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We study a multiuser detection system for code-division multiple access (CDMA). We show that applying multistage hard-decision parallel interference cancellation (HD-PIC) significantly improves performance compared to the matched filter system. In (multistage) HD-PIC, estimates of the interfering signals are used iteratively to improve knowledge of...
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... general , we cannot obtain a closed-form expression for . However, standard numerical packages allow us to compute for all and . In Fig. 1, is shown for . Observe that the optimal equals for . In probability language this means that, for large , the BEP caused by two or more bit errors in the first stage is negligible compared to the BEP caused by one bit error in the first stage. For , will give the minimal rate, meaning that “typ- ically” a bit error in the second stage (after HD-PIC) is caused by two bit errors in the first stage. This is further illustrated in Table I, where the optimal is given for ...
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Citations
... To improve the resilience to error propagation, one can employ the technique of soft decision as in [32], [36]. However, as is shown in [32], [37], the system performance can be improved effectively using hard decision so long as the initial detection error is moderate. In Section V, it will be shown by simulation that the system performance in terms of SINR and BER is substantially improved by introducing the second stage cancellation. ...
... In particular, the SINR achieved in the first stage of the proposed scheme is substan- tially lower than the SNR value when the path loss of far-end signal is small, while in the second stage cancellation this gap is reduced to about 8 dB. Note than this gap depends on the quantization error and phase noise, i.e., P PN and P QN in (37). Therefore the output SINR of the second stage cancellation can be further increased with higher Tx-Rx isolation, as can be seen in Fig. 6. ...
Full-duplex wireless communication is becoming an important research area because of its potential for increasing spectral efficiency. The challenge of such systems lies in cancelling the self-interference. In this paper, we focus on the design of digital cancellation schemes and use them to supplement RF/analog cancellation techniques. The performance of digital cancellation is limited by the non-ideal characteristics of different subsystems in the transceiver, such as analog/digital converter (ADC), power amplifier (PA), and phase noise. It is first shown that given the pre-cancellation achieved by existing RF/analog techniques, the effects of ADC, phase noise, and sampling jitter are not the bottleneck in the system. Instead, the performance of conventional digital cancellation approaches are mainly limited by nonlinearity of the PA and transmit I/Q imbalance. In addition, the output SINR of the desired signal is limited because the estimation precision of the self-interference channel is affected by the desired signal. To overcome these issues, we propose a two-stage iterative self-interference cancellation scheme based on the output signal of the power amplifier. Analytical and simulation results reveal that the proposed cancellation scheme substantially outperforms existing digital cancellation schemes for full-duplex wireless communication systems.
... Interference Cancellation (PIC) with hard limiters [28]. For decoding COW codes, we apply the Tensor Decoding Algorithm (which is ML) discussed in the previous section. ...
In this paper, we introduce a new class of codes for overloaded synchronous wireless and optical code-division multiple-access (CDMA) systems which increases the number of users for fixed number of chips without introducing any errors. Equivalently, the chip rate can be reduced for a given number of users, which implies bandwidth reduction for downlink wireless systems. An upper bound for the maximum number of users for a given number of chips is derived. Also, lower and upper bounds for the sum channel capacity of a binary overloaded CDMA are derived that can predict the existence of such overloaded codes. We also propose a simplified maximum likelihood method for decoding these types of overloaded codes. Although a high percentage of the overloading factor degrades the system performance in noisy channels, simulation results show that this degradation is not significant. More importantly, for moderate values of
E
b
/N
0
(in the range of 6-10 dB) or higher, the proposed codes perform much better than the binary Welch bound equality sequences.
... IHDIC using the serial and the parallel update scheme with several iterations is considered for DS-CDMA multiuser detection, for example, in23242526. Basic work on parallel interference cancellation in DS-CDMA systems with hard decision functions can also be found in [27, 28] whereas in [29] an information theoretic analysis is conducted. However, already Hopfield proposed to use a so-called sigmoid function instead of hard limiters for continuoustime neural networks in123. ...
The convergence of receivers performing iterative hard decision interference cancellation (IHDIC) is analyzed in a general framework for ASK, PSK, and QAM constellations. We first give an overview of IHDIC algorithms known from the literature applied to linear modulation and DS-CDMA-based transmission systems and show the relation to Hopfield neural network theory. It is proven analytically that IHDIC with serial update scheme always converges to a stable state in the estimated values in course of iterations and that IHDIC with parallel update scheme converges to cycles of length 2. Additionally, we visualize the convergence behavior with the aid of convergence charts. Doing so, we give insight into possible errors occurring in IHDIC which turn out to be caused by locked error situations. The derived results can directly be applied to those iterative soft decision interference cancellation (ISDIC) receivers whose soft decision functions approach hard decision functions in course of the iterations.
... There is one example where we can prove a partial convergence result, and that is when C ij = ±1 with equal probability. Indeed, in this case it is shown in [15, Section IV] that (71) holds for all t ≥ −1. This leads to the following result, which also implies that I k (α) > 0 for α = 1 in the case where Var(C 2 11 ) = 0 (recall also Theorem 2.1): ...
... Thus, this scheme is not linear, as SD-PIC is. In [15, 16], similar results as the above are obtained, and it is shown that the rate for a bit-error for a given user is asymptotic to s 8 s ...
We study sample covariance matrices of the form W = 1 n CC T , where C is a k × n matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of C are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞, or kn → ∞ with kn = o(n/ log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for k large are for the case where the entries of C are ±1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of C. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in mobile communication systems. In this example, k plays the role of the number of users in the system, and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.
... Thus, this scheme is not linear, as SD-PIC is. In [18] and [19], similar results to the above were obtained, and it was shown that the rate for a bit error for a given user is asymptotic to (s/8) s √ 4/k when s is fixed and k → ∞. This result is similar in spirit to the one in Theorem 5.2. ...
We study sample covariance matrices of the form W = (1 / n ) C C T , where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or k n → ∞ with k n = o ( n / log log n ), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C . This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.
... Interference Cancellation (PIC) with hard limiters [28]. For decoding COW codes, we apply the Tensor Decoding Algorithm (which is ML) discussed in the previous section. ...
In this paper we introduce a new class of codes for over-loaded synchronous wireless and optical CDMA systems which increases the number of users for fixed number of chips without introducing any errors. Equivalently, the chip rate can be reduced for a given number of users, which implies bandwidth reduction for downlink wireless systems. An upper bound for the maximum number of users for a given number of chips is derived. Also, lower and upper bounds for the sum channel capacity of a binary over-loaded CDMA are derived that can predict the existence of such over-loaded codes. We also propose a simplified maximum likelihood method for decoding these types of over-loaded codes. Although a high percentage of the over-loading factor degrades the system performance in noisy channels, simulation results show that this degradation is not significant. More importantly, for moderate values of Eb/N0 (in the range of 6-10 dB) or higher, the proposed codes perform much better than the binary Welch bound equality sequences.
... The upper bounds in (15)-(16) hold for every n. Furthermore, when E[e tX1 ] < ∞ for all t with |t| ≤ and some > 0, then the right-hand sides of (15) and (16) See e.g., [20, Theorem 1.1, pages 5-6 and Proposition 1.9, page 13] for this result, and see [6] and [12] for general introductions to large deviation theory. For the proof, we start by showing that I k (α) > 0 for all α = 1 when there exists > 0 such that E[e tC 2 11 ] < ∞ for all t < and when Var(C 2 11 ) > 0. For this, we note that, by the Cauchy-Schwarz inequality and (4), for every x with x 2 = 1, 2 11 ] k < ∞ whenever there exists > 0 such that E[e tC 2 11 ] < ∞ for all t < . ...
... Thus, this scheme is not linear, as SD-PIC is. In [15,16], similar results as the above are obtained, and it is shown that the rate for a bit-error for a given user is asymptotic to s 8 s 4 k when s is fixed and k → ∞. This result is similar in spirit as the one in Theorem 5.2 above. ...
We study sample covariance matrices of the form , where C is a matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of C are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and , or with , in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for k large are for the case where the entries of C are with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of C. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in telecommunications. In this example, k plays the role of the number of users in the system, and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.
... The second case is when we allow s to grow slowly with n, namely, proportionally with log n, and we allow the number of users to grow proportionally to n. All the results stated in this section are consequences of results in [12], [13] for HD-PIC and [7] for SD-PIC. For fixed s, our main result is the following: In words, applying s 0 1 stages of HD-PIC allows for a maximal number of users that is at least s times as large as for the MF system. ...
... For HD-PIC, it is shown in [12] that when < log 20 2 log 2 , then the BEP of the HD-PIC system is exponentially small when s is larger than M log n for M = M () sufficiently large. More specifically, M > C2 log 2 log 20 suffices when C is a fixed large enough constant. ...
... For example, for n = 256 the performance is virtually the same, no matter whether we take s = 4, s = 8 or s = 16 iterations. This phenomenon is proved in [12], where it is used to define the so-called optimal HD-PIC system. In general, for larger values of k, more stages of HD-PIC are necessary for this phenomenon to kick in (see [12] for details). ...
In this correspondence, we study a lightly loaded code-division multiple-access (CDMA) system with and without multistage hard- and soft-decision parallel interference cancelation (HD-PIC and SD-PIC). Throughout this paper we will only consider the situation of a noiseless channel, equal powers and random spreading codes. For the system with no or a fixed number of steps of interference cancelation, we give a lower bound on the maximum number of users such that the probability for the system to have no bit-errors converges to one. Moreover, we investigate when the matched filter system, where parallel interference cancelation is absent, has bit errors with probability converging to one. This implies that the use of HD-PIC and SD-PIC significantly enhances the number of users the system can serve
... Of course, in practice, one is also interested in letting k and n grow large simultaneously. In [11], Theorem II.5, this case was investigated, and one of the results reads as follows: ...
... In [12], also Theorems 2 and 3 are used to show that when k n = n γ log n with γ > 2 s , then the above probability converges to 0. This shows that we can increase the maximal number of users asymptotically by a factor of at least 2 without creating bit-errors when we use at least one stage of HD-PIC. We summarise the above extensions by concluding that the combined results in this paper, in [11] and in [12] shed light on the number of users that the system can maximally allow in the simple MF system, and how this number can be increased by using HD-PIC. ...
... Another interesting issue is the behaviour of the system when the number of users is fixed and s → ∞. In [11], this question is addressed, and it is shown that when we apply sufficiently many stages of HD-PIC, then the exponential rate becomes at least 1 2 log 2 − 1 4 for all k ≥ 2. Related results are shown when k ≤ δn with δ > 0 sufficiently small, again illustrating the relevance of using HD-PIC. From a practical point of view, however, it is not clear what is the relevant limiting regime, s fixed and k → ∞ or rather s → ∞ and then k → ∞. ...
The third-generation (3G) mobile communication system uses a technique called code division multiple access (CDMA), in which multiple users use the same frequency and time domain. The data signals of the users are distinguished using codes. When there are many users, interference deteriorates the quality of the system. For more efficient use of resources, we wish to allow more users to transmit simultaneously, by using algorithms that utilize the structure of the CDMA system more effectively than the simple matched filter (MF) system used in the proposed 3G systems. In this paper, we investigate an advanced algorithm called hard-decision parallel interference cancellation (HD-PIC), in which estimates of the interfering signals are used to improve the quality of the signal of the desired user. We compare HD-PIC with MF in a simple case, where the only two parameters are the number of users and the length of the coding sequences. We focus on the exponential rate for the probability of a bit-error, explain the relevance of this parameter, and investigate how it scales when the number of users grows large. We also review extensions of our results, proved elsewhere, showing that in HD-PIC, more users can transmit without errors than in the MF system.
... Since the PIC procedure involves addition and subtraction, the resulting variables are no longer independent. We know of similar examples ([8], Section II.F and [9], Section 4.2) where the Gaussian approximation leads to false conclusions. Simulation is a powerful method for obtaining more information about the BEP. ...
... For instance, our result immediately shows that for lightly and moderately loaded systems , SD-PIC is superior to HD-PIC in the infinite stage limit. The rate for infinite stage HD- PIC without noise has been derived in [8] and approaches 1 2 log 2 − 1 4 = 0.096 . . . as the number of users increases. ...
... k m=1 C mi converges to a standard normal distribution as k → ∞. For any k and x, the bound (12) is valid for all −1 ≤ tx 2 ≤ 1 ([8], Section IV). We use the above expression to arrive at: ...
We analytically compute a mea-sure of performance of various linear Paral-lel Interference Cancellation (PIC) decod-ing schemes in the infinite stage limit, for moderately loaded CDMA systems with-out AWGN, or with a sufficiently small amount of AWGN. This measure is the ex-ponential rate of the BEP, which does not involve Gaussian approximations. We ob-tain these rates using large deviation the-ory for the eigenvalues of the code corre-lation matrix. We find that the decorre-lator performs best, followed by infinite-stage SD-PIC, which is found to perform better than infinite stage HD-PIC.
