Analysis of the ratio of two moving average filters

Question

I have created a basic segmentation algorithm for 1-D signals (e.g. audio) based on a ratio of energy averages. The classification rule involves comparing the signal energy measured over a short time window divided by that over a longer window, with a fixed threshold.

More precisely, if $x[n] > 0$ is the instantaneous signal power at index $n$, then the quantity described above is:

$$ y[n] = \frac{\frac{1}{K} \displaystyle\sum_{i = 0}^{K-1} x[n-i]}{\frac{1}{L}\displaystyle\sum_{j = 0}^{L-1} x[n-j]} = \frac{L}{K} \left(1 + \frac{\displaystyle\sum_{j=K}^{L-1} x[n-j]}{\displaystyle\sum_{i=0}^{K-1} x[n-i]}\right)^{-1},$$ for fixed integers $K$, $L$ with $L \gg K > 1.$

I am interested in the behaviour of this nonlinear system, since it seems quite useful in practice. Can anyone suggest a way to analyse it theoretically?

Answer 1

This depends a bit on what you want to get our of the analysis

Step 1: Instantaneous power

You square the signal. That changes the spectrum considerably. In particularly you need to watch out for aliasing. Let's say your signal is a 15kHz sine wave sampled at 48 kHz. If you square this you get one component at DC (0 Hz) and another at 30 kHz. However, the 30 kHz component will alias down to 18 kHz, so your squared spectrum is outright wrong in this case. To be safe, you should either up-sample the signal by a factor of 2 before squaring OR lowpass filter it at a quarter of the sample rate. Many "natural" audio signals don't have a lot of energy at high frequencies, so could also choose to just live with the aliasing error, but that's an application specific decision.

The physical interpretation of instantaneous power is complicated. Let's say your input signal is the voltage at the terminals of a loudspeaker: Squaring it does NOT represent the instantaneous power. The actual instantaneous power is the product of the voltage and the current waveform. Since a loudspeaker is a reactive load, this will be quite different from the squared voltage. For example, it's perfectly normal for the instantaneous power to be negative which the square of a signal can never be. Negative instantaneous power simple means that the load has energy storage and that at times it can feed stored energy back into the source.

Going back to the sine wave example. A squared sine creates a component at DC and one at double the frequency. One possible interpretation: the DC component reflects the actual power consumed, the double frequency component represents power that gets exchanged between the source and the load but not consumed: it just gets shoved back and forth.

Provided that the signal is the "actual" instantaneous power, than the mean of it will represent the average power and the integral will represent the real energy consumed (exactly).

Step 2: Moving average

Moving average is just a low pass filter where the length of the averaging window determines the "cutoff" frequency. Unfortunately it's a rather poor one. It attenuates the "higher" frequency in the spectrum but very unevenly. Personally I would assume that in most cases a simple IIR filter would work much better.

A low pass filter will eliminate the "back and forth components" of the power signal but keep the "actual consumption". The choice of cutoff frequency (or time constant) should be tied to the physical process you want to observe. I.e. could be to time constant of the power supply storage or the thermal time constant of the system.

Step 3: Dividing averages for different time constants

This creates the ratio of "short term power " vs "long term power" where you can define "short" and "long" by the cutoff frequencies of your low pass filter. Whether that's useful or not depends on your application. It's a relative metric: it shows how "spikey" your energy consumption is.

Since real instantaneous power can be negative, it's conceivable that the lowpass filtered version can be negative or zero as well, which can make the division problematic.

Answer 2

This technique is similar to a classical signal processing in seismic being an intendance of "first break picking". It was proposed in First arrival picking on common-offset trace collections for automatic estimation of static corrections, Françoise Coppens, 1985 (a former colleagues of mine). You may find the name STA/LTA (short-term average/long-term average). Other details at History of First Break Picking.

I am providing this reference for two reasons:

• this may put you on track to other techniques applicable to your signal.
• as your signal is low-pass, you may be interested in their continuous formalism, that may better exhibit some features your are looking for: $$ F(\tau) = \frac{\int_{\tau-L}^{\tau}S^2(t)\mathrm{d}t}{\int_{0}^{\tau}S^2(t)\mathrm{d}t}$$

There are many variations like using different windows sizes, transforming the variables, for instance with powers.

$$ F(\tau) = \frac{\int_{\tau-L}^{\tau}|S|^p(t)\mathrm{d}t} {\int_{0}^{\tau}|S|^p(t)\mathrm{d}t}$$

I don't have in mind a theoretical analysis, but I can look it up. I would also suggest to invest in the literature on onset detection in audio or music, for instance the overview in:

• A Tutorial on Onset Detection in Music Signals (IEEE Transactions on speech and audio processing)

and on change detection.