Filter slope of multipass moving average

Question

Hey guys,
here is a picture of the frequency response of a multipass moving average filter. The number of stages tells you how often the data is passed through the filter.
As I am interested in the different slope behaviours of the different stages I wanted to ask you how to determine the slope.
Do I take the envelope to calculate the roll off (=slope) of the filter?
Actually it should be 1 pass (= 1 stage) is 20dB/Dekade, 2 passes = 40dB/dekade and so on...
I hope you can help me.
Kind regards
Slev1n
 

Answer 1

The equation for the envelope of the frequency response can be obtained by noting that the absolute value of the numerator sin(wM/2) varies between 0 and 1.  So its upper edge is 1, and you can replace sin(wM/2) with unity, to find the upper envelope of H(w). (This envelope only applies for wM/2>Pi, which is the first zero of the numerator.)  Thus you get
envelope of H(w)= 1/[M*sin(w/2)]^Mp. 
Note also that when w/2<<1, you can use the approximation sinx ~= x:
envelope of H(w) ~= 1/[M*w/2]^Mp
Therefore the envelope of H(w) is proportional to 1/w, 1/w^2, 1/w^3 when Mp=1,2,3.  Now you see why the envelope's slope is 20, 40, 60 dB/decade when Mp=1,2,3.  (Apply the equation for gain in dB, when w changes by 10X.)  The approximation above is decent only when 2Pi/M < w << 1.  If the number of moving average points, M, is small, for example a 5 point moving average, then there will be no frequencies which satisfy the preceding inequality for w.  But you chose M=500, so there is a range of frequencies where the 20 dB/decade rolloff is apparent.  If you look carefully at your plot, you will see that the frequency response starts to deviate from 20 dB/decade as frequency f (=w/2Pi) approaches 0.5 (the Nyquist frequency).  This is because you are getting into the range where sin(x)~=x is not accurate.
Final comment: Since you apparently care about the frequency response properties of your lowpass filter, you might want to use a different filter.  Other FIR and IIR filters have flatter passbands, sharper rolloff around the cutoff frequency, and lower sidebands.