Filter design with 8-tap latency on recursion I'm curious about the feasibility of designing a noise shaping filter (but might generalise to any recursive filter) with the constraint that the most recent output samples aren't available for several iterations of the filter.

The use case I have in mind is reducing the word length of an audio stream sampled at 48kHz, and the filter would try to keep most of the quantisation noise out of the 200Hz-8kHz band (where human hearing is more sensitive); but with the additional constraint that the most recent seven outputs are not available to the filter.

I'm applying this constraint so that the output can be produced several samples at a time using a mixture of SIMD and instruction-level parallelism -- working with a notional four-lane SIMD and two-fold unrolling.

There's a lot of work for each sample which could be performed in parallel (scaling, quantisation, dither) if not for the recursive factor.  Normally one might hope to use SIMD for the multiply-accumulate of the coefficients, but that's not a great proportion of the work if the filter has only a handful of taps.

So I'm wondering if it's actually possible to design a filter that doesn't get in the way, here, but is still effective for its intended purpose.

Stereo would obviously offer an opportunity to halve the delay, but avoiding that assumption is preferable, as it makes mono a failure case and multi-channel becomes complicated.
This is tricky. Any filter with 8 tab latency between coefficients can be represented in the z-domain as a rational function in $z^{-8}$ . That's basically the same as saying the impulse response has 7 zeros between each non zero tap. Any filter with this property is periodic in the frequency domain. You can pick any transfer function from -pi/8 to pi/8 but the regions from pi/8 to 3*pi/8,etc. will just be repetitions of that.

You may be able to do better with varying delays but this is a complicated design problem