Fourier Series Properties & Parseval's Theorem

Once you can compute a Fourier series, a handful of properties let you avoid recomputing it from scratch for related signals. Two of the most important are linearity and the time-shift property. These are analogous to properties you will encounter for the Fourier transform in the next lesson.

The Fourier series is a linear operation. If x₁(t) has coefficients cₙ⁽¹⁾ and x₂(t) has coefficients cₙ⁽²⁾, then a·x₁(t)+b·x₂(t) has coefficients a·cₙ⁽¹⁾+b·cₙ⁽²⁾. This follows directly from the linearity of the integral that defines cₙ.

In practice, this means you can build up the spectrum of a composite signal from the spectra of its parts, without doing any new integrals.

If you delay a signal by t₀ — forming x(t−t₀) — each exponential coefficient cₙ picks up a phase factor e^{−jnω₀t₀}. The magnitude |cₙ| does not change; only the phase rotates. Shifting a signal in time does not change which frequencies are present or how much energy each carries — it only changes the relative alignment (phase) of the components.

Parseval's theorem says: the average power of a periodic signal — computed by integrating |x(t)|² over one period — is exactly equal to the sum of the average powers in each harmonic component. The theorem is a direct consequence of the orthogonality of the Fourier basis functions: cross-terms between different harmonics integrate to zero, so the total power cleanly separates into a sum of individual harmonic powers.

In Module 8 you learned that the average power of a single sinusoid A cos(ωt) is A²/2. Parseval's theorem generalizes this: the total power is the DC power (a₀²) plus the power of each harmonic, where each cosine and sine term contributes (aₙ²+bₙ²)/2.

Recall from Module 8 that a single sinusoid A cos(ωt) has average power A²/2. In Parseval's formula, the n-th harmonic consists of aₙcos(nω₀t)+bₙsin(nω₀t): the cosine part contributes aₙ²/2 and the sine part contributes bₙ²/2, so the harmonic's total power is (aₙ²+bₙ²)/2. The DC term a₀ is a constant, so its power is simply a₀². Parseval's theorem is a direct extension of the single-sinusoid power result to an arbitrary sum of harmonics.

Parseval's theorem tells you that the Fourier coefficients cₙ (or aₙ, bₙ) carry complete information about the signal's power — no need to go back to the time domain. This is the basis for power spectral density analysis, filter design, and signal-to-noise ratio calculations you'll encounter later in the course and in communications systems. In the exponential form, Parseval's theorem becomes P=Σ|cₙ|², especially clean and reinforcing that each coefficient's squared magnitude represents the power at that frequency.