Proceso de Tejido

where 𝑦_𝑖 denotes the 𝑁_𝑖 × 1 received signal vector at the 𝑗𝑡ℎ _{receiver; 𝑧}

𝑖 denotes the 𝑁𝑖×

1 zero mean unit variance circularly symmetric AWGN noise vector at the 𝑖𝑡ℎ _receiver;

𝑥_𝑖 denotes the 𝑀_𝑖× 1 signal vector transmitted from the 𝑗𝑡ℎ_{transmitter; 𝐇}

𝑖𝑗 is the 𝑁𝑖×

𝑀_𝑖 matrix of the channel coefficients between the 𝑗𝑡ℎ transmitter and the 𝑖𝑡ℎ receiver; Also, 𝑃_𝑗 = 𝐸[𝑥_𝑗 𝑥_𝑗𝐻], where 𝑃_𝑗 is the transmit power of the 𝑗𝑡ℎ transmitter [20]. It should be noted that 𝑖 and 𝑗 are used as a generalization denoting each Rx and Tx pair.

4.3.PU Link Optimization

4.3.1. The Numerical Comparison

Most of the work on OIA in CR networks such as in [24], [97], [98] utilize SWF scheme for the PU link. The work in [75] makes use of the MEB scheme where the PUs Tx places all its power on the antenna that corresponds to the largest eigenmode of its channel matrix 𝐻_𝑝𝑝. Its

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advantage being that all other dimensions are clearly left unused for the opportunistic SUs to exploit. However, with the comparative study carried out in [76], [77], it is clear that the ST- WF offers improved SU performance for the same PU parameters. Most significantly though is the ST-WF’s higher capacity at low to moderate SNR regimes, which fits well with CR networks [76].

In order to implement the ST-WF algorithm, we take a look at the original approach for WF [70], [106] i.e. the SWF approach. For a single MIMO PU channel, recall that

max𝑄 𝑙𝑜𝑔 |𝐼 + 1 𝜎2𝐻𝑝𝑝𝑄𝐻𝑝𝑝 †_| 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑡𝑟(𝑄) ≤ 𝑃 (4.4)

where 𝑄 is the 𝑀 × 𝑀 input covariance matrix

where 𝐻_𝑝𝑝 is the MIMO channel, 𝑄 is the autocorrelation matrix of the input vector 𝑥, defined as 𝑄 = 𝐸[𝑥𝑥†], P is the instantaneous power limit, |𝐴| denotes the determinant of 𝐴, and 𝑡𝑟(𝐴) denotes the trace of matrix 𝐴 and 𝜎_𝑖 is the noise variance. Let the SVD on matrix 𝐻𝑝𝑝 be given as 𝐻𝑝𝑝 = 𝑈𝑝𝑝𝛴𝑉𝑝𝑝𝐻 where 𝑈𝑝𝑝 is 𝑀𝑝𝑝× 𝑀𝑝𝑝 and unitary while 𝑉𝑝𝑝 is

𝑁_𝑝𝑝× 𝑁_𝑝𝑝 and unitary, Σ is 𝑀_𝑝𝑝× 𝑁_𝑝𝑝 and diagonal with non-negative entries i.e. Σ = diag{𝜆₁, . . . 𝜆_𝑀}. The diagonal elements of the matrix Σ are the singular values of 𝐻_𝑝𝑝, 𝐻𝑝𝑝 has exactly 𝑅𝐻 positive singular values, where 𝑅𝐻 is the rank of 𝐻𝑝𝑝 which

satisfies 𝑅_𝐻 ≤ min (𝑀_𝑝𝑝, 𝑁_𝑝𝑝). The transmitter/receiver chooses precoding matrices as the columns of 𝑉_𝑝𝑝(𝑈_𝑝𝑝) that corresponds to a non-zero PA that is used to maximize the rate of the PU link under power constraints as shown (4.4).

SWF can be used to optimally allocate power to the parallel channels as defined by the following equation [70]:

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𝑃_i = (𝛽 −𝜎𝑖

2

𝜆_𝑖 )

+

; 1 ≤ 𝑖 ≤ 𝑅_𝐻 (4.5)

Where 𝑃_i is the power of 𝑥_𝑝𝑝. The WF level 𝛽 is chosen such that ∑𝑅𝐻 𝑃_i = P

𝑖=1 as defined in

equation (4.5).

Once the PA matrix using SWF is set up according to [12], the diagonal matrix Σ contains 𝑚₁ non-zero/used entries and 𝑁1− 𝑚1 zero/unused entries which crucially translate into a set of

𝑚1 used receive dimensions and a set of 𝑁1− 𝑚1 unused receive dimensions with no PU

signal.

4.3.2. Space-Time Water-filling

In terms of implementing the ST-WF algorithm, the PU-Tx also chooses precoding matrices as the columns of 𝑉_𝑝𝑝(𝑈_𝑝𝑝) that corresponds to a non-zero PA that is used to maximize the rate of the PU link under power constraints as shown in (4.6) below. It should be noted that for ST-WF, the function 𝐸[𝑡𝑟(𝑄)] is present in all MIMO channel realizations, implying that the symbol rate changes faster than the channel variation where 𝑄 can be computed from all symbols but within one channel realization.

max𝑄 𝐸 [𝑙𝑜𝑔 |𝐼 + 1 𝜎2𝐻𝑝𝑝𝑄𝐻𝑝𝑝 †_|] 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑡𝑟(𝑄) ≤ 𝑃 (4.6)

For computation of the diagonal PA matrix by applying the so-called ST-WF algorithm, 𝛽 can be found as follows

𝑃_i= (𝛽̅ −𝜎𝑖

2

𝜆_𝑖)

+

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University of Hertfordshire, Hatfield AL10 9AB United Kingdom 90

-100 -5 0 5 10 15 20 5 10 15 20 25 MEB SWF ST-WF A ve ra ge S u m R at e (b /s ) SNR (dB)

Fig. 4.2: Average sum rate versus the SNR at the PU’s link for both water-filling (SWF and ST- WF) and MEB algorithms

where 𝛽̅ is the mean water-level that can be solved by the equation given below

∑ ∫ (𝛽̅ −𝜎𝑛 2 𝜆_𝑖) 𝑓(𝜆𝑖) 𝑑𝜆𝑖 ∞ 𝜎𝑖2 𝛽 𝐿𝑝 𝑙=1 = 𝐏 (4.8)

where 𝑓(𝜆_𝑖) is the marginal probability density function (pdf) of the random variable(𝜆_𝑖). To gain more insight into this issue, Fig. 4.2 shows the average sum rate versus the SNR for a single user PU MIMO link with the SWF, ST-WF and MEB PA schemes.

For the Rayleigh channels in this work, Rayleigh fading is assumed to be pure due to the PU- Tx and PU-Rx assumed to be in close proximity and hence the shadowing effect is negligible. The ST-WF algorithm with no shadowing variance achieves higher spectral efficiency over SWF at low SNRs, and has the highest gain of 5dB SNR over equal power distribution at a

PhD Thesis by Idris Abdulkadir Yusuf

University of Hertfordshire, Hatfield AL10 9AB United Kingdom 91

spectral efficiency of 2.5bps/Hz/antenna. Furthermore, fig. 2 shows that the numerical results obtained from the Monte Carlo simulations support theoretical results [70], [77].

Implementing the MEB algorithm achieves a performance that is only close to that of the SWF scheme. The simulation results clearly shows that the ST-WF scheme outperforms the other schemes and shows the possibility of increased sum rates in two-tier CR networks when the PU participates in IA.