Puzzled by CIC filters

Question

I have been trying to wrap my head around CIC filters.

Say that I want to do filtered decimation by a factor D.

Option#1: boxcar filter. Each output sample is formed by taking the sum of D input samples, and multiplying/shifting to get a proper average. D-1 adds per output sample.

Option#2: 1-stage CIC filter with an accumulator running at the input rate, and a comb filter running at the output rate. D adds per output sample.

So what is the fuzz about? Does CIC only make sense for higher orders? If order N CIC is like convolving the rect() function N-1 times, that would be a B-spline, right? I have seen authors praising B-splines for their functional benefits. Being able to do them efficiently might be handy.

Thanks for any input Knut

Edit: pseudo matlab code showing the alternatives as I understand them.

N=100;   
x = randn(N,1);    
D = 5;

%% decimated moving average, ~100 adds
for k = 1:N/D    
  y(k) = (1/D)*(x(k*D) + x(k*D-1) + ... + x(k*D-D+1));
end

 %% recursive moving average, ~200 adds 
y(1) = x(1);

for k = 2:N   
  y(k) = x(k) + y(k-1); 
end

for k = D:N
    z(k) = y(k) - y(k-D);  
end

z = z(1:D:end);

%% CIC, ~120 adds
y(1) = x(1);

for k = 2:N
  y(k) = x(k) + y(k-1);  
end

z = y(1:D:end);

for k = 2:N/D    
  w(k) = z(k) - z(k-1);   
end

Answer 1

It seems you mixed up CIC filters and recursive moving-average

A recursive moving-average uses a comb-filter and an accumulator.

$$ H(z) = \frac{1}{N}\frac{1-z^{-N}}{1-z^{-1}} $$

The numerator corresponds to a comb filter (i.e a delay line of N elements and 1 adder). The denominator corresponds to an accumulator. The 1/N factor is to get a moving-average, otherwise you'd have a moving-accumulator. You only need 2 additions, a delay line, an accumulator and a divider. With a non-recursive moving-average filter, you'd need N additions and you'd need a delay line anyway. A recursive moving-average is much less computionally expansive than a non-recursive moving-average filter.

Now the CIC filter combines the recursive moving-average filter with decimation and interpolation.

For a CIC decimation filter, you insert the rate transition block between the accumulator and the comb filter. Say you have a 16-sample moving average-filter (N=16) and you want to decimate by 8 (R = 8). You'd need a delay line of 16 elements. However, by inserting the decimation between the accumulator and the comb filter, you can reduce the number of elements in the delay line. You'd only need N/R elements, i.e. 2. You'd still need to subtract 2 numbers, however you would do it at sampling rate of fs/8.

I suggest you consult this site

https://www.researchgate.net/publication/228648728_Understanding_cascaded_integrator-comb_filters

Update: see this nice write-up by Richard Lyons- https://www.dsprelated.com/showarticle/1337.php