.

The representation has two directions: synthesis builds x(t) from coefficients, and analysis extracts coefficients from x(t).

A. Fourier series representation of continuous time periodic signals

The exponential (complex) form is the compact core of the theory.

• Synthesis equation: reconstructs the signal from its spectrum.

x(t) = Σ (k = -∞ to +∞) aₖ e^{jkω₀t}

• Analysis equation: projects the signal onto each basis exponential.

aₖ = (1/T) ∫_T x(t) e^{-jkω₀t} dt

• Symbols: aₖ = complex Fourier coefficient of the k-th harmonic; T = fundamental period; ω₀ = 2π/T; ∫_T = integral over one full period.
• DC term: a₀ = (1/T) ∫_T x(t) dt is the average value of the signal over one period.

• Trigonometric form: an equivalent real-valued expansion.

x(t) = a₀ + Σ (k = 1 to ∞) [ aₖ cos(kω₀t) + bₖ sin(kω₀t) ]
aₖ = (2/T) ∫_T x(t) cos(kω₀t) dt
bₖ = (2/T) ∫_T x(t) sin(kω₀t) dt

• Link between forms: for real signals a_{-k} = aₖ* (conjugate symmetry), and the complex coefficient relates to trig ones by aₖ = ½(aₖ_trig − j bₖ).
• Polar/harmonic form: x(t) = C₀ + Σ Cₖ cos(kω₀t + θₖ), where Cₖ = 2|aₖ| is the harmonic amplitude and θₖ = ∠aₖ its phase.

Worked example — square wave: for a periodic square wave of amplitude A, period T, symmetric about the origin and equal to +A for |t| < T₁ and 0 elsewhere in the period:

aₖ = (A/T) · (2 sin(kω₀T₁)) / (kω₀)  =  (2A T₁ / T) · sinc(kω₀T₁/π)

The coefficients trace a sinc envelope, so amplitudes decay as 1/k and only certain harmonics dominate.

III. Convergence of the Fourier Series

The conditions under which the infinite sum actually reproduces the signal.

Because the series is an infinite sum, its equality to x(t) is not automatic; convergence sets the boundaries of validity.

A. Convergence of the fourier series

Convergence is guaranteed for physically reasonable signals through the Dirichlet conditions, with a characteristic overshoot at discontinuities.

• Dirichlet condition 1 — absolute integrability: over one period ∫_T |x(t)| dt < ∞. This bounds each coefficient since |aₖ| ≤ (1/T)∫_T|x(t)|dt.
• Dirichlet condition 2 — finite variation: the signal has a finite number of maxima and minima in one period, ruling out infinitely oscillating functions such as sin(1/t) near zero.
• Dirichlet condition 3 — finite discontinuities: a finite number of finite (jump) discontinuities per period, each bounded in size.
• Value at a discontinuity: where these hold, the series converges to x(t) at points of continuity and to the midpoint ½[x(t⁻) + x(t⁺)] at a jump.

Two convergence behaviours are worth contrasting:

1. Mean-square (energy) convergence: the total error energy vanishes, lim(N→∞) ∫_T |x(t) − x_N(t)|² dt = 0, whenever x(t) has finite energy per period. This holds even if pointwise values differ at isolated points.
2. Pointwise convergence and the Gibbs phenomenon: near a jump, the truncated sum x_N(t) overshoots by roughly 9% of the jump height regardless of how large N becomes. The overshoot narrows toward the discontinuity but never disappears in amplitude — it is an inherent artefact of truncating the series, not a numerical error.

• Significance: the 9% overshoot warns designers that band-limiting a sharp-edged signal (e.g. a clock pulse) rounds edges and introduces ringing, a practical limit in filter and reconstruction design.

IV. Properties of Continuous-Time Fourier Series

Structural rules that map time-domain operations to coefficient-domain operations.

Each property lets an operation on x(t) be predicted directly from its coefficients aₖ, avoiding re-integration.

A. Properties of continuous time fourier series

The properties assume x(t) ↔ aₖ and y(t) ↔ bₖ with the same period T.

• Linearity: A x(t) + B y(t) ↔ A aₖ + B bₖ. Scaling and summing signals scales and sums their spectra.
• Time shifting: x(t − t₀) ↔ aₖ e^{-jkω₀t₀}. A delay leaves magnitudes |aₖ| unchanged and adds a linear phase −kω₀t₀.
• Time reversal: x(−t) ↔ a_{-k}. The coefficient sequence is reflected about k = 0.
• Time scaling: x(αt) (with α > 0) keeps the same coefficients aₖ but changes the fundamental frequency to αω₀; the period becomes T/α.
• Multiplication: x(t) y(t) ↔ Σ_l a_l b_{k−l}, a discrete convolution of the two coefficient sequences.
• Conjugation: x*(t) ↔ a*_{-k}; for real x(t) this gives conjugate symmetry a_{-k} = aₖ*, so |aₖ| is even and ∠aₖ is odd.
• Differentiation: dx(t)/dt ↔ jkω₀ aₖ. Multiplying by jkω₀ emphasises high harmonics — differentiation sharpens edges.
• Integration: ∫ x(t) dt ↔ aₖ / (jkω₀) for k ≠ 0 (finite only when a₀ = 0); it attenuates high harmonics — integration smooths.
• Parseval's relation: links average power to coefficient energy.

(1/T) ∫_T |x(t)|² dt = Σ (k = -∞ to +∞) |aₖ|²

• Interpretation: total average power equals the sum of powers |aₖ|² carried by each harmonic, so the spectrum partitions power frequency by frequency.

• Symmetry shortcuts:
  • Even signal: bₖ = 0, the series is purely cosine.
  • Odd signal: a₀ = aₖ_cos = 0, the series is purely sine.
  • Half-wave symmetry (x(t) = −x(t ± T/2)): only odd harmonics survive.

V. Software Simulation of Frequency Spectrum of Periodic Signals

Numerical computation and plotting of Fourier coefficients.

Software estimates aₖ by sampling one period and approximating the analysis integral, then visualises the discrete line spectrum.

A. Software simulation of frequency spectrum of periodic signals

The workflow discretises the signal, computes coefficients by numerical integration or the FFT, and plots magnitude and phase against harmonic index.

• Sampling the period: choose N samples over one period T with spacing Δt = T/N; the discrete grid is t[n] = nΔt, n = 0 … N−1.
• Numerical analysis integral: approximate aₖ ≈ (1/T) Σ_{n=0}^{N−1} x(t[n]) e^{-jkω₀t[n]} Δt, which reduces to (1/N) Σ x[n] e^{-j2πkn/N} — this is the DFT scaled by 1/N.
• Frequency axis: harmonic k sits at physical frequency f = k f₀ = k/T Hz; the spectrum is a set of discrete lines, not a continuum.
• Magnitude and phase spectra: plot |aₖ| (stem plot of amplitudes) and ∠aₖ (phase in radians) versus k.

MATLAB example — spectrum of a periodic signal:

T  = 1;  N = 1000;  t = (0:N-1)*T/N;   % one period, N samples
x  = square(2*pi*t/T);                 % periodic square wave
X  = fft(x)/N;                         % scaled DFT -> Fourier coeffs a_k
k  = 0:20;                             % first 21 harmonics
stem(k, abs(X(k+1)));                  % magnitude line spectrum
xlabel('Harmonic k'); ylabel('|a_k|');

Python (NumPy) equivalent:

import numpy as np
T, N = 1.0, 1000
t = np.arange(N) * T / N
x = np.sign(np.sin(2*np.pi*t/T))       # square wave
ak = np.fft.fft(x) / N                 # Fourier coefficients
k  = np.arange(21)
mag = np.abs(ak[:21])                  # magnitude spectrum

• Verification against theory: for the square wave the stem plot shows non-zero lines only at odd k with amplitudes falling as 1/k, matching the analytic sinc result from Section II.
• Practical cautions:
  • Truncation: displaying finite k reproduces the Gibbs ripple when the signal is resynthesised from the plotted lines.
  • Sampling adequacy: N must be large enough that the highest harmonic of interest is well below N/2 (Nyquist) to avoid aliased coefficients.
  • Leakage: the sampled window must span exactly one period, otherwise energy smears across neighbouring bins and coefficient magnitudes are distorted.