I’m interested in sparse filters, in the sense that you can realize them with only a small number of taps to reduce the number of multiplications, and multiplication-free or multiplication-light filters, in the sense that the nonzero tap coefficients are numbers like 1, 2, 3, 4, or maybe 6 or 8, but not things like 1.03594513, except perhaps in a very few cases.
The Hogenauer cascaded integrator-comb filter is a well-known filter of this class, commonly used for sample-rate conversion. But here are a few other related ideas.
The context for this is that a lot of our filter design lore comes from the world of analog electronics, where multiplication is trivial and memory is hard. This means it is not a good fit for digital computation, where memory is trivial and multiplication is hard, although immense hardware effort has been devoted to papering over this for the benefit of DSP designers.
A CIC filter is low-pass but linear-phase, so you can invert it — subtracting the appropriately scaled low-pass signal from the input sample in the middle of the kernel — to get a high-pass filter. Or you can subtract the outputs of two such filters to get a bandpass filter. This may be particularly useful in combination with undersampling — decimating the bandpass-filtered signal to alias the band of interest down to IF or baseband, thus allowing you to detect a high-frequency signal without doing anything high-frequency except for running some integrators.
A unity-gain negative-feedback comb filter y(n) = x(n) - y(n-k) is an oscillator, and indeed it’s very close to the Karplus-Strong oscillator (which is, in its original form, y(n) = x(n) - ½y(n-k) - ½y(n-k-1), to gradually attenuate higher frequencies). If you compose it with a unity-gain feedforward comb filter in very much the same way you do in a CIC filter, its impulse response is a finite-length alternating impulse train, so it’s a bandpass filter, though it also detects harmonics of the target frequency. A cascade of a few of these approximates a Gaussian window; if you add an actual CIC filter, which you can tune to have nulls at the harmonics of the target frequency, you can get a very inexpensive high-Q filter. As one example, I got a bandpass Q of 17.8 and 38dB stopband attenuation to generate I and Q signals for oscillations with a period of 60 samples using 2.64 additions and subtractions per sample: a two-stage CIC filter with lags of 36 and 40 samples on the front end, three 30-sample feedback combs, and further feedforward combs of 300, 480, and 780 samples. This gives a kernel with a temporal response of about 1000 samples (FWHM) which is a reasonably good approximation to a Gabor filter with Q≈17.
(You’ll note that this is very similar to the previous technique, and the two may be alternatives; with the previous technique, for example, it may be useful to set up the low-pass filter you're inverting to have precise nulls at the harmonics of the signal you're trying to detect.)
More generally, if you can construct a filter whose impulse response is half a wave or more of some waveform you want, you can cascade it with the unity-gain negative-feedback comb filter and get an oscillator for that waveform, or more or less equivalently, a filter that matches it. And you can use the same trick described above to construct a filter whose impulse response is a specific number of oscillations of the waveform.
Anything you can construct by convolving, adding, and subtracting box filters and comb filters can be computed as a sparse filter. So, for example, you can get a difference of gaussians fairly easily.
Once we get into nonlinear filter territory, things get more interesting still. You can do a PLL as a pretty sparse filter, for example.