# Ritter's Noise Glossary

## Noise Characterization Terms

A reference for Experimental Characterization of Recorded Noise.

## Contents

A
Amplitude, Augmented Doubles, Autocorrelation, Autocorrelation Graph, Average Deviation
C
Correlation
D
dB, Decibel
E
Entropy, Expected Maxima
F
Filter, FFT, FFT Graph, Frequency
H
Highest Repetition
K
Kurtosis
M
Maxima, Maximum Value, Mean, Minima, Minimum Value
N
Noise Graph, Normal Distribution, Normal Graph
P
Peak, Pink Noise, Population
R
Range, RMS
S
Skew, Standard Deviation, Standard Error
U
Unique Values
V
Variance
W
.WAV File, White Noise

Amplitude
The signal level, or height.

Augmented Doubles
A term used in population estimation. If we take enough random samples from a population, eventually we will find some value which occurs again or repeats; this is a "double." As we continue to draw values, eventually we will get a triple, which is of course not a double. Augmentation converts triples and higher occurrences to the number of doubles which have the same probability. Then we can use the augmented doubles value to estimate population with better accuracy than we might otherwise have. See the article.

Autocorrelation
The similarity or correlation between a sequence of data and itself, at different offsets. Clearly, at zero offset, any data sequence is maximally correlated to itself. But if one version is rotated or given a circular shift of n positions, there may not be much correlation at all, especially in noise signals. The autocorrelation result is typically a correlation computed for every possible sequence offset, with the data considered repetitive or circular. Noise data are not repetitive, but the characteristics of the generation tend to be constant and we can treat the result as a circular array without much offense to the underlying requirements.

Autocorrelation Graph
In the autocorrelation graph, the horizontal axis represents the 512 possible unique offsets between two copies of the same sequence of 1024 values. (Note that a circular shift or "rotation" of 1 place in one copy produces the same comparison as a reverse 1 place rotation in the other copy.) The leftmost pictel column represents zero offset, which holds the expected "spike" from the maximum possible correlation.

The vertical axis is bipolar, with zero plotted in a light gray horizontal line. The "spike" at zero is scaled to 4096, which is about 32x the plotable range of -128..127. The spike is thus well "off scale" (plotted as 127 instead of 4096), and we see only the center 1/32 of the scaled graph. Normally this will include the structure we might want to see.

The autocorrelation values are computed here using FFT techniques. The resulting FFT blocks contain 1024 complex values in which each orthogonal or imaginary component is zero, and each real component is repeated twice in the array. The lowest 512 real results are averaged over each FFT block in the sample, which reduces the effect of random variations so we can see consistent patterns.

All of these .WAV recordings display an autocorrelation "ringing" which is clear through at least 8 elements, and presumably continues. I have variously attributed this result to an unknown digital filter in the sampling hardware or the apparent correlation inherent in the uneven distribution. Since lower values are most probable, we can expect that "clumps" of lower values may occur together, thus providing an apparent correlation between values.

We can investigate the alternative of an apparently flat distribution by taking values from a uniform random number generator, and then performing the same computation and plotting process. In this case we get a very peculiar "noise" distribution, and the autocorrelation spike is present only at offset zero. This is consistent with a correct computation and plot.

Average Deviation
The absolute value of the difference from the mean for each data value, summed, then divided by the number of values. A somewhat more robust estimator of the of the so-called "second moment" than standard deviation.

Correlation

Generally speaking, a similarity between data; the extent to which data are related. Typically, some sort of dependence relationship (not necessarily linear) between two data sequences, or mutual dependence on some other sequence. Also see the discussion under white noise.

dB

decibel.

Decibel
Ten times the base-10 logarithm of the ratio of two power values. Denoted by dB.

Entropy

The Shannon computation of the same name. The probability of finding a particular symbol, times the natural log of that probability, summed over all symbols, and negated. A measure of coding efficiency, in bits (binary digits) of information per bit of data. A distinctly different concept than the entropy of physics, where it represents disorder. In noise characterization we count every occurrence of all 65,536 possible bipolar 16-bit data values, and get the sample probability for each value out of the total number of data values.

The entropy computation measures coding efficiency, and not unknowable randomness as often claimed. For example, if we measure the entropy of a counting sequence, we will get a high value. But any counting sequence is controlled by a relatively tiny amount of arbitrary internal state. And a counting sequence is completely predictable given the design of the counter and any one result. Clearly, entropy does not reflect the essential weakness of a cryptographically poor generator.

In noise work, we expect noise sample values to occur in a normal distribution. That distribution amounts to a form of coding, thus limiting the amount of information per data bit. That sets an expected entropy value for good noise sequences, quite independent of other issues.

Expected Maxima
The number of maxima expected from white-noise theory. If we assume an upper frequency limit of 20 kHz, we expect about 15,492 maxima per second.

Ideally, we do expect that any sampling system will have a strong anti-alias filter to limit signal frequencies to below the Nyquist limit. For audio, we might well expect this to occur between the 20 kHz we use as the usual audio upper limit, and the 22 kHz Nyquist limit implied by a 44 ks/sec CD-quality sampling rate. Unfortunately, it appears that the sound card used here does not have an anti-aliasing filter, or at least not one with a strong cut-off. And while it would be easy enough to get a much better sound card, that would not be representative of the equipment available on most computers.

Filter

One interesting opportunity in these noise experiments was the possibility that some small "data fix-up" process could improve the resulting statistics. Various approaches were implemented, including:
• raw: the unmodified data from the .WAV file
• diff from mean: the mean is subtracted from each data value. Intended to force the subsequent mean to zero, thus eliminating simple positive or negative bias.
• diff from prev: the previous data value is subtracted from the current value. This gives the difference between samples, which reduces several forms of bias. But we can also see this as a sort of digital filter which necessarily affects frequency response.
• statistical RNG: a random value is acquired from a source which is known to be white and flat. We see, however, that this is very different from the usual noise source.
In the end, it appears that the differential mode (diff from prev) can provide a substantial improvement for many lower-quality sources.

FFT
Fast Fourier Transform. A numerically advantageous way of computing a discrete Fourier Transform. Basically a way of transforming information from a sequence of waveform amplitude values as sampled periodically through time, into the equivalent information in amplitude values at discrete frequencies. The FFT performs this transformation in time proportional to n log n, for some n a power of 2.

The basic FFT transformation is from complex-value (real plus orthogonal or imaginary) elements to complex frequency values. Various "tricks" can be used to decrease computation when only real-value data are used, or when the number of values is not a power of 2. Here we use 1024 values, each with an imaginary component of zero, and so ignore the various FFT "tricks."

One deceptive aspect of the FFT is that it produces results for typically arbitrary particular discrete frequencies. The result values are not, as one might expect, the simple accumulation of all energy near each particular frequency. Instead, the result values are how the original waveform could be reproduced using energy only at our selected frequencies. Significant signals between our selected frequencies can produce somewhat ambiguous results.

A restrictive requirement of the FFT is that the data need be repetitive or circular with exactly the length of the FFT block. This is often violated in practice, and various "windowing functions" are introduced to minimize resulting errors. The issue is much less important in noise work, and windowing functions have little effect on noise results.

FFT Graph
The FFT graph displays the amplitude of data at various discrete frequencies. In this case we use .WAV files which record a 16-bit bipolar noise amplitude at the CD rate of 44,100 samples (or data elements) per second.

Our FFT has a fixed block size of 1024 complex values, which each have a zero orthogonal or "imaginary" component. That block transforms into 511 complex values repeated twice, plus zero and the highest frequency each once. We convert each complex value to real magnitude and use 512 results.

We are measuring random-like data which will vary widely from one FFT block to another. To gain a systematic view of the results, we take the average magnitude for any particular frequency over all FFT blocks included in our sample size. Increasing the sample size thus tends to decrease the variation seen in the result (see standard error).

The leftmost column in the display (zero) corresponds to zero "frequency," which is the DC bias. Each successive column to the right represents a frequency increase of 44,100 / 1024 = 43.066 Hz. This makes column 511 correspond to 22,007 Hz.

Although frequency is traditionally plotted on a logarithmic scale, the frequency increments in FFT results are linear. If we plotted FFT results on a log scale, we might want to interpolate the few low-frequency values to fill the left-side pixels, and discard many of the high-frequency values which would occur in the same right-side pixels. That would be both deceptive and wasteful; accordingly, frequency is plotted linearly.

Frequency is calibrated with a series of vertical lines in a 1, 2, 5 pattern, in which red represents the start of a new decade (1) and green represents the next two steps (2 and 5). The three vertical red lines on the FFT graph represent 100 Hz at the left, then 1 kHz and 10 kHz to the right. The rightmost green line represents 20 kHz.

Amplitude is calibrated with a series of horizontal lines, also in a 1, 2, 5 pattern. The display is normalized to the average of the 443 magnitude averages from 1 kHz and 20 kHz, where we draw a light gray horizontal line. Two green lines surround this, representing +/- 0.5 dB change; then two red lines represent +/- 1.0 dB and another green pair represent +/- 2.0 dB.

Frequency
In general, the number of repetitions or cycles per second. Specifically, the number of repetitions of a sine wave signal per second: A signal of a single frequency is a sine wave of that frequency. Any deviation from the sine waveform can be seen as components of other frequencies, as described by an FFT. Now measured in Hertz (Hz); previously called cycles-per-second (cps).

Highest Repetition

In these noise measurements, the most times that any one 16-bit bipolar data value occurs.

Kurtosis

An expression of the so-called "fourth moment." A tendency for a distribution to form a sharp narrow peak in the center (or, when negative, a broad flat plateau). Also see standard deviation and skew.

Maxima

The number of local maxima. The count of data values where both the adjacent values (predecessor and successor) are lower.

Maximum Value
The largest bipolar 16-bit integer value in the data. Typically positive. Ideally, the maximum value would be close to -- but not quite reach -- the largest possible positive recorded value of +32,767.

Mean
The average; between the extremes. The sum of all values in the data, divided by the number of values. This is the sample mean, a value we can actually measure and compute, which approximates the population mean which we generally cannot measure. But we do expect the population mean to be within "a few multiples" of the standard error of the computed sample mean value.

The mean is a characterization of the distribution's "central value." It is most useful for those distributions which rise to a maximum in the center, as do most noise results. Also see average deviation and standard deviation. Since we have bipolar noise values, our mean is typically near zero.

Minima
The number of local minima. The count of data values where both the adjacent values (predecessor and successor) are higher.

Minimum Value
The smallest bipolar 16-bit integer value in the data. Typically negative. Ideally, the maximum value would be close to -- but not quite reach -- the largest possible negative recorded value of -32,768.

Noise Graph

Each graph contains exactly 512 horizontal "picture elements" (pictels) or columns (0..511), and 256 vertical pictels or rows (0..255). On some graphs, we plot bipolar horizontal data by on the range (-256..-1,0..255). Similarly, we may plot bipolar vertical data on (-128..-1,-..127). In most bipolar cases we plot a light gray line across the graph at zero.

There is also a single-pictel-width white border around the graph, making the overall display (with border) 514 x 258 pixels. This is the size of the display image.

Each graph has a dark-gray grid positioned every 8 pictels, starting at zero zero (the leftmost pictel column and the bottom pictel row). By using the grid, and a graphics program which can to expand the display, it is relatively easy to read the exact coordinates of any point plotted on the graph.

Also see: FFT graph, autocorrelation graph, and normal graph.

Normal Distribution
The usual "bell shaped" distribution which may or may not be due to Carl Friedrich Gauss 1777-1855. Called "normal" because it is similar to many real-world distributions. Note that real-world distributions can be similar to normal, and still differ from it in serious systematic ways. Also see the normal computation page.

"The" normal distribution is in fact a family of distributions, as parameterized by mean and standard deviation values. By computing the sample mean and standard deviation, we can reduce the whole family into a single curve. A value from any normal-like distribution can be "normalized" by subtracting the mean and dividing by the standard deviation; the result can be used to look up probabilities in standard normal tables. All of which of course assumes that the underlying distribution is in fact normal, which may or may not be the case.

Normal Graph
The normal graph displays the "sample distribution" or number of occurrences of each 16-bit data value over the entire sample. Ideally, these counts will occur in a manner similar to what we find in the "normal" statistical distribution.

The horizontal axis is bipolar and calibrated in standard deviations, which here range from -4 to +4. The mean (sd = 0) is given a light gray vertical line, while sd values 1..3 and -1..-3 are plotted as green vertical lines. The ideal normal distribution (for the computed mean and sd) is plotted in red.

Since the horizontal axis covers 8 standard deviations in 512 pictels, each standard deviation covers exactly 64 pictels. Each pictel-width is thus 1/64 sd; each pictel generally covers multiple data sample value counts. Although both the mean and sd are quite unlikely to be integer values, their basis is still the bipolar integer recorded data values. We can thus define the real-value range of a pictel as being between part of one value-count and part of another, the enclosed sum being what we plot. The pictel centered on the mean value (that is, +/- sd/128) is scaled to vertical height of 199.45, which is about 500 times the ideal normal curve at mean.

Peak

The maximum value. Here we specifically compare to RMS, and so average the maximum and minimum values to get the effective peak.

Pink Noise
A random-like signal in which -- ideally -- power is proportional to the inverse of frequency, or 1/f. At twice the frequency, we would expect half the power, which is a 3 dB decrease. This is a frequency-response slope of -3 dB / octave, or -10 dB / decade. As opposed to white noise, which has the same level at all frequencies, pink noise has more low-frequency or "red" components, and so is called "pink."

A common single-stage R-C low-pass filter has half the output voltage at twice the frequency. But this is actually one-quarter the power and a -6 dB / octave slope, which might be termed more "red" than "pink." For ideal pink noise, the desired voltage drop per octave is the square root of two, or about 0.707.

Population
The total number of unique values. When some values are more probable than others, the effective population approaches the number of the more probable values.

The unknown universe of values which we can only estimate by sampling.

Range

The number of possible values, from the minimum to the maximum.

RMS

The root-mean-square. The square root of the average (or mean) of the square of each data value. Because energy varies as the square of amplitude, RMS tracks the amount of energy in a complex signal. In noise data, the RMS value should be approximately the same as the standard deviation.

Skew

An expression of the so-called "third moment." A tendency for a distribution to lean to the right (or left, when negative), showing a fast dip from the central mean but having an extended tail. Also see standard deviation and kurtosis.

Standard Deviation
The square of the difference from the mean for each data value, summed, divided by one less than the number of values, then square-rooted. An expression of the so-called "second moment," which describes the "dispersion" or variability around the mean. With a normal distribution as we expect from most noise sources, about 68% of our data values should be within +/- 1 standard deviation about the mean. The square of the standard deviation is the variance. In noise data, the standard deviation should be approximately the same as the RMS.

Standard Error
The standard error of the mean. The standard deviation divided by the square root of the number of data values. The extent to which we expect the sample mean to differ (+/-) from the population mean. The more data we have, the smaller this range becomes; but to get 10x the precision, we need 100x as much data.

Unique Values

In these noise tests, the number of unique 16-bit data values found in the sample. This is not the number of data elements, but rather the number of different values in the sample. At most we could have 65536 different 16-bit values, but if that happened, we would be justifiably concerned that we did not capture peak values outside the recording range.

Variance

The square of the difference from the mean for each data value, summed and divided by one less than the number of values. An expression of the so-called "second moment" which describes the variability around the mean. The square root of the variance is the standard deviation.

.WAV File

A type of file used for sound storage. The structures in .WAV files can represent a wide range of data and sampling rates, but only simple "cannonic" files are used here. In these, the data are 16-bit bipolar integer samples (or data elements) starting at byte 44 in the file. These data represent monaural noise amplitude values sampled 44,100 times per second.

White Noise
A random-like signal with a flat frequency spectrum. Doubling the bandwidth doubles the noise power and increases RMS noise voltage by the square root of two. As opposed to pink noise, in which the frequency spectrum drops off with frequency. White noise is analogous to white light, which contains every possible color.

White noise is normally described as a relative power density in volts squared per hertz of bandwidth. White noise power varies directly with bandwidth, so white noise would have twice as much power in the next higher octave as in the current one. The introduction of a white noise audio signal can destroy high-frequency loudspeakers.

The definition of white noise as a random signal having a flat frequency spectrum is very common. However, frequency is defined in terms of a continuous sine wave, and that sort of correlation is something we do not expect in noise. Instead, we expect noise to be the result of multitudes of independent and unpredictable quantum pulses or actions. From simple random variation we do expect that any particular sample set or random sequence might seem to have some correlation. But we do not expect random effects to be repeatable or to have a clear and complex structure. So if we have structured results, or get similar results again, we are looking at true correlations. And if correlated frequency energy exists, individual noise samples cannot be considered completely independent.

Terry Ritter, his current address, and his top page.

Last updated: 2006-02-21
First updated: 1999-06-22