The (trigonometric) Fourier series of a periodic function \(f(t)\) is a representation that resolves \(f(t)\) into a DC component and an AC comprising an infinite series of harmonic sinusoids.
\(\boxed{f(t) = a_0 + \sum_{n = 1}^\infty\left[a_n\cos\left(n\omega_{0}t\right) + b_n\sin\left(n\omega_{0}t\right)\right]}\)
where \(\omega_0 = 2\pi / T\) is called the fundamental frequency in radians per second. The sinusoid \(\sin(n\omega_{0}t)\) or \(\cos(n\omega_{0}t)\) is called the \(n\mathrm{th}\) harmonic of \(f(t)\); it is an odd harmonic if \(n\) is odd and an even harmonic if \(n\) is even. The constants \(a_n\) and \(b_n\) are the Fourier coefficients. The coefficient \(a_0\) is the DC component or the average value of \(f(t)\) (since sinusoids have zero average values). For \(n \ne 0\), the coefficients \(a_n\) and \(b_n\) are the amplitudes of the sinusoids in the AC component.
The harmonic frequency \(\omega_n\) is an integral multiple of the fundamental frequency \(\omega_0\).
\(\boxed{\omega_n = n\omega_0}\)
where \(n > 0\) and an integer.
A function that can be represented by a Fourier series must meet certain requirements, because the infinite series may or may not converge. These conditions (Dirichlet conditions) on \(f(t)\) to yield a convergent Fourier series are as follows:
- \(f(t)\) is single-valued everywhere.
- \(f(t)\) has a finite number of finite discontinuities in any one period.
- \(f(t)\) has a finite number of maxima and minima in any one period.
- The integral \(\displaystyle\int_{t_0}^{t_0 + T}\left|f(t)\right|\,dt < \infty\) for any \(t_0\).
Determination of the Fourier coefficients \(a_0\), \(a_n\), and \(b_n\).
\(\boxed{a_0 = \frac{1}{T}\int_{0}^{T}f(t)\,dt}\)
\(\boxed{a_n = \frac{2}{T}\int_{0}^{T}f(t)\cos(n\omega_{0}t)\,dt}\quad\mathrm{n \ne 0}\)
\(\boxed{b_n = \frac{2}{T}\int_{0}^{T}f(t)\sin(n\omega_{0}t)\,dt}\quad\mathrm{n \ne 0}\)
The process of determining the coefficients is called Fourier analysis.
Fourier series in the amplitude-phase form.
\(\boxed{f(t) = a_0 + \sum_{n = 1}^\infty A_n\cos\left(n\omega_{0}t + \phi_n\right)}\)
where:
\(\displaystyle A_n = \sqrt{\left.a_n\right.^2 + \left.b_n\right.^2}\)
\(\displaystyle \phi_n = -\tan^{-1}\left(\frac{b_n}{a_n}\right)\)
\(\displaystyle a_n = A_n\cos(\phi_n)\)
\(\displaystyle b_n = -A_n\sin(\phi_n)\)
To avoid any confusion in determining \(\phi_n\), it may be better to relate the terms in complex form as
\(\boxed{A_n\angle\phi_n = a_n - jb_n}\)
The plot of the amplitude \(A_n\) of the harmonics versus \(n\omega_0\) is called the amplitude spectrum of \(f(t)\); the plot of the phase \(\phi_n\) versus \(n\omega_n\) is the phase spectrum of \(f(t)\). Both the amplitude and and phase spectra form the frequency spectrum of \(f(t)\).
The frequency spectrum of a signal consists of the plots of the amplitudes and phases of the harmonics versus frequency.
Exponential Fourier Series
A compact way of expressing the (trigonometric) Fourier series is to put it in exponential form using Euler’s identity.
\(\displaystyle e^{\displaystyle \pm{jn\omega_{0}t}} = \cos{(n\omega_{0}t)} \pm j\sin{(n\omega_{0}t)}\)
\(\displaystyle \cos{(n\omega_{0}t)} = \frac{1}{2}\left(e^{\displaystyle jn\omega_{0}t} + e^{\displaystyle -jn\omega_{0}t}\right)\)
\(\displaystyle \sin{(n\omega_{0}t)} = \frac{1}{j2}\left(e^{\displaystyle jn\omega_{0}t} - e^{\displaystyle -jn\omega_{0}t}\right)\)
then \(\displaystyle f(t) = a_0 + \sum_{n = 1}^\infty\left[a_n\cos\left(n\omega_{0}t\right) + b_n\sin\left(n\omega_{0}t\right)\right]\) becomes
\(\boxed{f(t) = a_0 + \frac{1}{2}\sum_{n = 1}^\infty\left[\left(a_n - jb_n\right)\,e^{\displaystyle jn\omega_{0}t} + \left(a_n + jb_n\right)\,e^{\displaystyle -jn\omega_{0}t}\right]}\)
The exponential Fourier series of a periodic function \(f(t)\) describes the spectrum of \(f(t)\) in terms of the amplitude and phase angle of AC components at positive and negative harmonic frequencies.
\(\boxed{f(t) = c_0 + \sum_{n = 1}^\infty\left[c_{n}e^{\displaystyle jn\omega_{0}t} + c_{-n}e^{\displaystyle -jn\omega_{0}t}\right] = \sum_{n = -\infty}^\infty c_{n}e^{\displaystyle jn\omega_{0}t}}\)
where:
\(\displaystyle \omega_0 = \frac{2\pi}{T}\)
\(\displaystyle c_0 = a_0\)
\(\displaystyle c_n = \frac{a_n - jb_n}{2}\)
\(\displaystyle c_{-n} = c_{n}^\ast = \frac{a_n + jb_n}{2}\)
The exponential Fourier series coefficients \(c_n\) can also be obtained directly from \(f(t)\).
\(\boxed{c_n = \frac{1}{T}\int_{0}^{T}f(t)\,e^{\displaystyle -jn\omega_{0}t}\,dt}\quad\mathrm{n \ne 0}\)
The coefficients of the three forms of Fourier series (sine-cosine form, amplitude-phase form, and exponential form) are related by
\(\boxed{A_n\angle\phi_n = a_n - jb_n = 2c_n}\)
The plots of the magnitude and phase of \(c_n\) versus \(n\omega_0\) are called the complex amplitude spectrum and complex phase spectrum of \(f(t)\), respectively. The two spectra form the complex frequency spectrum of \(f(t)\).
Parseval’s Relation
Since the Fourier series describes the periodic signal \(f(t)\) as a sum of sinusoids at different frequencies, its signal power also equals the sum of the power in each sinusoid.
\(\boxed{P = a_{0}^2 + \frac{1}{2} \sum_{n = 1}^{\infty} A_{n}^2 = a_{0}^2 + \frac{1}{2} \sum_{n = 1}^{\infty} \left[a_{n}^2 + b_{n}^2\right] = \sum_{n = -\infty}^{\infty} \left|c_n\right|^2 = \sum_{n = -\infty}^{\infty} c_{n}c_{n}^{\ast}}\)
where:
\(\displaystyle c_0 = a_0\)
\(\displaystyle c_n = \frac{a_n - jb_n}{2}\)
\(\displaystyle c_{-n} = c_{n}^\ast = \frac{a_n + jb_n}{2}\)
\(\displaystyle A_n = \sqrt{\left.a_n\right.^2 + \left.b_n\right.^2}\)
\(\displaystyle 2\left|c_n\right|^2 = \frac{a_{n}^2 + b_{n}^2}{2} = \frac{A_{n}^2}{2}\)
\(\displaystyle \left|c_n\right|^2 = c_{n} c_{n}^\ast\)
The equivalence of the time-domain and frequency-domain expressions for the signal power \(P\) forms the so called Parseval’s relation.
\(\boxed{P = \frac{1}{T}\int_{T} \left|f(t)\right|^2\,dt = \sum_{n = -\infty}^{\infty} \left|c_n\right|^2}\)
Parseval’s relation holds only if one period of \(f(t)\) is square integrable (an energy signal).
Finding the power in or up to a given number of harmonics \(N\) requires the harmonic coefficients.
The power \(P_N\) up to the \(N\)th harmonic is simply
\(\boxed{P_N = a_{0}^2 + \frac{1}{2} \sum_{n = 1}^{N} A_{n}^2 = a_{0}^2 + \frac{1}{2} \sum_{n = 1}^{N} \left[a_{n}^2 + b_{n}^2\right] = \sum_{n = -N}^{N} \left|c_n\right|^2}\)
This is a sum of positive quantities and is always less than the total signal power \(P\). As \(N\) increases, \(P_N\) approaches the true signal power \(P\).
Values for Integral Multiples of \(\pi\)
For integer values of \(n\).
| Function | Value |
|---|---|
| \(\displaystyle \cos(2n\pi)\) | \(\displaystyle 1\) |
| \(\displaystyle \sin(2n\pi)\) | \(\displaystyle 0\) |
| \(\displaystyle \cos(n\pi)\) | \(\displaystyle (-1)^n\) |
| \(\displaystyle \sin(n\pi)\) | \(\displaystyle 0\) |
| \(\displaystyle \cos\left(\frac{n\pi}{2}\right)\) | \(\displaystyle \begin{cases}(-1)^{n / 2}\,, & n = \textrm{even} \\ 0\,, & n = \textrm{odd}\end{cases}\) |
| \(\displaystyle \sin\left(\frac{n\pi}{2}\right)\) | \(\displaystyle \begin{cases}(-1)^{(n - 1) / 2}\,, & n = \textrm{odd} \\ 0\,, & n = \textrm{even}\end{cases}\) |
| \(\displaystyle e^{j2n\pi}\) | \(\displaystyle 1\) |
| \(\displaystyle e^{jn\pi}\) | \(\displaystyle (-1)^n\) |
| \(\displaystyle e^{jn\pi / 2}\) | \(\displaystyle \begin{cases}(-1)^{n / 2}\,, & n = \textrm{even} \\ j(-1)^{(n - 1) / 2}\,, & n = \textrm{odd}\end{cases}\) |
Effects of Symmetry on Fourier Coefficients
| Symmetry | \(a_0\) | \(a_n\) | \(b_n\) |
|---|---|---|---|
| Even \(\displaystyle f(t) = f(-t)\) | \(\displaystyle \frac{2}{T}\int_{0}^{T / 2}f(t)\,dt\) | \(\displaystyle \frac{4}{T}\int_{0}^{T / 2}f(t)\cos(n\omega_{0}t)\,dt\) | \(\displaystyle 0\) |
| Odd \(\displaystyle f(-t) = -f(t)\) | \(\displaystyle 0\) | \(\displaystyle 0\) | \(\displaystyle \frac{4}{T}\int_{0}^{T / 2}f(t)\sin(n\omega_{0}t)\,dt\) |
| Half-wave \(\displaystyle f\left(t - \frac{T}{2}\right) = -f(t)\) | \(\displaystyle 0\) | \(\displaystyle \begin{cases}\displaystyle\frac{4}{T}\int_{0}^{T / 2}f(t)\cos(n\omega_{0}t)\,dt\,, & n = \textrm{odd} \\ 0\,, & n = \textrm{even}\end{cases}\) | \(\displaystyle \begin{cases}\displaystyle\frac{4}{T}\int_{0}^{T / 2}f(t)\sin(n\omega_{0}t)\,dt\,, & n = \textrm{odd} \\ 0\,, & n = \textrm{even}\end{cases}\) |
Properties of odd and even functions:
- The product of two even functions is also an even function.
- The product of two odd functions is an even function.
- The product of an even function and an odd function is an odd function.
- The sum (or difference) of two even functions is also an even function.
- The sum (or difference) of two odd functions is an odd function.
- The sum (or difference) of an even function and an odd function is neither even nor odd.
Since odd symmetry and half-wave symmetry are destroyed by a nonzero DC offset, it is best to check for such symmetry only after removing the DC offset \(a_0\) from the signal.