Frequency shifting property of DTFS

Statement

If $x[n]$ is periodic with period $N$ and $x[n] \overset{FS}{\longleftrightarrow} a_k$, then for an integer $M$:

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

Multiplying by a complex exponential at the $M$th harmonic shifts the coefficient sequence by $M$.

Proof

$y[n] = e^{jM(2\pi/N)n}x[n]$ is also periodic with period $N$ (both factors are). Its coefficients are

$$\begin{aligned} b_k &= \frac{1}{N}\sum_{n=\langle N\rangle}e^{jM(2\pi/N)n}x[n]\,e^{-jk(2\pi/N)n} \\ &= \frac{1}{N}\sum_{n=\langle N\rangle}x[n]\,e^{-j(k-M)(2\pi/N)n} \\ &= a_{k-M} \end{aligned}$$

Example: for $N = 4$, multiplying by $e^{j\pi n/2}$ ($M = 1$) moves $a_0$ to position $k = 1$, $a_1$ to $k = 2$, and so on (cyclically, since $a_k$ has period 4).