# Beginning Statistical Signal Processing

The subject of *statistical signal processing*
requires a background
in probability theory, random variables, and stochastic processes
[201].
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.

## Random Variables & Stochastic Processes

For a full treatment of random variables and stochastic processes
(sequences of random variables), see, *e.g.*, [201]. For
practical every-day signal analysis, the simplified definitions and
examples below will suffice for our purposes.

### Probability Distribution

**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

(C.1) |

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.

### Independent Events

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.*,

(C.2) |

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

(C.3) |

### Random Variable

**Definition: **
A *random variable*
is defined as a real- or complex-valued
function of some random event, and is fully characterized by its
probability distribution.

**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.*,

(C.5) |

**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

(C.6) |

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

(C.7) |

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*(

(C.8) |

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

(C.9) |

where denotes an

*impulse*.

^{C.1}

### Stochastic Process

(Again, for a more complete treatment, see [201] or the like.)

**Definition: **
A *stochastic process*
is defined as a sequence of random
variables
,
.

A stochastic process may also be called a *random process*,
*noise process*, or simply *signal* (when the context
is understood to exclude deterministic components).

### Stationary Stochastic Process

**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.

### Expected Value

**Definition: **
The *expected value* of a continuous random variable
is denoted
and is defined by

(C.12) |

where denotes the

*probability density function*(PDF) for the random variable v.

**Example: **
Let the random variable
be uniformly distributed between
and
, *i.e.*,

(C.13) |

Then the expected value of is computed as

(C.14) |

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

(C.15) |

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.

### Mean

**Definition: **
The *mean* of a stochastic process
at time
is defined as
the expected value of
:

(C.16) |

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
.

### Sample Mean

**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:

(C.17) |

For a

*stationary stochastic process*, the sample mean is an

*unbiased estimator*of the mean,

*i.e.*,

(C.18) |

### Variance

**Definition: **
The *variance* or *second central moment* of a stochastic
process
at time
is defined as the expected value of
:

(C.19) |

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
.

### Sample Variance

**Definition: **
The *sample variance* of a set of
samples from a particular
realization of a *stationary stochastic process*
is defined
as *average squared magnitude* after removing the *known mean*:

(C.20) |

The sample variance is a

*unbiased estimator*of the true variance when the

*mean is known*,

*i.e.*,

(C.21) |

This is easy to show by taking the expected value:

When the mean is *unknown*, the sample mean is used in its place:

(C.23) |

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

Correlation analysis applies only to *stationary* stochastic
processes (§C.1.5).

### Cross-Correlation

**Definition: **The *cross-correlation* of two signals
and
may be defined by

(C.24) |

*I.e.*, it is the

*expected value*(§C.1.6) of the lagged products in random signals and .

### Cross-Power Spectral Density

The DTFT of the cross-correlation is called the *cross-power
spectral density*, or ``cross-spectral density,'' ``cross-power
spectrum,'' or even simply ``cross-spectrum.''

### Autocorrelation

The cross-correlation of a signal with itself gives the
*autocorrelation function* of that signal:

(C.25) |

Note that the autocorrelation function is Hermitian:

When is real, its autocorrelation is

*symmetric*. More specifically, it is

*real and even*.

### Sample Autocorrelation

See §6.4.

### Power Spectral Density

The Fourier transform of the autocorrelation function
is
called the *power spectral density* (PSD), or *power
spectrum*, and may be denoted

When the signal is real, its PSD is real and even, like its autocorrelation function.

### Sample Power Spectral Density

See §6.5.

## White Noise

**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).

### Making White Noise with Dice

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

(C.27) |

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}

### Independent Implies Uncorrelated

It can be shown that *independent* zero-mean random numbers are
also uncorrelated, since, referring to (C.26),

(C.28) |

For Gaussian distributed random numbers, being uncorrelated also implies independence [201]. For related discussion illustrations, see §6.3.

### Estimator Variance

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.

#### Sample-Mean Variance

The simplest case to study first is the *sample mean*:

(C.29) |

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

(C.30) |

or

(C.31) |

Now assume , and let denote the variance of the process ,

*i.e.*,

Var | (C.32) |

Then the variance of our sample-mean estimator can be calculated as follows:

where we used the fact that the time-averaging operator is linear, and denotes the unbiased autocorrelation of . If is white noise, then , and we obtain

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 .

#### Sample-Variance Variance

Consider now the *sample variance* estimator

(C.33) |

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

where

The autocorrelation of
need not be simply related to that of
. However, when
is assumed to be *Gaussian* white
noise, simple relations do exist. For example, when
,

(C.34) |

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

(C.35) |

so that the variance of our estimator for the variance of Gaussian white noise is

Var | (C.36) |

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.

As mentioned above, to obtain a grounding in statistical signal processing, see references such as [201,121,95].

**Next Section:**

Gaussian Function Properties

**Previous Section:**

Selected Continuous Fourier Theorems