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Water-Filling Algorithm approach for power Allocation in OFDM

Based Cognitive Radio System

Sakib.R.Mujawar1, A.N.Jadhav2 1Department of Electronics & telecommunication Engg, D.Y.Patil college of Engineering & Technology

Kolhapur,

Abstract— In this paper, a simple water-filling algorithm for OFDM based cognitive radio is

approach to solve the increasing demand for wireless multimedia service to creates a lack of

spectrum. A potential solution to this issue is to allocate the spectrum dynamically by means of

cognitive radio. Water-filling can offer a solution to this spectrum allocation. The proposed water-

filling power allocation algorithm cognitive radio user can achieve significantly higher transmission

capacity for given statistical interference constraints and a given budget compared to the classical

power allocation algorithm i.e. Uniform

Keywords—Cognitive Radio, Orthogonal frequency division multiplexing, Primary user, Secondary

user

I. INTRODUCTION

Now a day we are facing many problems regarding spectrums due to the growing application in

wireless fields. Lot of wireless applications is sharing the same medium and due to this there is

overload which leads to lack of spectrum in given frequency bands. On the other hand, measurement

show that[1] wide ranges of spectrum are rarely used most of the time, whereas other bands are used

heavily and as we know the radio frequency is scare natural resource and its efficient use is of the

utmost importance. Moreover, most spectrum bands are allocated to certain services but worldwide

spectrum occupancy measurements show that only portions of the spectrum band are fully used.

The utilization of radio spectrum can be improved significantly by using the cognitive radio (CR)

technology. In some case, the spectrum bands are not utilized because licensed user does not always

occupy their spectrum and unlicensed users are not allowed to operate in such spectrum bands.

Spectral efficiency can be increased significantly by giving Opportunistic access of the frequency

bands to a group of Potential users (referred to as secondary or CR users) for whom the band has not

been licensed. Cognitive radio (CR) has been proposed as a way to improve spectrum efficiency by

exploiting unused spectrum in dynamically changing environments. The CR design is an innovative

radio design philosophy which involves smartly sensing the swaths of spectrum and then

determining the transmission characteristics (e.g., symbol rate, power, bandwidth, latency) of a group

of secondary users based on the behavior of the users to whom the spectrum has been licensed

(referred to as primary users). Although opportunistic spectrum access would allow CR user to

identify and access available spectrum resources, one of the main concerns is to utilize the available

spectrum resources in an efficient manner.

OFDM base CR cognitive Radio:-OFDM is a multi-carrier modulation technique that can overcome

many problems that arise with high bit-rate communications, the most serious of which is time

dispersion. The data-bearing symbol stream is split into several lower-rate streams, and these streams

are transmitted on different carriers. Because this splitting increases the symbol duration by the

number of orthogonally overlapping carriers (subcarriers), multipath echoes affect only a small

portion of the neighboring symbols. The remaining inter-symbol interference (ISI) is removed by

extending the OFDM symbol with a cyclic prefix (CP). Other advantages of OFDM include high

spectral efficiency, robustness against narrowband interference (NBI). Orthogonal frequency

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 06; June - 2016 [ISSN: 2455-1457]


division multiplexing (OFDM), because of its flexibility in allocating the spectrum, has been

recognized as an air interface technology for CR systems. Because of the coexistence of CR and

primary users in side-by-side bands, mutual interference between these users is the limiting factor in

order to achieve a good performance for CR systems .Use of the classical power allocation

Algorithms e.g., well known water-filling algorithm for CR systems may result in higher interference

to the primary user (PU) receivers. In, we propose a power loading algorithm that maximized the

downlink transmission rate of a CR user while keeping the total interference introduced to different

PU receivers below a specified threshold .A distribute d algorithm for optimal resource allocation in

orthogonal frequency division multiplexing access (OFDMA)-based CR systems has been proposed.

When using orthogonal frequency Division Multiplexing in cognitive radio network, the power

allocation schemes for spectrum resources will be very flexible and convenient .However, it become

very challenging to allocate power to individual sub channels in the OFDM-based cognitive radio net

works. In traditional power allocation problems, water-filling algorithms are prevalent. Because

additional interference constraints must be considered in cognitive radio networks, the water-filling

algorithm is a real ways performed geometric to solve the power allocation problem [4].In this paper,

we study the properties of the water-filling algorithm and propose a linear algorithm to reduce

computational complexity.

II. SYSTEM MODEL

We consider a wireless system consisting of L sub-channels licensed to different primary users. Each

of these primary users behaves differently or has uncorrelated activity in their band. All the sub-

channels are divided into multiple subcarriers as shown in figure2.1 and they are opportunistically

available to some secondary or cognitive user which uses the band in OFDM fashion. the total

number of subcarriers are N with M sub-channels licensed to different primary users.

In this, first, we discuss the subcarrier grouping strategy that is used to maximize the transmission

rate and minimize interference. Second, we outline in details the system model used in this thesis.

This includes a description of the transmitter and receiver, the adaptive subcarrier allocation scheme

used in our work, as well as the channel model used in the analysis.

2.1 Underlay Model

We consider a downlink transmission scenario. It is assumed that the frequency bands of bandwidth

B1, B2... BL have been occupied by PU1, PU2... PUL. As in Figure 2.1, SUs can occupy either the

spectrum of PUs or the adjacent spectrum of PUs. The available bandwidth for CR transmission is

divided into N subcarriers based OFDM system, and the bandwidth for each subcarrier is ∆f H z.

In the downlink transmission scenario, there are three instantaneous fading gains: between the SUs

transmitter and SUs receiver for the ith subcarrier denoted as ℎ𝑖𝑠𝑠; between the SUs transmitter and

lth PU receiver denoted as ℎ𝑙𝑠𝑝

; between lth Pus transmitter and SUs receiver denoted as ℎ𝑙𝑝𝑠

.We

assume that these instantaneous fading gains are perfectly known at the SUs transmitter.



1 2 3 . . . . . . N

Figure 2.1 Illustration for underlay Model

2.1.1 Interference introduced to PU by SU

We assume that the signal transmitted on the subcarrier is an ideal Nyquist pulse. According to [6],

the power spectrum density of the ith subcarrier can be written as:

𝑋𝑖(𝑓)

= 𝑃𝑖𝑇𝑠(𝑠𝑖𝑛𝜋𝑓𝑇𝑠

𝜋𝑓𝑇𝑠)2 (1)

Where Pi is the total transmits power in the ith subcarrier and Ts is the symbol duration.

Then the interference introduced to the lth PU band by the ith subcarrier is:

𝐼𝑖(𝑙)

= 𝑃𝑖|ℎ𝑙𝑠𝑝

|2

𝑇𝑠 ∫ (𝑠𝑖𝑛𝜋𝑓𝑇𝑠

𝜋𝑓𝑇𝑠)

2𝑑𝑓

𝑑𝑖𝑙+𝐵𝑖2

𝑑𝑖𝑙−𝐵𝑖2

(2)

Where dil is the distance in frequency between the ith subcarrier and the lth PU band, and Bl

represents occupied bandwidth by the lth PU.

2.1.2 Interference introduced to SU by PU

According to [*], the power spectrum density of the PU signal after M-fast Fourier transform (FFT)

processing can be expressed as:

𝐸[𝐼𝑁(𝜔) =1

2𝜋𝑀∫ 𝑋𝑃𝑈(𝑒𝑗𝜔) (

sin(𝜔−𝑦)𝑀

2sin(𝜔−𝑦)

2

)

2

𝑑𝑦 (3) 𝜋

−𝜋

Where XPU (ejω) is the power spectrum density of the PU signal. The PU signal has been taken to be

an elliptically filtered white noise process with amplitude PPU. According to the interference

introduced to the ith subcarrier by the lth PU band can be written as:

𝐽𝑖(𝑙)

(𝑃𝑃𝑈) = |ℎ𝑙𝑠𝑝

|2

∫ 𝐸[𝐼𝑁(𝜔)]𝑑𝜔𝑑𝑖𝑙+

∆𝑓

2

𝑑𝑖𝑙−∆𝑓

2

(4)

PU1

PU1

PU2 PUL

B1 ∆f B2 BL

PU1

PU1



III. PROBLEM FORMULATION

The design goal is to find power value for each subcarrier, Pi (i=1, 2… N) For given instantaneous

fading gain ℎ𝑙𝑠𝑠 , given fading statistics of ℎ𝑙

𝑠𝑝 and the total transmit power budget PT. As such the

total transmission rate of the CR user, C is maximized while the probability that the interference

introduced to lth (l=1, 2… L) PU band is kept below the threshold 𝐼𝑡ℎ𝑙 (l=1, 2… L), respectively, with

the probability value α or above. Mathematically, the problem can be formulated as a constrained

optimization problem as follows.

𝐶 = ∑ 𝑙𝑜𝑔2 (1 +|ℎ𝑖

𝑠𝑠|2𝑝𝑖

𝜎2 + ∑ 𝐽𝑖(𝑙)𝐿

𝑙=1

)

𝑁

𝑖=1

𝑃𝑖𝑚𝑎𝑥 (5)

Subject to:

𝑃𝑟 (∑ 𝐼𝑖(𝑙)

𝑁

𝑖=1

(𝑑𝑖𝑙 , 𝑃𝑖) ≤ 𝐼𝑡ℎ(𝑙)

) ≥ 𝑎, ∀𝑙, (6)

𝑃𝑖 ≥ 0, ∀𝑙, (7)

∑ 𝑃𝑖

𝑁

𝑖=1

≤ 𝑃𝑇 (8)

Where Pr. denotes the probability. Now the probabilistic interference constraint in Eq. (6) can be

written as.

𝑃𝑟 (|ℎ𝑙𝑠𝑝

|2

∑ 𝐾𝑖(𝑙)

𝑃𝑖 ≤ 𝐼𝑡ℎ(𝑙)

𝑁

𝑖=𝑙

) ≥ 𝑎, ∀𝑙, (9)

Where 𝐾𝑖(𝑙)

= 𝑇𝑠 ∫ (𝑠𝑖𝑛𝜋𝑓𝑇𝑠𝜋𝑓𝑇𝑠

)2𝑑𝑓𝑑𝑖𝑙+𝐵𝑖/2

𝑑𝑖𝑙−𝐵𝑖/2. Since |ℎ𝑙

𝑠𝑝|is assumed to be Rayleight distributed with known

parameter𝜆𝑙, the distribution of |ℎ𝑙𝑠𝑝

|2 corresponds to an exponential distribution with the

parameter𝜆𝑙2. The Eq.(9) can be evaluated in closed form for the Rayleigh fading case as follows

1 − 𝑒−

𝐼𝑡ℎ(𝑙)

2𝜆𝑙2 ∑ 𝑃𝑖𝐾

𝑖(𝑙)𝑁

𝑖=1

,

≥ 𝑎, ∀𝑙, (10)

After some mathematical manipulations, Eq.(10) can be written as

∑ 𝑃𝑖

𝑁

𝑖=1

𝐾(𝑙) ≤𝐼𝑡ℎ

(𝑙)

2𝜆𝑙2(− ln(1 − 𝑎))

, ∀𝑙 (11)

The optimal power of the ith subcarrier is given by:

𝑃𝑖∗ = [𝑤𝑖 −

𝜎2 + ∑ 𝐽𝑖(𝑙)𝐿

𝑙=1

|ℎ𝑖𝑠𝑠|

2 ] ∀𝑙, (12)



Where 𝑤𝑖 = 1

𝛽+∑ 𝛾𝑙𝐾𝑖(𝑙)𝐿

𝑙=1

and 𝛽 and 𝛾𝑙 are deterministic Lagrange parameters.

IV. WATER-FILLING AND UNIFROM POWER LOADING SCHEMES

In this section, we proposed a water-filling algorithm and also describe classical algorithm namely

uniform power loading algorithm which are used for conventional OFDM system.

4.1 Water-Filling Loading Algorithm

In water-Filling algorithm, which is optimal power allocation algorithm in conventional OFDM

system, we use the total power allocation by uniform loading as the power constraint. The allocated

power in the ith subcarrier because of the 𝑖th subcarrier because of the 𝑙th interference constraint is

written as

𝑃𝑖(𝑙)

= 𝑃 𝐾𝑖(𝑙)⁄ , ∀𝑙, (13)

By using Eq.(11), after nulling (2L-1) subcarriers P can calculated by assuming strict equality in the

equality in the lth interference in Eq. (11). Using Eq. (13), this equality constraint can be written as

∑ 𝑃𝑖𝐾𝑖(𝑙)

𝑁

𝑖=1

=𝐼𝑡ℎ

(𝑙)

2𝜆𝑙2(− ln(1 − 𝑎))

(14)

𝑖 ≠ (𝑁

𝐿,𝑁

𝐿+1,….,𝑁)

i=1

We can derive

𝑃 =𝐼𝑡ℎ

(𝑙)

2(𝑁 − 2𝐿 + 1)𝜆𝑙2(− ln(1 − 𝑎))

(15)

Through Eq. (4.1) and (4.3), we can get 𝑃𝑖(𝑙)

as following

𝑃𝑖(𝑙)

=𝐼𝑡ℎ

(𝑙)

2𝐾𝑖(𝑙)(𝑁 − 2𝐿 + 1)𝜆𝑙

2(− ln(1 − 𝑎)) (16)

Now we need to calculate power values𝑃𝑖(𝐿+1)

due to the totalpower constraint. In order to meet the

total power constraint, we use the standard water-filling algorithm to distributetotal power 𝑃𝑇among

𝑁 CR subcarriers. According to the water-filling algorithm with a total power constraint 𝑃𝑇, the

power values can be written as

𝑃𝑖(𝐿+1)

= 𝑚𝑎𝑥 {0,1

𝛼−

𝜎2 + ∑ 𝐽𝑖(𝑙)𝐿

𝑙=1

|ℎ𝑖𝑠𝑠|

2 } ∀𝑖, (18)

Where the Lagrange constant 𝛼 can be calculated from the following eqution

∑ 𝑚𝑎𝑥 {0,1

𝛼−

𝜎2 + ∑ 𝐽𝑖(𝑙)𝐿

𝑙=1

|ℎ𝑖𝑠𝑠|

2 } = 𝑃𝑇 (19)

𝑁

𝑖=1



i

The power value for 𝑖th subcarrier, denoted by 𝑃𝑖(𝑊𝐹)

, are obtained using the standard water-filling

algorithmas mentioned in Eq (18) and (19) considering the total power constraint equal to the total

power allocated by uniform loading algorithm. The power values will satisfy the total power

constraint given in Eq. (8) however it is checked that if the power values satisfy the interference

constraints specified in Eq. (11). If a particular interference constraint is not satisfied, the power

value in each subcarrier 𝑃𝑖(𝑊𝐹)

is reduced such that the all interference constraints are satisfied. Also,

if none of these interference constraints is met strictly, the power value 𝑃𝑖(𝑊𝐹)

is increased until one

of these interference constraints is met strictly.

4.2 Uniform Loading Algorithm

In uniform power loading algorithm, which is used in the Conventional OFDM systems due to its

reduced complexity, equal amount of power is allocated in each subcarrier such that all 𝐿

+1constraints in Eqs.(8) And (11) can be satisfied. By assuming equal power in each subcarrier and

solving Eq. (11) to satisfy the strict equality on 𝑙th interference constraint (𝐼𝑡ℎ(𝑙)

), the corresponding

power for 𝑖th subcarrier can be written as

𝑃𝑖(𝑙)

=𝐼𝑡ℎ

(𝑙)

2 ∑ .𝑁𝑖=1 𝐾𝑖

(𝑙)𝑁𝜆𝑙

2l𝑛1

(1−𝑎)

, ∀𝑖. (20)

The power allocation for constraint in Eq. (8) can be written as

𝑃𝑖(𝐿+1)

= 𝑃𝑇 𝑁.⁄ (21)

The final power allocation to each subcarrier is done according

𝑃𝑖𝑇 = 𝑚𝑖𝑛{𝑃𝑖

(1), 𝑃𝑖

(2), … … 𝑃𝑖

(𝐿+1)} ∀𝑖 (22)

It should be noted that at least one of these 𝐿+1constraints will be met strictly and hence, scaling of

power value is not required.

V SIMULATION RESULTS

In this section we present simulation results where we assume that there are three PU bands (L=3),

and there are twelve OFDM subcarriers (N = 12) for the CR user. The values of Ts , ∆f, B1, B2, and

B3 have been assigned to be 4 seconds, 0.3125 MHz, 1MHz, 2 MHz, and 5 MHz, respectively.

AWGN variance, (σ2) is assumed to be equal to10−8W and the channel fading gains are assumed to

follow Rayleigh distribution. The average channel power gains for |ℎ𝑖𝑠𝑠|2, |ℎ1

𝑠𝑝|

2, |ℎ2

𝑠𝑝|

2 and

|ℎ3𝑠𝑝

|2are assumed to be -10 dB, -5 dB, -7 dB, and -10 dB, respectively. The values of J (l) are

generated randomly with an average value of 1×10−6W. The values of 𝐼𝑡ℎ(1)

, and 𝐼𝑡ℎ(3)

have been

assumed to be 1×10−6W, and 5×10−6W , respectively.

In Figure 5.1, we plot the achievable maximum transmission rate for the CR user versus the total

power budget for various algorithms. The value of 𝐼𝑡ℎ(2)

has been fixed to 2×10−6W, and the value of

α has been considered to be 0.95.The second curve is made by water-filling method; the lowest curve

is made by using uniform power allocation method. From this figure, we observe that the proposed

water-filling algorithm is able to achieve higher transmission rate for a given power budget than. It

should be noted that as we increase the power budget for CR user, the interference constraint

becomes dominant and the transmission rate of CR user does not increase as the power budget



increases. This is expected as in this region the CR system operates in an interference limited

scenario.

Figure 5.1: Maximum transmitted data rate vs. power budget (Total Power Constraint) for CR.

In Figure 5.2, we plot the achievable transmission data rate for the CR user versus interference

threshold for second PU band, (𝐼𝑡ℎ(2)

) for all the algorithms under consideration. The value of total

transmits power, PT has been assumed to be 5×10-4W. Again, we observe that the proposed

suboptimal algorithm achieves highest transmission rate over other algorithms. Further, water-filling

algorithm achieves higher transmission rate than the uniform algorithm. The transmission rate

versus interference threshold curve saturates after a certain value of (𝐼𝑡ℎ(2)

)). The reason is that

although (𝐼𝑡ℎ(2)

) is relaxed by increasing its value, other constraints ((𝐼𝑡ℎ(1)

), ( 𝐼𝑡ℎ(3)

), and PT) becomes

dominant.

Figure 5. 2: Maximum transmitted data rate vs. Interference threshold for 2nd PU band, 𝑰𝒕𝒉

(𝟐)



In Figure 5.3, we plot achievable transmission rate for the CR user versus probability α, the values of

PT , and (𝐼𝑡ℎ(2)

), is assumed to be 5×10−4 W , and 2×10−6W , respectively.

Figure 5.3: Transmission rate of the CR user vs. probability, α

In Figure 5.4, we plot the achievable maximum transmission rate for the CR user versus individual

power constraints for various algorithms. The value of 𝐼𝑡ℎ(2)

has been fixed to 2×10−6W, the total

power PT has been fixed to be 5 × 10−4W, and the value of α has been considered to be 0.95. From

this figure, we observe that the proposed water-Filling algorithm is able to achieve higher

transmission rate for a given power budget than the Uniform algorithm.

Figure 5.4: Maximum transmission rate for the CR user Vs individual power constraints.



VI. CONCLUSION

In this letter, We have developed a Water-filling allocation algorithm for orthogonal frequency

division multiplexing (OFDM) based cognitive radio CR system As such the transmission rate of the

cognitive radio user is maximized for given power budget and different probabilistic interference

constraints imposed by different PU receiver. The result has shown that our proposed Water-filling

allocation can achieve significantly higher transmission rate for CR user compared to the classical

power allocation algorithm namely the Uniform power allocation algorithm that are used for

conventional OFDN-based system

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3. D. Ngo, C. Tellumbra, andH. H. Nguyen,“Resource allocation for OFDM-based cognitive radio multicast

networks,”in Proc.IEEE Wireless Commun. And Networking Conf.,pp.1–6,Apr.2009.

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