Maximal-ratio combining detection in massive multiple-input multiple-output systems with accurate probability distribution function

This letter derives the probability distribution function (PDF) of received symbols for orthogonal frequency-division multiplexing (OFDM)basedmassivemultiple-input,multiple-output(MMIMO)sys-tems,whichusesmaximal-ratiocombining(MRC)detection.Theef-fectsofnoiseandinterferencesareevaluatedthroughrandomvariables,andthePDFisthenderivedfromtheirjointprobabilityandcharacter-isticfunctions.Thebiterrorrate(BER)usingBinaryandQuadraturePhaseShiftKeying(BPSK,QPSK),and M -ary Quadrature Amplitude Modulation(QAM)waveformsisthenanalyzedbyusingthisPDF.Sim-ulationandanalyticalresultsconﬁrmthatthederivedequationspro-videdaccuratePDFandBER,andtherefore,canbeusedefﬁcientlytoevaluateperformanceofOFDM-MMIMOsystems.

where N t and N r represent the number of transmit and receive antennas, respectively. H n ∈ C Nr×Nt is channel frequency response matrix with components, H i, j , that are complex-valued, zero-mean, Gaussian random variables, that is, CN (0, 2σ 2 h ). Their variance per dimension σ 2 h is 0.5. W n ∈ C Nr×1 represents complex-valued, zero-mean, additive white Gaussian noise (AWGN), that is, CN (0, 2σ 2 w ), where the variance σ 2 w is defined as where E s is the average symbol energy and M is the number of signal constellation points. γ b = E b /N 0 represents the signal-to-noise ratio per bit. X n ∈ C Nt ×1 represents the transmitted symbols that are uniformly distributed and modulated. Let us define the index λ = I, Q and let X I n and X Q n be the in-phase and quadrature components of X n . The PDF of these variables for M-QAM waveform can be expressed as where is the number of symbols per dimension in the signal constellation, andX λ v is the constellation point per dimension, that is, {−1, 1} for BPSK and QPSK symbols.
Classical MRC detection: In order to detect the transmitted symbols X n , the MRC detector multiplies the received symbols with a weight matrix H † n , to obtain an estimateX n as Substituting Y n from (1) in (4), this equation can be rewritten as where G n = H † n H n is the Gram matrix with components denoted as G i, j . According to (5), the i th component inX contains information for the i th user and can be expressed as Before detection, we need to normalize the estimated symbols due to the multiplication with the Gram matrix, thus, the symbols for the i th user is then evaluated fromX i asX Finally, by substitutingX i in (6), the received symbols for MRC detection can be expressed aŝ Definition of random variables: Undoubtedly, the second and third term in (8) represent the effect of co-channel interferences and enhanced noise in the received symbols for MRC detection. We proceed now with the derivation of the PDF of the received symbol for MRC detection, where utilize M-QAM modulation. A number of random variables are defined to simplify the explanation. According to the MMIMO channel model in (1), the i th component of Y n can be expressed as X k is information which is transmitted by k th user. Let α i be Nt j=1, j =k H i, j X j , (9) is rewritten as Let β i be α i + W i , this equation is become According to the definition of Y i in (11),X i in (8) can be rewritten aŝ Let Z i in (13) represents the effect of co-channel interference and noise in the received symbols. Since α i , β i , η i , and ξ i are from functions of independent random variables, the PDF of Z i can be then determined from the joint probabilities and characteristic functions of these variables.
The PDF of interferences α i : For M-QAM symbols, the in-phase and quadrature components of α i can be expressed as If the PDFs of H λ i, j and X λ j are substituted in (15), the result from this integral operation is The characteristic function of U λ i is then evaluated from p u (U λ i ), and the result is given as If u (ω) in (17) is substituted in (18), this equation becomes where k 1 , k 2 , . . . k is an integer which varies from 0 to (2N t − 2). The PDF of α λ i for M-QAM symbols can be then evaluated from α (ω), i.e.
The PDF of noise and interference β i : β λ i is sum of α λ i and w λ i , and the characteristic function of this random variable is determined by Therefore, the PDF of β λ i can be determined by w λ i is N (0, σ 2 w ), and w (ω) = exp(−σ 2 w ω 2 /2). If we substitute α (ω) and w (ω) in (22), p β (β λ i ) becomes Evidently, (23) is a sum of zero-mean Gaussian functions, which are multiplied by constant coefficient. Their variance σ 2 β,{k} depends on k 1 , k 2 , . . . k , and is determined by Deriving PDF of η i : η λ i is a summation of (2N r ) products of H λ i, j and β λ i . Generally, the PDF of summation of random variables is evaluated through convolution of their PDFs, and the PDF of product of random variables is determined through integral operation. However, p β (β λ i ) in (23) is a sum of scaled Gaussian functions and, p h (H λ i, j ) is a Gaussian distributed too. Therefore, we use Equation (6.9) in [5] to simplify the evaluation of the PDF of sum of product of two zero-means Gaussian random variables. Coefficients in (23) are evaluated separately. If the PDF of H λ i, j and β λ i are substituted in (6.9) of [5], p η (η λ i ) becomes The PDF of received symbols Z i : The PDF of ξ i was previously derived as [2] p Since Z λ i is the ratio of η λ i and ξ i , its PDF is evaluated from their joint probability as [4] If the PDFs of η λ i and ξ i in (25) and (26) are substituted in (27), the result from this integration is Equation (28) represents the PDF of Z λ i for M-QAM waveform. It is worth to point out that the PDF of Z λ i for BPSK and QPSK waveform can be derived from the PDF of α λ i , β λ i , and η λ i using the same approach as for M-QAM. Since = 2, the PDF of Z λ i for BPSK and QPSK waveforms is given as The parameter σ 2 β for BPSK and QPSK waveform is defined as  compared using a 2-sample Kolmogorov-Smirnov test, which has significance level of 5%, thus, confirming that there is no significant difference between analytical and empirical results.
BER analysis: BER of OFDM-MMIMO systems can be directly analyzed through p z (Z λ i ). For BPSK and QPSK and 16-QAM, there are only two constellation points per dimension, and two error events in each dimension. Since the occurrence of these constellation points is uniformly distributed, their symbol error rate P s can be evaluated by considering only one error event as However, solving this equation with p z (Z λ i ) directly is complex. In order to solve this issue, the definition of p z (Z λ i ) in (27) is substituted into (31) as [2] If p η (η λ i ) for QPSK and p ξ (ξ i ) in (26) are substituted in (32), we obtain If the system uses Gray-coded mapping to produce BPSK or QPSK symbols, their BER P e is as same as P s in (33). Similarly, 16-QAM has four constellation points per dimension, that is, {−3, −1, 1, 3}, and six error events. The probability of each error event is also determined by substituting p η (η λ i ) and p ξ (ξ i ) in (25) and (26) in (32) to obtain The BER for 16-QAM waveform is then evaluated as (36) Figure 2 compares simulation and analytical results of BER for the MMIMO systems using QPSK and 16-QAM, where N t = 10, while N r = {64, 128, 256}. The block size utilized was 1024 bits. Evidently, both results are closely matched. The small deviation is due to the fact that for the systems with small N r , the PDF of Z λ i is derived from the joint probability of η λ i and ξ i , by assuming the variables to be independent random variables, which is only fulfilled if N r is large enough.
In Figure 3, the analytical results from (36) are now compared to that of the BER analysis in [3]. The block size utilized and N r were as described in the previous results with N t = 5. The results confirm that the BER from the proposed analysis were closer to the simulation results than that of [3]. It is worth noting that for low E b /N 0 , the P s in (32) Conclusion: This letter derived the PDF of uncoded OFDM-based MMIMO systems, which uses MRC detection, as well as, the BER for BPSK, QPSK and 16-QAM modulations. Theoretical results using the derived BER were shown to closely match the simulation results when N r is large.