Open Access

Space Alignment for Cognitive Transmission in MIMO Uplink Channels

EURASIP Journal on Wireless Communications and Networking20102010:465157

DOI: 10.1155/2010/465157

Received: 5 July 2010

Accepted: 2 November 2010

Published: 3 November 2010

Abstract

This paper investigates a new transmission technique for cognitive access in multiple-input multiple-output (MIMO) uplink channels with waterfilling power allocation- (WPA-) based primary transmission. The proposed technique allows a cognitive node to access the common destination simultaneously with the primary node, without affecting the MIMO primary performance. By using an appropriate precoding design, the cognitive node aligns its transmitted signal to the unused primary eigenmodes and ensures an orthogonality between the primary and the cognitive links. In order to overcome the limitation of the conventional WPA at high signal-to-noise ratios, a modified WPA enables the primary node to release some eigenmodes in order to provide a positive cognitive rate for all the cases.

1. Introduction

Cognitive radio (CR) is introduced as an efficient technique in order to use the radio spectrum more efficiently [1]. It is characterized by the capability of the cognitive radios (secondary or unlicensed nodes) to coexist with the spectrum owners (primary or licensed nodes) and share the same frequency band in an opportunistic fashion. The opportunistic access enables the cognitive nodes to use the spectrum when it is idle (spectrum holes) and makes the CR network transparent to the primary network. However, for scenarios with high primary traffic, the number of spectrum holes becomes limited, and thus a positive cognitive data rate cannot be supported.

In order to overcome this limitation of the conventional CR concept, several approaches have been proposed in the literature. In [2], a dirty-paper coding (DPC) approach allows both the primary and the secondary nodes to simultaneously access the channel (interference channel) and protects the primary receiver from interference. However, the DPC design requires a knowledge of the primary signal at the CR as well as a global instantaneous channel knowledge which correspond to a high implementation complexity. On the other hand, in [3] the primary node exchanges its transmission silence with a cooperative (relaying) assistance from the cognitive node in order to increase both the primary and the cognitive performance. However, this approach requires a cooperation between primary and CR nodes which is not always possible in a "strict" cognitive context where the primary network is not aware of the CR nodes. Another approach incorporates the interference alignment concept [4, 5] with a multiple-input multiple-output (MIMO) waterfilling power allocation (WPA) [6, 7] in order to achieve orthogonality between primary and cognitive networks. This approach uses an appropriate precoding technique in order to align the cognitive transmitted signal to the unused eigenmodes of the primary channel. However, the original work in [6, 7] focuses on an interference channel (2 Tx-2 Rx) and cannot guarantee a cognitive transmission at high signal-to-noise ratios (SNRs), where the number of the unused eigenmodes is limited.

In this paper we extend the technique presented in [6] for a MIMO uplink channel with CR. An appropriate space alignment design enables the CR node to access the common destination simultaneously with the primary node without affecting the primary performance by using the unused primary eigenmodes. We show that due to the uplink topology the cognitive space alignment corresponds to a parallel symmetric Gaussian channel where the number of subchannels is equal to the number of unused primary eigenmodes. In addition, in order to ensure a positive cognitive data rate for high SNRs, a modified WPA that allows the primary node to control its transmitted power for each subchannel is proposed. We show that an appropriate power threshold can release some primary eigenmodes for cognitive transmission without significantly affecting the primary performance. The average achievable rate for both the primary and the secondary networks is evaluated via theoretical and analytical results. To the best of our knowledge the space alignment design for CR uplink channels as well as the modified WPA scheme are reported in this paper for the first time.

The rest of this paper is organized as follows. In Section 2 we present the system model, and we introduce the main assumptions required for our analysis. In Section 3 we describe the proposed CR space alignment design, and we discuss its achievable rate performance. Numerical results are shown and discussed in Section 4, followed by concluding remarks in Section 5.

Notation 1.

Upper case and lower case bold symbols denote matrices and vectors, respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq1_HTML.gif denotes the trace of a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq3_HTML.gif denotes the identity matrix of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq5_HTML.gif denotes the logarithm of base 2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq6_HTML.gif represents the expectation operator, and the superscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq7_HTML.gif denotes hermitian transposition operation.

2. System Model

We assume a three-node cognitive topology consisting of one primary node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq8_HTML.gif , one secondary (cognitive) node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq9_HTML.gif , and a common destination https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq10_HTML.gif (e.g., base station) as shown in Figure 1 (a similar uplink configuration is assumed in [3]). All the nodes are equipped with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq11_HTML.gif antennas, and both nodes operate in the same frequency band following the rules of the cognitive radio. More specifically, the node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq12_HTML.gif (licensed node) exclusively uses the channel and "enjoys" a point-to-point MIMO channel while the node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq13_HTML.gif (unlicensed node) is looking for transmission opportunities and can access the channel subject to the constraint of not affecting the primary link. The common receiver has been optimized (designed) for the primary network and cannot handle Multiple-access interference (MAI) by using advanced interference cancelation schemes. (The goal of most cognitive radio applications is to allow unlicensed nodes to opportunistically use a licensed band. Thus, in many cases, the signaling format of the cognitive nodes may not be known, and thus interference mitigation will generally not be possible.) All the nodes always have data to transmit, and therefore the primary link never becomes idle which results in no-transmission opportunities for the cognitive node. However, an appropriate linear precoding enables the cognitive node to simultaneously access the channel and ensures a positive cognitive rate without affecting the primary network. The received signal at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq14_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ1_HTML.gif
(1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq15_HTML.gif denotes the transmit vector for the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq16_HTML.gif th node with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq17_HTML.gif , the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq18_HTML.gif denotes the linear precoder used at the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq19_HTML.gif th node (unitary matrix with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq20_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq21_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq22_HTML.gif channel matrix for the link https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq23_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq24_HTML.gif indicates an additive white Gaussian noise (AWGN) vector at the receiver with a covariance matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq25_HTML.gif . Both nodes are subject to a power constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq26_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq27_HTML.gif denotes the covariance matrix of the transmit vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq28_HTML.gif and indicates the power allocation at each node. The entries of the channel matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq29_HTML.gif are independently and identically distributed (i.i.d.) complex Gaussian circularly symmetric random variables with unit variance (without loss of generality); this means that both channel matrices are almost surely of full rank (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq30_HTML.gif ). In addition, the channel matrices remain constant for the whole transmission and change to an independent realization for the next transmission. At the receiver https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq31_HTML.gif , the received signal is linearly processed with the postprocessing matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq32_HTML.gif which results in the output signal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq33_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Fig1_HTML.jpg
Figure 1

Cognitive transmission in an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq34_HTML.gif MIMO channel via SVD and space alignment.

As for the channel side information (CSI), we assume that both the primary and the secondary networks have a global knowledge of their channels (a similar assumption is considered in [2, 6, 7]). More specifically, we assume that the instantaneous primary channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq35_HTML.gif is perfectly known at the nodes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq36_HTML.gif , and the receiver https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq37_HTML.gif while the instantaneous secondary channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq38_HTML.gif is perfectly known at the cognitive node https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq39_HTML.gif . It is worth noting that although this assumption provides an upper bound on the achievable rate for both the primary and the cognitive networks, several techniques make this assumption reasonable: (a) in some contexts channel reciprocity can be exploited to acquire CSI at the transmitters (Time Division Duplex (TDD) mode), (b) feedback channels are often available in wireless communications (in several modern wireless standards, e.g., Long Term Evolution (LTE)), and (c) learning mechanisms [8] can be exploited to iteratively track the required CSI. The impact of an imperfect channel knowledge on the achievable rates is beyond the scope of this paper and will be considered for future investigation.

3. A Space Alignment Technique for Cognitive Access

In this section we introduce a space alignment technique that enables the cognitive node to communicate with the common destination https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq40_HTML.gif , simultaneously with the primary node, without affecting the MIMO primary network. The performance of the system is determined in terms of the achievable data rate for both the primary and the secondary networks.

3.1. Primary Network

According to the principles of the cognitive radio, the primary network has an exclusive use of the spectrum and ignores the existence of the cognitive network. Given that both the primary node and the common receiver have a knowledge of the instantaneous channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq41_HTML.gif , the combination of a spatial multiplexing transmission based on the single value decomposition (SVD) of the channel, matrix with a WPA maximizes the primary achievable rate [9, Section 7.1]. The SVD technique allows the primary node to send parallel data streams (without interference between them) along the eigenmodes of the channel and the WPA allocates the power based on the instantaneous strength of the subchannels. More specifically, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq42_HTML.gif denotes the SVD of the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq43_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq44_HTML.gif are (rotation) unitary matrices and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq45_HTML.gif is a rectangular matrix whose diagonal elements https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq46_HTML.gif are the ordered singular values of the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq47_HTML.gif , the SVD design requires
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ2_HTML.gif
(2)
and transforms the initial MIMO channel to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq48_HTML.gif scalar parallel channels defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ3_HTML.gif
(3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq49_HTML.gif is an AWGN with covariance matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq50_HTML.gif . If we ignore the interference-related term (the proposed technique forces this interference to zero), the maximum average achievable primary rate is given as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ4_HTML.gif
(4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq51_HTML.gif denotes the expectation operator and the power allocation covariance matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq52_HTML.gif is given by the modified WPA defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ5_HTML.gif
(5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq53_HTML.gif , the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq54_HTML.gif is chosen to satisfy the total power constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq56_HTML.gif is a constant, and the noise power is equal to 1. In contrast to the conventional WPA technique, which corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq57_HTML.gif and maximizes the achievable rate, the introduced modified waterfilling policy gives the primary node the flexibility to control the power allocated at each eigenmode. More specifically, the system parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq58_HTML.gif depends on the degradation tolerance (maximum degradation level without affecting the required system quality of service (QoS)) that characterizes the system and enables the primary node to release some spatial directions without significantly affecting its performance (e.g., if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq59_HTML.gif , the release of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq60_HTML.gif th spatial channel does not modify the achievable performance).

Parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq61_HTML.gif . From a CR point of view, the maximization of the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq62_HTML.gif in respect to the primary performance degradation tolerance, gives more opportunities for a secondary transmission (as the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq63_HTML.gif is increased, the primary eigenmodes are released with a higher probability). Therefore, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq64_HTML.gif denotes the primary performance degradation tolerence (its value depends on the QoS of the application), the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq65_HTML.gif is expressed by the following optimization problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ6_HTML.gif
(6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ7_HTML.gif
(7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq66_HTML.gif denotes the maximum achievable primary rate corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq67_HTML.gif . Due to the iterative nature of the WPA [9], the above optimization problem can be solved by using a simple iterative algorithm that continuously increases https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq68_HTML.gif (with a constant step) until the constraint in (7) is satisfied. Although a further theoretical analysis of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq69_HTML.gif is beyond the scope of this paper, an interesting remark holds for the high SNR regime. More specifically, given that the WPA converges to a uniform PA scheme at high SNRs, a parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq70_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq71_HTML.gif (where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq72_HTML.gif ) ensures the release of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq73_HTML.gif primary eigenmodes for all the cases. The parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq74_HTML.gif is introduced as a critical system parameter, and its importance is evaluated via simulation results in the next section.

3.2. Primary Network

The proposed technique allows the cognitive node to access the channel simultaneously with the primary node by using the unused primary spatial directions. An appropriate space alignment concentrates the cognitive transmission to the unused primary spatial eigenmodes (with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq75_HTML.gif ) and enables a positive cognitive rate without affecting the primary performance. The proposed space alignment requirement is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ8_HTML.gif
(8)
where the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq76_HTML.gif models the unused primary subspace (eigenmodes) and is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq77_HTML.gif diagonal matrix with entries
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ9_HTML.gif
(9)
Based on (8) and (9), the secondary precoding matrix that aligns the cognitive signal to the available primary subspace is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ10_HTML.gif
(10)
(The cognitive node perfectly senses the available primary eigenmodes, or a simple primary signaling feeds the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq78_HTML.gif to the cognitive node.)The average rate achieved by the secondary node becomes equal to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ12_HTML.gif
(12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq79_HTML.gif is the number of the available primary eigenmodes (e.g., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq80_HTML.gif and the upper bound in (12) yields by Jensen's inequality for the concave function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq81_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq82_HTML.gif is a constant. The above expressions demonstrate that the cognitive transmission is transformed to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq83_HTML.gif symmetric parallel Gaussian channels (without a channel fading degradation), and thus a PA scheme, which symmetrically allocates the available power https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq84_HTML.gif among the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq85_HTML.gif subchannels, maximizes the instantaneous achievable rate. In addition, the average number of the unused primary eigenmodes can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ13_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Equ14_HTML.gif
(14)

where the above expression is based on the ordered singular values of the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq86_HTML.gif as well as the WPA applied on the primary network. The probability in (14) can be evaluated numerically, and its general closed form is beyond the scope of this paper. (For small values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq87_HTML.gif a closed form expression is possible; for example, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq89_HTML.gif the joint probability density function of the ordered eigenvalues is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq90_HTML.gif [10] which results in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq91_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq92_HTML.gif .) It is worth noting that, due to the channel inversion that is involved in the cognitive precoder matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq93_HTML.gif in (10), the achievable cognitive rate is independent of the channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq94_HTML.gif and the related fading/shadowing effects.

4. Numerical Results

Computer simulations were carried out in order to evaluate the performance of the proposed scheme. The simulation environment follows the system model of Section 2 and the adopted performance metric is the average data rate expressed in bits per channel use (BPCU).

In Figure 2 we plot the achievable data rate for both the primary (4) and the secondary (11) network versus the SNR ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq95_HTML.gif ) for different numbers of antennas. (The SNR is similar for both the primary and the secondary networks due to the normalized assumed system model and the precoding process at the cognitive node (the primary channel matrix is a unitary matrix and the cognitive link is independent of the channel fading degradation.) A conventional WPA ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq96_HTML.gif ) is assumed for the primary link while the primary performance with a uniform power allocation ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq97_HTML.gif ) is used as a reference curve. The first important observation is that the proposed scheme enables a positive cognitive data rate without any modification at the primary network. More specifically, the cognitive data rate achieves its maximum value at the intermediate SNRs ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq98_HTML.gif 10 dB) where the number of the unused eigenmodes as well as their SNR strength provides the best combination. On the other hand, at low SNRs, although the number of the unused eigenmodes is the maximum one (the primary transmitted power is concentrated to the eigenmode with the maximum singular value), the channel quality results in a poor cognitive performance. For high SNRs, the conventional WPA spreads the transmitted power symmetrically along the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq99_HTML.gif eigenmodes of the channel, and therefore the cognitive data rate converges to zero, as no eigenmode is available to convey the cognitive data [9, Section https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq100_HTML.gif ]. In order to validate this remark we can see that the primary performance matches the one achieved by a uniform PA at high SNRs. Furthermore, it can be seen that as the number of antennas increases, the performance is improved for both the primary and the secondary networks. An increase in the number of antennas increases the number of eigenmodes ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq101_HTML.gif ) which further increases the number of unused eigenmodes.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Fig2_HTML.jpg
Figure 2

Average data rate for both the primary and the secondary networks versus SNR for different numbers of antennas; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq102_HTML.gif and, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq103_HTML.gif (conventional WPA).

Figure 3 demonstrates the impact of the PA parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq104_HTML.gif on the performance of the proposed scheme. The simulation setup is similar to the one used in Figure 2 with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq105_HTML.gif , which is motivated by the behavior of the conventional WPA at high SNRs. More specifically, given that the conventional WPA becomes equivalent to a uniform PA for the high SNR regime, a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq106_HTML.gif ensures that at least one primary eigenmode will always be available in order to convey cognitive data. As can be seen in Figure 3, the cognitive data rate does not converge to zero for high SNRs and continues to increase as the SNR increases. An important observation is that for intermediate SNRs the cognitive data rate significantly increases without significantly affecting the primary data rate (i.e., for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq108_HTML.gif 15 dB, the cognitive rate increased from 3 BPCU to 7 BPCU while the primary performance remains almost equal to 24.5 BPCU). On the other hand, for high SNRs, the release of some eigenmodes (due to the constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq109_HTML.gif ) yields a performance degradation for the primary link. This performance degradation should satisfy the tolerance of the system and is the cost for the cognitive transmission. However, it can be seen that as the number of antennas increases, the number of the released eigenmodes becomes negligible in comparison to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq110_HTML.gif , and thus the performance degradation decreases.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Fig3_HTML.jpg
Figure 3

Average data rate for both the primary and the secondary networks versus SNR for different numbers of antennas; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq111_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq112_HTML.gif .

Finally, Figures 4 and 5 plot the average data rate for both the primary and the secondary nodes ((4) and (11)) versus the power allocation level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq114_HTML.gif  dB and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq115_HTML.gif  dB, respectively, with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq116_HTML.gif antennas. As it can be seen, the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq117_HTML.gif significantly affects the performance for both nodes and introduces a trade-off between them. More specifically, as the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq118_HTML.gif is increased, the primary eigenmodes are released with a higher probability which results in a degradation of the primary performance while it increases the CR performance. An interesting observation is that the average performance for both nodes is divided in some rate areas where the achievable data rate is almost constant. For high SNRs (Figure 5) this observation is in line with the discussion in Section 3.1, and therefore a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq119_HTML.gif that takes values in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq120_HTML.gif (with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq121_HTML.gif ) ensures that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq122_HTML.gif spatial directions are used for CR transmission while the remaining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq123_HTML.gif spatial directions are used for primary transmission. On the other hand, at low SNRs and for high values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq124_HTML.gif (e.g., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq125_HTML.gif ) the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq126_HTML.gif does not guarantee a continuous release of the corresponding primary eigenmodes as the WPA levels https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq127_HTML.gif strongly depend on the related single values of the primary channel ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq128_HTML.gif is of the same order with the inverse of the single values). However, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq129_HTML.gif increases the WPA levels converge to a uniform PA ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq130_HTML.gif becomes significantly larger than the inverse of the single values), and therefore the system performance follows our remarks for the high SNR regime.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Fig4_HTML.jpg
Figure 4

Average data rate for both the primary and the secondary networks versus the power allocation level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq131_HTML.gif (linear scale); https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq132_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq133_HTML.gif dB.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_Fig5_HTML.jpg
Figure 5

Average data rate for both the primary and the secondary networks versus the power allocation level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq134_HTML.gif (linear scale); https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq135_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F465157/MediaObjects/13638_2010_Article_1914_IEq136_HTML.gif dB.

5. Conclusion

A new cognitive transmission technique for MIMO uplink channels has been proposed. The new technique incorporates space alignment with WPA and results in an orthogonality between primary and cognitive transmissions. We have shown that a conventional WPA provides an efficient cognitive performance for intermediate SNRs but results in a zero cognitive data rate for high SNRs. A modified WPA that allows the primary node to release some eigenmodes without affecting its required QoS and ensures a positive cognitive data rate for all the cases has been also investigated.

Authors’ Affiliations

(1)
Department of Electrical and Computer Engineering, University of Cyprus

References

  1. Haykin S: Cognitive radio: brain-empowered wireless communications. IEEE Journal on Selected Areas in Communications 2005, 23(2):201-220.View ArticleGoogle Scholar
  2. Devroye N, Mitran P, Tarokh V: Achievable rates in cognitive radio channels. IEEE Transactions on Information Theory 2006, 52(5):1813-1827.MathSciNetView ArticleMATHGoogle Scholar
  3. Kourdi TE, Simeone O: An information-theoretic view of spectrum leasing via secondary cooperation. Proceedings of the IEEE International Conference on Communications, 2010, Cape Town, South Africa 1-6.Google Scholar
  4. Cadambe VR, Jafar SA: Interference alignment and degrees of freedom of the K -user interference channel. IEEE Transactions on Information Theory 2008, 54(8):3425-3441.MathSciNetView ArticleMATHGoogle Scholar
  5. Lee N, Lim J-B: A novel signaling for communication on MIMOY channel: signal space alignment for network coding. Proceedings of IEEE International Symposium on Information Theory, 2009, Seoul, Republic of Korea 2892-2896.Google Scholar
  6. Perlaza SM, Debbah M, Lasaulce S, Chaufray J-M: Opportunistic interference alignment in MIMO interference channels. Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '08), 2008, Cannes, France 1-5.Google Scholar
  7. Perlaza SM, Fawaz N, Lasaulce S, Debbah M: From spectrum pooling to space pooling: opportunistic interference alignment in MIMO cognitive networks. IEEE Transactions on Signal Processing 2010, 58(7):3728-3741.MathSciNetView ArticleGoogle Scholar
  8. Gomadam K, Cadambe V, Jafar S: Approaching the capacity of wireless networks through distributed interference alignment. Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '08), 2008, New Orleans, La, USA 1-6.Google Scholar
  9. Tse D: Fundamentals of Wireless Communication. Cambridge University Press, New York, NY, USA; 2005.View ArticleMATHGoogle Scholar
  10. Ordóñez LG, Palomar DP, Fonollosa JR: Ordered eigenvalues of a general class of Hermitian random matrices with application to the performance analysis of MIMO systems. IEEE Transactions on Signal Processing 2009, 57(2):672-689.MathSciNetView ArticleGoogle Scholar

Copyright

© Ioannis Krikidis. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.