The subject of statistical signal processing requires a background in probability theory, random variables, and stochastic processes . However, only a small subset of these topics is really necessary to carry out practical spectrum analysis of noise-like signals (Chapter 6) and to fit deterministic models to noisy data. For a full textbook devoted to statistical signal processing, see, e.g., [121,95]. In this appendix, we will provide definitions for some of the most commonly encountered terms.
For a full treatment of random variables and stochastic processes (sequences of random variables), see, e.g., . For practical every-day signal analysis, the simplified definitions and examples below will suffice for our purposes.
Definition: A probability distribution may be defined as a non-negative real function of all possible outcomes of some random event. The sum of the probabilities of all possible outcomes is defined as 1, and probabilities can never be negative.
Example: A coin toss has two outcomes, ``heads'' (H) or ``tails'' (T), which are equally likely if the coin is ``fair''. In this case, the probability distribution is
where denotes the probability of outcome . That is, the total ``probability mass'' is divided equally between the two possible outcomes heads or tails. This is an example of a discrete probability distribution because all probability is assigned to two discrete points, as opposed to some continuum of possibilities.
Two probabilistic events and are said to be independent if the probability of and occurring together equals the product of the probabilities of and individually, i.e.,
where denotes the probability of and occurring together.
Example: Successive coin tosses are normally independent. Therefore, the probability of getting heads twice in a row is given by
Example: A random variable can be defined based on a coin toss by defining numerical values for heads and tails. For example, we may assign 0 to tails and 1 to heads. The probability distribution for this random variable is then
Example: A die can be used to generate integer-valued random variables between 1 and 6. Rolling the die provides an underlying random event. The probability distribution of a fair die is the discrete uniform distribution between 1 and 6. I.e.,
Example: A pair of dice can be used to generate integer-valued random variables between 2 and 12. Rolling the dice provides an underlying random event. The probability distribution of two fair dice is given by
This may be called a discrete triangular distribution. It can be shown to be given by the convolution of the discrete uniform distribution for one die with itself. This is a general fact for sums of random variables (the distribution of the sum equals the convolution of the component distributions).
Example: Consider a random experiment in which a sewing needle is dropped onto the ground from a high altitude. For each such event, the angle of the needle with respect to north is measured. A reasonable model for the distribution of angles (neglecting the earth's magnetic field) is the continuous uniform distribution on , i.e., for any real numbers and in the interval , with , the probability of the needle angle falling within that interval is
Note, however, that the probability of any single angle is zero. This is our first example of a continuous probability distribution. Therefore, we cannot simply define the probability of outcome for each . Instead, we must define the probability density function (PDF):
To calculate a probability, the PDF must be integrated over one or more intervals. As follows from Lebesgue integration theory (``measure theory''), the probability of any countably infinite set of discrete points is zero when the PDF is finite. This is because such a set of points is a ``set of measure zero'' under integration. Note that we write for discrete probability distributions and for PDFs. A discrete probability distribution such as that in (C.4) can be written as
where denotes an impulse.C.1
(Again, for a more complete treatment, see  or the like.)
Definition: A stochastic process is defined as a sequence of random variables , .
Definition: We define a stationary stochastic process , as a stochastic process consisting of identically distributed random variables . In particular, all statistical measures are time-invariant.
When a stochastic process is stationary, we may measure statistical features by averaging over time. Examples below include the sample mean and sample variance.
where denotes the probability density function (PDF) for the random variable v.
Example: Let the random variable be uniformly distributed between and , i.e.,
Then the expected value of is computed as
Thus, the expected value of a random variable uniformly distributed between and is simply the average of and .
For a stochastic process, which is simply a sequence of random variables, means the expected value of over ``all realizations'' of the random process . This is also called an ensemble average. In other words, for each ``roll of the dice,'' we obtain an entire signal , and to compute , say, we average together all of the values of obtained for all ``dice rolls.''
For a stationary random process , the random variables which make it up are identically distributed. As a result, we may normally compute expected values by averaging over time within a single realization of the random process, instead of having to average ``vertically'' at a single time instant over many realizations of the random process.C.2 Denote time averaging by
Then, for a stationary random processes, we have . That is, for stationary random signals, ensemble averages equal time averages.
We are concerned only with stationary stochastic processes in this book. While the statistics of noise-like signals must be allowed to evolve over time in high quality spectral models, we may require essentially time-invariant statistics within a single frame of data in the time domain. In practice, we choose our spectrum analysis window short enough to impose this. For audio work, 20 ms is a typical choice for a frequency-independent frame length.C.3 In a multiresolution system, in which the frame length can vary across frequency bands, several periods of the band center-frequency is a reasonable choice. As discussed in §5.5.2, the minimum number of periods required under the window for resolution of spectral peaks depends on the window type used.
Definition: The mean of a stochastic process at time is defined as the expected value of :
where is the probability density function for the random variable .
For a stationary stochastic process , the mean is given by the expected value of for any . I.e., for all .
Definition: The sample mean of a set of samples from a particular realization of a stationary stochastic process is defined as the average of those samples:
For a stationary stochastic process , the sample mean is an unbiased estimator of the mean, i.e.,
where is the probability density function for the random variable .
For a stationary stochastic process , the variance is given by the expected value of for any .
The sample variance is a unbiased estimator of the true variance when the mean is known, i.e.,
This is easy to show by taking the expected value:
When the mean is unknown, the sample mean is used in its place:
The normalization by instead of is necessary to make the sample variance be an unbiased estimator of the true variance. This adjustment is necessary because the sample mean is correlated with the term in the sample variance expression. This is revealed by replacing with in the calculation of (C.22).
Correlation analysis applies only to stationary stochastic processes (§C.1.5).
Definition: The cross-correlation of two signals and may be defined by
I.e., it is the expected value (§C.1.6) of the lagged products in random signals and .
Cross-Power Spectral Density
Note that the autocorrelation function is Hermitian:
When is real, its autocorrelation is symmetric. More specifically, it is real and even.
When the signal is real, its PSD is real and even, like its autocorrelation function.
Definition: To say that is a white noise means merely that successive samples are uncorrelated:
where denotes the expected value of (a function of the random variables ).
In other words, the autocorrelation function of white noise is an impulse at lag 0. Since the power spectral density is the Fourier transform of the autocorrelation function, the PSD of white noise is a constant. Therefore, all frequency components are equally present--hence the name ``white'' in analogy with white light (which consists of all colors in equal amounts).
An example of a digital white noise generator is the sum of a pair of dice minus 7. We must subtract 7 from the sum to make it zero mean. (A nonzero mean can be regarded as a deterministic component at dc, and is thus excluded from any pure noise signal for our purposes.) For each roll of the dice, a number between and is generated. The numbers are distributed binomially between and , but this has nothing to do with the whiteness of the number sequence generated by successive rolls of the dice. The value of a single die minus would also generate a white noise sequence, this time between and and distributed with equal probability over the six numbers
To obtain a white noise sequence, all that matters is that the dice are sufficiently well shaken between rolls so that successive rolls produce independent random numbers.C.4
It can be shown that independent zero-mean random numbers are also uncorrelated, since, referring to (C.26),
For Gaussian distributed random numbers, being uncorrelated also implies independence . For related discussion illustrations, see §6.3.
As mentioned in §6.12, the pwelch function in Matlab and Octave offer ``confidence intervals'' for an estimated power spectral density (PSD). A confidence interval encloses the true value with probability (the confidence level). For example, if , then the confidence level is .
This section gives a first discussion of ``estimator variance,'' particularly the variance of sample means and sample variances for stationary stochastic processes.
The simplest case to study first is the sample mean:
Here we have defined the sample mean at time as the average of the successive samples up to time --a ``running average''. The true mean is assumed to be the average over any infinite number of samples such as
Now assume , and let denote the variance of the process , i.e.,
Then the variance of our sample-mean estimator can be calculated as follows:
We have derived that the variance of the -sample running average of a white-noise sequence is given by , where denotes the variance of . We found that the variance is inversely proportional to the number of samples used to form the estimate. This is how averaging reduces variance in general: When averaging independent (or merely uncorrelated) random variables, the variance of the average is proportional to the variance of each individual random variable divided by .
Consider now the sample variance estimator
where the mean is assumed to be , and denotes the unbiased sample autocorrelation of based on the samples leading up to and including time . Since is unbiased, . The variance of this estimator is then given by
by the independence of and , and when , the fourth moment is given by . More generally, we can simply label the th moment of as , where corresponds to the mean, corresponds to the variance (when the mean is zero), etc.
When is assumed to be Gaussian white noise, we have
so that the variance of our estimator for the variance of Gaussian white noise is
Again we see that the variance of the estimator declines as .
The same basic analysis as above can be used to estimate the variance of the sample autocorrelation estimates for each lag, and/or the variance of the power spectral density estimate at each frequency.
Gaussian Function Properties
Selected Continuous Fourier Theorems