Fourier series: coefficients, convergence and signal spectra
Orthogonality and the coefficient formulas, the square wave and its harmonics, Dirichlet's theorem, the Gibbs phenomenon, Parseval's identity and the complex form.
The Taylor series approximates a function with powers, and it is excellent near a point. For periodic functions, such as the signals of a circuit in AC steady state, sines and cosines are far more natural. Fourier proposed this in 1807 to solve the heat equation; today it is the foundation of signal analysis, filters, audio compression and communications.
Trigonometric polynomials and Fourier series
We want to write a function 
of period 
as
![\[ f(x) \sim \frac{a_0}{2} + \sum_{k=1}^{\infty}\big(a_k\cos kx + b_k\sin kx\big) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-65849316afe5cfdd776ed1eb0823a892_l3.png "Rendered by QuickLaTeX.com")
The term 
is the mean value, the 
terms are the fundamental, those with 
the harmonics. For any period 
, replace 
with 
, where 
.
Proof: the coefficient formulas
The key is the orthogonality relations: for integers 
![\[ \int_{-\pi}^{\pi}\cos kx\cos mx\,dx = \int_{-\pi}^{\pi}\sin kx\sin mx\,dx = \begin{cases}\pi & k = m\\ 0 & k \ne m\end{cases},\qquad \int_{-\pi}^{\pi}\sin kx\cos mx\,dx = 0 \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-dd15123a21e3ffdcecdae215177a82b3_l3.png "Rendered by QuickLaTeX.com")
They are proved with product-to-sum formulas: for example ![\cos kx\cos mx = \tfrac12[\cos(k - m)x + \cos(k + m)x]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-c2b4946589dea41f122655e24e70115c_l3.png "Rendered by QuickLaTeX.com")
, and the integral over a period of 
is zero for every 
.
Multiply the series by 
and integrate over ![[-\pi, \pi]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-367cdb53682a8af536ed9d44534d5253_l3.png "Rendered by QuickLaTeX.com")
, assuming we may integrate term by term. Every term vanishes except the one with , which gives 
. Hence
![\[ a_k = \frac1\pi\int_{-\pi}^{\pi}f(x)\cos kx\,dx,\qquad b_k = \frac1\pi\int_{-\pi}^{\pi}f(x)\sin kx\,dx \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-3b394e82c2f83e6a6fad4f6fe8062134_l3.png "Rendered by QuickLaTeX.com")
It is exactly the computation of coordinates in an orthogonal basis from the Geometry and Algebra lesson: the functions 
and 
are an orthogonal basis for the inner product 
, and the Fourier coefficients are the projections of onto it.
Useful symmetries. If is even, all 
are zero (a cosine series); if it is odd, all 
are zero (a sine series).
Example: the square wave
Let 
for 
and 
for 
, extended periodically. It is odd, so 
, and
![\[ b_k = \frac2\pi\int_0^\pi\sin kx\,dx = \frac{2}{k\pi}(1 - \cos k\pi) = \begin{cases}\dfrac{4}{k\pi} & k \text{ odd}\\ 0 & k \text{ even}\end{cases} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-529d160bc37d998a5c8ca1fdc85c062b_l3.png "Rendered by QuickLaTeX.com")
![\[ f(x) = \frac4\pi\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \dots\right) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-28cc947624e3aaf303af51c294c9ebef_l3.png "Rendered by QuickLaTeX.com")
A square wave contains only odd harmonics, with amplitude falling as 
. That is why a 100 MHz digital clock produces interference at 300, 500, 700 MHz, and why the edges of fast signals are sometimes deliberately slowed down.
Convergence
Theorem (Dirichlet). If is periodic and piecewise smooth (with finitely many jumps and corners in a period), the Fourier series converges at every point:
- to
where is continuous; - to the average
![\tfrac12[f(x^+) + f(x^-)]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-7d5dfa4b141c76dc0ff58f7822940233_l3.png "Rendered by QuickLaTeX.com")
at jumps.
For the square wave, at 
the series gives 0, the average of 
and 
.
Near the jumps the convergence is not uniform: the partial sums overshoot the function by about 9% of the size of the jump, and this peak moves towards the jump but does not shrink as terms are added. This is the Gibbs phenomenon, visible in the figure. If instead is continuous and piecewise smooth, the convergence is uniform.
Parseval’s identity and energy
![\[ \frac1\pi\int_{-\pi}^{\pi}f(x)^2\,dx = \frac{a_0^2}{2} + \sum_{k=1}^{\infty}\big(a_k^2 + b_k^2\big) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-a943b86a1425ca72d6aa1cbed4a1e95d_l3.png "Rendered by QuickLaTeX.com")
It is Pythagoras’ theorem in infinite dimensions: the squared norm is the sum of the squares of the coordinates. Physically, the power of a periodic signal is the sum of the powers of its harmonics. For the square wave the left side is 2, and a surprising result follows: 
, that is 
.
It also follows that the fundamental alone carries 
of the square wave’s power: this is the starting point for computing a signal’s total harmonic distortion (THD).
Complex form
With Euler’s formula the series becomes more compact:
![\[ f(x) = \sum_{k=-\infty}^{\infty}c_k\,e^{ikx},\qquad c_k = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,e^{-ikx}\,dx \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-d7f7cdf2b818a9d5460b6c0e076fd6f2_l3.png "Rendered by QuickLaTeX.com")
This is the form used in electronics and telecommunications. Letting the period tend to infinity, the sum over discrete frequencies 
becomes an integral over all frequencies: the Fourier transform, which describes the spectrum of non-periodic signals. The version for sampled signals, computed with the FFT algorithm, is inside every digital oscilloscope, modem and audio codec. The Z transform, linked to this subject, generalises it to discrete-time systems.
