The sampling rate of a digital signal can be easily be increased by an integer multiple--say, M--by adding M-1 zeros between each sample, then low-pass filtering with a digital cutoff frequency of π/M and a gain of M.

Derivation

Let x(n) be the signal we wish to upsample (interpolate), and define

y(k) = { x(k/M) , when m/M is an integer
       { 0        otherwise

Evaluating the z-transform of y(k),

Y(z) = Σ_ky(k)*z^-m

but y(m) is zero for all non-integer k/I, so

Y(z) = Σ_kx(k)*z^-kM = X(z^M)

The DTFT of y is computed by evaluating Y(z) on the unit circle (z = e^jΩ). Therefore, Y(Ω) = X(ΩM).

Let's look at what's going on so far in the frequency domain. Suppose M=3, and X has the spectrum given below with maximum value 1.

               |X(Ω)|
                  |
                 *1*
              *** | ***
           ***    |    ***      
        ***       |       ***
     ***          |          ***
  ***             |             ***
 -|-------|-------+-------|-------|--> Ω
-π      -π/2      0      π/2      π

               |Y(Ω)|
                 |
      *          1          *
     * *        *|*        * *
    *   *      * | *      *   *
   *     *    *  |  *    *     *
  *       *  *   |   *  *       *
 *         **    |    **         *
 |-------|-------+-------|-------|--> Ω
-π     -π/2      0      π/2      π

Y(Ω) contains M spectral images of X. What we want, however, is just the middle image, the one shown in bold in the figure above. This can clearly be extracted by low-pass filtering Y to remove frequencies above π/M, yielding:

               |Y(Ω)|
                 |
                 1     
                *|*    
               * | *   
              *  |  *  
             *   |   * 
            *    |    *
 |-------|-------+-------|-------|--> Ω
-π     -π/2      0      π/2      π

One step remains: adjusting the scaling of y. It is convenient to compare y(0) to x(0):

y(0) = ∫_ΩY(Ω) dΩ = (1/M) ∫_ΩX(Ω) dΩ = x(0)/M

Thus, to keep y(k) scaled correctly, we must multiply it by M:

               |Y(Ω)|
                 |
                 *- M     
                *|*    
               * | *   
              *  |  *  
             *   |   * 
            *    |    *
 |-------|-------+-------|-------|--> Ω
-π     -π/2      0      π/2      π