Down-sampling: 

Consider that x(n) is calculate from an underlying continuous-time signal x(t) by sampling at Fx Hz. Suppose that x(t) was originally band limited to Fx/2 Hz. On the digital frequency (ω) scale that values to x(n) being band limited to π.

We now wish to create a signal y(n) by down-sampling x(n) by a factor of M, that is, we are replacing the sampling rate by a factor of M. That values to:

1. Changing x(n) to x(t) using a D/A converter.

2. Band limiting x(t) to Fx/2M Hz. Suppose that no information is lost because of this band limiting.

3. Resampling x(t) at Fx/M Hz. to provide y(n).

Equivalently the above task is completed entirely in the digital part by

1. Band limiting x(n) to π/M. Suppose that no information is lost because of that step.

2. Down-sampling the above x(n) by a type of M to give y(n).

We can view y(n) as though it were prepared by sampling an underlying analog signal y(t) at a rate Fy = Fx/M Hz.  Provided the signal x(n) that was calculated at a certain sampling rate the new signal y(n), the down-sampled version of x(n), with a sampling rate that is (1/M) of that of x(n), calculated from x(n), is provided by:

 y(n) = x(Mn)

and is build up of every M^th sample value of x(n); the intervening (M-1) sample numbers of x(n) are changed. That amounts to

 y(0) = x(0), y(1) = x(M), y(2) = x(2M), y(3) = x(3M), ...

The time between samples of y(.) is M times that between samples of x(.), or the sampling frequency of y(.) is replaced by a factor of M from that of x(.). The block diagram of a down sampler is define below.

Example As an example, if x(n) = aⁿu(n) , a < 1, is the sequence:

n = 0 1 2 3 4 5 6 7 8

x(n) = {1 a² a³ a⁴ a⁵ a⁶ a⁷ a⁸ a⁹ . . .}

then y(n) = x(2n), with M = 2, is its 2-fold down-sampled version and is obtained by keeping every other sample of x(n) and dropping the samples in between:

n = 0 1 2 3 4 5

y(n) = {1 a² a⁴ a⁶ a⁸ a¹⁰ . . .}

In this example it is understood that the time between samples of y(n) is twice that between samples of x(n), or, the sampling rate of y(n) is one-half of that of x(n).

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