The Fourier Expansion

Posted on Edited on In Views:

Sources:

  1. B. P. Lathi & Roger Green. (2021). Chapter 3: Signal Representation by Fourier Series. Signal Processing and Linear Systems (2nd ed., pp. 261-277). Oxford University Press.

Trigonometric Fourier Expansion

After showing that (see my past post) the set of sinusoids (called the trigonometric set) in \[ \begin{aligned} & \left\{1, \cos \omega_0 t, \cos 2 \omega_0 t, \ldots, \cos n \omega_0 t, \ldots,\right. \\ & \left.\sin \omega_0 t, \sin 2 \omega_0 t, \ldots, \sin n \omega_0 t, \ldots\right\} \end{aligned} \] is a complete set over any contiguous interval of duration \(T_0\).

We can therefore express an arbitary[^3] signal \(x(t)\) by a trigonometric Fourier series1 over any interval of duration \(T_0\) seconds as \[ \begin{aligned} x(t)= & a_0+a_1 \cos \omega_0 t+a_2 \cos 2 \omega_0 t+\cdots \\ & +b_1 \sin \omega_0 t+b_2 \sin 2 \omega_0 t+\cdots \end{aligned} \] or

\[ \color{red} {x(t)= {a_0}+\sum_{n=1}^{\infty} a_n \cos n \omega_0 t+ \sum_{n=1}^{\infty} b_n \sin n \omega_0 t}, \quad t_1<t<t_1+T_0 \] where \[ \omega_0=2 \pi f_0=\frac{2 \pi}{T_0} \] and \[ a_0=\frac{1}{T_0} \int_{T_0} x(t) d t, \quad a_n=\frac{2}{T_0} \int_{T_0} x(t) \cos n \omega_0 t d t, \quad \text { and } \quad b_n=\frac{2}{T_0} \int_{T_0} x(t) \sin n \omega_0 t d t \]

  • Terminologies:
    • A sinusoid of frequency \(n \omega_0\) is called the \(n\)th harmonic of the sinusoid of frequency \(\omega_0\) when \(n\) is an integer. In this set, the sinusoid of frequency \(\omega_0\), called the fundamental, serves as an anchor of which all the remaining terms are harmonics.

NOTE: In signal processing, we use \(x(t)\) to represent the original signal under any interval duration \(T_0\) (Not the original signal, just one interval duration of it!), and we also use it to represent the Fourier series of the former. This is not confusing since they are equal. Unless otherwise notified, in my posts, the \(x(t)\) always refers to the Fourier series. In this sense, \(x(t)\) is always periodic (proved later), while the original signal over its whole duration may not be periodic.

NOTE: See the existence of FE.

The Effect of Symmetry

Recall from Even and Odd Functions that every signal \(x(t)\) can be expressed as a sum of even and odd components because \[ x(t)=\underbrace{\frac{1}{2}[x(t)+x(-t)]}_{\text {even }}+\underbrace{\frac{1}{2}[x(t)-x(-t)]}_{\text {odd }} \] For a Fourier series \[ x(t)= {a_0} +\sum_{n=1}^{\infty} a_n \cos n \omega_0 t+b_n \sin n \omega_0 t , \] we obtain

  1. Even part \(x_e(t)\) : \[ \color{green} {x_e(t)= {a_0} +\sum_{n=1}^{\infty} a_n \cos n \omega_0 t} . \]
  2. Odd part \(x_o(t)\) : \[ \color{brown} {x_o(t)=\sum_{n=1}^{\infty} b_n \sin n \omega_0 t} . \]

For this reason, we can easily see that

  1. If \(x(t)\) itself is even, then \(x(t) = x_e(t)\), we have \[ x(t)={a_0}+\sum_{n=1}^{\infty} a_n \cos n \omega_0 t . \]

  2. If \(x(t)\) itself is odd, then \(x(t) = x_o(t)\), we have \[ x(t) = \sum_{n=1}^{\infty} b_n \sin n \omega_0 t . \]

Periodicity of the Trigonometric Fourier Series

We now use \(x(t)\) to represent the trigonometric Fourier series of the original signal under any interval duration \(T_0\). Now we prove that \(x(t)\) is periodic with the same period as \(T_0\).

(Note that, the original signal, usually also denoted as \(x(t)\), may not be periodic. Here we only talk about its trigonometric Fourier series.)

Proof: \[ \begin{aligned} x\left(t+T_0\right) & =a_0+\sum_{n=1}^{\infty} a_n \cos n \omega_0\left(t+T_0\right)+b_n \sin n \omega_0\left(t+T_0\right) \\ & =a_0+\sum_{n=1}^{\infty} a_n \cos \left(n \omega_0 t+n \omega_0 T_0\right)+b_n \sin \left(n \omega_0 t+n \omega_0 T_0\right) \end{aligned} \]

We know that \(n \omega_0 T_0=2 \pi n\), and \[ \begin{aligned} x\left(t+T_0\right) & =a_0+\sum_{n=1}^{\infty} a_n \cos \left(n \omega_0 t+2 \pi n\right)+b_n \sin \left(n \omega_0 t+2 \pi n\right) \\ & =a_0+\sum_{n=1}^{\infty} a_n \cos n \omega_0 t+b_n \sin n \omega_0 t=x(t) \end{aligned} \] Since \(x(t)\) is periodic, it's integral \(\color {orange} {\int_{T_0} f(x) d x}\) of duration \(T_0\) can start from any instant \(t\) (see the proof).

Compact Trigonometric Fourier Series

# TODO

The results derived so far are general and apply whether \(x(t)\) is a real or a complex function of \(t\). However, when \(x(t)\) is real, coefficients \(a_n\) and \(b_n\) are real for all \(n\), and the trigonometric Fourier series can be expressed in a compact form, using the results in Eq. (B.16): \[ \begin{equation} \label{eq_6_9} x(t)= {C_0} +\sum_{n=1}^{\infty} C_n \cos \left(n \omega_0 t+\theta_n\right) \end{equation} \] where \(C_n\) and \(\theta_n\) are related to \(a_n\) and \(b_n\), as [see Eq. (B.17)] \[ C_0=a_0, \quad C_n=\sqrt{a_n^2+b_n{ }^2}, \quad \text { and } \theta_n=\arctan \left(\frac{-b_n}{a_n}\right) \]

img

The Fourier Spectrum

The compact trigonometric Fourier series in Eq. \(\eqref{eq_6_9}\) indicates that:

A periodic signal \(x(t)\)2 can be expressed as a sum of sinusoids of frequencies \[ 0(\mathrm{dc}), \omega_0, 2 \omega_0, \ldots, n \omega_0, \ldots , \] whose amplitudes are \[ C_0, C_1, C_2, \ldots, C_n, \ldots , \]

and whose phases are \[ 0, \theta_1, \theta_2, \ldots, \theta_n, \ldots , \]

respectively.

Here, we use the term spectrum to refer the representation of a signal in terms of its constituent sinusoidal signals (computed by the Fourier transform) in frequency domain.

In frequency domain, a sinusoidal signal of frequency \(n \omega_0\), say \(C_n \cos \left(n \omega_0 t+\theta_n\right)\), is charaterized by its amplitude \(C_n\) and phase \(\theta_n\).

We use two charts to show the signal:

  1. The amplitude spectrum, which is the amplitude \(C_n\) versus \(n\) (the amplitude spectrum).
  2. The phase spectrum, which is the \(\theta_n\) versus \(n\) (the phase spectrum).

The two plots together are the frequency spectra of \(x(t)\).

Appendix: Addition of Sinusoids

Two sinusoids having the same frequency but different phases add to form a single sinusoid of the same frequency. This fact is readily seen from the well-known trigonometric identity \[ C \cos \theta \cos \omega_0 t-C \sin \theta \sin \omega_0 t=C \cos \left(\omega_0 t+\theta\right) \]

Setting \(a=C \cos \theta\) and \(b=-C \sin \theta\), we see that \[ a \cos \omega_0 t+b \sin \omega_0 t=C \cos \left(\omega_0 t+\theta\right) \]

From trigonometry, we know that \[ C=\sqrt{a^2+b^2} \quad \text { and } \quad \theta=\tan ^{-1}\left(\frac{-b}{a}\right) \]

Equation (B.17) shows that \(C\) and \(\theta\) are the magnitude and angle, respectively, of a complex number \(a-j b\). In other words, \(a-j b=C e^{j \theta}\). Hence, to find \(C\) and \(\theta\), we convert \(a-j b\) to polar form and the magnitude and the angle of the resulting polar number are \(C\) and \(\theta\), respectively.

The process of adding two sinusoids with the same frequency can be clarified by using phasors to represent sinusoids. We represent the sinusoid \(C \cos \left(\omega_0 t+\theta\right)\) by a phasor of length \(C\) at an angle \(\theta\) with the horizontal axis. Clearly, the sinusoid \(a \cos \omega_0 t\) is represented by a horizontal phasor of length \(a(\theta=0)\), while \(b \sin \omega_0 t=b \cos \left(\omega_0 t-\pi / 2\right)\) is represented by a vertical phasor of length \(b\) at an angle \(-\pi / 2\) with the horizontal (Fig. B.7). Adding these two phasors results in a phasor of length \(C\) at an angle \(\theta\), as depicted in Fig. B.7. From this figure, we verify the values of \(C\) and \(\theta\) found in Eq. (B.17). Proper care should be exercised in computing \(\theta\), as explained on page 8 ("A Warning About Computing Angles with Calculators").

  1. This post mainly involves the trigonometric Fourier expansion. There are other forms of Fourier expansion. Like the exponential Fourier expansion.↩︎

  2. As mentioned before, unless otherwise told, the notation "\(x(t)\)" always refers to the trigonometric Fourier series of the original signal under any interval duration \(T_0\). Meanwhile, we have proved that this \(x(t)\) is always periodic.↩︎