# Estimating change in RMS of noise given FIR filter coefficients

Filtering noise leads to a change in the overall RMS of the noise as demonstrated by the following code-block. which uses scipy's firwin2 to calculate the taps. The first set of taps (taps1) generates a large notch from 1 to 40 kHz whereas the second set of taps (taps2) generates a small notch from 4 to 7 kHz.

import matplotlib.pyplot as plt
import numpy as np
from scipy import signal

def rms_db(x):
rms = np.sqrt(np.mean(x ** 2))
return 20 * np.log10(rms)

ntaps = 10001

freq1 = [0,  1e3, 2e3,  40e3, 41e3, 50e3]
gain1 = [1,  1,   1e-2, 1e-2, 1,    1   ]
taps1 = signal.firwin2(ntaps, freq1, gain1, fs=100e3)

freq2 = [0,  4e3, 5e3,  6e3,  7e3,  50e3]
gain2 = [1,  1,   1e-3, 1e-3, 1,    1   ]
taps2 = signal.firwin2(ntaps, freq2, gain2, fs=100e3)

# Generating uniform noise from -sqrt(3) to sqrt(3) results in RMS value of 1
noise = np.random.uniform(-np.sqrt(3), np.sqrt(3), size=100000)

# Trim off the first ntaps to eliminate boundary conditions
noise1 = signal.lfilter(taps1, 1, noise)[ntaps:]
noise2 = signal.lfilter(taps2, 1, noise)[ntaps:]
noise = noise[ntaps:]

print('Unfiltered noise RMS (dB re 1Vrms): {:.2f}'.format(rms_db(noise)))
print('Filtered noise1 RMS (dB re 1Vrms): {:.2f}'.format(rms_db(noise1)))
print('Filtered noise2 RMS (dB re 1Vrms): {:.2f}'.format(rms_db(noise2)))


As expected, filtering the noise leads to a reduction in RMS of approx. -6.65 dB for the large notch filter and approx. -0.19 dB for the small notch filter. Running the code gives us the following values.

Unfiltered noise RMS (dB re 1Vrms): 0.01
Filtered noise1 RMS (dB re 1Vrms): -6.65
Filtered noise2 RMS (dB re 1Vrms): -0.19


Since noise is inherently random, the actual reduction, in dB, of the noise will vary each time I run the code. I'd like to estimate the theoretical reduction in RMS that results from a particular when it's applied to a broadband signal (i.e., a signal containing equal power at all frequencies).

I thought I might be able to use freqz to estimate this:

w, h1 = signal.freqz(taps1, fs=fs)
h1 = np.abs(h1)

w, h2 = signal.freqz(taps2, fs=fs)
h2 = np.abs(h2)

plt.plot(w, 20 * np.log10(h1))
plt.plot(w, 20 * np.log10(h2))
plt.xlabel('Frequency (Hz)')
plt.ylabel('Gain (dB)')

average_db1 = 20 * np.log10(h1.mean())
average_db2 = 20 * np.log10(h2.mean())

print('Expected change in RMS for filter 1 (dB): {:.2f}'.format(average_db1))
print('Expected change in RMS for filter 2 (dB): {:.2f}'.format(average_db2))


However, this gives me an expected change in RMS that does not match the empirically-calculated change in RMS above.

Expected change in RMS for filter 1: -12.85
Expected change in RMS for filter 2: -0.35


Here is a plot of the frequency response of each filter.

# Update

Per @Hilmar, I was doing the averaging incorrectly. I needed to average power, not amplitude. The correct calculation:

average_db1 = 20 * np.log10(np.sqrt(np.square(h1).mean()))
average_db2 = 20 * np.log10(np.sqrt(np.square(h2).mean()))

• Re your final update, just simply do average_db1 = 10 * np.log10(np.square(h1).mean()) . No need to sqrt and then multiply by 20. (minor but it jumps out at me like saying 4 is the square of the square root of 4 :) ). ( @TimWescott you were searching for examples where you would use 10log) Jan 6 at 5:48
• Thanks! A very valid point @DanBoschen.
Jan 6 at 18:28
• I hope I didn't come across as being nit-picky! Jan 6 at 20:00
• No. It's a good reminder about the difference between power and amplitude. Thanks!