# Confused about filter's cutoff frequency in rational sampling rate converter

A interpolator which increase sampling rate by L times is followed by an anti-imaging filter, the filter is said to have a cutoff frequency of Fs/2L. Is Fs here is the frequency before or after interpolation?

Also during rational sampling rate conversion, the filter in between the interpolator and decimator is said to have cutoff at min(Fs/2L, Fs/2M). But clearly Fs isn't same for both of them. The interpolator changes the sampling rate before the signal passes through the decimator.

## 1 Answer

I assume you'r talking about digital interpolators.

An interpolator is basically an L times expander followed by an anti-imaging lowpass filter with gain L. The cutoff frquency of the lowpass filter is defined in the discrete-time frequency domain as

$$w_c = \frac{\pi}{L}.$$

The analog equivalent of this cutoff frequency is

$$f_c = F'_s \frac{ w_c }{ 2\pi} = L \times F_s \frac{ \pi / L}{ 2\pi} = F_s / 2.$$

This frequency refers to the increased sampling rate $$F'_s = L \times F_s$$.

A similar argument holds of the rational sample rate conversion system where an expander (by L) is followed by a lowpass filter and then followed by a decimator (by M).

The cutoff frequency of the intermediate lowpass filter is $$\omega_c = \min(\pi/L, \pi/M).$$ Then the analog equivalent of this lowpass filter will be $$f_c = L \times F_s \frac{ \min(\pi/L, \pi/M)}{ 2\pi} .$$