An Efficient Bit Allocation Algorithm for Multicarrier Modulation
`
`Peter Kabal
`Fabrice Labeau
`Alexander M. Wyglinski
`Department of Electrical & Computer Engineering
`McGill University, Montr´eal, Canada H3A 2A7
`{alexw, flabeau, kabal}@TSP.ECE.McGill.CA
`
`Abstract
`
`In this paper we present an efficient bit allocation al-
`gorithm for multicarrier systems operating in frequency-
`selective environments. The proposed algorithm strives to
`maximize the overall throughput while guaranteeing that
`the mean bit error rate (BER) remains below a prescribed
`threshold. The algorithm is compared with several other al-
`gorithms found in literature in terms of the overall through-
`put, mean BER, and relative computational complexity.
`Furthermore, the algorithms are compared with an exhaus-
`tive search routine to determine the optimal bit allocation in
`terms of maximizing throughput given the constraint on er-
`ror performance. No power allocation is performed by the
`algorithms. Results show the proposed algorithm has ap-
`proximately the same throughput and mean BER as the op-
`timal solution while possessing a significantly lower com-
`putational complexity relative to the other algorithms with
`similar performance. When compared to algorithms which
`employ approximations to waterfilling, the computational
`complexity is comparable while the overall throughput is
`closer to the optimum.
`
`Keywords: Wireless Local Area Networks, Bit Loading,
`Wireless Multicarrier Transmission, Adaptive Modulation
`
`1 Introduction
`
`Several high-speed data transmission systems, such as
`wireless local area networks (WLAN) [1, 2] and xDSL
`modems [3], employ multicarrier modulation at the core
`of their design. Multicarrier modulation operates by trans-
`mitting data in parallel subcarriers at a lower data rate, ef-
`fectively transforming a frequency selective fading channel
`into a collection of flat fading subchannels. Thus, simple
`techniques can be employed at the receiver to reverse the
`effects of the channel.
`Conventional wireless multicarrier systems use a fixed
`signal constellation across all subcarriers, thus the overall
`error probabilities are dominated by the subcarriers with the
`
`This research was partially funded by the Natural Sciences and Engineer-
`ing Research Council of Canada (NSERC) and Le Fonds de Recherche sur
`la Nature et les Technologies du Qu´ebec.
`
`worst performance. To improve the system error perfor-
`mance, adaptive bit allocation can be employed such that
`the information is redistributed across the subcarriers in or-
`der to minimize the overall error probability. This redistri-
`bution is achieved by varying the signal constellation size
`across the subcarriers according to the measured signal-to-
`noise ratio (SNR) values. In extreme cases, poorly perform-
`ing subcarriers can be “turned off” or nulled [4].
`Most bit allocation algorithms can be classified into three
`categories:
`incremental (i.e. “greedy”) allocation [5–8],
`channel capacity approximation-based allocation [9, 10],
`and bit error probability expression-based allocation [11,
`12]. The first type of algorithm incrementally allocates an
`integer number of bits while the other two types use closed-
`form expressions of performance measures in order to de-
`termine a non-integer bit allocation and then round the re-
`sults. On the other hand, bit allocation algorithms can also
`be classified according to the objective functions they are at-
`tempting to optimize. Common choices are the maximiza-
`tion of the overall throughput given a total power constraint,
`known as rate-adaptive loading [10], and the minimization
`of the energy given a fixed throughput, known as margin-
`adaptive loading [9]. Both cases also employ an error rate
`constraint. Although some algorithms may have certain ad-
`vantages over others in terms of how close they come to the
`optimum allocation or how quickly they reach their final
`allocation, in this work, we present a rate-adaptive alloca-
`tion algorithm that tries to balance these two criteria while
`attempting to maximize the throughput over a set of mod-
`ulation schemes given that the mean bit error rate is below
`some prescribed threshold.
`In Section 2, an
`The paper is organized as follows.
`overview of the three current types of bit allocation algo-
`rithms is presented. In Section 3, the proposed algorithm is
`described in detail. In Section 4, the simulation results are
`presented and comparisons between the proposed algorithm
`and four other algorithms are made with respect to through-
`put, mean BER, and execution time. Several concluding
`remarks are made in Section 5.
`
`2 Previous Work
`Before presenting the proposed loading algorithm, sev-
`eral bit allocation algorithms are briefly covered in order to
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`the target bit rate is exceeded. Since the number of bits is
`non-integer, the allocation is rounded to the nearest integer
`value. After the bit allocation, the transmission power levels
`are then adjusted in order to achieve the same subcarrier bit
`error rate, Pi, per non-nulled subcarrier.
`The allocation algorithm presented by Leke and
`Cioffi [10] assigns energy to different subcarriers in order
`to maximize the data rate for a given SNR margin. A sort
`and search is performed in order to find which subcarriers
`should be left on while others shut off. The bits are then al-
`located to each subcarrier using the SNR gap approximation
`of Eq. (1).
`
`2.3 Probability of Error-Based Allocation
`Bits can also be allocated using closed-form expressions
`for the probability of error, given the target probability of
`error and the SNR. For instance, the probability of error
`(cid:2)
`(cid:2)
`(cid:1)
`(cid:1)
`expression for Mi-QAM on subcarrier i is given by [14]
`3γi
`1− 1√
`PMi,i(γi) =4
`Mi − 1
`Q
`(cid:1)
`(cid:1)
`(cid:2)(cid:2)
`(cid:2)
`(cid:1)
`Mi
`3γi
`1− 1√
`·
`1−
`Mi − 1
`Q
`Mi
`(Mi) gives the (non-integer) number of bits to
`where log2
`represent a signal constellation point, and Q(·) is the Q-
`(cid:3) ∞
`function, defined as
`Q(x) = 1√
`−t2/2dt.e
`2π
`x
`Making several simplifying approximations, Mi can be de-
`termined and discretized.
`For instance, Fischer and Huber [11] distribute the bits
`and power across the subcarriers in order to minimize the
`error probability on each subcarrier. Using the union bound
`as an equality for the symbol error probabilities of QAM
`modulation, the algorithm iteratively distributes the bits and
`power until the probability of error on all subcarriers are
`equal. This algorithm is subjected to a total rate and total
`power constraint.
`
`(2)
`
`(3)
`
`
`
`define the current state-of-the-art for each of the three com-
`mon types of allocation algorithms. The values of the sub-
`carrier signal-to-noise ratios (SNR) are available to these
`algorithms, obtained via data-aided channel estimation at
`the receiver during the initialization phase of the system as
`well as during transmission, and sent to the transmitter us-
`ing feedback.
`
`2.1 Incremental Allocation
`Most incremental allocation algorithms are greedy algo-
`rithms, where the algorithm allocates one bit at a time to
`the subcarrier that will do the most good for the current par-
`tial allocation. The algorithm is called greedy since it only
`maximizes the reduction in distortion for each step without
`regard to the global effects of its choice [13].
`One example of a bit and power allocation algorithm for
`multicarrier systems was developed by Hughes-Hartog [5].
`Starting from an all-zero allocation, this algorithm allocates
`an additional bit to the subcarrier requiring the smallest in-
`cremental energy until either the total power or aggregate
`bit error rate constraints are not violated. Another alloca-
`tion algorithm that can be applied to bit allocation is by
`Fox [6], where bits are allocated incrementally to the sub-
`carriers which maximize the ratio of the change in through-
`put to the change in bit error rate (BER). Finally, the bit al-
`location algorithm by Wyglinski, Kabal, and Labeau [7, 8]
`starts off with all the subcarriers allocated with the maxi-
`mum number of bits, and then incrementally removes bits
`from the subcarriers with the worst BER values until the
`mean BER constraint is satisfied.
`Although this type of algorithm may approach the op-
`timal allocation in terms of maximizing the throughput
`given a BER constraint, they can have a high computational
`complexity, requiring numerous complex iterations before
`reaching a final allocation.
`
`2.2 Channel Capacity Approximation-Based Allocation
`A solution to the complexity problem is to perform bit
`allocation based on closed-form expressions of some error
`performance criteria. One approach uses an approximation
`of the Shannon capacity expression to determine the num-
`ber of bits to be allocated per subcarrier.
`For instance, to find the number of bits, bi, for subcarrier
`i, the allocation algorithm of Chow, Cioffi, and Bingham [9]
`(cid:2)
`(cid:1)
`uses the expression
`
`γi
`1 +
`bi = log2
`,
`(1)
`Γ
`where γi is the SNR of subcarrier i, and Γ is the difference in
`SNR values corresponding to the maximum number of bits
`the system can sustain, given a target probability of error
`PT , and the capacity normalized by the signal bandwidth.
`The parameter Γ is also known as the SNR Gap. Assuming
`equal energy across all used subcarriers, Γ is adjusted until
`
`3 Proposed Algorithm
`From the previous section, several allocation algorithms
`have been introduced. However, they are either too com-
`plex, such as the incremental algorithms, or do not approach
`the optimal allocation, such as the capacity approximation
`and probability of bit error-based algorithms. Due to the
`trade-off between the complexity of the bit allocation algo-
`rithm and its effectiveness at maximizing the throughput,
`what is needed is an algorithm which accurately maps the
`subcarrier SNR values to some final bit allocation in a low
`complexity fashion.
`
`3.1 Algorithm Description
`One solution to this problem is to limit the maximum
`BER, ˆP, allowed per subcarrier across all the N subcarri-
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`¯P =
`
`biPi
`
`,
`
`bi
`
`ers. Therefore, the modulation scheme with the largest sig-
`nal constellation for which its BER is below ˆP is chosen
`for each subcarrier. Given the probability of bit error and
`number of bits for subcarrier i, Pi and bi, the mean of the
`subcarrier BER values, ¯P, defined as
`N∑
`i=1
`N∑
`i=1
`is then computed and compared against the BER threshold,
`PT . In this work, closed-form expressions of the probability
`(cid:2)
`(cid:1)(cid:4)
`of bit error are used, namely [14]
`2γi
`P2,i(γi) = Q
`for BPSK and Eq. (2) for QPSK, rectangular 16-QAM, and
`rectangular 64-QAM. Although these closed form expres-
`sions were derived for the AWGN channel case, they can be
`employed on a subcarrier basis since the subbands are nar-
`row enough that the channel is spectrally flat across each
`subband.
`If ¯P is below PT , the target probability of error, ˆP is in-
`creased by an amount δ. On the other hand, if ¯P is above
`PT , ˆP is reduced by an amount δ. This continues until the
`algorithm crosses over PT , in which case the δ is reduced
`and the algorithm converges fast to a final allocation.
`The complete operation of the proposed algorithm is de-
`scribed as follows:
`1. Given γi, i = 1, . . . , N, compute the Pi values for all the
`subcarriers.
`2. Calculate ¯P for the case when all subcarriers employ
`the largest signal constellation.
`3. If the resulting ¯P is below PT , set the final allocation to
`the largest signal constellation for all subcarriers and
`end the algorithm.
`
`(4)
`
`(5)
`
`8. If both current and previous ¯P values are above PT ,
`reduce ˆP by a factor δ and go to Step 5, else increase
`ˆP by a factor δ and go to Step 5.
`9. If the previous and current allocations differ by one
`signal constellation level, make the allocation with ¯P
`below PT the final allocation and end the algorithm,
`else go to Step 10.
`10. Reduce δ.
`11. If the current allocation gives a ¯P that is above PT , re-
`duce ˆP by a factor δ and go to Step 5, else increase ˆP
`by a factor δ and go to Step 5.
`In the case that the previous and current ¯P values straddle
`PT , as in Step 9, the allocations are compared in order to
`see of they differ by one signal constellation. If they do, it
`is obvious that the additional bit(s) is/are the cause of the
`violation of the mean BER constraint. Otherwise, we re-
`duce δ until the case of one differing signal constellation is
`achieved.
`
`3.2 Channel Characterization for Initial Peak Bit
`Error Rate Threshold
`
`The speed at which the algorithm in Section 3.1 reaches
`its final allocation depends on the choice of the initial ˆP and
`δ it uses. Therefore, it is desirable to estimate the initial
`values for ˆP and δ before starting the iterations using the
`available information, i.e. subcarrier SNR values.
`One approach to this problem is to determine how much
`any given subcarrier can individually exceed PT while ¯P re-
`mains below it. Given that a subcarrier can support one of
`five possible modulation schemes, resulting in five possible
`values for Pi, we define the largest Pi value that is below PT
`as βi while the smallest value of Pi above PT as αi. Using
`this leeway, ∆P, which is the difference between PT and the
`sum of βi, it can be determined how many subcarriers can
`have a Pi which exceeds PT on an individual basis while still
`maintaining a ¯P below it.
`The algorithm for finding the peak BER estimate is as
`follows:
`1. Given the subcarrier SNR values, γi, calculate Pi for all
`the different modulation schemes which could poten-
`tially be employed in the system.
`2. Find βi, the largest Pi that does not exceed PT .
`3. Find αi, the smallest Pi that exceeds PT .
`4. Find all values of βi that are within an order of magni-
`tude of its largest βi and assign their indices to a set S
`(βi not within an order of magnitude can be neglected).
`5. Given βi, i ∈ S , we need to determine ∆P to have sev-
`eral subcarriers exceed PT on an individual basis while
`
`The previous step provides for a quick exit from the algo-
`rithm when the subcarrier SNR values are large enough.
`4. Calculate ¯P for the case when the subcarrier with the
`largest γi employs the smallest signal constellation and
`the other subcarriers are nulled.
`5. If the resulting ¯P is above PT , turn off all subcarriers
`and end the algorithm.
`
`The previous step provides a quick exit from the algorithm
`if the subcarrier SNR values are too low.
`
`5. Find the largest signal constellation for subcarrier i for
`which Pi is below ˆP.
`6. Compute the current value of ¯P.
`7. If the current and previous values of ¯P are either both
`above or both below PT , go to Step 8, else go to Step 9.
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`10
`
`20
`
`40
`30
`Subcarrier Index
`
`50
`
`60
`
`−35
`
`−40
`
`−45
`
`−50
`
`−55
`
`−60
`
`Magnitude [dB]
`
`Fig. 1 Frequency response of an indoor channel environment in
`the 5.15–5.25 GHz U-NII band with transmitter/receiver distance
`of 50 m. Note that for the case of 8 subcarriers, only a portion of
`the channel (the boundaries of which are indicated by the dotted
`lines) is used.
`
`4.2 Channel Model
`
`Since one of the target applications of these algorithms is
`WLAN, a channel impulse response that adequately mod-
`elled an indoor environment was required. The statistical
`indoor propagation modelling technique devised by Saleh
`and Valenzuela [15] is used.
`
`In these experiments, WLAN systems, such as IEEE Std.
`802.11a [1] or HiperLAN/2 [2], operate at around 5 GHz,
`such as in the lower portion of the unlicensed national in-
`formation infrastructure (UNII) band at 5.15–5.25 GHz [16]
`for IEEE Std. 802.11a. The transmitter-receiver separation
`was varied between between 1 m and 60 m and the signal,
`which is composed of 52 subcarriers, was transmitted over
`a 16.6 MHz bandwidth. Furthermore, there was no line-of-
`sight and the channel was assumed to be quasi-stationary,
`thus time-invariant during the adaptation phase of the al-
`location algorithm. Finally, only a single pair of antennas
`were employed. An example of a typical channel frequency
`response is shown in Fig. 1.
`It can be observed that the
`channel experiences frequency selective fading, with nulls
`as deep as 20 dB.
`
`For each channel realization, the algorithms were operat-
`ing at 70 different TSNR values ranging from 70 dB to 140
`dB. The TSNR is defined here as the nominal transmitted
`power divided by the noise power in the signal bandwidth.
`When measured this way, the TSNR values tend to be large
`relative to SNR values measured at the receiver due to the
`channel attenuation. For instance, the channel attenuation is
`approximately 80 dB across a 50 m distance at 5 GHz. The
`trials were repeated for 10000 different channel realizations
`and the results averaged. Furthermore, the change in TSNR
`corresponds to the change in transmitter/receiver separation
`distance.
`
`having ¯P below PT . In this case, we have
`biβi + ∆P
`∑
`i∈S
`∑
`i∈S
`
`PT =
`
`bi
`
`(6)
`
`(7)
`
`biβi.
`
`or equivalently
`bi − ∑
`∆P = PT · ∑
`i∈S
`i∈S
`6. Add up the values of αi, from smallest to largest, until
`the sum is greater than ∆P. Once exceeded, the last
`value of αi added to the sum is chosen as the initial ˆP
`for the algorithm described in Section 3.1.
`The initial value of δ is directly proportional to the av-
`erage SNR of the system, ¯γ. As a result, using empirical
`measurements, the values for δ as a function of ¯γ was de-
`termined. Using these values of δ in conjunction with the
`initial ˆP algorithm, the number of iterations required to find
`the final ˆP can be reduced by as much as half when com-
`pared to a scheme using an initialization that is independent
`from the channel.
`
`4 Simulations
`
`4.1 System Configuration
`
`In this work, we refer to the IEEE Std. 802.11a [1],
`a wireless local area network (WLAN) standard employ-
`ing conventional multicarrier modulation, in order to ob-
`tain realistic system parameters, such as the number of
`subcarriers, the frequency band of operation, and available
`modulation schemes. The signal constellations used are
`BPSK, QPSK, rectangular 16-QAM, and rectangular 64-
`QAM. The subcarrier can also be nulled, depending on sub-
`carrier SNR values. Unlike the standard, where the same
`modulation scheme is employed across all subcarriers, the
`allocation algorithms can use a different modulation scheme
`per subcarrier. As in other studies, we consider only un-
`coded systems for the sake of straightforward comparison.
`However, the introduction of coding would improve the
`performance relative to an uncoded system and can be ac-
`counted for by a nonlinear modification of the SNR value,
`in relationship with the coding gain.
`In this work we evaluated the proposed algorithm along-
`side with the algorithms which solely perform bit alloca-
`tion, namely, Fox [6], Wyglinski, Kabal, & Labeau [7, 8],
`and Leke & Cioffi [10], where the multicarrier system em-
`ploys 52 subcarriers (as in IEEE Std. 802.11a) and has PT
`
`−3 and 10values of 10 −5. Furthermore, an exhaustive search
`
`algorithm was also employed with a reduced number of sub-
`carriers over a portion of the band, to keep the complexity
`manageable, in order to determine how close the various
`methods were to the optimal solution.
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`Table 1 Number of PT Violations by the bit allocation algorithm
`of Leke & Cioffi.
`TSNR
`−3
`N = 8, PT = 10
`8.23% 3.53% 0.66% 0.00% 0.00%
`−5 54.95% 96.84% 99.94% 19.62% 1.99%
`N = 52, PT = 10
`
`86 dB
`
`91 dB 106 dB 116 dB
`
`81 dB
`
`4.3 Results and Discussion
`
`In Fig. 2, the overall throughput of the five bit alloca-
`tion algorithms are presented for the case of 8 subcarriers.
`The algorithm of Leke and Cioffi does not reach the same
`throughput as the other algorithms until high TSNR val-
`ues of 130 dB. As for the other methods, the difference in
`throughput between them is small. The largest throughput
`is produced by the exhaustive search algorithm, followed by
`both Fox’s and Wyglinski, Kabal, & Labeau’s algorithms,
`and finally by the proposed algorithm. Since the objective
`function is not concave and the constraint function is not
`strictly convex, there is no guarantee that Fox’s algorithm
`would reach the optimal allocation [6].
`
`The ¯P values corresponding to the throughputs in Fig. 2
`are shown in Fig. 3. It can be observed that all the algo-
`rithms, except for Leke and Cioffi, have approximately the
`same values as the exhaustive search algorithm. The algo-
`rithm by Leke & Cioffi possesses values of ¯P that are sig-
`nificantly lower than the other algorithms at the expense of
`lower throughput. Since the algorithm of Leke and Cioffi
`does not check if the bit allocation exceeds PT , there is a
`possibility that PT may be violated. In such cases, the re-
`sults of that allocation were not considered. Table 1 shows
`the number of violations as a percentage of the total number
`of channel realizations per TSNR value.
`
`The results are similar when 52 subcarriers are employed,
`as shown in Fig. 4. All the algorithms, except for Leke and
`Cioffi, achieve nearly the same throughput with some small
`differences. The throughput of the algorithm of Leke and
`Cioffi is substantially less than that of the other methods,
`only reaching the other algorithms at high TSNR values.
`Note how at low TSNR values, the algorithm of Leke and
`Cioffi goes to zero. This is mostly due to the algorithm pro-
`ducing allocations that exceed PT . Table 1 shows the num-
`ber of violations. The corresponding ¯P values are shown in
`Fig. 5. As in the 8 subcarrier case, except for Leke & Cioffi,
`all the algorithms have approximately the same values.
`
`As seen in Fig. 2, the throughput of the proposed algo-
`rithm is very close to that of the optimal algorithm.
`In
`Fig. 4, the proposed algorithm also has one of the largest
`throughput values. However, as for the execution times, the
`proposed algorithm executes much more quickly relative to
`either Fox or Wyglinski, Kabal, & Labeau. Although Leke
`& Cioffi may execute at the same speed as the proposed al-
`gorithm, the latter achieves far greater throughput. A sum-
`mary of mean and worst-case computation times for a 52
`−5 is shown in Table 2 for
`subcarrier system with a PT of 10
`several TSNR values. For a fair comparison, all algorithms
`were programmed in C and executed on the same worksta-
`tion (Intel Pentium IV 2 GHz processor).
`
`Table 2 Mean (Worst) computation times in milliseconds at dif-
`−5
`ferent TSNR values, 52 subcarriers, PT = 10
`Algorithm
`91 dB
`106 dB
`121 dB
`
`136 dB
`
`1.13 (3.23) 1.48 (5.01) 1.41 (5.00) 1.37 (4.40)
`Fox
`0.94 (2.78) 0.96 (4.98) 0.93 (4.24) 0.90 (4.66)
`Leke & Cioffi
`Wyglinski et al. 1.09 (2.86) 0.91 (4.10) 0.84 (2.09) 0.80 (2.62)
`0.91 (2.96) 0.91 (2.71) 0.86 (3.98) 0.82 (4.54)
`Proposed
`
`5 Conclusion
`An efficient bit allocation scheme was presented which
`achieves a throughput that is close to the optimal solution
`while possessing a relatively small computation time. The
`algorithm uses a peak BER threshold which the subcarriers
`cannot exceed. The mean BER and overall throughput are
`computed and the peak BER is adjusted iteratively until the
`throughput is maximized while guaranteeing that the mean
`bit error rate is less than some specified threshold. The pro-
`posed algorithm is compared with three practical bit alloca-
`tion algorithms as well as an exhaustive search algorithm.
`The results show that the proposed algorithm approaches
`the optimal solution while achieving a low computational
`complexity.
`
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`

`[9] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical dis-
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`[12] L. Goldfeld, V. Lyandres, and D. Wulich, “Minimum BER power
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`
`[13] A. Gersho and R. M. Gray, Vector Quantization and Signal Com-
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`pp. 128–137, Feb. 1987.
`[16] Federal Communications Commission, “Part 15 – Radio frequency
`devices, subpart E – Unlicensed national information infrastructure
`devices, section 403 – Definitions.” Code of Federal Regulations,
`FCC 47CFR15.403, vol. 47, pp. 752-753, Oct. 2000.
`
`Fox
`Leke & Cioffi
`Wyglinski et al.
`Proposed
`
`210
`208
`206
`
`100.2 100.3
`
`350
`
`300
`
`250
`
`200
`
`150
`
`100
`
`50
`
`Overall Throughput (Bits per Symbol)
`
`Exhaustive
`Fox
`Leke & Cioffi
`Wyglinski et al.
`Proposed
`
`38.65
`
`38.6
`100.2
`
`100.22
`
`50
`
`40
`
`30
`
`20
`
`10
`
`Overall Throughput (Bits per Symbol)
`
`0
`70
`
`80
`
`90
`
`110
`100
`TSNR (dB)
`
`120
`
`130
`
`140
`
`0
`70
`
`80
`
`90
`
`110
`100
`TSNR (dB)
`
`120
`
`130
`
`140
`
`Fig. 2 Throughput of systems with 8 subcarriers satisfying
`−3. The range of points plotted corresponds to
`a PT of 10
`transmitter/receiver separation distances of 1 m to 60 m.
`
`Fig. 4 Throughput of systems with 52 subcarriers satisfy-
`−5. The range of points plotted corresponds to
`ing a PT of 10
`transmitter/receiver separation distances of 1 m to 60 m.
`
`Fox
`Leke & Cioffi
`Wyglinski et al.
`Proposed
`
`−5.67
`
`10
`
`10
`
`−5.73
`116.1 116.2 116.3
`
`80
`
`90
`
`110
`100
`TSNR (dB)
`
`120
`
`130
`
`140
`
`−4
`
`10
`
`−6
`
`10
`
`−8
`
`10
`
`−10
`
`10
`
`−12
`
`10
`
`Mean Bit Error Rate
`
`10
`
`−14
`70
`
`Exhaustive
`Fox
`Leke & Cioffi
`Wyglinski et al.
`Proposed
`
`−3.6
`
`10
`
`10
`
`−3.7
`106
`
`106.5
`
`80
`
`90
`
`110
`100
`TSNR (dB)
`
`120
`
`130
`
`140
`
`−2
`
`10
`
`−4
`
`10
`
`−6
`
`10
`
`−8
`
`10
`
`Mean Bit Error Rate
`
`−10
`
`10
`
`10
`
`−12
`70
`
`Fig. 3 Mean BER of systems with 8 subcarriers satisfying
`−3. Except for the curve corresponding to Leke &
`a PT of 10
`Cioffi, all the curves are superimposed.
`
`Fig. 5 Mean BER of systems with 52 subcarriers satisfying
`−5. Except for the curve corresponding to Leke &
`a PT of 10
`Cioffi, all the curves are superimposed.
`
`WCNC 2004 / IEEE Communications Society
`
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`
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