White noise is a random signal whose power spectral density (PSD) is flat across all frequencies -- meaning every frequency band of equal width carries equal power. The term is analogous to white light, which contains all visible frequencies at roughly equal intensity.
In practice
In embedded signal processing, white noise is the idealized model for quantization noise from an ADC, thermal noise in analog front-ends, and the noise floor of a sensor. Whether a real noise source is actually white depends on the bandwidth of interest: ADC quantization noise is well-approximated as white when the input is not heavily correlated and dithering is applied, but at very low frequencies many real noise sources deviate toward 1/f (pink) behavior.
When designing filters for sensor data, assuming white noise allows straightforward application of Wiener filter theory and lets you predict how much a moving-average or FIR low-pass filter will reduce noise variance. A brick-wall filter of bandwidth B reduces white noise power in proportion to B, so halving the bandwidth cuts noise power by half. This relationship drives filter design tradeoffs between noise rejection and signal bandwidth in applications like IMU fusion, ADC oversampling, and analog sensor conditioning.
A practical pitfall is treating colored noise as white. If your noise PSD rises at low frequencies (1/f noise, common in MEMS sensors and op-amp inputs) or at high frequencies (aliased out-of-band interference), a filter sized for white noise will underperform. Examining the actual PSD via spectral averaging -- as described in "Take Control of Noise with Spectral Averaging" -- before committing to a filter design avoids this mistake. The post "An Interesting Fourier Transform - 1/f Noise" also illustrates how quickly real noise departs from the white noise ideal.
In oversampling ADC systems, white noise enables a direct relationship between oversampling ratio (OSR) and ENOB improvement: every factor-of-4 increase in OSR yields approximately 1 additional bit of resolution (6 dB SNR gain) when the noise is white and the decimation filter is matched appropriately. Noise shaping techniques, covered in "Noise shaping," push quantization noise energy away from the signal band, which only works predictably when the in-band noise is close to white.
Discussed on DSPRelated
Frequently asked
Is real ADC quantization noise actually white?
It is a useful approximation under the right conditions. When the input signal exercises many quantization levels and changes significantly between samples -- or when analog
dither is added -- quantization noise is largely decorrelated and its
PSD is approximately flat. For slowly varying or nearly DC signals, quantization noise becomes strongly correlated (periodic, in the worst case), and the white noise model breaks down. Most
oversampling and noise-shaping analyses assume the white-noise approximation holds.
How does white noise differ from pink (1/f) noise in embedded sensor work?
White noise has equal power per hertz across frequency; pink noise has equal power per octave, meaning its
PSD falls as 1/f and low-frequency components dominate. MEMS gyroscopes, accelerometers, and precision op-amp inputs often exhibit a 1/f noise floor below a corner frequency (sometimes called the flicker corner) and white noise above it. A filter or averaging window sized purely for white noise will leave more residual noise than expected if 1/f content is significant.
How much does a simple moving-average filter reduce white noise?
A moving average of N samples reduces white noise power by a factor of N (standard deviation by sqrt(N)). This is because a moving average is a low-pass
FIR filter whose noise
bandwidth is approximately
fs/(2N) -- narrowing the bandwidth linearly with N. For non-white noise the reduction will differ, which is why 'Filtering Noise: The Basics (Part 1)' cautions against applying moving-average intuition directly to colored noise sources.
Why does white noise enable the 'every 4x oversampling = 1 extra bit' rule for ADCs?
Oversampling spreads the fixed total quantization noise power over a wider frequency band (0 to
fs/2). If the noise is white -- uniformly distributed across that band -- then a
decimation low-pass filter removes the out-of-band portion, keeping only the fraction that falls in the signal
bandwidth. Each doubling of OSR halves the in-band noise power, and since
SNR improves by 3 dB per doubling, you need 4x oversampling for 6 dB (1 bit). If the noise is not white, this simple scaling no longer applies.
How can you verify whether noise in your system is white before choosing a filter?
Capture a sufficiently long noise record (with the signal of interest removed or held constant), then compute its
PSD using an
FFT-based method with averaging across multiple segments to reduce variance. A flat PSD across your band of interest is consistent with white noise. Spectral averaging techniques are covered in 'Take Control of Noise with Spectral Averaging'. Slopes or peaks in the PSD indicate colored noise or interference that may require a different filter strategy.
Differentiators vs similar concepts
White noise is often contrasted with pink noise (1/f noise), where
power spectral density rises as frequency decreases, giving lower frequencies disproportionate power. Brown (Brownian) noise falls even more steeply, at 1/f^2. In embedded sensor and
ADC work, these distinctions matter because filter designs and noise-reduction formulas (e.g., the
oversampling SNR gain rule) assume white noise; applying them to 1/f-dominated signals produces overly optimistic predictions. Gaussian white noise (white PSD, Gaussian amplitude distribution) is an additional qualification sometimes assumed in statistical signal processing -- white noise can technically have any amplitude distribution, and ADC quantization noise, while approximately white spectrally, has a roughly uniform (not Gaussian) amplitude distribution.
