# Practical vs ideal lowpass interpolator

Consider a signal with a sample rate $f_s = 44.1$ kHz. Let us upsample the signal by a factor of $L = 2$ and interpolate the zeros.

An ideal lowpass interpolator would have a gain of $L$ and a cutoff frequency of:

$$f_c = \frac{f_s}{L}$$

An ideal lowpass filter has an infinitesimally small transition band.

In practice I see real lowpass interpolators have a small transition band centred around $f_c$.

The transition band can be quite large, say, $0.45 f_s$ to $0.55 f_s$.

My question is: why do we centre the transition band of a practical lowpass interpolator around the ideal cutoff frequency? By doing that the practical lowpass stopband is above the ideal cutoff which does not make sense to me as that will allow a small unwanted spectral image from the $0.45 f_s$ to $0.50 f_s$ region to creep into the new signal. The obvious alternative is to make the stopband of the practical lowpass $0.5 f_s$ and put up with a passband starting at $0.4 f_s$ assuming we can't make the transition band steeper. There must be some reason this isn't the way it's done.

When designing a filter, you really care about its behavior in two regions:

1. Passband: You want little attenuation in this region, and maybe other properties as well, like linear phase, depending upon your application.

2. Stopband: You want as much attenuation as needed in this region.

Between these two is the transition region. This is treated as somewhat of a "don't-care" band. You don't typically constrain the response too tightly in this area, so you can't really count it being usable. In your example, the passband lies below $0.45 f_s$; after the interpolator, you only plan on using frequency content below this threshold.

This means that you can allow some aliasing in order to simplify your filter design. Your transition region starts at $0.45 f_s$; everything above that frequency in your filter's output will either have a response that is unpredictable given your filter specs (if it lies in the transition band) or one that is highly attenuated (if it lies in the stopband). The takeaway:

You can't rely upon any frequency content past the passband edge in your filter output anyway. So if it makes the filter design cheaper, why not allow the frequencies above the passband edge in the filter output to contain aliased garbage?

This technique is commonly used in multirate filters as you've noticed, as it allows savings in the required filter order in order to meet a given set of passband/stopband specifications.