optimal channel switching for average capacity maximization

1. Outline of Presentation
 Introduction
 Literature survey
 Problem identification with description
 Proposed methods for problem solution
 Detailed description of proposed methods
 Tools required
 Simulation results
 Extension work
 Conclusion
 References

2. Introduction
 Time sharing among different power levels, detectors,
or channels can provide performance improvements for
communication systems that operate under average
power constraints in the presence of additive time-
invariant noise.

3. Channel Switching
 In the presence of multiple channels between a
transmitter and a receiver, performing time sharing among
different channels is called channel switching.

4.  Channel switching provides certain performance
improvements
Need for channel switching?
Example:
Number of channels= K = 4,
Standard deviation= σ = [0.4 0.6 0.8 1],
Cost= C = [7 5 3 1],
Average cost limit= Ac = 2.

5. Power and Cost Constraints
 The constraint on average transmitted power varies according
to the application.
 In practical systems, there is constraint on the average power
of the signals, which can be expressed as
 Random power control in wireless ad hoc networks [15]
Cost constraint
 In practical systems, each channel can be associated with a
certain cost depending on its quality.
Example
 A channel that presents high SNR conditions has a high cost
(price) compared to channels with low SNRs.

6. Convex optimization
 Convex function?
Example: f(x)=x2 is convex since f’(x)=2x,
f’’(x)=2>0
Convex hull?
x
xa xb
f(x)
 f x( ) 0

7. Literature survey
 convexity properties in binary detection problems [1]
The channel switching problem is proposed under an
average power constraint for the optimal detection of binary
antipodal signals over a number of channels that are subject
to additive unimodal noise.
 Error rates of the maximum-likelihood detector for arbitrary
constellations: convex/concave behavior and applications [2]
This discussion is extended from binary modulations to
arbitrary signal constellations by concentrating on the
maximum likelihood (ML) detection over additive white
Gaussian noise (AWGN) channels.
 Optimal signaling and detector design for M-ary
communication systems in the presence of multiple additive
noise channels [3]
The channel switching problem is investigated for M-ary
communication systems in the presence of additive noise

8. •Optimum power allocation for average power constrained jammers in
the presence of non-Gaussian noise [4]
The study investigates the optimum power allocation policy for an
average power constrained jammer operating over an arbitrary additive
noise channel, where the aim is to minimize the detection probability of
an instantaneously and fully adaptive receiver that employs the
Neyman-Pearson criterion.
Detector randomization
•Optimal noise benefits in Neyman-Pearson and inequality-constrained
signal detection [5]
An average power constrained binary communication system is
considered, and the optimal time sharing between two antipodal signal
pairs and the corresponding maximum a posteriori probability (MAP)
detectors is investigated.
•Detector randomization and stochastic signaling for minimum
probability of error receivers [6 &8]
The results are generalized by considering an average power
constrained M-ary communication system that can employ time

9. •Optimal detector randomization for multiuser communications
systems [7]
Investigates the benefits of time sharing among multiple detectors
for the downlink of a multiuser communication system and
characterizes the optimal time sharing strategy.
•It is proved that the optimal solution is achieved via randomization
among at most min{K, Nd} detector sets.
•where K is the number of users and
Nd is the number of detectors at each receiver.
•Lower and upper bounds are derived on the performance of optimal
detector randomization, and it is proved that the optimal detector
randomization approach can reduce the worst-case average
probability of error of the optimal approach that employs a single
detector for each user by up to K times.
•Optimal randomization of signal constellations on the downlink of a
multiuser DS-CDMA system [9]
The joint design of optimal randomization of signal constellations is
performed in the downlink of a multiuser system for given receiver

10. •Performance Analysis of Generalized Selection Combining with
Switching Constraints [10]
Generalized selection combining (GSC), in which the best ‘g’ out of
‘L’ independent diversity channels are linearly combined has been
proposed and analyzed for Rayleigh fading channels.
•Optimal Channel Switching For Average Capacity Maximization [11]
Optimal single channel algorithm
When switching is required?
•To Stay Or To Switch: Multiuser Dynamic Channel Access [12]
Main focus is on the effect of collective switching decisions by the
users, and how their decision process, in particular their channel
switching decisions, are affected by increasing congestion levels in
the system.
Upon finding out the channel condition, user faces the following
choices:
oSTOP

11. •Motivation behind assigning cost to different
channels?
•Opportunistic spectrum access with channel switching cost for
cognitive radio networks [13]
we investigate the channel access problem by taking into account
the channel switching cost due to the change from one frequency
band to another. Such channel switching cost is non-negligible in
terms of delay packet loss and protocol overhead .In such context,
it is crucial to design channel access policies reluctant to switch
channels unless necessary.
•Signal Recovery With Cost-Constrained Measurements [14]
properties that any plausible cost function must possess
i. c(p) must be a nonnegative, monotonically increasing function
of p, with c(1)=0 since a device with one measurement level
gives no useful information.
ii. For any integer L ≥ 1 , we must have L c(p) ≥ c (pL ).
•A function possessing these properties is the logarithm function

12.  Caratheodory’s theorem [17]
In convex geometry, Caratheodory’s theorem states that
if a point x of Rd lies in the convex hull of a set p, there
is subset p| of p consisting of d+1 or fewer points such
that x lies in the convex hull of p|.
 The error probability for channel i is expressed as
where Q denotes the Q-function, P is the average
symbol energy, η and κ are some positive constants
that depend on the modulation type and order. [18]

13. Problem identification with description
 Optimal channel switching provides the highest
performance over a set of Gaussian channels with
variable utilization costs is in the presence of average
power and average cost constraints.
 The proposed optimization problem can be expressed
as
subject to ,
- channel switching factor
- average power of transmitted signal over channel
‘i’

14. Proposed methods for problem solution
 First, generic cost values are considered for the
channels and the optimal channel switching strategy is
characterized.
 Then, logarithmic cost functions are employed to relate
the cost of a channel to its average noise power, and
specific results are obtained about the optimality of
channel switching between two channels or among
three channels.
 Also, for channel switching between two channels,
relations between the optimal power levels are
obtained depending on the average power limit, and it
is proved that the ratio of the optimal power levels is
upper bounded by the ratio of the larger noise variance

15. Detailed description of proposed
methods
• Which implies that

16.  If Ac = Ck, the optimal solution is to transmit over
channel K exclusively with power Ap.
 If Ac ≥ C1, the optimal solution is to transmit over
channel 1 exclusively with power Ap.
>

17. Consider a scenario for which the average cost limit satisfies Ck < Ac< C1
Proposition 1: The optimal channel switching strategy utilizes the
maximum average power and the maximum average cost.
i. A solution that operates at an average power below Ap cannot be
optimal
ii. A solution that employs at least two channels and operates below Ac
cannot be optimal
In case of single channel
Suppose that an optimal solution employs a single channel (say, channel i)
and operates below Ac; that is, channel i is employed exclusively with
power pi and its cost Ci is strictly smaller than Ac.
Then

18. Proposition 2: The optimal channel switching strategy is to
switch among at most min{K, 3} channels.
Hence, channel switching among up to 3 different channels is optimal.
•Based on Proposition 1 and Proposition 2, the optimal channel
switching corresponds to one of the following three strategies:
1. Transmission over a Single Channel
2. Channel Switching Between Two Channels
The optimal solution for Strategy 2 requires a search over a one-
dimensional space only (for each possible channel pair).
The objective is strictly convex. Therefore, convex optimization
algorithms can be employed to obtain the result in polynomial time.
3. Channel Switching Among Three Channels
Advantage: optimization over a three-dimensional space instead of six
by utilizing the three equality constraints.

19. Proposition 3: Suppose that , &
define
,then, the optimal solution of
startegy2 denoted by , satisfies the following
relations depending on the average power limit:
i)
ii)
iii)
In addition, the ratio between the optimal power levels
cannot exceed , that is,
Ci >Ac> Cj

20. Logarithmic cost functions
 Noise power of each channel to a cost value can be related
as
 Proposition 4 states that if the average and peak power
limits are larger than certain values, then the optimal solution
for Strategy 2 is to switch between the two channels, one of
which has the lowest cost among the channels with costs
higher than Ac, and the other has the highest cost among
the channels with costs lower than Ac.
 Remark : Under the condition in Proposition 4, transmission
over a single channel with cost Ac at the maximum power
level Ap achieves a smaller average probability of error than
performing optimal channel switching between two channels.

21.  Proposition 5 states that in the absence of peak power
constraints, if the average power limit is larger than a
certain value, then the optimal channel switching strategy
is to use a single channel exclusively or to switch between
two channels; that is, Strategy 3 is not optimal.
 Solution:
a. To transmit over a single channel with cost Ac if such a
channel exists or
b. To switch between channel i and channel j as specified
in Proposition 4 if there exists no channels with cost Ac.

22. Summary:
If Ac = Ck, the optimal channel switching strategy is to transmit
over channel K exclusively with power Ap
• If Ac ≥ C1, the optimal channel switching strategy is to
transmit over channel 1 exclusively with power Ap
• If Ck < Ac< C1,
— if the cost function is the logarithmic cost With
certain Ap value and no peak power constraints exist,
∗ if there exists a channel with cost Ac,
transmission over that channel at the maximum power level Ap
is the optimal
∗ otherwise, the optimal strategy is to perform
time sharing between channels and the optimal solution can
be obtained based on strategy 2
— otherwise, the optimal channel switching strategy
is obtained based on the optimization problem under strategy
3 (strategy 1 & 2 are special cases).

23. Tools required
 To implement optimal channel switching over Gaussian
channels under average power and cost constraints
MATLAB R2012b version is required with
 Signal processing,
 Communications system,
 Global optimization toolboxes.

24. Simulation results
 The error probability for channel i is expressed as
where Q denotes the Q-function, P is the average
symbol energy, η and κ are some positive constants
that depend on the modulation type and order. [18]
 Cost function can be expressed as

25. 0 5 10 15
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average Power Limit Ap
AverageProbabilityofError
Average probability of error versus Ap
for Ac
=2
Optimal Channel Switching
Optimal Single Channel
Fig 1: Average probability of error versus Ap for the optimal single
channel and optimal channel switching strategies, where K = 4, σ =
[0.4 0.6 0.8 1], C = [7 5 3 1], Ac = 2

26. 0.1
0.2
2
4
6
8
10
0.1568 0.8432
0.1568 0.8432
0.25 0.75
0.5 0.5
0.5 0.5
0.5 0.5
0.5 0.5
0.1632 ‒ ‒ 0.939
0.2324 ‒ ‒ 0.1332
‒ 1.3567 ‒ 2.2097
‒ ‒ 3.3057 4.5673
‒ ‒ 4.9898 7.0907
‒ ‒ 6.5102 9.4804
‒ ‒ 8.1937 11.594
Table 1: Parameters of the Optimal Channel Switching Strategy in Fig.2

27. Fig 1: Average probability of error versus Ap for the optimal single channel
and optimal channel switching strategies, where K = 4, σ = [0.4 0.6 0.8
1], C = [7 5 3 1], Ac = 5
0 5 10 15
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average Power Limit Ap
AverageProbabilityofError
Average probability of error versus Ap
for Ac
=5
Optimal Channel Switching
Optimal Single Channel

28. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
-2
10
-1
10
0
Average Power Limit Ap
AverageProbabilityofError
Closer view - Average probability of error versus Ap
for Ac
=5
Optimal Channel Switching
Optimal Single Channel
Fig 3:A closer look at fig for Ap [0,1]

29. 1 2 3 4 5 6 7 8
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average Cost Limit Ap
AverageProbabilityofError
Average probability of error versus Ac
Ap
=1 Channel Switching
Ap
=2 Channel Switching
Ap
=3 Channel Switching
Ap
=1 Single Channel
Ap
=2 Single Channel
Ap
=3 Single Channel
Fig 4:Average probability of error versus Ac for the optimal single channel
and optimal channel switching strategies, where K = 4, σ = [0.4 0.6 0.8
1], and C = [7 5 3 1]

30. 0 5 10 15
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average Power Limit Ap
AverageProbabilityofError
Average probability of error versus Ap
for Ac
=0.9
Optimal Channel Switching
Optimal Single Channel
Fig 5:Average probability of error versus Ap for the optimal single channel and
optimal channel switching strategies, where K = 5, σ = [0.6 0.7 0.8 0.9 1], C
= [1.329 1.112 0.941 0.804 0.6931].

31. Fig 6:Average probability of error versus Ap for the optimal single channel and
optimal channel switching strategies, where K = 5, σ = [0.6 0.7 0.8 0.9 1], C =
[1.329 1.112 0.941 0.804 0.6931], Ac = 0.9
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
10
-5
10
-4
10
-3
10
-2
10
-1
Average Cost Limit Ac
AverageProbabilityofError
Average probability of error versus Ac for the optimal single channel and optimal channel switching strategies,
Ap
=1 Channel Switching
Ap
=2 Channel Switching
Ap
=3 Channel Switching
Ap
=1 Single Channel
Ap
=2 Single Channel
Ap
=3 Single Channel

32. Extension work
 Optimal channel switching is proposed for average
capacity maximization in the presence of average
and peak power constraints.
 A necessary and sufficient condition is derived in
order to determine when the proposed optimal
channel switching approach can or cannot
outperform the optimal single channel approach,
which performs no channel switching.

33. Capacity Maximization
 The capacity of a channel i can be expressed as
 can be defined as

34. Subject to ,
If the first order derivative of is continuous at average power
constraint and condition is satisfied then there is no need of channel
switching.
Where
•Otherwise, the optimal solution involves time sharing between two
channels

35. Fig 7:Capacity of each channel versus power and Average capacity
versus average power limit for the optimal channel switching and the
optimal single channel approaches.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
1
2
3
4
5
6
7
8
9
10
P(mW)
CapacityinMbps
Capacity of each channel versus power
Optimal Single Channel
Optimal Channel Switching
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
1
2
3
4
5
6
7
8
9
10
P(mW)
CapacityinMbps
Capacity of each channel versus power
Channel 1
Channel 2
Channel 3

36. Conclusion
 Joint optimization of channel switching factors and
signal powers provides minimum average
probability of error under average power and cost
constraints. Channel switching can provide certain
performance improvements and is for important for
cognitive radio systems in terms of performance
optimization of secondary users under realistic
constraints. Optimal channel switching provides
average capacity maximization in the presence of
average and peak powers.
 Future work involves the incorporation of switching
delays in the design of optimal channel switching

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41. THANK YOU