Library / Signal processing and digital filters / DSP-32
Signal processing and digital filters
Output variance of an FIR driven by white noise
Output variance of an FIR filter (up to 8 taps, unused ones zero) fed by zero-mean, uncorrelated input noise of variance σx².
- Ref
- DSP-32
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- σy2 [V²]
White noise through an FIR filter y[n] = Σ h[j]x[n − j] gives an output variance σy² = σx² Σ|h[j]|², because the samples of the input are uncorrelated, so their variances add after weighting by the squared taps. The sum Σh² is the noise power gain, and for a unit-DC-gain smoother it is below 1.
Output variance of an FIR filter (up to 8 taps, unused ones zero) fed by zero-mean, uncorrelated input noise of variance σx². The factor is the sum of squared tap magnitudes, the noise power gain of the filter.
Calculator
- Noise power gain, Σh²0.375
- Output RMS noise1.225 V
- DC gain, Σh1
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The filter passes 37.5 % of the input noise power: 4 V² in, 1.5 V² out.
- With a DC gain of 1, a slowly varying signal passes unchanged while the noise falls, which is the point of smoothing.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σx2 Input noise variance | 0.001 to 1,000 V² | 3.75e-4 | 375 | 375 | +10% | +10 % |
| h0 Filter tap 0 | −2 to 2 | 1.25 | 17.25 | 16 | +0.08 | +12.4 % |
| h1 Filter tap 1 | −2 to 2 | 0.5 | 16.5 | 16 | +0.08 | +23 % |
| h2 Filter tap 2 | −2 to 2 | 1.25 | 17.25 | 16 | +0.08 | +12.4 % |
| h3 Filter tap 3 | −2 to 2 | 1.5 | 17.5 | 16 | +0.08 | +1.71 % |
| h4 Filter tap 4 | −2 to 2 | 1.5 | 17.5 | 16 | +0.08 | +1.71 % |
| h5 Filter tap 5 | −2 to 2 | 1.5 | 17.5 | 16 | +0.08 | +1.71 % |
| h6 Filter tap 6 | −2 to 2 | 1.5 | 17.5 | 16 | +0.08 | +1.71 % |
| h7 Filter tap 7 | −2 to 2 | 1.5 | 17.5 | 16 | +0.08 | +1.71 % |
Sweep of Input noise variance
| Input noise variance (V²) | Output variance (V²) |
|---|---|
| 0.001 | 3.75e-4 |
| 0.005623 | 0.002109 |
| 0.03162 | 0.01186 |
| 0.1778 | 0.06669 |
| 1 | 0.375 |
| 5.623 | 2.109 |
| 31.62 | 11.86 |
| 177.8 | 66.69 |
| 1,000 | 375 |
Sweep one variable
Variables and how each one contributes
Output variance
What the equation gives you. Everything else feeds it.
Input noise variance
The input noise variance, in V². Raises the output variance in direct proportion. Raising it by 10 % (to 4.4) moves the result by +10 %, from 1.5 to 1.65 V².
Filter tap 0
The filter tap h[0]. Adds h[0]² to the noise power gain. Raising it by 10 % (to 0.275) moves the result by +3.5 %, from 1.5 to 1.552 V².
Filter tap 1
The filter tap h[1]. Adds h[1]² to the noise power gain. Raising it by 10 % (to 0.55) moves the result by +14 %, from 1.5 to 1.71 V².
Filter tap 2
The filter tap h[2]. Adds h[2]² to the noise power gain. Raising it by 10 % (to 0.275) moves the result by +3.5 %, from 1.5 to 1.552 V².
Filter tap 3
The filter tap h[3]. Adds h[3]² to the noise power gain. Raising it by 10 % (to 0) moves the result by +0 %, from 1.5 to 1.5 V².
Filter tap 4
The filter tap h[4]. Adds h[4]² to the noise power gain. Raising it by 10 % (to 0) moves the result by +0 %, from 1.5 to 1.5 V².
Filter tap 5
The filter tap h[5]. Adds h[5]² to the noise power gain. Raising it by 10 % (to 0) moves the result by +0 %, from 1.5 to 1.5 V².
Filter tap 6
The filter tap h[6]. Adds h[6]² to the noise power gain. Raising it by 10 % (to 0) moves the result by +0 %, from 1.5 to 1.5 V².
Filter tap 7
The filter tap h[7]. Adds h[7]² to the noise power gain. Raising it by 10 % (to 0) moves the result by +0 %, from 1.5 to 1.5 V².
Worked example with the values above
- Squared taps: 0.25² + 0.5² + 0.25² = 0.375.
- Input variance: σx² = 4 V².
- σy² = 4 × 0.375 = 1.5 V².
Assumptions and limits
- Equally spaced samples, and noise that is uncorrelated from sample to sample.
- A signal that is smooth over the filter span, so a local quadratic fit describes it.
Common mistakes
- Using these filters at the ends of a record, where the window is incomplete.
- Forgetting that centred filters add a delay of half the window length in real-time use.
- Using a moving average where the sinc sidelobes let high frequencies leak through.
Statistics and uncertainty
Smoothing reduces random error and adds bias, which cannot both be small. A Savitzky–Golay fit lowers the noise variance by the factor Σc² (here 0.49) but flattens any curvature of higher order than a quadratic.
Where it is used
- Smoothing sensor signals and estimating velocity from position records.
- Predicting the noise reduction and frequency response of simple FIR smoothers.
Maths you need
- Moving average: the mean of the latest M samples
- Least-squares fit: the polynomial that minimises squared error
- Noise power gain: Σh² for an FIR filter