Forecasting Spectrum Usage

In this section, we describe the forecasting techniques used to predict the activity of primary and secondary users in the channel of interest.

One approach for forecasting is to determine the likely state at the next time instant given that we have the state estimate for the current instant. For this, we need to calculate the probability of transitioning to another state at time (m + 1)Tm. The transitioning

probabilities can be determined starting from the following differential equation, ˙

Pi = QiPi, for i = p, s (2.25)

within the time interval mTm < t < (m + 1)Tm. The Pi and Qi are as defined in Sec. 2.1. Equation (2.25) governs the evolution of state transition probabilities. The solution of Eq. (2.25) gives the state transition probabilities from the m th measurement instant. The existence of the solution of Eq. (2.25) depends on two conditions. The first condition is the diagonalizable property of matrix Q_i. The second one is non-positive definiteness of matrix Q_i. The matrix Q_i satisfies both conditions. It is reducible to a diagonal form

Q_i = EiΓiE−1_i , where Γi is a diagonal matrix with eigenvalues of Qi, and Ei is the matrix

for r = 0,· · · , Ni. Hence, the solution, for t ∈ (mTm, (m + 1)Tm], is given by

Pi = EieΓitFi, for i = p, s; (2.26)

where Fi is a constant vector determined from the initial condition (i.e, bxi(m|m)) as Fi = (eΓimTm)−1E−1i PmTm(i), (2.27)

where PmTm(i) is a vector with all 0’s except the bxi(m|m)) th element which is 1. Now, we

compute state transitioning probability values for the instant (m + 1)Tm by integrating the

time varying state transitioning probability expressions (i.e., Eq. (2.26)) [86] as e Pi = 1 Tm ∫ (m+1)Tm mTm Pidt = 1 Tm Ei (∫ (m+1)Tm mTm eΓit_dt ) Fi = 1 Tm

[pre_i,0 pre_i,1 · · · epr_i,Ni]T, for i = p, s. (2.28)

In the above integration notation, we have used the fact that the integral of a matrix is the integral of each element of the matrix. The elements pre_i,k of the vector ePi denote the

probabilities of transitioning to state k at instant (m+1)Tm, for i = p, s. Based on the state

transitioning probabilities, we can now forecast the number of primary and secondary users. For ease in presentation, the forecasting method for spectrum usage of SU is described first.

2.3.1

Forecasting Spectrum Usage of SU

Forecasting spectrum usage by other secondary users is critical for the following reasons. Each SU can now determine if a particular channel is overcrowded with secondary users. If it is, the channel may be avoided as it may degrade the QoS. Additionally, by forecasting the number of secondary users in a channel, each SU can also determine how much power it needs to transmit without violating spectral emission limits (while maintaining its QoS). In this dissertation, we propose an upper bound forecasting based on state transitioning probability matrix of SU similar to the approach taken for forecasting the number of flows in

Internet traffic [86]. We discuss the case when the number of PU is 0 and we are interested in forecasting the number of secondary users at the next instant. The optimal estimate of the number of secondary users i.e., bxs(m|m), at time instant m is used to forecast the

number of secondary users at (m + 1) th instant. Based on estimated number of secondary users, bxs(m|m) at time mTm, state transitioning probability values is computed from Eq.

(2.28) and then prediction for (m + 1) th instant is done. The predicted state of SU for (m + 1) th instant at time mTm corresponds to [86],

exs(m) = min xs∈[bxs(m|m), Ns]

xs s.t. pres,k < β. (2.29)

Here, β is pre-set probability value. Equation (2.29) can be understood with the help of an example. Suppose, bxs(m|m) is obtained as 3 and Ns is 8. Based on the value of bxs(m|m),

a state transition probability vector i.e., ePs = _T1_m[pres,0 pres,1 · · · eprs,8]T is obtained, which

corresponds to the possible states of SU with number of users [0 1 · · · 8]T_{, respectively.}

By observing this state transition probability vector, one can determine multiple states for which pre_s,k < β. All states with number of users greater than 5 might satisfy this condition.

This effectively suggests that the probability of xs(m + 1) being 5 or more is going to be

negligible. Therefore, the upper bound for the xs(m + 1) should be 5. In general, the chosen

state is the state with minimum number of user satisfying Eq. (2.29). As a result, Eq. (2.29) serves as a good upper bound for the number of secondary users at time (m + 1) based on measurements up to time m.

2.3.2

Forecasting Spectrum Usage of PU

We propose two ways to forecast the presence or absence of a primary user. The first method is Kalman filter (KF) based prediction. In this method, the number of primary user obtained from the prediction stage (Eq. (2.20)) is used as the forecasted number of primary user. For example, from the estimated value at m th instant, bxp(m|m), the predicted value

for (m + 1) th instant at m th instant is taken as, exp(m) = bxp(m + 1|m)

= Apbxp(m|m) + Bp. (2.30)

The second method is based on state transitioning probabilities as given in Eq. (2.28). In this method, state transition probabilities from the present estimated state bxp(m|m)

is computed in the same way as it is described for SU above and state with higher state transition probability is taken as predicted state exp(m). For the proposed system model in

this chapter, Markov chain of PU shows that it follows two states - state 0 (number of PU is 0) and state 1 (number of PU is 1). Suppose, bxp(m|m) is obtained as 1 and based on this,

a state transition probability vector i.e., ePp = _T1_m[prep,0 prep,1]T is computed. The predicted

state of PU for (m + 1) th instant at time mTm is proposed as,

exp(m) =

{

1, if pre_p,1 >pre_p,0,

0, otherwise . (2.31)