Show that one particular delta-correlated process is the stationary "shot noise" defined by F(t)= ΣFj δ(t-Tj) in which {T1, T2,... , Tj,...} with 0 ≤ T1 ≤ T2 ≤ ... ≤ Tj are the random arrival times for a Poisson process {Z(t)} (see Example 4.13), and {F1, F2,..., Fj,...} is a sequence of

Question

Show that one particular delta-correlated process is the stationary "shot noise" defined by

[latex] F(t)= \Sigma F_{j} \delta (t-T_{j}) [/latex]

in which [latex]

\left\{T_{1},T_{2},... ,T_{j},... \right\} [/latex] with [latex] 0\leq T_{1}\leq T_{2}\leq ...\leq T_{j} [/latex] are the random arrival times for a Poisson process [latex] \left\{Z(t)\right\} [/latex] (see Example 4.13), and [latex] \left\{F_{1},F_{2},... ,F_{j},... \right\} [/latex] is a sequence of identically distributed random variables that are independent of each other and of the arrival times. Also show that the process is not Gaussian and evaluate the cumulant functions for [latex] \left\{F(t) \right\} [/latex] .

Accepted Answer

The nature of the Poisson process is such that knowledge of past arrival times gives no information about future arrival times. Specifically, the interarrival time [latex] T_{j+1}-T_{j} [/latex] is independent of [latex] \left\{T_{1},...,T_{j} ight\} [/latex] . Combined with the independence of the [latex] F_{j} [/latex] pulse magnitudes, this ensures that [latex] F(t) [/latex] must be independent of [latex] F(s) [/latex] for [latex] t eq s [/latex] For [latex] s\lt t [/latex], knowledge of [latex] F(s) [/latex] may give some information about past arrival times and pulse magnitudes, but it gives no information about the likelihood that a pulse will arrive at future time [latex] t [/latex] or about the likely magnitude of such a pulse if it does arrive at time [latex] t [/latex]. Thus, [latex] \left\{F(t) ight\} [/latex] must be delta-correlated.

In order to investigate the cumulants, we first define a process [latex] \left\{Q(t) ight\} [/latex] that is the integral of [latex] \left\{F(t) ight\} [/latex] :

[latex] Q(t)=\int_{0}^{t}{F_{(s)}ds=\sum{F_{j}}U(t-T_{j}) } [/latex]

then we will use these results in describing [latex] \left\{F(t) ight\} =\left\{Q(t) ight\} [/latex] . The cumulant functions are an ideal tool for determining whether a stochastic process is Gaussian, because the cumulants of order higher than two are identically zero for a Gaussian process. Thus, we will proceed directly to seek the cumulant functions of [latex] \left\{Q(t) ight\} [/latex] and [latex] \left\{F(t) ight\} [/latex] It turns out that this is fairly easily done by consideration of the log-characteristic function of [latex] \left\{Q(t) ight\} [/latex] as in Section 3.7.

The nth-order joint characteristic function of [latex] \left\{Q(t) ight\} [/latex] for times [latex] \left\{t_{1},...,t_{n} ight\} [/latex] can be written as

[latex] M_{Q(t_{1})...Q(t_{n})}( heta _{1},..., heta _{n})\equiv E\left(\exp \left[i\sum\limits_{j=1}^{n}{ heta _{j}Q(t_{j})} ight] ight) [/latex]

We can take advantage of the delta-correlated properties of shot noise by rewriting [latex] Q(t_{j})[/latex] as

[latex] Q(t_{j})=\sum\limits_{l=1}^{j}{\Delta Q_{l} } [/latex] , with [latex] {\Delta Q_{l} } \equiv Q(t_{l})-Q(t_{l-1}) [/latex]

in which to [latex] t_{0}\lt t_{1}\lt t_{2}\lt ...\lt t_{n} [/latex] and [latex] t_{0}=0, [/latex] so [latex] Q(t_{0})=0, [/latex] . Now we can rewrite the characteristic function as

[latex] M_{Q(t_{1})...Q(t_{n})}( heta _{1},..., heta _{n})= E\left(\exp \left[i\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{j}{ heta _{j}\Delta Q_{l} } ight] ight) = E\left(\exp \left[i\sum\limits_{l=1}^{n}\sum\limits_{j=1}^{n}{ heta _{j}\Delta Q_{l} } ight] ight) [/latex]

The fact that [latex] F(t)[/latex] and [latex] F(s) [/latex] are independent random variables for [latex] t eq s [/latex] allows us to say that [latex] \left\{\Delta Q_{1},\Delta Q_{2},...,\Delta Q_{n} ight\} [/latex] is a sequence of independent random variables. This allows us to write the characteristic function as

[latex] M_{Q(t_{1})...Q(t_{n})}( heta _{1},..., heta _{n})=\prod\limits_{l=1}^{n}{E} \left(\exp \left[i\Delta Q_{l}\sum\limits_{j=l}^{n}{ heta _{j}} ight] ight)=\prod\limits_{l=1}^{n}{M_{\Delta Q_{l} }( heta _{l}+...+ heta _{n})} [/latex]

Thus, we can find the desired joint characteristic function from the characteristic function of [latex] \Delta Q_{l} [/latex] .

For simplicity, we will derive the characteristic function for the increment from time zero to time [latex] t_{1} [/latex] , that is, for the random variable [latex] \Delta Q_{1} \equiv Q(t_{1}) [/latex] . This can most easily be evaluated in two steps. First, we evaluate the conditional expected value

[latex] E\left(\exp \left[i heta Q(t_{1}) ight] |Z(t)=r ight) =E\left(\exp \left[i heta \sum\limits_{j=1}^{r}{F_{j}} ight] ight) =E\left(e^{i heta F_{1}}...e^{i heta F_{r}} ight) =M^{r}_{F}( heta ) [/latex]

in which the final form follows from the fact that [latex] \left\{F_{1},F_{2},...,F_{j},... ight\} [/latex] is a sequence of independent, identically distributed random variables, and [latex] M_{F}( heta ) [/latex] is used to designate the characteristic function of any member of the sequence. Using the probability values for the Poisson process from Example 4.13 then gives the characteristic function of interest as

[latex] M_{\Delta Q_{1} }( heta )=\sum\limits_{r=1}^{\infty }{E\left(\exp \left[i heta Q(t_{1}) ight]|Z(t_{1})=r ight) } P\left[Z(t_{1})=r ight] =\sum\limits_{r=1}^{\infty }{M^{r}_{F}}( heta ) \frac{\mu ^{r}_{Z}(t_{1})}{r!} e^{-\mu _{Z}(t_{1})} [/latex]

The result can be simplified as

[latex] M_{\Delta Q_{1} }( heta )=e^{-\mu _{Z}(t_{1})}\sum\limits_{r=1}^{\infty }{\frac{1}{r!} } \left[M_{F}( heta )\mu _{Z}(t_{1}) ight]^{r}= \exp \left[-\mu _{Z}(t_{1})+M_{F}( heta )\mu _{Z}(t_{1}) ight] [/latex]

because the summation is exactly the power series expansion for the exponential function. The log-characteristic function of [latex] \Delta Q_{1} [/latex] is then

[latex] \log \left[M_{\Delta Q_{1} }( heta ) ight]=\log \left[M_{Q(t_{1}) }( heta ) ight] =\mu _{Z}(t_{1})\left[M_{F}( heta )-1 ight] [/latex]

The corresponding function for any arbitrary increment [latex] \Delta Q_{l} [/latex] is obtained by replacing [latex] \mu _{Z}(t_{1}) [/latex] with [latex] \mu _{Z}(t_{l})-\mu _{Z}(t_{l-1}) [/latex]

The log-characteristic function for [latex] \Delta Q_{1} \equiv Q(t_{1}) [/latex] provides sufficient evidence to prove that [latex] \left\{F(t) ight\} [/latex] is not Gaussian. In particular, we can find the cumulants of [latex] \left\{Q(t) ight\} [/latex] by taking derivatives of [latex] \log \left[M_{Q(t)}( heta ) ight] [/latex] and evaluating them at [latex] heta =0 [/latex] .

Inasmuch as

[latex] \frac{d^{j}}{d heta ^{j}}\log \left[M_{Q(t)}( heta ) ight] =\mu _{Z}(t)\frac{d^{j}}{d heta ^{j}}M_{F}( heta ) [/latex]

though, this relates the cumulants of [latex] Q(t)[/latex] to the moments of [latex] F [/latex] , because the derivatives of a characteristic function evaluated at [latex] heta =0 [/latex] always give moment values (see Section 3.7). It is not possible for the higher-order even-order moment values of [latex] F [/latex] , such as [latex] E(F^{4}) [/latex] , to be zero, so we can see that the higherorder even cumulants of [latex] Q(t) [/latex] are not zero. This proves that [latex] Q(t) [/latex] is not a Gaussian random variable, which then implies that [latex] \left\{Q(t) ight\} [/latex] cannot be a Gaussian stochastic process. If [latex] \left\{Q(t) ight\} [/latex] is not a Gaussian process, though, then its derivative, [latex] \left\{F(t) ight\} [/latex] , is also not a Gaussian process.

Using the preceding results, we can evaluate the joint log-characteristic function for [latex] \left\{Q(t) ight\} [/latex] as

[latex] \log \left[M_{Q(t_{1})...Q(t_{n})}( heta _{1},..., heta _{n}) ight] =\sum\limits_{l=1}^{n}{\left[\mu _{Z}(t_{l})-\mu _{Z}(t_{l-1}) ight] } \left[M_{F}( heta _{l}+...+ heta _{n})-1 ight] [/latex]

The joint cumulant function for [latex] \left\{Q(t_{1}),...,Q(t_{n}) ight\} [/latex] is then

[latex] K_{n}\left[{Q(t_{1}),...,Q(t_{n})} ight]=i^{-n}\left(\frac{\partial^{n}}{\partial heta _{1}...{\partial heta _{n}}}\log \left[M_{Q(t_{1})...Q(t_{n})}( heta _{1},..., heta _{n}) ight] ight) _{ heta _{1}=...= heta _{n}=0} [/latex]

However, there is only one term in our expression for the log-characteristic function that gives a nonzero value for the mixed partial derivative. In particular, only the first term, with [latex] l=1 [/latex] , contains all the arguments [latex] ( heta _{1},..., heta _{n}) [/latex] . Thus, only it contributes to the cumulant function, and we have

[latex] K_{n}\left[{Q(t_{1}),...,Q(t_{n})} ight]=i^{-n}\mu _{Z}(t_{1})\left(\frac{\partial^{n}}{\partial heta _{1}...{\partial heta _{n}}}M_{F}( heta _{1}+...+ heta _{n}) ight) _{ heta _{1}=...= heta _{n}=0} [/latex]

[latex] =\mu _{Z}(t_1)E\left[F^{n} ight] [/latex]

Recall that the only way in which [latex] t_{1} [/latex] was distinctive in the set [latex] \left(t_{1},...,t_{n} ight) [/latex] was that it was the minimum of the set. Thus, the general result for [latex] \left\{Q(t) ight\} [/latex] is

[latex] K_{n}\left[Q(t_{1}),...,Q(t_{n}) ight] =\mu _{Z}\left(min\left[t_{1},...,t_{n} ight] ight) E\left[F^{n} ight] [/latex]

The linearity property of cumulants (see Eq. 3.44):

[latex]\kappa _{n+1}\left(X_{1}, ... , X_{n}\sum{a_{j}Y_{j}} ight) =\sum{a_{j}\kappa _{n+1}\left(X_{1}, ...,X_{n},Y_{j} ight) }[/latex]

then allows us to write the corresponding cumulant function for the [latex] \left\{F\left(t ight) ight\} [/latex] process as

[latex] K_{n}\left[F(t_{1}),...,F(t_{n}) ight] =E\left[F^{n} ight] \frac{\partial ^{n}}{\partial t_{1}...\partial t_{n}} \mu _{Z}\left(min\left[t_{1},...,t_{n} ight] ight) =E\left[F^{n} ight] \frac{\partial ^{n}}{\partial t_{1}...\partial t_{n}} \sum\limits_{j=1}^{n}{\mu _{Z}(t_{j})}\underset{l eq j}{\prod\limits_{l=1}^{n}} {U(t_{l}-t_{j})} [/latex]

After performing the differentiation and eliminating a number of terms that cancel, this expression can be simplified to

[latex] K_{n}\left[F(t_{1}),...,F(t_{n}) ight] =E\left[F^{n} ight] \dot \mu _{Z}(t_{n})\prod\limits_{l=1}^{n-1}{\delta (t_{l}-t_{n})} [/latex]

which confirms the fact that [latex] \left\{F\left(t ight) ight\} [/latex] is a delta-correlated process. One may also note that choosing [latex] n=2 [/latex] in this expression gives the covariance function for shot noise as

[latex] K_{FF}(t,s)\equiv K_{2}\left[F\left(t ight), F\left(s ight) ight] =E\left[F^{2} ight] \dot\mu _{Z}(t)\delta (t-s) [/latex]

One could have derived this covariance function without consideration of characteristic functions, but it would be very difficult to do that for all the higher-order cumulants.

Question 5.6: Show that one particular delta-correlated process is the sta...

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