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.

Answer 2

Here's how I like to think about the expression

$$ \displaystyle \frac{1}{D} \sum_{k=0}^{D-1} e^{j 2 \pi k m / D}. $$

1. For any value of $D$, the first term is 1, and all of the terms lie on the unit circle.
2. Assuming $D > 1$, each term after the first is found by rotating the previous term through an angle of $2 \pi m / D$ radians.
3. If $m$ is not a multiple of $D$, then the resulting terms will be $D$ complex numbers that are evenly distributed about the unit circle, and will sum to zero. This can be shown explicitly using the finite geometric series. $$ \sum_{k=0}^{D-1} e^{j 2 \pi k m / D} = \frac{1 - e^{j 2 \pi D m / D}}{1 - e^{j 2 \pi m / D}} = \frac{1 - 1}{1 - e^{j 2 \pi m / D}} = 0. $$
4. If $m$ is a multiple of $D$, all of the terms in the sum will be 1, so the sum will be $D$, and the expression evaluates to 1.

Therefore, this sum is equivalent to $s_d[m]$, and as the previous answer correctly stated, it is the discrete-time Fourier series representation for a periodic impulse train or comb function.