Derivation of downsampling in the frequency domain

Question

I'm having some trouble with this derivation

$$s_d = \begin{cases}1&\text{for}\quad m\quad\text{multiples of}\quad D\\ 0 &\text{otherwise} \end{cases}$$

then $s_d$ is somehow rewritten to:

$$\frac{1}{D}\sum_{k=0}^{D-1}e^{jk2\pi \frac{m}{D}} $$

Can someone explain this?

EDIT:

Basically the same derivation as done here Frequency Representation of Downsampled Signal

Answer 1

The expression

$$s_d[m] = \frac{1}{D}\sum_{k=0}^{D-1}\exp(j2\pi \frac{km}{D})$$

is the Fourier series expansion of the periodic function $s_d[m]$ in the discrete case (i.e. only up to a certain amount of frequencies).

$s_d[m]$ is a also called comb function, and I have written more about it (including the relation you ask for) in one of my articles.

Furthermore, here's some example Python code that verifies the correctness:

m = np.arange(64)
D = 16
sdm = 1/D*sum(np.exp(2j*np.pi*k*m/D) for k in range(D))
plt.plot(m, sdm.real)

The picture shows the Comb function, where every 16 samples the values becomes one, else zero.