signal to meet specified constraints, OFDM provides straight- forward spectral shaping abilities through notching out sub- carriers that may not be used for ...
Dynamic Spectral Shaping in Cognitive Radios With Quality of Service Constraints Deepak R. Joshi and Dimitrie C. Popescu
Octavia A. Dobre
Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia
Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John’s, Canada
Abstract—In this paper we consider cognitive radios (CR) that employ orthogonal frequency division multiplexing (OFDM) modulation at the physical layer and present an algorithm for transmitter adaptation subject to specified quality of service (QoS) constraints. The proposed algorithm explicitly considers multipath channels between the CR transmitter and its intended receiver, and performs joint spectral shaping and transmitter power control in order to avoid spectral bands used by licensed users of the spectrum while maintaining specified target signalto-interference+noise-ratios (SINR) at the CR receiver. The algorithm is illustrated with numerical examples obtained from simulations in which dynamic scenarios are considered with licensed transmitters getting in and out of the system.
I. I NTRODUCTION Cognitive Radio (CR) [1] is an emerging concept designed to enable dynamic access to the frequency spectrum and the reuse of licensed frequencies under specific conditions that no harmful interference be caused to the incumbent licensed users of the spectrum. Over the past several years OFDM modulation has emerged as a preferred choice for the physical layer of CR due to many attractive features such as simplicity of implementation and scalability, or its ability to transmit over non-contiguous frequency bands that can be easily adapted to use idle licensed spectrum [2], [3]. OFDM-based modulation schemes are currently present in wireless standards like the IEEE 802.11 standard for wireless local area networking (WLAN) systems and the short distance wireless networks standard IEEE 802.15 for wireless personal area network (WPAN) systems. When it is desired to shape the spectrum of a transmitted signal to meet specified constraints, OFDM provides straightforward spectral shaping abilities through notching out subcarriers that may not be used for transmission, and various spectral shaping procedures have been proposed recently for OFDM-based systems [4], [5], [6]. We note that these procedures are applied in static scenarios where, once the desired spectral shape is achieved, any changes in the spectral pattern require renewed applications of the spectral shaping procedure. CRs are dynamic systems in which the spectrum utilized by licensed primary users (PU) changes while non-licensed secondary users (SU) are also actively utilizing the spectrum. In such a scenario it is desirable for the SU to dynamically adapt its transmission to conform to the new spectral requirements imposed by the PU rather than dropping out of
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the system to update it. This motivates the work presented in this paper where dynamic spectral shaping is proposed for transmitter adaptation in CR systems that operate subject to QoS constraints. The paper is organized as follows: in Section II we introduce the system model and formally state the problem. In Section III we discuss the spectral shaping concept for OFDM-based systems followed by presentation of the proposed spectral shaping algorithm through precoder adaptation in Section V. In Section V we present numerical results obtained from simulations that illustrate the proposed algorithm for a dynamic scenario with realistic parameters where the available spectrum is utilized by two licensed systems: a WiMax system and a digital TV system. Final remarks and conclusions are given in Section VI. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT We consider a CR that uses a linearly transformed (LT) OFDM system similar to the one considered in [7] where a precoding matrix is used to map frames of incoming symbols from the digital modulator into OFDM symbols. The block diagram of the system is sketched in Figure 1. Unlike [7] the Transmitter Adaptation Mechanism Symbols from digital
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Block diagram of the LT-OFDM CR system.
precoding matrix is no longer fixed and restricted to a square Walsh-Hadamard matrix, but can be adjusted to obtain desired spectral characteristics for the signal transmitted by the CR as well as to satisfy QoS constraints specified in terms of target SINRs that must be met at the CR receiver.
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The N -dimensional OFDM block symbol is x = SP1/2 b,
(1)
where S is the N × M precoding matrix (N is the length of the OFDM symbol equal to the number of sub-carriers used for transmission and M is the length of the block of information symbols transmitted as one OFDM symbol), b = [b1 , . . . , bM ] is the M -dimensional vector representing the block of information symbols to be transmitted as one OFDM block symbol, and P = diag{p1 , . . . , pM } is an M × M diagonal matrix containing the power values at which information symbols are transmitted by the CR transmitter. After the usual analog-to-digital and serial-to-parallel conversion, removal of the cyclic prefix, and FFT processing, the received signal is given by y = ΛSP1/2 b + n + i,
(2)
where Λ is the N × N diagonal matrix containing the N point discrete Fourier transform (DFT) of the channel impulse response, n is the additive Gaussian noise that corrupts the signal at the receiver with covariance matrix Rn and i is the interference from licensed transmitters with correlation matrix Ri . We assume that matrices Rn and Ri are identified by the CR system in a preliminary spectrum sensing operation and are known. The received signal is first equalized by multiplying it with the inverse channel matrix to obtain ˜ = Λ−1 y = SP1/2 b + w, y
(3)
where w = Λ−1 (n + i) is the interference+noise vector ˜ is then processed by a bank after equalization. The signal y of matched filters (MF) which yields the vector of decision variables ˆ = SH y ˜ = SH SP1/2 b + SH w, (4) b where (·)H denotes the Hermitian operator (transpose and complex conjugate), such that for a given symbol m the SINR expression at the output of its corresponding MF is pm , m = 1, . . . , M, (5) γm = Im where pm represents its corresponding power in P and Im = s(m)H Rm s(m) is the effective interference corrupting ˆ expressed in terms of the the m-th decision variable ˆbm in b interference+noise correlation matrix Rm corrupting ˆbm and the m-th column s(m) of the precoding matrix S. Matrix Rm is obtained by subtracting the contribution corresponding to symbol m from the correlation matrix of the equalized signal, Ry˜, and is expressed as [8], [9] Rm = Ry˜ − pm s(m) s(m)H ,
(6)
where Ry˜ is given as ˜ H ] = SPSH + W, Ry˜ = E[˜ yy
(7)
H
and W = E[ww ], is the correlation matrix of interference+noise after channel equalization. We note that matrix
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W is related to the original interference and noise correlation matrices by (8) W = Λ−1 (Ri + Rn )Λ−1 Our goal in this setup is to present an algorithm for joint adaptation of the precoder and power matrices, S and P, subject to some total received power constraint such that all symbols are received with a specified target SINR γ ∗ . We note that the target SINR is implied by the QoS requirements of the application that must be supported by the CR system and should satisfy the admissibility condition [10] γ∗ N , < ∗ 1+γ M
(9)
III. DYNAMIC S PECTRAL S HAPING In the LT-OFDM system, the frequencies over which a specific information symbol bm in the frame b is transmitted, are determined by the m-th column, s(m) , of the precoding (m) matrix. A given element n of this column sn determines the fraction of total power pm of symbol m that is transmitted over (m) subcarrier n. Thus, when sn = 0 symbol m does not use for transmission the n-th subcarrier and this property can be used to dynamically shape the spectrum of the signal transmitted by the CR in response to the spectrum patterns of the active licensed users. To illustrate the dynamic spectral shaping capabilities of the LT-OFDM system we consider an example where the total number of sub-carriers available to the CR system is N = 128 with a sub-carrier spacing of Δf = 78.125 kHz which implies a total bandwidth of, BW = 10 MHz, available for the CR transmitter. We assume that the frame of information symbols has length M = 50 such that the precoding matrix is of dimension 128 × 50. By taking the columns of the precoding matrix to be equal to the first M columns of the identity matrix of order N the spectrum of the resulting LT-OFDM signal is shown in Figure 2. By changing rows 10 ÷ 20 and 70 ÷ 80 of the precoding matrix to all zeros the new spectrum of the LT-OFDM signal displays notches in the bandwidths corresponding to the nulled sub-carriers that is from 14.218 to 15 MHz and from 19.218 to 20 MHz as shown in Figure 3. IV. J OINT P RECODER A DAPTATION AND P OWER C ONTROL Our proposed approach to dynamic spectral shaping is inspired by [8], [9] and is based on joint adaptation of the precoder and power matrices, S and P, through a distributed incremental algorithm that adjusts the columns of the precoder matrix to minimize the interference function corresponding to each symbol followed by incremental adaptation of its power to meet the specified target SINR. From the SINR equation (5) for a given symbol m we note that, for fixed symbol power, minimization of the interference function Im is equivalent to the maximization of the symbol SINR. Thus, the optimal s(m) can be found by solving the constrained optimization problem min Im = s(m)H Rm s(m) ,
s(m) ∈Sm
subject to γm = γ ∗
(10)
We note that the updates implied by (12)–(14) may result in abrupt changes of s(m) and/or pm which are not desirable in practical implementations, and similar to [8], [9] the proposed algorithm uses the following incremental updates:
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At step j of the algorithm one column m of the precoding matrix is updated to,
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Spectrum of LT-OFDM signal with all available sub-carriers active. •
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s(m) (j) + zβvm (j) , s(m) (j) + zβvm (j)
(15)
where, vm (j) is the minimum eigenvector of interference+noise correlation matrix Rm (j), β is a parameter that limits the how far (in terms of Euclidean distance) the updated vector is from the previous, and z = sgn[s(m)H (j)vm (j)]. Power for symbol m at step of j of the algorithm is updated to,
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pm (j + 1) = pm (j) − μ[pm (j) − γ ∗ Im (j)],
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Following [8], [9], we also define the cost function corresponding to a given symbol m to be the product between its power and its corresponding interference function
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um = pm Im = pm s(m)H Rm s(m)
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Fig. 3. Spectrum of LT-OFDM signal with some of the available sub-carriers nulled out by the precoding matrix.
with
Sm = s(m) |s(m) ∈ RN , s(m) = 1
(11)
being the N dimensional sphere with radius 1 where s(m) , m = 1, . . . , M, take values. As discussed in [8], [9] the potential updates for s(m) are implied by a greedy interference avoidance (IA) procedure [11] which replaces it with the eigenvector corresponding to the minimum eigenvalue λm of the Rm matrix1 , that is after the update we have Rm s(m) = λm s(m) ∀m = 1, . . . , M
(12)
In addition, the optimal update satisfies also the determinant condition [8] 2pm (Rm − γ ∗ IN ) 2s(m) s > 0. (13) Dm = (−1) 2s(m)H 0 Once s(m) is replaced by the minimum eigenvector of Rm , the effective interference experienced by symbol m is minimized and becomes equal to λm , and the optimal power update is obtained by matching the specified target SINR [8], [9], that is (14) pm = γ ∗ λm ∀m = 1, . . . , M 1 This
(16)
is also referred to as the minimum eigenvector of Rm .
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m = 1, . . . , M,
(17)
Using the incremental updates (15)–(16) the proposed algorithm for dynamic spectral shaping for OFDM-based CR is formally stated here: 1) Input data: • Precoding matrix, S, power matrix, P, channel matrix, Λ, and target SINR γ ∗ . • Spectrum sensing information: noise and PU correlation matrices Rn and Ri . • Constants β, μ, and tolerance . 2) If there is idle licensed spectrum GO TO Step 3. ELSE, wait until spectrum becomes available. 3) Notch out unavailable sub-carriers by putting zeros in the corresponding rows of the precoder matrix S. 4) If admissibility condition in equation (9) is satisfied GO TO Step 5. ELSE STOP, the desired target SINR is not admissible. 5) FOR each symbol m = 1, 2, · · · , M DO a) Compute corresponding Rm (j) using equation (6) and determine its minimum eigenvector vm (j) corresponding to the minimum eigenvalue λm (j), b) Update s(m) (j) using equation (15). c) Update pm (j) using equation (16). 6) IF change in cost function is larger than for any symbol then GO TO Step 5. ELSE STOP, a fixed point has been reached. 7) IF optimality condition equation (13) is true, STOP, an optimal equilibrium has been reached. ELSE GO TO Step 5.
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This algorithm sequentially adapts the corresponding precoder column and power for each symbol to be transmitted until a fixed point is reached. Numerically, a fixed point of the algorithm is reached when the updates result in changes of the symbol cost functions that are smaller than the specified tolerance . We note that the convergence speed of the algorithm depends on the values of the corresponding increments specified by the algorithm constants μ and β. We also note that existence of a fixed point of this algorithm is guaranteed by the existence of a Nash equilibrium for the corresponding separable game which is discussed in detail in [8], [9]. In addition, the check of the optimality condition equation (13) in Step 7 ensures that the optimal equilibrium is reached and that the algorithm does not stop in a sub-optimal fixed point.
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V. S IMULATIONS AND N UMERICAL R ESULTS −60
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TABLE I PU1(W I MAX) S IMULATION PARAMETERS [12] Parameter Bandwidth Number Of Subcarrier Subcarrier Spacing Guard Band Data Subcarrier Modulation Symbol Duration
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We study the proposed algorithm in a dynamic scenario with realistic parameters where the available spectrum is initially utilized by two licensed systems as PU and an unlicensed one as SU. The PU are a WiMax system and a digital TV system and the SU is a CR using the LT-OFDM scheme. The corresponding parameters for the two PU systems are shown in the table I and table II. The other simulation parameters
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Value 4MHz 256 15.625 KHz 1/4 QAM 72usec
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Fig. 5. Simulation scenario with 1 primary user (DTV) and the CR system transmitting in non-interfering bands.
TABLE II PU2(DTV) S IMULATION PARAMETERS
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Value 7.61MHz 1705 4.46 KHz 1/4 QAM 280usec
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are: the algorithm constants μ = 0.1 and β = 0.1, algorithm tolerance, = 10−2 , target SINR, γ ∗ = 0.67 and maximum power, Pmax = 5. The proposed algorithm is applied to obtain the optimal precoding and power matrices that allow the CR system to operate alongside the PU without interfering with them. In Figure 4 the initial condition of dynamic scenario is shown in which both PU and the SU are transmitting in nonoverlapping frequency bands. At some time instant the WiMax system stops transmitting and the CR is able to take advantage of the spectrum vacated by it (see Figure 5) by optimizing its precoding and power matrices for the new scenario. The precoding and power matrices are optimized one more time
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Fig. 6. Simulation scenario with a different primary user (WiMax) and the CR system transmitting in non-interfering bands.
when the WiMax system starts transmitting again while the DTV system stops its transmission. In this case the CR must vacate the spectrum required by the WiMax system while taking advantage of the spectrum released by the DTV system (see Figure 6).
VI. C ONCLUSIONS
7
In this paper we presented a novel algorithm for joint precoder adaptation and power control for OFDM-based CR systems that operate in multipath channels under specified target SINRs at the receiver. The proposed algorithm can be used for dynamic spectral shaping to avoid frequency bands that are actively used by licensed transmitters, and is illustrated with numerical results obtained from simulations in realistic scenarios where the licensed transmissions come from a WiMax system and a digital TV system.
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ACKNOWLEDGMENT 0
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This work was supported in part by the National Sciences and Engineering Research Council of Canada.
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SINR variation for the M data symbols.
R EFERENCES
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Figures 7 and 8 illustrate the transmitter adaptation process implied by the proposed algorithm for the dynamic scenario described by Figure 4–6. From Figure 7 we note that the SINRs corresponding to all symbols vary until the desired target value is reached. When the system configuration changes the SINRs decrease but the proposed algorithm adapts the CR transmission to the new available spectrum such that, when the fixed point is reached, the desired target SINR value continues to be maintained. Figure 8 shows a similar behavior for the power values of all symbols. We note that while initial power values are high, the algorithm ensures that the desired target SINRs are achieved with minimum transmitted power. We also note that power values increase when a change in the system configuration occurs and the available spectrum changes, but they return to minimal values upon successful adaptation by the proposed algorithm.
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