Fourier Series — Trigonometric & Exponential Form

In Module 9 you learned that complex exponentials are eigenfunctions of LTI systems: if the input is x(t)=e^st, the output is y(t)=H(s)e^st. The signal shape doesn't change — it just gets scaled by a complex number. This motivates one of the most useful ideas in signal processing: if we can write an arbitrary input as a sum of complex exponentials, the output of an LTI system is simply the sum of those same exponentials, each individually scaled by its own H value.

Jean-Baptiste Joseph Fourier showed in the early 1800s that any periodic signal — no matter how irregular it looks in the time domain — can be decomposed into a sum of sinusoids (or complex exponentials) at frequencies that are integer multiples of a fundamental frequency. This decomposition is the Fourier series, the bridge between the time-domain and frequency-domain views of signals.

A signal x(t) is periodic with fundamental period T₀ if x(t)=x(t+T₀) for all t. The fundamental frequency is ω₀=2π/T₀. The harmonics are at frequencies nω₀ for n=1,2,3,…. The Fourier series says x(t) is a sum of sinusoids at exactly these harmonic frequencies.

There are two common forms — the trigonometric form and the exponential form. They are mathematically equivalent, but the exponential form is more compact and is used for most derivations throughout the rest of this course.

The coefficient a₀ is the DC component — the average value of the signal over one period. The coefficients aₙ and bₙ tell you how much cosine and sine energy is present at the n-th harmonic. Each is computed by integrating the signal against the corresponding basis function over one full period.

The integrals work because of orthogonality: over one period, the integral of cos(nω₀t)·cos(mω₀t) is zero when n≠m, and T₀/2 when n=m. This means each coefficient can be extracted independently — multiplying x(t) by cos(nω₀t) and integrating picks out exactly the aₙ component, with all other terms integrating to zero.

Using Euler's formula (e^jθ=cos θ+j sin θ), the trigonometric form can be rewritten using complex exponentials. Instead of three separate coefficients (a₀, aₙ, bₙ), there is a single coefficient cₙ defined for all integers n, positive and negative.

For n>0, the exponential coefficients connect to the trigonometric ones as cₙ=(aₙ−jbₙ)/2 and c₋ₙ=(aₙ+jbₙ)/2, while c₀=a₀. For real-valued signals, c₋ₙ is the complex conjugate of cₙ — conjugate symmetry. This is why the exponential form, despite running over both positive and negative n, carries the same information as the trigonometric form.

If x(t)=Σcₙe^jnω₀t and the system is LTI with frequency response H(jω), then by the eigenfunction property each exponential e^jnω₀t is simply scaled by H(jnω₀): y(t)=Σcₙ H(jnω₀)e^jnω₀t. No convolution integral needed — you multiply each Fourier coefficient by the system's response at that harmonic frequency, sum them back up, and you have the output. Every harmonic of the input passes through the system independently, scaled by the system's gain and phase at that frequency. This is exactly what makes the Fourier series so central to circuit analysis, communications, and control theory.