Signal processing Part 10 pot

iii Using the stationarized observation data ˆyk, the signal detection is based on the model ˆyk = ˆsk +wk, 4 where ˆskis the modified signal.. Stationarization of Observation Data Recal

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Ohsumi & Yamaguchi (2006) to estimate the time-delay of signals in nonstationary random

noise, incorporated with the Wigner distribution-based maximum likelihood estimation

In this paper the signal detection problem is investigated using the stationarization approach

to nonstationary data The model of the corrupting noise is given by an ARMA(p, q) model

with unknown time-varying coefficients These coefficient parameters are estimated from the

(original) observation data by the Kalman filter

2 Problem Statement

Let{ y(k)} be the (scalar) observation data taken at sampling time instant t k(k = 1, 2,· · · ),

and assume that it can be expressed as

y(k) =s(k) +n(k) (k=1, 2,· · · ), (1)

where s(·)is a signal to be detected, whose form is surely known, and is assumed to exist

in a brief interval if it exists; and n(·)is the nonstationary random noise In consequence,

the observation data{ y(k)}becomes nonstationary, but its trend time series is assumed to be

removed by the process

where Y(k)is the original data received by the receiver; ∆Y(k) = Y(k)− Y(k −1); and d

indicates the order

In this paper the random noise n(k) is assumed to be given as the output of ARMA(p, q)

model with time-varying coefficient parameters:

where w(·) is the white Gaussian noise with zero-mean and variance parameter σ2;{ α i(·)}

and{ β j(·)}are slowly and smoothly varying parameters to be specified

Then our purpose is to propose a method of detecting the signal s(k)from the noisy

observa-tion data{ y(k)}

The procedure taken in this paper is as follows:

(i) First, based on the noise model (3), coefficient functions{ α i(·)}and{ β j(·)}are estimated

using Kalman filter from the observation data{ y(k)}

(ii) Using the estimates{ ˆα i(·)}and{ ˆβ j(·)} obtained in (i), the observation data y(k)is

modi-fied to become stationary This procedure is called the stationarization of observation data.

(iii) Using the stationarized observation data ˆy(k), the signal detection is based on the model

ˆy(k) = ˆs(k) +w(k), (4)

where ˆs(k)is the modified signal Equation (4) is familiar in the conventional signal detection

problem where the noise is stationary

3 Stationarization of Observation Data

Recalling the assumption that the duration of the signal s(k)is short, neglect the signal in the

observation data and consider the signal-free case, i.e., y(k) =n(k), then the observation data

y(k)is expressed by (1) and (3) as follows:

In order to estimate the time-varying parameters{ α i(k)}and{ β j(k)}in (5), suppose that they

change from step k − 1 to k under random effects { e ·(k)} Define vectors

− α p(k)

β1(k)

− e p(k)

e p+1(k)

1,· · · , τ2

p+q.Then, Eq (5) is expressed formally as

ν () = y () − Hˆ() ˆx ( | −1) (12)and

Here, ˆx ( | −1)and P ( | −1)are the one-step prediction and its covariance matrix computed

by Kalman filter for the past interval

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Ohsumi & Yamaguchi (2006) to estimate the time-delay of signals in nonstationary random

noise, incorporated with the Wigner distribution-based maximum likelihood estimation

In this paper the signal detection problem is investigated using the stationarization approach

to nonstationary data The model of the corrupting noise is given by an ARMA(p, q) model

with unknown time-varying coefficients These coefficient parameters are estimated from the

(original) observation data by the Kalman filter

2 Problem Statement

Let{ y(k)} be the (scalar) observation data taken at sampling time instant t k(k = 1, 2,· · · ),

and assume that it can be expressed as

y(k) =s(k) +n(k) (k=1, 2,· · · ), (1)

where s(·)is a signal to be detected, whose form is surely known, and is assumed to exist

in a brief interval if it exists; and n(·)is the nonstationary random noise In consequence,

the observation data{ y(k)}becomes nonstationary, but its trend time series is assumed to be

removed by the process

where Y(k) is the original data received by the receiver; ∆Y(k) = Y(k)− Y(k −1); and d

indicates the order

In this paper the random noise n(k) is assumed to be given as the output of ARMA(p, q)

model with time-varying coefficient parameters:

where w(·) is the white Gaussian noise with zero-mean and variance parameter σ2;{ α i(·)}

and{ β j(·)}are slowly and smoothly varying parameters to be specified

Then our purpose is to propose a method of detecting the signal s(k)from the noisy

observa-tion data{ y(k)}

The procedure taken in this paper is as follows:

(i) First, based on the noise model (3), coefficient functions{ α i(·)}and{ β j(·)}are estimated

using Kalman filter from the observation data{ y(k)}

(ii) Using the estimates{ ˆα i(·)}and{ ˆβ j(·)} obtained in (i), the observation data y(k)is

modi-fied to become stationary This procedure is called the stationarization of observation data.

(iii) Using the stationarized observation data ˆy(k), the signal detection is based on the model

ˆy(k) = ˆs(k) +w(k), (4)

where ˆs(k)is the modified signal Equation (4) is familiar in the conventional signal detection

problem where the noise is stationary

3 Stationarization of Observation Data

Recalling the assumption that the duration of the signal s(k)is short, neglect the signal in the

observation data and consider the signal-free case, i.e., y(k) =n(k), then the observation data

y(k)is expressed by (1) and (3) as follows:

In order to estimate the time-varying parameters{ α i(k)}and{ β j(k)}in (5), suppose that they

change from step k − 1 to k under random effects { e ·(k)} Define vectors

− α p(k)

β1(k)

− e p(k)

e p+1(k)

1,· · · , τ2

p+q.Then, Eq (5) is expressed formally as

ν () = y () − Hˆ() ˆx ( | −1) (12)and

Here, ˆx ( | −1)and P ( | −1)are the one-step prediction and its covariance matrix computed

by Kalman filter for the past interval

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It is a simple exercise to show that the statistical properties of ν m(·) is the same as that of w(·),

i.e., E { ν m(k)} = 0 and E {| ν m(k)|2} = σ2(for proof, see Appendix) Then, instead of (8) we

have the expression,

y(k) =Hˆ(k)x(k) +w(k) (14)The procedure for computing ˆH(k)is stated as follows:

(i) Preliminaries: Assume for the past k (<0)that{ ν m(−1), ν m(−2),· · · , ν m(− q)}are set

appro-priately (may be set all zero), and preassign ˆx(0| −1), ˆP(0| −1)and ˆH(0)as initial values

Repeat Steps (ii) and (iii) for = k − q, k − q+1,· · · , k −1 to obtain ˆH(k) In computing (12)

and (13), ˆx ( | −1)and P ( | −1)are computed by the Kalman filter (e.g., Jazwinski, 1970):

Kalman filter constructed for (7) and (14) (whose form is the same as (15)-(19) replacing

by the present k) Under the basic assumption that the coefficient parameters vary slowly and

smoothly, they can be treated like constants in an interval I k around the current time k Write

them as ˆα ik and ˆβ jk in I k Replacing the past{ w(k − j)}in (5) by the statistically equivalent

sequence{ ν m(k − j)} , define the sequence ˆy(k)by

which implies that the sequence{ ˆy(k)} is stationary because w(k) is the stationary white

noise

4 Signal Detection

After obtained the estimates of coefficient parameters, the observation process (14) may be

written using estimates as

Note that (4)bisis familiar to us as the mathematical model for the detection problem of signals

in stationary noise (e.g., Van Trees, 1968).

Now, consider the binary hypotheses: H1: ˆy(k) =ˆs(k) +w(k), and H0: ˆy(k) =w(k), and let ˆY k

be the stationarized observation data taken up to k, ˆY k={ ˆy (), =1, 2,· · · , k } Since the

ad-ditive noise w(k)is white Gaussian sequence with zero-mean and variance σ2, the

likelihood-ratio function Λ(k) =p { ˆY k | H1} / ˆY k | H0}is evaluated as follows:

Λ(k) =

k

∏

=1(2π)−1exp

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It is a simple exercise to show that the statistical properties of ν m( ·) is the same as that of w(·),