# Design of efficient digital interpolation filter

I came across this paper entitled "Design of Efficient Digital Interpolation Filters and Sigma-Delta Modulator for Audio DAC" where the author oversamples an input frequency, fsig = 1kHz with ratio L = 128 and update frequency, fsi = 64kHz. The interpolation filter specification is given by:

• passband ripple = 0.001dB for frequency <0.45*fsi and
• stopband attenuation = 174dB for frequency > 0.55*fsi.

For two-stage system using IIR Butterworth:

• L1 = 2 and L2 = 64, how to get filter order of N1 = 77 and N2 = 21 respectively?
• L1 = 16 and L2 = 8, how to get filter order of N1 = 120 and N2 = 7 respectively?
• ah thanks Olli, I was blind. – Marcus Müller Aug 5 '19 at 9:08
• is it this paper? – Marcus Müller Aug 5 '19 at 9:29

Butterworth low-pass filters are not going to work for this, as demonstrated in the following. You'd need to use other types of filters, for example linear-phase finite impulse response (FIR).

The magnitude frequency response of an order $$N$$ discrete-time Butterworth low-pass filter can be approximated by that of an analog prototype (from Wikipedia, with a small notation change):

$$G(\omega) = {\frac{1}{\sqrt{1+{\omega}^{2N}}}},$$

with normalized frequency $$\omega$$. The Butterworth filter magnitude frequency response is strictly decreasing. For a pass-band edge gain of $$a = 10^{-0.001/20}$$, that is, -0.001 dB, we'd employ a further gain factor of 0.001 dB to have a gain of 0.001 dB, that is, $$\frac{1}{a}$$ at 0 Hz. With this gain factor the magnitude frequency response is $$\frac{1}{a}G(\omega)$$. Then the transition band begins at the frequency $$\omega_0$$ found by solving:

$$\frac{1}{a}G(\omega_0) = a$$ $$\Rightarrow \omega_0 = \left(\frac{1 - a^4}{a^4}\right)^{\textstyle\frac{1}{2N}}$$

Then we plot the magnitude frequency response at the beginning of the stop band (the end of the transition band) at frequency $$\omega_1 = \frac{0.55}{0.45}\omega_0$$ as function of filter order $$N$$:

Figure 1. $$\frac{1}{a}G(\omega_1)$$ as function of $$N$$.

Based on the graph, -174 dB of stop band corner gain would require a filter order of about 120. I don't think such a high order is feasible for a recursive filter as it will lead to numerical problems. Filter types other than Butterworth will be able to meet the specification with a lower filter order.

• FIY 120 dB is feasible for an IIR filter provided you split it in second-order-sections. – Ben Aug 5 '19 at 13:58
• I meant 174 dB is feasible. – Ben Aug 5 '19 at 14:09
• @Ben: "x dB is feasible" doesn't really mean much if you don't specify the width of the transition band. A first order IIR low pass can have $\infty$ dB attenuation at Nyquist. – Matt L. Aug 5 '19 at 22:11
• Sorry I misread, you're right. – Ben Aug 5 '19 at 23:58