FS coefficients; time scaling and conjugation

Significance of Fourier series coefficients

The coefficients $a_k = \frac{1}{T}\int_T x(t)e^{-jk\omega_0 t}dt$ describe the signal in the frequency domain:

• $|a_k|$ gives the amplitude and $\angle a_k$ the phase of the $k$th harmonic (frequency $k\omega_0$); plotting them gives the magnitude and phase spectra.
• $a_0$ is the dc (average) value of the signal.
• $|a_k|^2$ is the power in the $k$th harmonic, and by Parseval $\sum|a_k|^2$ is the total average power.
• They show the bandwidth of the signal: how fast $|a_k|$ falls tells how many harmonics are needed (smooth signals need fewer).
• An LTI system acts on each harmonic separately: the output coefficients are $a_k H(jk\omega_0)$, which makes system analysis simple.

Time scaling property

Statement: if $x(t)$ has period $T$, $\omega_0 = 2\pi/T$, and $x(t) \overset{FS}{\longleftrightarrow} a_k$, then for $\alpha > 0$, $x(\alpha t)$ has period $T/\alpha$ and

$$x(\alpha t) = \sum_{k=-\infty}^{\infty}a_k e^{jk(\alpha\omega_0)t}$$

i.e. the coefficients are unchanged but the fundamental frequency becomes $\alpha\omega_0$.

Proof: let $y(t) = x(\alpha t)$, period $T' = T/\alpha$; put $\tau = \alpha t$:

$$\begin{aligned} b_k &= \frac{1}{T'}\int_0^{T'}x(\alpha t)e^{-jk\alpha\omega_0 t}dt \\ &= \frac{\alpha}{T}\int_0^{T}x(\tau)e^{-jk\omega_0\tau}\frac{d\tau}{\alpha} = a_k \end{aligned}$$

Conjugation property

Statement:

$$x^*(t) \overset{FS}{\longleftrightarrow} a_{-k}^*$$

Proof: take the conjugate of the synthesis equation:

$$\begin{aligned} x^*(t) &= \left[\sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0 t}\right]^* = \sum_{k=-\infty}^{\infty}a_k^* e^{-jk\omega_0 t} \\ &= \sum_{m=-\infty}^{\infty}a_{-m}^*\,e^{jm\omega_0 t} \qquad (m = -k) \end{aligned}$$

So the coefficient of $e^{jm\omega_0 t}$ in $x^*(t)$ is $a_{-m}^*$.

Consequence (conjugate symmetry): if $x(t)$ is real, $x(t) = x^*(t)$, so $a_k = a_{-k}^*$. Hence $|a_k| = |a_{-k}|$ (even magnitude spectrum) and $\angle a_{-k} = -\angle a_k$ (odd phase spectrum). If $x(t)$ is also even, $a_k$ is real and even.