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Joint Sensing and Resource Allocation in Underlay CRs

  • Universitetet i Agder, Grimstad, Norway
Luis M. Lopez-Ramos, Antonio G. Marques, and Javier Ramos
King Juan Carlos University (Madrid, Spain), Dept. of Signal Theory and Communications
Effective operation of cognitive radios (CRs) requires sensing the
spectrum and dynamic adaptation of the available resources accord-
ing to the sensed information. Although sensing and resource al-
location are coupled, most existing designs optimize each of the
tasks separately. This work optimizes them jointly for an under-
lay CR paradigm. The formulation considers that secondary users
adapt their power and rate based on the available imperfect chan-
nel state information, while taking into account the cost associated
with acquiring such an information. The objective of the optimiza-
tion is twofold: maximize the (sum-rate) performance of the CR and
protect the primary users through an average interference constraint.
Designing the sensing in our underlay paradigm amounts to decide
what channel/frequency slots are sensed at every time instant. Par-
tial observability of the channel state (due to noisy and outdated in-
formation) calls for (Bayesian) sequential estimators to keep track
of the interference channel gains, as well as for dynamic program-
ming tools to design the optimal schemes. Together with the optimal
schemes, a simple approximate solution is also developed.
Index TermsCognitive radio, underlay paradigm, sensing,
dual decomposition, sequential estimation, dynamic programming.
Cognitive radios (CRs) are a key technology to alleviate spectrum
scarcity. When CRs are deployed, secondary users (SUs) have to
sense their radio environment to optimize their communication per-
formance while controlling the interference to primary users (PUs)
[1]. In underlay CRs, sensing amounts to acquire the channel state
information (CSI) needed to limit the power transmitted by the SU,
so that the interference inflicted to the PU is kept under a prescribed
limit. This typically amounts to acquiring the (fading) gains of the
SU-to-PU channels (also known as interference channels). Due to
lack of collaboration mechanisms between PU and CR systems, ac-
curately estimating the SU-to-PU channels requires considerable ef-
fort [2]. However, the cost of acquiring the CSI is often not taken
into account in the modeling; see [3, 4, 5, 6, 7] for some excep-
tions. As a result, a careful design of the sensing policy is critical
to guarantee an efficient operation of the CR. This paper optimizes
the sensing and resource allocation (RA) tasks for an underlay CR
model jointly. Uncertainties on the sensed CSI and sensing cost will
be taken into account during the RA, while the actual benefit of the
CSI for the SUs will be taken into account during the sensing phase.
Some important challenges to optimize the sensing and RA
jointly are: (C1) the need of the RA algorithms to deal with im-
perfect CSI (noisy and outdated) that renders the exact interfer-
ence caused to PUs uncertain; (C2) the inability to estimate the
totality of the PU-SU-channel-time grid, due to the scarcity of
This work was partially supported by the Spanish Ministry of Science
(FPU Grant AP2010-1050) and the EU-FP7 (ICT-2011-9-TUCAN3G).
resources (power, time or hardware availability); and (C3) the cou-
pling between sensing and transmission resources. To deal with
(C1), several works aim to control the average interference using a
probabilistic representation of the state information of the primary
network (SIPN) [2, 7]. Adaptive stochastic algorithms provide ro-
bustness to non-stationarities and lack of knowledge of the channel
distributions. See also [8] for a different approach to cope with
estimation errors. To deal with (C2), advanced sensing schemes
aim at optimally selecting the subset of sensed channels [9, 10, 7].
Moreover, when the SIPN exhibits time correlation, the information
acquired can be reused ahead on time (accounting for the fact that
information gets outdated). These schemes are usually designed
using dynamic programming (DP) tools such as partially observable
Markov decision processes (POMDPs) [7, 9]. Regarding (C3), RA
in underlay CRs has been extensively investigated. In [11] an RA
framework that considers both interference constraints for PUs and
QoS constraints for SUs is presented; power is optimized jointly
with admission control. The optimal RA strategies to achieve the
ergodic and outage capacity of the SU fading channel is studied in
[12] under different types of power constraints and fading channel
models. See, e.g., [13, 8, 14, 2, 15] for other relevant setups. All
those works consider that the sensing is given and, at best, account
for the SIPN uncertainties (quantized, noisy, outdated) by making
the RA aware of such imperfections. The number of works that
aim to globally optimum RA and sensing by implementing a joint
optimization is much smaller; see, e.g., [9, 16, 10, 6, 7, 17], all
for interweave setups. When a joint design is implemented, the
decision of what time instants/users/channels to sense has to take
into account what the RA is going to do with such information, as
well as the impact on CR performance for current and future time
instants. As a result, the analytical complexity of the problem and
the computational burden to obtain the optimal schemes increase
To the best of our knowledge, no previous work has addressed
the joint design of sensing and RA for underlay CRs. The un-
derlay setup is more challenging than the interweave setup, which
only requires to know whether a frequency band is occupied or not.
Not only the variables to estimate are continuous in the underlay
setup, but their number is much higher (all SU-to-PU pairs in all fre-
quency bands). Uncertainty and time correlation in the SIPN call
for (Bayesian) sequential estimators to keep track of the interference
channel and DP/POMDP tools to design the optimal schemes.
Our design approach is similar to one followed in our previous
work [7, 17] for interweave CRs. We first design the RA for any
sensing scheme and, then, design the optimal sensing taking into ac-
count the optimal RA. Since the modifications in the RA to account
for the sensing cost are relatively simple, the main novelty is on the
design of the sensing schemes. The main contributions of this pa-
per are: the formulation of a joint optimization of RA and sensing
for an underlay CR; the design of an algorithm that, leveraging dual
decomposition and DP/POMDP tools, solves the joint optimization;
and the design of a low-complexity algorithm that, using a greedy
(myopic) approach, approximates the optimal solution. Our paper
must be viewed as a first step to developing low-complexity approx-
imations to the optimal solution.
The paper is organized as follows. Section 2 presents the system
setup, SIPN and state information of the secondary network (SISN)
models, design variables, and the constraints to be satisfied. The
problem is formulated in Section 3. Section 4 solves the problem
and analyzes its complexity. Section 5 presents a simple suboptimal
solution. Preliminary numerical simulations evaluating the perfor-
mance of the algorithms are provided in Section 6.
A CR with MSUs (indexed by m) is considered. The frequency
band used by the CR is divided into Kfrequency-flat orthogonal
subchannels (indexed by k), so that if a SU is transmitting, no other
SU can be active in the same subchannel. No constraints are im-
posed on the number of channels that can be accessed by a user.
For simplicity, we assume that there is always exactly one active
PU per channel. Extensions to scenarios where these assumption(s)
do not hold can be handled with a moderate increase in complex-
ity. Each SU can obtain (imperfect) measurements of the channel
gain between itself and the PUs. More precisely, at every time slot
(indexed by n) the following three tasks are run sequentially by the
CR: T1) the SISN is acquired; T2) based on the output of T1 (and
previous measurements) a set of users are selected to measure their
interference links; T3) the outputs of T1 and T2 are used to find the
optimal RA for instant n. This section describes the model for the
SISN and SIPN; the variables to be designed; and the constraints that
such variables need to satisfy.
Starting with the SISN, the instantaneous fading coefficient of
the channel between the mth secondary transmitter-receiver pair in
the kth channel at time nis denoted as hm
k,2[n]. This variable is
normalized with respect to noise and PU interference. Regarding
the SIPN, the noise-normalized instantaneous fading coefficient of
the interference channel between the mth SU and the kth PU is de-
noted as hm
k,1[n]. Every time that the mth SU is required to obtain
measurements from its interference channels, it has to pay a power
cost denoted by qm(other sensing costs can also be accommodated
into our formulation [7]). The instantaneous value of hm
not be assumed perfectly known because of: i) outdated information
(to save power, the interference channels are not sensed at every n);
and ii) errors due to noisy measurements. As a consequence, instead
of the true value of the channel gain (perfect SIPN), only statistical
information about it is available (probabilistic SIPN).
Let ˜
k,1[n]denote the observation (output of the sensing task,
possibly corrupted by noise) of hm
k,1[n]. The CR relies on the dy-
namics of hm
k,1[n]to track the SIPN. Let us define the Boolean vari-
able sm[n], which is 1 if at time nthe mth SU takes measurements
k,1[n], and 0 otherwise. While sm[n]will depend on measurements
acquired in the past [cf. T1], the power allocation and transmission
scheduling will also leverage the newly available measurements [cf.
T3]. Let fm
k,1[n]|n1) denote the information about hm
sensing (prediction density). Similarly, let fm
the information about hm
k,1after sensing (filtering density)1. To use
a compact notation, these densities will be assumed to belong to the
same family and will be represented by their parameters. Let ˆ
be the parameter vector of the prediction density at time n, and
1The prediction and filtering densities play the role of the pre-decision
(prior) and post-decision (posterior) beliefs, respectively [17, 7, 18, 19].
k[n]that of the filtering density. This way, fm
k,1[n]|n1) :=
k,1[n]; ˆ
k[n]) and fm
k,1[n]|n) := f(hm
k,1[n]; Fm
k[n]). The
stochastic filter that tracks the SIPN works as follows [19]. The pre-
diction density parameters at time nare deterministically computed
at the prediction step from the previous filtering density parameters:
k[n] := P(Fm
The filtering density at time nwill depend on sm[n]. If sm[n]=0,
then Fm
k[n] = ˆ
k[n]; if sm[n] = 1, then
k[n] := U(ˆ
There exist different alternatives to model the stochastic process
k,1. Here, the time dynamics of the complex-valued secondary-
primary channel gain hm
k,1[n]are described by an auto-regressive
(AR) model with circularly-symmetric complex normal (CSCN)
innovations and CSCN noise. As a result, the parameter vectors cor-
respond to the mean and variance of the densities, and the prediction
and correction steps of the channel estimation can be effected by a
standard Kalman filter [19]. Since the time variability of the SISN is
considered faster than that of the SIPN, hm
k,2[n]will be considered
i.i.d. across time. As stated in [15], such a heterogeneous system
information model is well suited for scenarios where the mobility of
the PUs is low and sensing the SIPN is more difficult than sensing
the SISN.
Next, we introduce the design variables wm
k[n](scheduling co-
efficients), pm
k[n](transmit power), and sm[n](sensing decision, al-
ready described). Coefficients wm
k[n]effect the orthogonal access
among SUs. Specifically, wm
k[n]is 1 if the mth SU is scheduled to
transmit into the kth band at time nand 0 otherwise. Moreover, if
k[n] = 1,pm
k[n]denotes the instantaneous nominal power trans-
mitted over the kth band by the mth SU. This means that power
k[n]is consumed when wm
k[n]=1. Under bit error rate or capac-
ity constraints, instantaneous rate and power variables are coupled.
This rate-power coupling will be represented by the non-decreasing
function Cm
k[n], pm
k[n]) and βmwill denote the benefit (price)
associated with the rate.
The last step is to describe the constraints that the aforemen-
tioned variables need to satisfy. The sensing decision variable is
binary, so that sm[n]∈ {1,0}. Powers are non-negative, so that
k[n]0. Moreover, orthogonal access requires
k[n]∈ {0,1}and Pmwm
The average (long-term) power the mth SU can consume (including
the power devoted to transmit and the power devoted to estimate the
interference channel gains) is upper bounded, that is, m
γnE"qmsm[n] +X
where 0< γ < 1is a discount factor that is typically included in
infinite horizon formulations to facilitate the design of the optimal
schemes and accommodate potential non-stationarities [18]. Note
also that the right hand side of (4) is equivalent to ˇpm
1γ. The allocated
power will generate interference to PUs. Since an underlay setup is
considered, each time a SU transmits in channel k, the interference
generated at the PU receiver is hm
k[n]. To protect the PUs, a
limit on the average (long-term) interference at each PU is enforced.
This amounts to require for all k
Here, we controlled interference by limiting the average interfering
power at the PU receiver [11]. This keeps the modeling simple and
leads to a convex constraint. Alternative metrics can be used to con-
trol interference (e.g. outage probability [15]), provided that the in-
crease in computational complexity can be afforded.
The last step to formulate the optimization problem is to identify
the metric to be maximized. In this work, the average sum rate
achieved by the secondary network will be maximized. With X:=
{sm[n], wm
k[n], pm
k[n]|∀m, k, n}, the optimal joint design is then
γnEhXk,m βmwm
k,2[n], pm
k[n])i(6a) : (3),(4),(5), pm
k[n]0, sm[n]∈ {0,1}.(6b)
The two main issues that render this problem challenging to
solve are: i) The design variables wm
k[n]and sm[n]are binary, so
that the complexity to optimize over them is combinatorial; and
ii) The value of some design variables at time nhas an impact on
the state variables at instants n0n(specifically, sm[n]has an im-
pact on future beliefs through Fm
k[n]) – as a consequence, solving
(6) optimally requires using DP tools.
Regarding the first challenge, the combinatorial complexity as-
sociated with optimizing over wm
k[n]can be bypassed by relaxing
the binary constraint to its convex counterpart wm
k[n][0,1]. Such
a relaxation can be shown optimal because {wm
k[n]}are present only
in linear terms and because {wm
k[n]}do not have an impact on the
future state variables; see, e.g., [15] for details. Unfortunately, that
is not true for sm[n]and, hence, the associated complexity remains
combinatorial. The optimal solution is presented in the next section,
while Section 5 presents a low-complexity approximation.
After dualizing the long-term constraints (4) and (5), the opti-
mization of {wm
k[n]}and {pm
k[n]}can be separated across time
and channels. This fact, together with other properties of (6)
will be leveraged to decrease the computational complexity re-
quired to solve the DP. The critical step is to tackle the op-
timization in two stages: i) finding the optimal {wm
k[n]}for any sensing policy; and ii) substituting the output
of (i) into (6) and solving for the optimal {sm[n]}. Note that
this does not entail a loss of optimality because the solution in
(i) is a function of the sensing policy, which is later optimized in
(ii). Mathematically, for a generic function f(x, y), the approach
amounts to find (x, y) = arg minx,y f(x, y)as follows: i)
x(y) := arg minxf(x, y)and ii) y= arg minyf(x(y), y).
4.1. Optimal RA
The optimization carried out in the first step yields a problem of
the same form than that solved in [7]; for this reason, the optimal
solution is given here directly. The optimal solution to the problem
at hand consists in defining a link quality indicator (LQI) ϕm
optimizing it with respect to the power for every user-channel pair,
and selecting for transmission the SU with the highest LQI in each
channel. The LQI for the problem at hand is:
k(p) := βmC(hm
k,2[n], p)(πm+θkµm
k[n]) p(7)
where µm
k[n] := E|hm
k[n]is the expected power gain
of the interference channel (according to the post-decision be-
lief); and πmand θkare the Lagrange multipliers associated with
constraints (4) and (5), respectively2. To optimize the RA for in-
stant n, select pm,?
k[n] := arg maxpϕm
k(p), and wm,?
k[n] :=
k(p)=maxq,p ϕq
k(p)}, where {·} is the indicator function. Note
that (7) can be expressed in closed form for several choices of C(·).
For example, if C(·)is Shannon’s capacity, then pm,?
ktakes the form
of the water-filling solution [20, 12].
4.2. Optimal sensing
Leveraging the expressions for the optimal RA, we now solve for
the optimal sm[n]. First, we define the instantaneous reward R[n],
which accounts for the terms at time nthat depend on sm[n]:
R[n] := Xkmax
k[n]) Xmπmqmsm[n],(8)
where ϕm,?
k[n]) is the optimal value of ϕm
k[n]for a given
k[n]. Note that ϕm,?
k[n]) depends on s[n]because the SIPN
k[n]depends on s[n][cf. (2)].
After substituting the optimal RA and (8) into the Lagrangian of
(6), the maximization boils down to
To stress that the sensing decisions of all users have to be jointly
optimized, the notation sm[n]∈ {0,1} ∀mhas been replaced with
s[n]M. The coupling exists because the sensing decision for
user maffects its probability (and hence, also the other users’ prob-
abilities) of being scheduled.
The main differences between (9) and the original formulation in
(6) are that now: i) as a result of the Lagrangian relaxation of the DP,
the objective has been augmented with the terms accounting for the
dualized constraints; ii) the only remaining optimization variables
are s[n]; and iii) because the optimal RA fulfills the constraints (3)-
(5) and pm
k[n]0, the only remaining constraint is s[n]M
(standard DP algorithms usually assume countable action spaces).
The problem falls into the class of POMDP because state tran-
sitions and average rewards only depend on the current state-action
pair, and the system state is not known perfectly. Only an observa-
tion (affected by noise or missing data) of the state is available in-
stead [21]. In this model, the partially observable variable is hm
The belief variable required to solve this POMDP is constituted by
the prediction and filtering densities associated with hm
To solve for s[n], we derive the Bellman equations [18] associ-
ated with (9). The objective is split into present and future rewards,
To account for the effect of current actions in future instants, the
value function V(h2[n],ˆ
F[n]) is introduced (where ˆ
k[n](k, m)and h2[n]collects hm
k,2[n](k, m)). It quantifies the
expected sum reward for all future instants. Since the latter is an
infinite horizon DP with γ < 1, the optimal value function is sta-
tionary and its existence is guaranteed [18]. Similarly to [7], since
2Since, after relaxation, the RA problem has zero duality gap, there exists
a constant (stationary) optimal value for each multiplier [7]. In this work, the
optimal values of {πm, θk}are assumed to be known.
the hm
k,2[n]are considered i.i.d. across time and independent of
sm[n], the Bellman equations that drive the optimal sensing can be
expressed in terms of ¯
F[n]) := Eh2[V(h2[n],ˆ
s[n] = arg max
hhR[n]+ γ¯
F[n+1])s[n]=sio (11)
hhR[n]+ γ¯
where E˜
his the expectation over the distribution of {˜
k,1}∀(m, k).
The only remaining step to design the sensing scheme is to design
an algorithm to compute ¯
F[n]). There exist different alternatives
that exploit the recursive definition in (12) to accomplish this task
[18]. Space limitations prevent us to delve into the details of such
algorithms, but it is important to stress that (even after leveraging the
problem structure) their computational complexity is very large.
The two main sources of complexity to find s[n]are: i) during the
initialization phase, the multidimensional function ¯
V(·)needs to be
estimated iteratively using a Monte Carlo approach and ii) at every
time instant, an exhaustive search over Mneeds to be implemented.
Since (i) is run off line only once, we focus on reducing the online
complexity in (ii). In particular, we use a greedy approach under
which users are sequentially selected to measure the channel. We
start by supposing that no user senses the channel and sequentially
set sm[n] = 1 for the SU that yields the highest (positive) expected
reward. The algorithm stops either when none of the remaining SUs
yields a positive reward, or when all users are scheduled to sense
the channel. The approximation is well justified because channels
across SUs are not correlated. Algorithm 1 lists the main steps of the
algorithm, with 0and 1denoting the all-zeros and all-ones vectors
and emthe mth canonical M×1vector.
Algorithm 1 Greedy approximation to the optimal sensing policy.
1: ˜s 0and M ← {1,...,M}
2: repeat
3: m?arg maxm∈M E[R[n] +γ¯
F[n+1])|s[n] = ˜s +em]
4: RE[R[n] +γ¯
F[n+1])|s[n] = ˜s +em?]E[R[n] +
F[n+1])|s[n] = ˜s]
5: if R > 0,then ˜s ˜s +em?and M ← M \ {m}
6: until R < 0or ˜s =1
7: return s[n] := ˜s
The expectations in line 3 (which are taken over ˜
h) can be run
efficiently using a Monte Carlo method. Since the imperfections in
k,1[n]are independent across (m, k), each of the Mexpectations
in line 3 can be implemented with complexity O(MKN ), where
Nis the number of random realizations per ˜
k,1[n]. Then, the on-
line complexity of the overall algorithm is O(M3KN ), because the
repeat loop is executed at most Mtimes. Note that Algorithm 1 as-
sumes that ¯
V(·)has been computed off line. If the associated burden
cannot be afforded, further simplified algorithms can be developed
either by approximating ¯
V(·), or just by dropping it (myopic pol-
3A myopic policy ignores the impact of the sensing decision on future
time instants, focusing only on maximizing the instantaneous reward. Math-
ematically this amounts to setting ¯
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Av era ge I nt er fe re nc e Pow er G ai n
Av er ag e s p ec t ra l e c ie nc y (b ps /H z)
Algorithm 1
Always Sense
Never Sense
Upper Bound
Fig. 1.1. Results for Test Case 1
Nu mbe r of c han ne ls
Av er age s pe ctr al ec ie nc y (b ps /Hz )
Fig. 1.2. Results for Test Case 2
A CR with M= 4 SUs is simulated. The SISN follows a Rayleigh
fading model with an average SNR of -5 dB. All users have the same
priority, so that βm= 1 m. The SIPN follows an AR-1 model
with a coefficient of 0.95. The observed SIPN is corrupted by ad-
ditive gaussian noise with a SNR of 3 dB. The power constraint at
the SU transmitters is set to [ˇp1, ...ˇp4] = [6.0,7.2,9.0,12.0]. The
interference power constraint at the PU receivers is ˇok= 2.0k.
The sensing cost parameter [cf. (4)] is qm= 5 m. The Lagrange
multipliers [cf. (7)] are computed using the method in [15].
Since we focus on the sensing policy, all tested schemes imple-
ment the optimal RA policy in Section 4.1. We are interested in
comparing the performance of the myopic policy using the follow-
ing schemes: i) an exhaustive search over s[n](combinatorial com-
plexity); ii) Algorithm 1 (proposed, polynomial complexity); iii) a
round-robin scheme that sequentially selects a single different user
at each n; iv) a scheme that randomly selects sm[n]mimicking the
distribution of sm[n]at (ii); deterministic schemes that v) always
sense, vi) never sense; and vii) an upper bound on the system perfor-
mance (using the algorithm in (v) and setting qm= 0 m).
Two test cases are run: TC1) the average power gain of the in-
terference is fixed to -3 dB and the spectral efficiency is plotted vs.
Kin Fig. 1.1; TC2) K= 4 and the spectral efficiency is plotted vs.
the average power gain of the interference in Fig 1.2. The average
power and interference constraints are tightly satisfied in all cases.
Results show close performance of Algorithm 1 and exhaustive
search for the simulated test cases. This suggests that Algorithm 1
can be a good option when Mis large. Further, this motivates using
the greedy approach to compute a suboptimal estimation of ¯
Such schemes will be addressed in future work.
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Efficient design of cognitive radios (CRs) calls for secondary users implementing adaptive resource allocation schemes that exploit knowledge of the channel state information (CSI), while at the same time limiting interference to the primary system. This paper introduces stochastic resource allocation algorithms for both interweave (also known as overlay) and underlay cognitive radio paradigms. The algorithms are designed to maximize the weighted sum-rate of orthogonally transmitting secondary users under average-power and probabilistic interference constraints. The latter are formulated either as short- or as long-term constraints, and guarantee that the probability of secondary transmissions interfering with primary receivers stays below a certain pre-specified level. When the resultant optimization problem is non-convex, it exhibits zero-duality gap and thus, due to a favorable structure in the dual domain, it can be solved efficiently. The optimal schemes leverage CSI of the primary and secondary networks, as well as the Lagrange multipliers associated with the constraints. Analysis and simulated tests confirm the merits of the novel algorithms in: i) accommodating time-varying settings through stochastic approximation iterations; and ii) coping with imperfect CSI.
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This is an overview of partially observable Markov decision processes (POMDPs). We describe POMDP value and policy iteration as well as gradient ascent algorithms. The emphasis is on solution methods that work directly in the space of policies.
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A cognitive radio network (CRN) is formed by either allowing the secondary users (SUs) in a secondary communication network (SCN) to opportunistically operate in the frequency bands originally allocated to a primary communication network (PCN) or by allowing SCN to coexist with the primary users (PUs) in PCN as long as the interference caused by SCN to each PU is properly regulated. In this paper, we consider the latter case, known as spectrum sharing, and study the optimal power allocation strategies to achieve the ergodic capacity and the outage capacity of the SU fading channel under different types of power constraints and fading channel models. In particular, besides the interference power constraint at PU, the transmit power constraint of SU is also considered. Since the transmit power and the interference power can be limited either by a peak or an average constraint, various combinations of power constraints are studied. It is shown that there is a capacity gain for SU under the average over the peak transmit/interference power constraint. It is also shown that fading for the channel between SU transmitter and PU receiver is usually a beneficial factor for enhancing the SU channel capacities.
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Cognitive radio (CR) networks can re-use the RF spectrum licensed to a primary user (PU) network, provided that the interference inflicted to the PUs is carefully controlled. However, due to lack of explicit cooperation between CR and PU systems, it is often difficult for CRs to acquire CR-to-PU channels accurately. In fact, if the PU receivers are off, the sensing algorithms cannot obtain the channels for the PU receivers, although they have to be protected nevertheless. In order to achieve aggressive spectrum re-use even in such challenging scenarios, power control algorithms that take channel uncertainty into account are developed. Both log-normal shadowing and small-scale fading effects are considered through suitable approximations. Accounting for the latter, centralized network utility maximization (NUM) problems are formulated, and their Karush-Kuhn-Tucker points are obtained via sequential geometric programming. For the case where CR-to-CR channels are also uncertain, a novel outage probability-based NUM formulation is proposed, and its solution method developed in a unified fashion. Numerical tests verify the performance merits of the novel design.
A Dynamic Programming Example: A Shortest Path Problem The Three Curses of Dimensionality Some Real Applications Problem Classes The Many Dialects of Dynamic Programming What is New in this Book? Bibliographic Notes
Multiple-input multiple-output (MIMO) technology constitutes a breakthrough in the design of wireless communication systems, and is already at the core of several wireless standards. Exploiting multi-path scattering, MIMO techniques deliver significant performance enhancements in terms of data transmission rate and interference reduction. This book is a detailed introduction to the analysis and design of MIMO wireless systems. Beginning with an overview of MIMO technology, the authors then examine the fundamental capacity limits of MIMO systems. Transmitter design, including precoding and space-time coding, is then treated in depth, and the book closes with two chapters devoted to receiver design. Written by a team of leading experts, the book blends theoretical analysis with physical insights, and highlights a range of key design challenges. It can be used as a textbook for advanced courses on wireless communications, and will also appeal to researchers and practitioners working on MIMO wireless systems.
We consider a wideband spectrum sharing system where a secondary user can access a number of orthogonal frequency bands each licensed to a distinct primary user. We address the problem of optimum secondary transmit power allocation for its ergodic capacity maximization subject to an average sum (across the bands) transmit power constraint and individual average interference constraints on the primary users. The major contribution of our work lies in considering quantized channel state information (CSI) (for the vector channel space consisting of all secondary-to-secondary and secondary-to-primary channels) at the secondary transmitter as opposed to the prevalent assumption of full CSI in most existing work. It is assumed that a central entity called a cognitive radio network manager has access to the full CSI information from the secondary and primary receivers and designs (offline) an optimal power codebook based on the statistical information (channel distributions) of the channels and feeds back the index of the codebook to the secondary transmitter for every channel realization in real-time, via a delay-free noiseless limited feedback channel. A modified Generalized Lloyds-type algorithm (GLA) is designed for deriving the optimal power codebook, which is proved to be globally convergent and empirically consistent. An approximate quantized power allocation (AQPA) algorithm is presented, that performs very close to its GLA based counterpart for large number of feedback bits and is significantly faster. We also present an extension of the modified GLA based quantized power codebook design algorithm for the case when the feedback channel is noisy. Numerical studies illustrate that with only 3-4 bits of feedback per band, the modified GLA based algorithms provide secondary ergodic capacity very close to that achieved by full CSI and with only as little as 4 bits of feedback per band, AQPA provides a comparable performance, thus making it an attractive choice for prac- - tical implementation.
We consider joint optimization for sensing-channel selection and ensuing power control problem with cognitive radios over time-varying fading channels. It is shown that this joint design can be judiciously formulated as a convex optimization problem. Optimal joint sensing-channel selection and power control scheme is then derived in closed-form under the constraints of average power budget and maximum allowable probability of collisions with the primary communications. In addition, we develop a stochastic optimization algorithm that can operate without a-priori knowledge of the fading channel statistics. It is rigourously established that the proposed stochastic scheme is capable of dynamically learning the intended wireless channels on-the-fly to approach the optimal strategy almost surely. Numerous results are also provided to evaluate the proposed schemes for cognitive transmissions over block fading channels.