Four properties of DTFS

Let $x[n]$ and $y[n]$ be periodic with the same period $N$, $\omega_0 = 2\pi/N$, with $x[n] \overset{FS}{\longleftrightarrow} a_k$ and $y[n] \overset{FS}{\longleftrightarrow} b_k$, where $a_k = \frac{1}{N}\sum_{n=\langle N\rangle}x[n]e^{-jk\omega_0 n}$.

1. Linearity

$$A\,x[n] + B\,y[n] \overset{FS}{\longleftrightarrow} A a_k + B b_k$$

The analysis sum is linear, so the coefficients of a weighted sum are the same weighted sum of coefficients. Example: $x[n] = 2\cos(\omega_0 n) + 3$ has $a_0 = 3$, $a_{\pm1} = 1$.

2. Time shifting

$$x[n - n_0] \overset{FS}{\longleftrightarrow} a_k e^{-jk(2\pi/N)n_0}$$

Proof: with $m = n - n_0$, $\frac{1}{N}\sum_n x[n-n_0]e^{-jk\omega_0 n} = e^{-jk\omega_0 n_0}\frac{1}{N}\sum_m x[m]e^{-jk\omega_0 m}$. A shift leaves $|a_k|$ unchanged and adds a linear phase.

3. Frequency shifting

$$e^{jM(2\pi/N)n}x[n] \overset{FS}{\longleftrightarrow} a_{k-M}$$

Multiplying by the $M$th harmonic exponential shifts the coefficients by $M$ (used in modulation).

4. Parseval's relation

$$\frac{1}{N}\sum_{n=\langle N\rangle}|x[n]|^2 = \sum_{k=\langle N\rangle}|a_k|^2$$

The average power in one period equals the sum of the powers of the $N$ harmonics.

Other properties (for reference)

A property special to the DTFS is that the coefficients are themselves periodic: $a_{k+N} = a_k$.