Numpy Root-Mean-Squared (RMS) smoothing of a signal

Question

I have a signal of electromyographical data that I am supposed (scientific papers' explicit recommendation) to smooth using RMS.

I have the following working code, producing the desired output, but it is way slower than I think it's possible.

#!/usr/bin/python
import numpy
def rms(interval, halfwindow):
    """ performs the moving-window smoothing of a signal using RMS """
    n = len(interval)
    rms_signal = numpy.zeros(n)
    for i in range(n):
        small_index = max(0, i - halfwindow)  # intended to avoid boundary effect
        big_index = min(n, i + halfwindow)    # intended to avoid boundary effect
        window_samples = interval[small_index:big_index]

        # here is the RMS of the window, being attributed to rms_signal 'i'th sample:
        rms_signal[i] = sqrt(sum([s**2 for s in window_samples])/len(window_samples))

    return rms_signal

I have seen some deque and itertools suggestions regarding optimization of moving window loops, and also convolve from numpy, but I couldn't figure it out how to accomplish what I want using them.

Also, I do not care to avoid boundary problems anymore, because I end up having large arrays and relatively small sliding windows.

Thanks for reading

Answer 1

It is possible to use convolution to perform the operation you are referring to. I did it a few times for processing EEG signals as well.

import numpy as np
def window_rms(a, window_size):
  a2 = np.power(a,2)
  window = np.ones(window_size)/float(window_size)
  return np.sqrt(np.convolve(a2, window, 'valid'))

Breaking it down, the np.power(a, 2) part makes a new array with the same dimension as a, but where each value is squared. np.ones(window_size)/float(window_size) produces an array or length window_size where each element is 1/window_size. So the convolution effectively produces a new array where each element i is equal to

(a[i]^2 + a[i+1]^2 + … + a[i+window_size]^2)/window_size

which is the RMS value of the array elements within the moving window. It should perform really well this way.

Note, though, that np.power(a, 2) produces a new array of same dimension. If a is really large, I mean sufficiently large that it cannot fit twice in memory, you might need a strategy where each element are modified in place. Also, the 'valid' argument specifies to discard border effects, resulting in a smaller array produced by np.convolve(). You could keep it all by specifying 'same' instead (see documentation).

Answer 2

I found my machine struggling with convolve, so I propose the following solution:

Compute Moving RMS Window Quickly

Suppose we have analog voltage samples a0 ... a99 (one hundred samples) and we need to take moving RMS of 10 samples through them.

The window will scan initially from elements a0 to a9 (ten samples) to get rms0.

    # rms = [rms0, rms1, ... rms99-9] (total of 91 elements in list):
    (rms0)^2 = (1/10) (a0^2 + ...         + a9^2)            # --- (note 1)
    (rms1)^2 = (1/10) (...    a1^2 + ...  + a9^2 + a10^2)    # window moved a step, a0 falls out, a10 comes in
    (rms2)^2 = (1/10) (              a2^2 + ... + a10^2 + a11^2)     # window moved another step, a1 falls out, a11 comes in
    ...

Simplifying it: We havea = [a0, ... a99] To create moving RMS of 10 samples, we can take sqrt of the addition of 10 a^2's and multiplied by 1/10.

In other words, if we have

    p = (1/10) * a^2 = 1/10 * [a0^2, ... a99^2]

To get rms^2 simply add a group of 10 p's.

Let's have an acummulator acu:

    acu = p0 + ... p8     # (as in note 1 above)

Then we can have

    rms0^2 =  p0 + ...  p8 + p9 
           = acu + p9
    rms1^2 = acu + p9 + p10 - p0
    rms2^2 = acu + p9 + p10 + p11 - p0 - p1
    ...

we can create:

    V0 = [acu,   0,   0, ...  0]
    V1 = [ p9, p10, p11, .... p99]          -- len=91
    V2 = [  0, -p0, -p1, ... -p89]          -- len=91

    V3 = V0 + V1 + V2

if we run itertools.accumulate(V3) we will get rms array

Code:

    import numpy as np
    from   itertools import accumulate

    a2 = np.power(in_ch, 2) / tm_w                  # create array of p, in_ch is samples, tm_w is window length
    v1 = np.array(a2[tm_w - 1 : ])                  # v1 = [p9, p10, ...]
    v2 = np.append([0], a2[0 : len(a2) - tm_w])     # v2 = [0,   p0, ...]
    acu = list(accumulate(a2[0 : tm_w - 1]))        # get initial accumulation (acu) of the window - 1
    v1[0] = v1[0] + acu[-1]                         # rms element #1 will be at end of window and contains the accumulation
    rmspw2 = list(accumulate(v1 - v2))

    rms = np.power(rmspw2, 0.5)

I can compute an array of 128 Mega samples in less than 1 minute.