## White Noise

*White*noise may be defined as a sequence of

*uncorrelated*random values, where correlation is defined in Appendix C and discussed further below. Perceptually, white noise is a wideband ``hiss'' in which all frequencies are equally likely. In Matlab or Octave, band-limited white noise can be generated using the

`rand`or

`randn`functions:

y = randn(1,100); % 100 samples of Gaussian white noise % with zero mean and unit variance x = rand(1,100); % 100 white noise samples, % uniform between 0 and 1. xn = 2*(x-0.5); % Make it uniform between -1 and +1True white noise is obtained in the limit as the sampling rate goes to infinity and as time goes to plus and minus infinity. In other words, we never work with true white noise, but rather a finite time-segment from a white noise which has been band-limited to less than half the sampling rate and sampled.

In making white noise, it doesn't matter how the amplitude values are distributed probabilistically (although that amplitude-distribution must be the

*same*for each sample--otherwise the noise sequence would not be

*stationary*,

*i.e.*, its statistics would be

*time-varying*, which we exclude here). In other words, the relative probability of different amplitudes at any single sample instant does not affect whiteness, provided there is

*some*zero-mean distribution of amplitude. It only matters that successive samples of the sequence are

*uncorrelated*. Further discussion regarding white noise appears in §C.3.

#### Testing for White Noise

To test whether a set of samples can be well modeled as white noise, we may compute its*sample autocorrelation*and verify that it approaches an

*impulse*in the limit as the number of samples becomes large; this is another way of saying that successive noise samples are

*uncorrelated*. Equivalently, we may break the set of samples into successive blocks across time, take an FFT of each block, and average their squared magnitudes; if the resulting average magnitude spectrum is

*flat*, then the set of samples looks like white noise. In the following sections, we will describe these steps in further detail, culminating in

*Welch's method*for noise spectrum analysis, summarized in §6.9.

**Next Section:**

Sample Autocorrelation

**Previous Section:**

Spectral Characteristics of Noise