# Derivation of downsampling in the frequency domain

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

• Where in the derivations are you having trouble specifically? – Dan Boschen Apr 18 '17 at 10:42

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.

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.