Outline of Presentation IntroductionLiterature surveyProblem identification with descriptionProposed methods for problem solutionDetailed description of proposed methodsTools requiredSimulation resultsExtension work ConclusionReferences

IntroductionTime 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.

Channel SwitchingIn the presence of multiple channels between a

transmitter and a receiver, performing time sharing among different channels is called channel switching.

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.

Power and Cost ConstraintsThe 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.

Convex optimizationConvex function?

Example: f(x)=x2 is convex since f’(x)=2x, f’’(x)=2>0

Convex hull?

xxa xb

f(x)

f x( ) 0

Literature surveyconvexity 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 channels with arbitrary probability distributions and by facilitating time sharing among multiple signal constellations over each channel.

•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 sharing among both signal levels and detectors over an additive noise channel with some known distribution.

•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 structures.

•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:

oSTOPoSTAYoSWITCH

•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

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]

Problem identification with descriptionOptimal 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’ Ac& Ap - average power and cost limits respectively

Proposed methods for problem solutionFirst, 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 to the smaller one.

Detailed description of proposed methods

• Which implies that

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.

>

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

A solution that employs single channel and operates below Ac cannot be optimal

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 Channel2. Channel Switching Between Two ChannelsThe 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 ChannelsAdvantage: optimization over a three-dimensional space instead of six by utilizing the three equality constraints.

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

Logarithmic cost functionsNoise 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.

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 orb. To switch between channel i and channel j as

specified in Proposition 4 if there exists no channels with cost Ac.

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).

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.

Simulation resultsThe 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

0 5 10 1510

-6

10-5

10-4

10-3

10-2

10-1

100

Average Power Limit Ap

Ave

rage

Pro

babi

lity

of E

rror

Average probability of error versus Ap for Ac=2

Optimal Channel SwitchingOptimal 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

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

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 1510

-6

10-5

10-4

10-3

10-2

10-1

100

Average Power Limit Ap

Ave

rage

Pro

babi

lity

of E

rror

Average probability of error versus Ap for Ac=5

Optimal Channel SwitchingOptimal Single Channel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-2

10-1

100

Average Power Limit Ap

Ave

rage

Pro

babi

lity

of E

rror

Closer view - Average probability of error versus Ap for Ac=5

Optimal Channel SwitchingOptimal Single Channel

Fig 3:A closer look at fig for Ap [0,1]

1 2 3 4 5 6 7 810

-6

10-5

10-4

10-3

10-2

10-1

100

Average Cost Limit Ap

Ave

rage

Pro

babi

lity

of E

rror

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]

0 5 10 1510

-6

10-5

10-4

10-3

10-2

10-1

100

Average Power Limit Ap

Ave

rage

Pro

babi

lity

of E

rror

Average probability of error versus Ap for Ac=0.9

Optimal Channel SwitchingOptimal 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].

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 1.610

-5

10-4

10-3

10-2

10-1

Average Cost Limit Ac

Ave

rage

Pro

babi

lity

of E

rror

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

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.

Capacity MaximizationThe capacity of a channel i can be expressed as

can be defined as

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

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 0.10

1

2

3

4

5

6

7

8

9

10

P(mW)Cap

acity

in M

bps

Capacity of each channel versus power

Optimal Single ChannelOptimal Channel Switching

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

P(mW)

Cap

acity

in M

bps

Capacity of each channel versus power

Channel 1Channel 2Channel 3

ConclusionJoint 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 strategies.

References [1] M. Azizoglu, “Convexity properties in binary detection

problems,” IEEE Trans. Inf. Theory, vol. 42, no. 4, pp. 1316–1321, Jul. 1996.

[2] S. Loyka, V. Kostina, and F. Gagnon, “Error rates of the maximumlikelihood detector for arbitrary constellations: Convex/concave behavior and applications,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1948–1960, Apr. 2010.

[3] B. Dulek, M. E. Tutay, S. Gezici, and P. K. Varshney, “Optimal signaling and detector design for M-ary communication systems in the presence of multiple additive noise channels,” Digit. Signal Process., vol. 26, pp. 153–168, Mar. 2014.

[4] S. Bayram, N. D. Vanli, B. Dulek, I. Sezer, and S. Gezici, “Optimum power allocation for average power constrained jammers in the presence of non-Gaussian noise,” IEEE Commun. Lett., vol. 16, no. 8, pp. 1153–1156, Aug. 2012.

[5] A. Patel and B. Kosko, “Optimal noise benefits in Neyman-Pearson and inequality-constrained signal detection,” IEEE Trans. Signal Process., vol. 57, no. 5, pp. 1655–1669, May 2009.

[6] B. Dulek and S. Gezici, “Detector randomization and stochastic signaling for minimum probability of error receivers,” IEEE Trans. Commun., vol. 60, no. 4, pp. 923–928, Apr. 2012

[7] M. E. Tutay, S. Gezici, and O. Arikan, “Optimal detector randomization for multiuser communications systems,” IEEE Trans. Commun., vol. 61, no. 7, pp. 2876–2889, Jul. 2013.

[8] C. Goken, S. Gezici, and O. Arikan, “Optimal signaling and detector design for power-constrained binary communications systems over non- Gaussian channels,” IEEE Commun. Lett., vol. 14, no. 2, pp. 100–102, Feb. 2010.

[9] M. Tutay, S. Gezici, and O. Arikan, “Optimal randomization of signal constellations on the downlink of a multiuser DS-CDMA system,” IEEE Trans. Wireless Commun., vol. 12, no. 10, pp. 4878–4891, Oct. 2013.

[10] J. A. Ritcey and M. Azizoglu, “Performance analysis of generalized selection combining with switching constraints,” IEEE Commun. Lett., vol. 4, no. 5, pp. 152–154, May 2000.

[11] A. Sezer, S. Gezici, and H. Inaltekin, “Optimal channel switching for average capacity maximization,” in Proc. IEEE ICASSP, May 2014, pp. 3503–3507.

[12] Y. Liu and M. Liu, “To stay or to switch: Multiuser dynamic channel access,” in Proc. IEEE INFOCOM, Apr. 2013, pp. 1249–1257.

[13] L. Chen, S. Iellamo, andM. Coupechoux, “Opportunistic spectrum access with channel switching cost for cognitive radio networks,” in Proc. IEEE ICC, Kyoto, Japan, Jun. 2011, pp. 1–5.

[14] A.Ozcelikkale, H. M. Ozaktas, and E. Arikan, “Signal recovery with cost constrained measurements,” IEEE Trans. Signal Process., vol. 58, no. 7, pp. 3607–3617, Jul. 2010.

[15] T.-S. Kim and S.-L. Kim, “Random power control in wireless ad hoc networks,” IEEE Commun. Lett., vol. 9, no. 12, pp. 1046–1048, Dec.2005.

[16] Mehmet Emin Tutay, Sinan Gezici, Hamza Soganci, and Orhan Arikan, “optimal channel switching over gaussian channels under average power and cost constraints,” IEEE Trans. Commun., vol. 63, no. 5, pp. 1907–1922, May. 2015.

[17] R. T. Rockafellar, Convex Analysis. Princeton, NJ, USA: Princeton Univ. Press, 1968.

[18] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005

[19] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.

[

[20] F. Gaaloul, H.-C. Yang, R. Radaydeh, and M.-S. Alouini, “Switch based opportunistic spectrum access for general primary user traffic model,” IEEE Wireless Commun. Lett., vol. 1, no. 5, pp. 424–427, Oct. 2012.[21] G. Strang, Introduction to Linear Algebra, 4th ed. Cambridge, MA, USA: Wellesley-Cambridge, 2009.[22] Thanh Le, Csaba Szepesv´ari, and Rong Zheng, “Sequential Learning for Multi-channel Wireless Network Monitoring with Channel Switching Costs”, IEEE Wireless Commun. Lett., vol. 1, no. 5, pp. 424–427, Oct. 2014.[23] Sergey Loyka, Victoria Kostina and Francois Gagnon, “Bit Error Rate is Convex at High SNR”.

THANK YOU