An Introduction to Statistical Signal Processing

6.5. SPECIFICATION OF CONTINUOUS TIME ISI PROCESSES 355

Consider next the autocorrelation function RY (t, s). Choose t > s and write Yt as the sum of two increments as

Yt = (Yt − Ys) + Ys ,

and hence

RY (t, s) = E[YtYs] = E[(Yt − Ys)Ys] + E[Ys2]

using the linearity of expectation. The left term on the right is, however, the expectation of the product of two independent random variables since the increments Yt −Ys and Ys −Y0 are independent. Thus from theorem 4.3 the expectation of the product is the product of the expectations. Furthermore, the expectation of the increment Yt − Ys is the same as the expectation of the increment Yt−s − Y0 = Yt−s since the increments are stationary. Thus we have from this, (6.14), and (6.17) that

RY (t, s) = (t − s)msm + sσ2 + (sm)2 = sσ2 + (tm)(sm) .

Repeating the development for the case t ≤ s then yields

which yields the covariance

which extends (6.8) to the continuous time case.

6.5Specification of Continuous Time ISI Processes

The specification of processes with independent and stationary increments is almost the same in continuous time as it is in discrete time, the only real di erence being that in continuous time we must consider more general collections of sample times. In discrete time the specification was constructed using the marginal probability function of the underlying iid process, which implies the pmf of the increments. In continuous time we have no underlying iid process so we instead assume that we are given a formula for the cdf (pdf or pmf) of the increments; that is, for any t > s we have a cdf

or, equivalently, the corresponding pmf pYt−Ys (y) for a continuous amplitude process.

To specify a continuous time process we need a formula for the joint probability functions for all n and all ordered sample times t1, t2, . . . , tn (that is, ti < tj if i < j). As in the discrete time case, we consider conditional probability functions. To allow both discrete and continuous alphabet, we first focus on conditional cdf’s and find the conditional cdf

P (Ytn ≤ yn|Ytn−1 = yn−1, Ytn−2 = yn−2, . . . ). Then, using (6.10) we can apply the techniques used in discrete time by simply replacing the sample

times i by ti for i = 0, 1, . . . , n. That is, we define the random variables

{Xn} by

Then the {Xn} are independent (but not identically distributed unless the times between adjacent samples are all equal), and

n

This conditional cdf can then be used to evaluate the conditional pmf or pdf as

respectively. These can then be used to find the joint pmf’s or pdf’s as before as

n

fYt1 ,... ,Ytn (y1, . . . , yn) = fYti −Yti−1 (yi − yi−1) i=1

or

n

pYt1 ,... ,Ytn = pYti −Yti−1 (yi − yi−1) , i=1

6.5. SPECIFICATION OF CONTINUOUS TIME ISI PROCESSES 357

respectively. Since we can thus write the joint probability functions for any finite collection of sample times in terms of the given probability function for the increments, the process is completely specified.

The most important point of these relations is that if we are told that a process has independent and stationary increments and we are given a cdf or pmf or pdf for Yt = Yt − Y0, then the process is completely defined via the specification just given! Knowing the probabilistic description of the jumps and that the jumps are independent and stationary completely describes the process.

As in discrete time, a continuous time random process {Yt} is called a Markov process if and only if for all n and all ordered sample times t1 < t2 < . . . < tn we have for all yn, yn−1, . . . that

P (Ytn ≤ yn|Ytn−1 = yn−1, Ytn−2 = yn−2, . . . ) =

or equivalently,

fYtn |Ytn−1 ,... ,Yt1 (yn|yn−1, . . . , y1) = fYtn |Ytn−1 (yn|yn−1)

for continuous alphabet processes and

pYtn |Ytn−1 ,... ,Yt1 (yn|yn−1, . . . , y1) = pYtn |Ytn−1 (yn|yn−1)

for discrete alphabet processes. Analogous to the discrete time case, continuous time independent increment processes are Markov processes.

We close this section with the two most famous examples of continuous time independent increment processes.

[6.1 ] The Continuous Time Wiener process

The Wiener process is a continuous time independent increment process with stationary increments such that the increment densities are Gaussian with zero mean; that is, for t > 0,

e− y22 2tσ

fYt (y) = √2πtσ2 ; y R .

The form of the variance follows necessarily from the previously derived form for all independent increment processes with stationary increments. The specification for this process and the Gaussian form of the increment pdf’s imply that the Wiener process is a Gaussian process.

[6.2] The Poisson counting process is a continuous time discrete alphabet independent increment process with stationary increments such that the increments have a Poisson distribution; that is, for t > 0,

pYt (k) = (λtk!)k e−λt ; k = 0, 1, 2, . . . .

6.6Moving-Average and Autoregressive Processes

We have seen in the preceding sections that for discrete time random processes the moving-average representation can be used to yield the secondorder moments and also can be used to find the marginal probability function of independent increment processes. The general specification for independent increment processes, however, was found using the autoregressive representation. In this section we consider results for more general processes using virtually the same methods.

First assume that we have a moving-average process representation described by (6.1). We can use characteristic function techniques to find a simple form for the marginal characteristic function of the output process. In particular, assuming convergence conditions are satisfied where needed and observing that Yn is a weighted sum of independent random variables, the characteristic function of the output random process marginal distribution is calculated as the product of the transforms

MYn (ju) = MhkXn−k (ju) .

k

The individual transforms are easily shown to be

MhkXn−k (ju) = E 6ejuhkXn−k 7 = MXn−k (juhk) = MX(juhk) .

Thus

k

where the product is, as usual, dependent on the index sets on which {Xn} and {hk} are defined.

Equation (6.27) can be inverted in some cases to yield the output cdf and pdf or pmf. Unfortunately, however, in general this is about as far as one can go in this direction, even for an iid input process. Attempts to find joint or conditional distributions of the output process by distributions this

6.6. MOVING-AVERAGE AND AUTOREGRESSIVE PROCESSES 359

or other techniques will generally be frustrated by the complexity of the calculations required.

Part of the di culty in finding conditional distributions lies in the moving-average representation. The techniques used successfully for the independent increment processes relied on an autoregressive representation of the output process. We will now show that the methods used work for more general autoregressive process representations. We will consider specifically causal autoregressive processes represented as in (6.3) so that

Yn = Xn − akYn−k .

k>0

By the independence and causality conditions, the {Yn−k} in the sum are independent of Xn. Hence we have a representation for Yn as the sum of two independent random variables, Xn and the weighted sum of the Y ’s. The latter quantity is treated as if it were a constant in calculating conditional probabilities for Yn. Thus the conditional probability of an event for Yn can be specified in terms of the marginal probability of an easily determined event for Xn. Specifically, the conditional cdf for Yn is

k≥0

where a0 = 1. The conditional pmf or pdf can now be found. For example, if the input random process is continuous alphabet, the conditional output pdf is found by di erentiation to be

k

Finally, the complete specification can be obtained by a product of pmf’s or pdf’s by the chain rule as in (3.144) or (3.154). The discrete time independent increment result is obviously a special case of this equation. For more general processes, we need only require that the sum converge in (6.29) and that the corresponding conditional pdf’s be appropriately defined (using the general conditional probability approach). We next consider an important example of the ideas of this section.

6.7The Discrete Time Gauss-Markov Process

As an example of the development of the preceding section, consider the filter given in example [5.1]. Let {Xn} be an iid Gaussian process with mean m and variance σ2. The moving-average representation is

∞

that is, a Gaussian random variable with mean mY = m rk = m/(1 − r)

k

and variance σY2 = σX2 r2k = σX2 /(1 − r2), the moments found by the

k

second-order theory in example [5.1].

To find a complete specification for this process, we now turn to an autoregressive model. From (6.30) it follows that Yn must satisfy the difference equation

Hence {Yn} is a first-order autoregressive source with a0 = 1 and a1 = −r. Note that as with the Wiener process, this process can be represented as a first-order autoregressive process or as an infinite-order, moving average process. In fact, the Wiener process is the one-sided version of this process with r = 1.

Application of (6.29) yields

fY n (yn) = fYn (yn|yn−1, yn−2, . . . )fYn−1 (yn−1|yn−2, yn−3, . . . ) . . . fY1 (y1)

6.8Gaussian Random Processes

We have seen how to calculate the mean, covariance function, or spectral density of the output process of a linear filter driven by an input random process whose mean, covariance function, or spectral density is known. In general, however, it is not possible to derive a complete specification of the output process. We have seen one exception: The output random process of an autoregressive filter driven by an iid input random process can be specified through the conditional pmf’s or pdf’s, as in equation (6.29). In this section we develop another important exception by showing that the output process of a linear filter driven by a Gaussian random process — not necessarily iid — is also Gaussian. Thus simply knowing the output mean and autocorrelation or covariance functions — the only parameters of a Gaussian distribution — provides a complete specification. The underlying idea is that of theorem 4.4: a linear operation on a Gaussian vector yields another Gaussian random vector. The output vector mean and matrix covariance of the theorem are in fact just the vector and matrix versions of the linear system second-moment I/O relations (5.3) and (5.7)). The output of a discrete time FIR linear filter can be expressed as a linear operation on the input as in (4.26), that is, a finite dimensional matrix times an input vector plus a constant. Therefore we can immediately extend theorem 4.4 to FIR filtering and argue that all finite dimensional distributions of the output process are Gaussian and hence the process itself must be Gaussian. It is also possible to extend theorem 4.4 to include more general impulse responses and to continuous time by using appropriate limiting arguments. We will not prove such extensions. Instead we will merely state the result as a corollary:

Corollary 6.1 If a Gaussian random process {Xt} is passed through a linear filter, then the output is also a Gaussian random process with mean and covariance given by (5.3) and (5.7).

6.9The Poisson Counting Process

An engineer encounters two types of random processes in practice. The first is the random process whose probability distribution depends largely on design parameters: the type of modulation used, the method of data coding used, etc. The second type of random processes have probability distributions that depend on naturally occurring phenomena over which the engineer has little if any control: noise in physical devices, speech waveforms, the number of messages in a telephone system as a function of time, etc. Central limit theorems provide one example of such processes. This

chapter is devoted towards another example: the Poisson process. Here the basic Poisson counting process is derived from physical assumptions and a variety of properties are developed. Gaussian and Poisson processes provide classes of random processes that characterize (at least approximately) the majority of naturally occurring random processes. The development of Poisson processes provides further examples of many of the techniques developed so far.

Our intent here is to remove some of the mystery of the functional forms of two important distributions by showing how these apparently complicated distributional forms arise from nature. Therefore, the development presented is somewhat brief, without consideration of all the mathematical details.

The Poisson counting process was introduced as an example of specification of an independent and stationary increment process. In this section the same process is derived from a more physical argument.

Consider modeling a continuous time counting process {Nt; t ≥ 0} with the following properties:

1.N0 = 0 (the initial condition).

2.The process has independent and stationary increments. Hence the changes, called jumps, during nonoverlapping time intervals are independent random variables. The jumps in a given time interval are memoryless, and their amplitude does not depend on what happened before that interval.

3.In the limit of very small time intervals, the probability of an increment of 1, that is, of increasing the total count by 1, is proportional to the length of the time interval. The probability of an increment greater than 1 is negligible in comparison, e.g., is proportional to powers greater than 1 of the length of the time interval.

These properties well describe many physical phenomena such as the emission of electrons and other subatomic particles from irradiated objects (remember vacuum tubes?), the arrival of customers at a store or phone calls at an exchange, and other phenomena where events such as arrivals or discharges occur randomly in time. The properties naturally capture the intuition that such events do not depend on the past and that for a very tiny interval, the probability of such an event is proportional to the length of the interval. For example, if you are waiting for a phone call, the probability of its happening during a period of τ seconds is proportional to τ. The probability of more than two phone calls in a very small period τ is, however, negligible in comparison.

The third property can be quantified as follows: Let λ be the proportionality constant. Then for a small enough time interval ∆t,

Pr(Nt+∆t − Nt = 1) λ∆t

=

Pr(Nt+∆t − Nt = 0) 1 − λ∆t

=

Pr(Nt+∆t − Nt > 1) 0 . (6.33)

=

The relations of (6.33) can be stated rigorously by limit statements, but we shall use them in the more intuitive form given.

We now use the properties 1 to 3 to derive the probability mass function pNt−N0 (k) = pNt (k) for an increment Nt − N0, from the starting time at time 0 up to time t > 0 with N0 = 0. For convenience we temporarily change notation and define

p(k, t) = pNt−N0 (k) ; t > 0 .

Let ∆t be a di erentially small interval as in (6.33), and we have that

p(k, t + ∆t) =

k

Pr(Nt = n) Pr(Nt+∆t − Nt = k − n|Nt = n) .

n=0

Since the increments are independent, the conditioning can be dropped so that, using (6.33),

p(k, t + ∆t) =

k

Pr(Nt = n) Pr(Nt+∆t − Nt = k − n)

n=0

p(k, t)(1 − λ∆t) + p(k − 1, t)λ∆t ,

=

which with some algebra yields

p(k, t + ∆t) − p(k, t) = p(k − 1, t)λ − p(k, t)λ .

∆t

In the limit as ∆t → 0 this becomes the di erential equation

dtd p(k, t) + λp(k, t) = λp(k − 1, t) , t > 0 .

The initial condition for this di erential equation follows from the initial condition for the process, N0 = 0; i.e.,

since this corresponds to Pr(N0 = 0) = 1. The solution to the di erential equation with the given initial condition is

(This is easily verified by direct substitution.)

The pmf of (6.34) is the Poisson pmf, and hence the given properties produce the Poisson counting process. Note that (6.34) can be generalized using the stationarity of the increments to yield the pmf for k jumps in an arbitrary interval (s, t), t ≥ s as

As developed in chapter 6, these pmf’s and the given properties yield a complete specification of the Poisson counting process.

Note that sums of Poisson random variables are Poisson. This follows from the development of this section. That is, for any t > s > r, all three of the indicated quantities in (Nt −Ns)+(Ns −Nr) = Nt −Nr are Poisson. Thus the Poisson distribution is infinitely divisible in the sense defined at the end of the preceding section. Of course the infinite divisibility of Poisson random variables can also be verified by characteristic functions as in (4.101). Poisson random variables satisfy the requirements of the central limit theorem and hence it can be concluded that with appropriate normalization, the Poisson cdf approaches the Gaussian cdf asymptotically.

6.10Compound Processes

So far the various processes with memory have been constructed by passing iid processes through linear filters. In this section a more complicated construction of a new process is presented which is not a simple linear operation. A compound process is a random process constructed from two other random processes rather than from a single input process. It is formed by summing consecutive outputs of an iid discrete time random process, but the number of terms included in the sum is determined by a counting random process, which can be discrete or continuous time. As an example where such processes arise, suppose that on a particular telephone line the number of calls arriving in t minutes is a random variable Nt. The resulting Nt calls have duration X1, X2, . . . , XNt . What is the total amount of time occupied by the calls? It is the random variable

Nt

Yt = Xk.

k=1