# Two stage interpolation : Optimum Order

Started by January 28, 2008
```Hello Rick,

I worked out the math and here is the solution. For two stage
interpolation I = I1 x I2, the optimum value of I1 can be given
by

I1 = (fp + fs + sqrt(I*Fs*(Fs - fs - fp)))/Fs

where various quantities are:

fp = passband freq in hz
fs = stopband freq in hz
Fs = sampling freq at the input side.
I  = overall interpolation ratio.

For example if we take

fp = 1khz, fs = 3khz, Fs = 8khz and I = 128 and substitute in above
equation we get I1 = 8.5, (nearest integer value and also a value
which is ratio of 128 is 8). Hence 1st stage interpolation should be
by a factor of 8 and second stage interpolation should be by a factor
0f 128/8 = 16.

I1 = 8;
I2 = 16;

This will minimize the number of taps for both filters put together.
The above equation has been arrived using Bellangers formula to
compute
filter order. More accurate equation can be derived using Hermann
order
equation but that procedure will be very cumbersome and might not be
useful for hand calculations. Sometimes approximations help.

Regards
Bharat Pathak

Arithos Designs
~dsp simplified.

```