# Bit Error Rate Performance of a MIMO-CDMA System Employing Parity-Bit-Selected Spreading in Frequency Nonselective Rayleigh Fading

Authors: Claude D'Amours; Adel Omar Dahmane
Journal: International Journal of Antennas and Propagation
 (2011)

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## Abstract

We analytically derive the upper bound for the bit error rate (BER) performance of a single user multiple input multiple output code division multiple access (MIMO-CDMA) system employing parity-bit-selected spreading in slowly varying, flat Rayleigh fading. The analysis is done for spatially uncorrelated links. The analysis presented demonstrates that parity-bit-selected spreading provides an asymptotic gain of10log(Nt) dB over conventional MIMO-CDMA when the receiver has perfect channel estimates. This analytical result concurs with previous works where the (BER) is determined by simulation methods and provides insight into why the different techniques provide improvement over conventional MIMO-CDMA systems.


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## Body

## 1. Introduction

The object of much research in wireless communications is to enable users to transmit and receive at high and variable data rates to support the growing number of applications that involve such transfer of data [1]. Code Division Multiple Access (CDMA) systems employ spread spectrum (SS) technology and were developed for second and third generation (2G, 3G) wireless communications. For example, IS-95 and Wideband CDMA (WCDMA) systems are based on direct sequence SS techniques.Multiple access interference (MAI) is present in CDMA systems due to the nonzero cross-correlation between the different users' spreading codes [2]. The MAI that each user's signal creates in all other users' signals results in increased bit error rate (BER). The overall system capacity is determined by the number of simultaneous transmitters that can be supported before the BER increases to an unacceptable level [3]. Much research presented in the literature has concentrated on making systems more power efficient as a means to increase the overall spectral efficiency of the CDMA system [4–6]. Other techniques, such as multiuser detection, have also been considered to increase the capacity of CDMA systems [2, 7, 8].Recent research has shown that combining DS-CDMA systems with Multiple Input Multiple Output (MIMO) techniques can achieve high gains in capacity, reliability and data transmission speed [9–14]. This is achieved by exploiting the spatial diversity made possible by multiple antennas at the transmitter and the receiver, allowing more degrees of freedom when the complex channel gains between different transmit and receive antenna pairs are sufficiently uncorrelated. MIMO-CDMA systems are also more robust to multiple access interference (MAI) than their single input single output (SISO) DS-CDMA counterparts. Currently, MIMO-CDMA is considered for many beyond 3G (B3G) applications [13, 14].In [15], the concept of parity-bit-selected spreading for direct sequence spread spectrum (DS-SS) is introduced. In [16], the parity-bit-selected spreading technique is extended to code division multiple access (CDMA) systems using multiple input multiple output (MIMO) techniques. Also in [16], the parity-bit-selected spreading technique is modified to create the so-called permutation spreading technique. The permutation spreading technique combines spreading and space time coding to produce the effect of transmit diversity without retransmitting data from different antennas.Compared to a conventional MIMO-CDMA system, which assigns a unique spreading waveform from a set of mutually orthogonal waveforms to each antenna, the MIMO-CDMA system employing either parity-bit-selected or permutation spreading provides significant power gains. This is demonstrated in [16] through the use of Monte Carlo simulations.The BER performance of the aforementioned spreading techniques for MIMO-CDMA is also determined for systems encountering multiple access interference (MAI), spatial correlation, and/or channel estimation errors [17, 18]. What is lacking in [16–18] is an analysis of the BER performance of MIMO-CDMA systems, employing these techniques. Such an analysis permits us to better understand why the new spreading techniques provide improvements over conventional MIMO-CDMA systems and this can, in turn, permit us to improve upon the new spreading techniques. In this paper, we provide, for the first time, an analytical expression for the upper bound of the BER for a MIMO-CDMA system employing parity-bit-selected or permutation spreading.


## 2. MIMO-CDMA System

A MIMO-CDMA system hasNt transmit and Nr receive antennas. The serial data, whose bit rate is Rb, is converted into Nt parallel data streams, each with bit rate Rb/Nt. The ith data stream of user m is spread by spreading waveform wmi(t), which is an antipodal signal with chip rate Rc and is selected from a set of mutually orthogonal spreading waveforms Cm={cm1(t),cm2(t),cmN(t)}. In other words, (1)1T∫0Tcmi(t)cmj(t)dt={1,i=j,0,i≠j,
where T=Nt/Rb is the signaling interval. In this paper, we consider the use of short spreading waveforms. A short spreading waveform is one whose duration is equal to one signaling interval. Thus on the interval 0≤t≤T, the ith spreading waveform from the set Cm can be described mathematically by (2)cmi(t)=∑j=1Lcmi(j)p(t-jTc),
where cmi(j)=±1 is the jth bipolar chip of the mth user's ith spreading waveform, Tc=1/Rc is the chip interval, L is the number of chips in the spreading waveform and is given by L=T/Tc = RcNt/Rb, and p(t) is the rectangular chip pulse shape given by (3)p(t)={1,0≤t≤Tc,0,otherwise.Different users are assigned unique sets of spreading waveforms. Therefore,Cm∩Cl=∅ for m≠l.The transmitter, receive antennas and link gains are shown in Figure1. On interval (n-1)T≤t≤nT (signaling interval n), the data to be transmitted is m(n)=[m1(n),m2(n),…,mNt(n)], where mk(n) is either 0 or 1 with equal probability and they are independent of one another. In this paper, we assume that binary phase shift keying (BPSK) modulation is used. Therefore, we define b(n)=2m(n)-1=[b1(n),b2(n),…,bNt(n)], where bk(n) for our baseband model is ±1 with equal probability and E[bk(n)bg(n)]=0, if k≠g, where E[·] is the expectation operator. Each independent identically distributed (iid) bit to be transmitted on the nth signaling interval is assigned to a different transmit antenna, therefore, bk(n) is transmitted on transmit antenna k. The mth user's data bit, transmitted by transmit antenna k on the nth signaling interval, is multiplied by spreading waveform wmk(n)(t). The complex channel gain between transmit antenna k and receive antenna j on the nth signaling interval is αij(n).Figure 1
MIMO-CDMA transmitter and link gains.The receiver for MIMO-CDMA is shown in Figure2. Here, the signal received by each receive antenna is correlated with each spreading waveform, and the contributions from the different antennas are combined according to the spreading technique used. The kth matched filter output on receive antenna j and signaling interval n is ujk(n). The expressions for these decision variables and how they are combined depend on the spreading method used by the transmitter.Figure 2
MIMO-CDMA receiver.


## 3. Parity-Bit-Selected Spreading

Parity-bit-selected spreading for spread spectrum systems is introduced in [1]. The technique therein is modified slightly for MIMO-CDMA. the vector m(n) is input to a parity-bit generator. The parity-bit generator produces a vector p(n)=[p1(n),p2(n),...,pN(n)], where N is the number of spreading codes used and is given by N=2length(p). There are as many parity bit patterns as there are spreading waveforms in each user's set. Each spreading waveform is assigned to one of the parity bit vectors. Therefore, if p(n)=px, then (4)wmi(n)(t)=cmx(t-nT)fori=1,2,…,Nt.For the spreading strategy described by (4), the decision variables of Figure 2 are given by (5)ujk(n)={∑i=1Nt(Ebbi(n)αij(n))+njk(n),k=x,njk(n),k≠x,
where Eb is the received energy per bit.Letu(n)=[u11(n),u12(n),…,u1N(n),u21(n),…,u2N(n),…,uNrN(n)] which is a 1×NrN vector. The xth channel gain matrix, Hx(n) is an Nt×NrN matrix, is defined as(6)Hx(n)=[0Nt,(x-1)h1(n)0Nt,(N-1)h2(n)0Nt,(N-1),h3(n)⋯hNr(n)0Nt,(N-x)],
where 0a,b is an a×b all zero matrix and hi(n)=[α1i(n),α2i(n),…,αNti(n)]T. The channel matrix used on signaling interval n depends on the spreading waveform used during that interval. For example, if the transmitter employs cm1(t-nT), then H1(n) is the appropriate channel gain matrix to use.We can now expressu(n) as (7)u(n)=Ebb(n)Hb(n)+n(n),
where Hb(n) is the channel matrix associated with data vector b(n) and n(n)=[n11(n),n12(n),…,n21(n),…,nNrN(n)] is a 1×NrN noise vector. The elements of n(n) are uncorrelated zero mean complex Gaussian random variables with variance σn2.In [15], a suboptimum detection scheme is presented in which the detector determines which spreading code has been used and then detects the data only by observing the outputs matched to that code and comparing them to the possible messages associated with that code. In this paper, maximum likelihood (ML) detection is considered. The receiver does not try to determine which code has been used. It compares u(n) to all possible receive vectors EbbiHbi(n), where Hbi is the channel matrix associated with transmitted message bi. The receiver selects b̂(n) as the message vector that minimizes the square of the Euclidean distance as shown in: (8)b̂(n)=min⁡bi∈B‖u(n)-EbbiHbi(n)‖2,
where B is the set of all possible transmitted data vectors.We consider the four transmit antenna case of [2]. Table 1 shows the allocation of spreading waveforms to message vectors as well as the corresponding channel gain matrix that is used in (7) and (8).Table 1
The allocation of spreading waveforms in parity bit selected spreading technique forNt=4.
MessagesSpreading waveformChannel gain matrixm(n)usedused in (7) and (8)0000cm1(t-nT)H1(n)11110001cm2(t-nT)H2(n)11100010cm3(t-nT)H3(n)11010011cm4(t-nT)H4(n)11000100cm5(t-nT)H5(n)10110101cm6(t-nT)H6(n)10100110cm7(t-nT)H7(n)10010111cm8(t-nT)H8(n)1000From Table1, we see that the two message vectors m = 0000 and m = 1111 (which correspond to b = −1 −1 −1 −1 and b = 1111) form a subset of all possible binary message vectors of length 4. This subset is spread using spreading waveform cm1(t-nT). The messages associated with other spreading waveforms are simply cosets of the subset {0000,1111}. Therefore, we can state that each coset is assigned a unique spreading waveform.As previously mentioned, ML detection is used in this paper. Therefore, the squared Euclidean distance betweenu(n) and each of the vectors in Table 1 is computed, and the message vector corresponding to the smallest squared distance is selected as the most likely transmitted message. In terms of BER performance, this is equivalent to correlating u(n) to each of the vectors in Table 1 and selecting the one with the highest correlation value. The correlation technique is in fact the least computationally complex method as many terms in the vectors in Table 1 are 0. However, if additional coding is used, the Euclidean distances are useful for calculating log likelihood ratios for the decoder.


## 4. BER Performance of MIMO-CDMA Using Parity-Bit-Selected Spreading

In CDMA systems, bit error rate (BER) performance is a very important parameter. Not only does BER determine the quality of transmission, but also does it determine the amount of data that can be transmitted per unit of bandwidth. As every user contributes to the interference levels at the receiver, the BER of each user increases as more users access the channel. Thus, the maximum number of users is determined by the amount of interference that can be tolerated.We will determine the BER performance of MIMO-CDMA employing parity-bit-selected spreading analytically using the following assumptions.(1)
The MIMO-CDMA system under consideration uses four transmit antennas and eight spreading waveforms as in [2] and as detailed in Table 1.(2)
The fading process is assumed to be frequency nonselective. In other words, the multipath spread is zero and there is no channel induced intersymbol interference (ISI).(3)
The channel gains are independent slowly varying circularly-symmetric complex Gaussian random variables with zero mean and unit variance.(4)
The channel gains are known to the receiver, thus coherent detection is performed.When determining the BER of communication systems operating in Rayleigh fading channel with diversity orderL, typically we are required to integrate a Q-function multiplied by the probability density function (pdf) of the energy per bit to noise spectral density ratio (Eb/No). The pdf of Eb/No is a chi-square distribution with 2L degrees of freedom. The result of this integral takes to form of p(L,x) [19] which is shown in:(9)p(L,x)=(1-μ2)L∑k=0L-1(L-1+kk)(1+μ2)k,
where μ=x/(x+1).


### 4.1. Single Receive Antenna,Nr=1

We start by assuming that there is only one receive antenna. Thus, the received vector is(10)u(n)=[u11(n),u12(n),…,u1N(n)].In the absence of noise, the set of all received vectors form an 8 dimensional constellation. The points of the constellation are given in Table2. The variable Eb is the average energy per bit.Table 2
The received vectors for the parity-bit-selected spreading technique forNt=4 and Nr=1.
MessagesReceived vectorm(n)u(n)0 : 0000[-Eb(α11(n)+α21(n)+α31(n)+α41(n)),0,0,0,0,0,0,0]15 : 1111[Eb(α11(n)+α21(n)+α31(n)+α41(n)),0,0,0,0,0,0,0]1 : 0001[0,-Eb(α11(n)+α21(n)+α31(n)-α41(n)),0,0,0,0,0,0]14 : 1110[0,Eb(α11(n)+α21(n)+α31(n)-α41(n)),0,0,0,0,0,0]2 : 0010[0,0,-Eb(α11(n)+α21(n)-α31(n)+α41(n)),0,0,0,0,0]13 : 1101[0,0,Eb(α11(n)+α21(n)-α31(n)+α41(n)),0,0,0,0,0]3 : 0011[0,0,0,-Eb(α11(n)+α21(n)-α31(n)-α41(n)),0,0,0,0]12 : 1100[0,0,0,Eb(α11(n)+α21(n)-α31(n)-α41(n)),0,0,0,0]4 : 0100[0,0,0,0,-Eb(α11(n)-α21(n)+α31(n)+α41(n)),0,0,0]11 : 1011[0,0,0,0,Eb(α11(n)-α21(n)+α31(n)+α41(n)),0,0,0]5 : 0101[0,0,0,0,0,-Eb(α11(n)-α21(n)+α31(n)-α41(n)),0,0]10 : 1010[0,0,0,0,0,Eb(α11(n)-α21(n)+α31(n)-α41(n)),0,0]6 : 0110[0,0,0,0,0,0,-Eb(α11(n)-α21(n)-α31(n)+α41(n)),0]9 : 1001[0,0,0,0,0,0,Eb(α11(n)-α21(n)-α31(n)+α41(n)),0]7 : 0111[0,0,0,0,0,0,0,-Eb(α11(n)-α21(n)-α31(n)-α41(n))]8 : 1000[0,0,0,0,0,0,0,Eb(α11(n)-α21(n)-α31(n)-α41(n))]For ML detection, the probability of symbol error, as well as the BER, can be determined by the Euclidean distance between the constellation points. Let us assume that the transmitted message is message 0 (0000). The probability that message 15 (1111) is detected, given that message 0 is transmitted, is a function of the distance between their constellation points,d15,0. This distance is given by(11)d15,02=‖u1111-u0000‖2=4Eb|α11(n)+α21(n)+α31(n)+α41(n)|2.We letX=α11(n)+α21(n)+α31(n)+α41(n) and Z=|X|2. Because of assumption (3) above, X is a zero mean complex Gaussian random variable with variance 4. Thus, Z has a chi-square probability density function (pdf) with 2 degrees of freedom [19]. It has a mean of 4. Thus, the pdf of Z is given by(12)fZ(z)=14e-z/4,0<z<∞.The probability of incorrectly detecting message 15 when message 0 is transmitted is(13)P(15∣0)=12erfc⁡(d15,024No)=12erfc⁡(EbZNo),
where No is the single-sided noise spectral density. We see that P(15∣0) is a function of the random variable Z. Thus, E[P(15∣0)]=p15∣0 is found by multiplying P(15∣0) by the pdf of Z and integrating from 0 to ∞. This is shown in [19] to be(14)p15|0=∫0∞18erfc⁡(EbZNo)e-z/4dz=12[1-4γb1+4γb]=p(1,4γb),
where γb=Eb/No is the average energy per bit to noise spectral density ratio.Next we consider the probability that message 1 (0001) is detected when message 0 is transmitted. The distanced1,0 is given by(15)d1,02=‖u0001-u0000‖2=Eb|α11(n)+α21(n)+α31(n)+α41(n)|2+Eb|α11(n)+α21(n)+α31(n)-α41(n)|2.It can be shown that|A+B|2+|A-B|2=2|A|2+2|B|2, thus d1,02 becomes(16)d1,02=2Eb(|α11(n)+α21(n)+α31(n)|2+|α41(n)|2).LetYA=α11(n)+α21(n)+α31(n) and YB=α41(n). Also, let WA=|YA|2 and WB=|YB|2. Finally, let W=WA+WB. Although W has the same mean as Z, it does not have the same higher order statistics, nor does it have the same pdf. The pdfs of WA and WB are(17)fWA(w)=13e-w/3,0<w<∞,fWB(w)=e-w,0<w<∞.
Since the channel gains are i.i.d., WA and WB are independent. Thus the pdf of W=WA+WB is given by(18)fW(w)=fWA(w)*fWB(w)=12e-w/3-12e-w,0<w<∞,
where * denotes convolution. It can be seen that fW(w) is a weighted sum of chi-square distributions with two degrees of freedom. Therefore, the expected probability of detecting message 1 when message 0 is transmitted is [19](19)p1∣0=∫0∞12erfc⁡(EbW2No)12erfc⁡(EbW2No)fW(w)dw=34[1-3γb2+3γb]-14[1-γb2+γb]=32p(1,32γb)-12p(1,12γb).We can show that distances between messages 2, 4, 7, 8, 11, 13, and 14 and message 0 all have the same statistics as the distance between message 1 and message 0. Thereforep1∣0=p2∣0=p4∣0=p7∣0=p8∣0=p11∣0=p13∣0=p14∣0.The distance between message 3 and message 0 is given by(20)d3,02=‖u0011-u0000‖2=Eb|α11(n)+α21(n)+α31(n)+α41(n)|2+Eb|α11(n)+α21(n)-α31(n)-α41(n)|2=2Eb(|α11(n)+α21(n)|2+|α31(n)+α41(n)|2).Here, we letV=VA+VB, where VA=|α11(n)+α21(n)|2 and VB=|α31(n)+α41(n)|2 are independent chi-square distributed random variables. The pdfs of VA and VB are(21)fVA(v)=fVB(v)=12e-v/2,0<v<∞.
The pdf of V is fV(v)=fVA(v)*fVB(v), which is(22)fV(v)=14ve-v/2,0<v<∞,
which is a chi-square distribution with 4 degrees of freedom. The expected probability of detecting message 3 when message 0 is transmitted is [19](23)p3∣0=∫0∞12erfc⁡(γbV2)fV(v)dv=p(2,γb).It can be shown that the distance between messages 5, 6, 9, 10, and 12 and message 0 have the same statistics as the distance between message 3 and message 0. Thus,p3∣0=p5∣0=p6∣0=p9∣0=p10∣0=p12∣0.Using a union bound, we can estimate the probability of detection error when message 0 is transmitted as(24)P(E∣0)<∑i=115pi∣0<p(1,4γb)+8p(1,32γb)-4p(1,12γb)+6p(2,γb).We can show that the probability of detection error is independent of which message was transmitted; therefore,P(E∣0)=P(E). Similarly, the probability of bit error is independent of the transmitted message; therefore, we can use the conditional expressions that we derived previously to determine an upper bound. It is given by(25)Pb<p(1,4γb)+4p(1,32γb)-2p(1,12γb)+3p(2,γb).


### 4.2. Multiple Receive Antennas,Nr>1

In the single user case, the use of multiple receive antennas provides BER improvement through receive diversity. The received vector is a1×8Nr vector. As an example, u0000 is(26)u0000=[u00001∣u00002∣⋯∣u0000Nr],
where u0000i=[-Eb/Nr(α1i(n)+α2i(n)+α3i(n)+α4i(n)),0,…,0].The probability that message 15 is detected when message 0 is sent is given by(27)p15∣0=∫0∞12erfc⁡(γbZ(Nr)Nr)fZ(Nr)(z)dz,
where Z(Nr)=Z1+Z2+⋯+ZNr. Each of the Zi's has a pdf that is given by (12). Assuming no spatial correlation, the Zis are mutually independent; therefore fZ(Nr)(z) is the convolution of (12) with itself Nr-1 times. Thus, as shown in [19], fZ(Nr)(z) is(28)fZ(Nr)(z)=1(Nr-1)!4NrzNr-1e-z/4,0<z<∞,
which is a chi-square distribution with 2Nr degrees of freedom. Therefore, from [19], we can show that p15∣0 is(29)p15∣0=p(Nr,4γbNr).The probability of detecting message 3 when message 0 is transmitted is given by(30)p3|0=∫0∞12erfc⁡(γbV(Nr)Nr)fV(Nr)(v)dv,
where V(Nr)=V1+V2+⋯+VNr, and each Vi has distribution fVi(v) that is given by (22). Therefore, fV(Nr)(v) is the convolution of (22) with itself Nr-1 times. We can show that fV(Nr)(v) is given by(31)fV(Nr)(z)=1(2Nr-1)!22Nrv2Nr-1e-v/2,0<v<∞,
which is a chi-square distribution with 4Nr degrees of freedom. Therefore, p3|0 is(32)p3∣0=p(2Nr,γbNr).Lastly, the probabilityp(1∣0) is given by(33)p1∣0=∫0∞12erfc⁡(γbW(Nr)2Nr)fW(Nr)(w)dw,
where W(Nr)=W1+W2+⋯+WNr. Each Wi has distribution fWi(w) given by (18). To find the pdf of W(Nr) we must convolve (18) with itself Nr-1 times. Therefore:(34)fW(Nr)(w)=14we-w/3+14we-w-34e-w/3+34e-w,0<w<∞
for Nr=2. For Nr=3 it is given by: (35)fW(Nr)(w)=(e-w/3-e-w)(116w2+2716)-916w(e-w/3+e-w),0<w<∞.ForNr=4 the pdf is given by: (36)fW(Nr)(w)=(e-w/3+e-w)(196w3+4532w)-(e-w/3-e-w)×(316w2+13532),0<w<∞.Using these pdfs and the results in [19], we can find p1∣0. For Nr=2,p1∣0 is given by:(37)p1∣0=94p(2,3γb4)+14p(2,γb4)-94p(1,3γb4)+34p(1,γb4).ForNr=3, p1∣0 is given by: (38)p1|0=278p(3,γb2)-18p(3,γb6)-8116p(2,γb2)-916p(2,γb6)+8116p(1,γb2)-2716p(1,γb6).ForNr=4, p1∣0 is given by: (39)p1|0=8116p(4,3γb8)+116p(4,γb8)-818p(3,3γb8)+38p(3,γb8)+40532p(2,3γb8)+40532p(1,3γb8)+4532p(2,γb8)-13532p(1,γb8).
As is the case for Nr=1, the probability of bit error is given by:(40)Pb<p15∣0+4p1∣0+3p3∣0.


### 4.3. BER Performance of MIMO-CDMA Using Parity-Bit-Selected Spreading

The upper bounds on the BER performance of MIMO-CDMA employing parity-bit-selected spreading that were derived in the previous subsections are shown in Figure3. We have also simulated the BER performance using the same assumptions that were made in the derivation of the equations.Figure 3
Comparison of BER upper bound to simulated performance of MIMO-CDMA employing parity-bit-selected spreading forNt=4 and Nr=1,2,3, and 4.We see from Figure3 that the upper bound is tight as Eb/No increases as we would expect. This is due to the probability of the intersection of events (the distance between the received vector and desired vector is greater than two or more incorrect vectors) tends towards zero as Eb/No increases.We also noticed that asEb/No increases, Pb of (25) and (40) tends towards p15∣0 of (14) and (29), respectively. This means that as Eb/No increases, the message being inverted is the most likely detection error. In other words, the most likely error is to detect the other message from the same coset. Let us consider the coset {0000,1111}. If we were to place 1111 in a different coset, for Nr=1, the distance squared d15,02 would decrease to 2Eb|α11+α21+α31+α41|2, thus increasing the probability of this error. If we were to include other messages in this coset, for example, 0011, then the distance squared between 0000 and 0011 would be 4Eb|α31+α41|2 which is less than d15,02, making it the dominant error. Thus, the pairing of a message with its one's complement is the optimum coset pairing with respect to the BER for this scheme.It can be shown that the BER of a conventional single-user MIMO-CDMA system operating in frequency nonselective Rayleigh fading has a probability of bit error given by:(41)Pb=p(Nr,1NrEbNo),
when the fading on the different links is independent and the receiver has perfect knowledge of the channel gains. The conventional system assigns each antenna a unique spreading waveform, and the set of Nt waveforms are mutually orthogonal. From the above discussion, we know that as Eb/No tends towards infinity, the probability of bit error for a single-user MIMO-CDMA system using the parity-bit-selected spreading technique has a bit error rate of(42)Pb=p(Nr,NtNrEbNo).
Therefore, the MIMO-CDMA system employing the parity-bit-selected spreading technique provides an asymptotic gain of 10log(Nt) over the conventional MIMO-CDMA system under the conditions discussed in this paper. As Eb/No increases, the actual gain approaches but is less than 10log(Nt). This gain can be traded off against additional users accessing the common channel.


## 4.1. Single Receive Antenna,Nr=1

We start by assuming that there is only one receive antenna. Thus, the received vector is(10)u(n)=[u11(n),u12(n),…,u1N(n)].In the absence of noise, the set of all received vectors form an 8 dimensional constellation. The points of the constellation are given in Table2. The variable Eb is the average energy per bit.Table 2
The received vectors for the parity-bit-selected spreading technique forNt=4 and Nr=1.
MessagesReceived vectorm(n)u(n)0 : 0000[-Eb(α11(n)+α21(n)+α31(n)+α41(n)),0,0,0,0,0,0,0]15 : 1111[Eb(α11(n)+α21(n)+α31(n)+α41(n)),0,0,0,0,0,0,0]1 : 0001[0,-Eb(α11(n)+α21(n)+α31(n)-α41(n)),0,0,0,0,0,0]14 : 1110[0,Eb(α11(n)+α21(n)+α31(n)-α41(n)),0,0,0,0,0,0]2 : 0010[0,0,-Eb(α11(n)+α21(n)-α31(n)+α41(n)),0,0,0,0,0]13 : 1101[0,0,Eb(α11(n)+α21(n)-α31(n)+α41(n)),0,0,0,0,0]3 : 0011[0,0,0,-Eb(α11(n)+α21(n)-α31(n)-α41(n)),0,0,0,0]12 : 1100[0,0,0,Eb(α11(n)+α21(n)-α31(n)-α41(n)),0,0,0,0]4 : 0100[0,0,0,0,-Eb(α11(n)-α21(n)+α31(n)+α41(n)),0,0,0]11 : 1011[0,0,0,0,Eb(α11(n)-α21(n)+α31(n)+α41(n)),0,0,0]5 : 0101[0,0,0,0,0,-Eb(α11(n)-α21(n)+α31(n)-α41(n)),0,0]10 : 1010[0,0,0,0,0,Eb(α11(n)-α21(n)+α31(n)-α41(n)),0,0]6 : 0110[0,0,0,0,0,0,-Eb(α11(n)-α21(n)-α31(n)+α41(n)),0]9 : 1001[0,0,0,0,0,0,Eb(α11(n)-α21(n)-α31(n)+α41(n)),0]7 : 0111[0,0,0,0,0,0,0,-Eb(α11(n)-α21(n)-α31(n)-α41(n))]8 : 1000[0,0,0,0,0,0,0,Eb(α11(n)-α21(n)-α31(n)-α41(n))]For ML detection, the probability of symbol error, as well as the BER, can be determined by the Euclidean distance between the constellation points. Let us assume that the transmitted message is message 0 (0000). The probability that message 15 (1111) is detected, given that message 0 is transmitted, is a function of the distance between their constellation points,d15,0. This distance is given by(11)d15,02=‖u1111-u0000‖2=4Eb|α11(n)+α21(n)+α31(n)+α41(n)|2.We letX=α11(n)+α21(n)+α31(n)+α41(n) and Z=|X|2. Because of assumption (3) above, X is a zero mean complex Gaussian random variable with variance 4. Thus, Z has a chi-square probability density function (pdf) with 2 degrees of freedom [19]. It has a mean of 4. Thus, the pdf of Z is given by(12)fZ(z)=14e-z/4,0<z<∞.The probability of incorrectly detecting message 15 when message 0 is transmitted is(13)P(15∣0)=12erfc⁡(d15,024No)=12erfc⁡(EbZNo),
where No is the single-sided noise spectral density. We see that P(15∣0) is a function of the random variable Z. Thus, E[P(15∣0)]=p15∣0 is found by multiplying P(15∣0) by the pdf of Z and integrating from 0 to ∞. This is shown in [19] to be(14)p15|0=∫0∞18erfc⁡(EbZNo)e-z/4dz=12[1-4γb1+4γb]=p(1,4γb),
where γb=Eb/No is the average energy per bit to noise spectral density ratio.Next we consider the probability that message 1 (0001) is detected when message 0 is transmitted. The distanced1,0 is given by(15)d1,02=‖u0001-u0000‖2=Eb|α11(n)+α21(n)+α31(n)+α41(n)|2+Eb|α11(n)+α21(n)+α31(n)-α41(n)|2.It can be shown that|A+B|2+|A-B|2=2|A|2+2|B|2, thus d1,02 becomes(16)d1,02=2Eb(|α11(n)+α21(n)+α31(n)|2+|α41(n)|2).LetYA=α11(n)+α21(n)+α31(n) and YB=α41(n). Also, let WA=|YA|2 and WB=|YB|2. Finally, let W=WA+WB. Although W has the same mean as Z, it does not have the same higher order statistics, nor does it have the same pdf. The pdfs of WA and WB are(17)fWA(w)=13e-w/3,0<w<∞,fWB(w)=e-w,0<w<∞.
Since the channel gains are i.i.d., WA and WB are independent. Thus the pdf of W=WA+WB is given by(18)fW(w)=fWA(w)*fWB(w)=12e-w/3-12e-w,0<w<∞,
where * denotes convolution. It can be seen that fW(w) is a weighted sum of chi-square distributions with two degrees of freedom. Therefore, the expected probability of detecting message 1 when message 0 is transmitted is [19](19)p1∣0=∫0∞12erfc⁡(EbW2No)12erfc⁡(EbW2No)fW(w)dw=34[1-3γb2+3γb]-14[1-γb2+γb]=32p(1,32γb)-12p(1,12γb).We can show that distances between messages 2, 4, 7, 8, 11, 13, and 14 and message 0 all have the same statistics as the distance between message 1 and message 0. Thereforep1∣0=p2∣0=p4∣0=p7∣0=p8∣0=p11∣0=p13∣0=p14∣0.The distance between message 3 and message 0 is given by(20)d3,02=‖u0011-u0000‖2=Eb|α11(n)+α21(n)+α31(n)+α41(n)|2+Eb|α11(n)+α21(n)-α31(n)-α41(n)|2=2Eb(|α11(n)+α21(n)|2+|α31(n)+α41(n)|2).Here, we letV=VA+VB, where VA=|α11(n)+α21(n)|2 and VB=|α31(n)+α41(n)|2 are independent chi-square distributed random variables. The pdfs of VA and VB are(21)fVA(v)=fVB(v)=12e-v/2,0<v<∞.
The pdf of V is fV(v)=fVA(v)*fVB(v), which is(22)fV(v)=14ve-v/2,0<v<∞,
which is a chi-square distribution with 4 degrees of freedom. The expected probability of detecting message 3 when message 0 is transmitted is [19](23)p3∣0=∫0∞12erfc⁡(γbV2)fV(v)dv=p(2,γb).It can be shown that the distance between messages 5, 6, 9, 10, and 12 and message 0 have the same statistics as the distance between message 3 and message 0. Thus,p3∣0=p5∣0=p6∣0=p9∣0=p10∣0=p12∣0.Using a union bound, we can estimate the probability of detection error when message 0 is transmitted as(24)P(E∣0)<∑i=115pi∣0<p(1,4γb)+8p(1,32γb)-4p(1,12γb)+6p(2,γb).We can show that the probability of detection error is independent of which message was transmitted; therefore,P(E∣0)=P(E). Similarly, the probability of bit error is independent of the transmitted message; therefore, we can use the conditional expressions that we derived previously to determine an upper bound. It is given by(25)Pb<p(1,4γb)+4p(1,32γb)-2p(1,12γb)+3p(2,γb).


## 4.2. Multiple Receive Antennas,Nr>1

In the single user case, the use of multiple receive antennas provides BER improvement through receive diversity. The received vector is a1×8Nr vector. As an example, u0000 is(26)u0000=[u00001∣u00002∣⋯∣u0000Nr],
where u0000i=[-Eb/Nr(α1i(n)+α2i(n)+α3i(n)+α4i(n)),0,…,0].The probability that message 15 is detected when message 0 is sent is given by(27)p15∣0=∫0∞12erfc⁡(γbZ(Nr)Nr)fZ(Nr)(z)dz,
where Z(Nr)=Z1+Z2+⋯+ZNr. Each of the Zi's has a pdf that is given by (12). Assuming no spatial correlation, the Zis are mutually independent; therefore fZ(Nr)(z) is the convolution of (12) with itself Nr-1 times. Thus, as shown in [19], fZ(Nr)(z) is(28)fZ(Nr)(z)=1(Nr-1)!4NrzNr-1e-z/4,0<z<∞,
which is a chi-square distribution with 2Nr degrees of freedom. Therefore, from [19], we can show that p15∣0 is(29)p15∣0=p(Nr,4γbNr).The probability of detecting message 3 when message 0 is transmitted is given by(30)p3|0=∫0∞12erfc⁡(γbV(Nr)Nr)fV(Nr)(v)dv,
where V(Nr)=V1+V2+⋯+VNr, and each Vi has distribution fVi(v) that is given by (22). Therefore, fV(Nr)(v) is the convolution of (22) with itself Nr-1 times. We can show that fV(Nr)(v) is given by(31)fV(Nr)(z)=1(2Nr-1)!22Nrv2Nr-1e-v/2,0<v<∞,
which is a chi-square distribution with 4Nr degrees of freedom. Therefore, p3|0 is(32)p3∣0=p(2Nr,γbNr).Lastly, the probabilityp(1∣0) is given by(33)p1∣0=∫0∞12erfc⁡(γbW(Nr)2Nr)fW(Nr)(w)dw,
where W(Nr)=W1+W2+⋯+WNr. Each Wi has distribution fWi(w) given by (18). To find the pdf of W(Nr) we must convolve (18) with itself Nr-1 times. Therefore:(34)fW(Nr)(w)=14we-w/3+14we-w-34e-w/3+34e-w,0<w<∞
for Nr=2. For Nr=3 it is given by: (35)fW(Nr)(w)=(e-w/3-e-w)(116w2+2716)-916w(e-w/3+e-w),0<w<∞.ForNr=4 the pdf is given by: (36)fW(Nr)(w)=(e-w/3+e-w)(196w3+4532w)-(e-w/3-e-w)×(316w2+13532),0<w<∞.Using these pdfs and the results in [19], we can find p1∣0. For Nr=2,p1∣0 is given by:(37)p1∣0=94p(2,3γb4)+14p(2,γb4)-94p(1,3γb4)+34p(1,γb4).ForNr=3, p1∣0 is given by: (38)p1|0=278p(3,γb2)-18p(3,γb6)-8116p(2,γb2)-916p(2,γb6)+8116p(1,γb2)-2716p(1,γb6).ForNr=4, p1∣0 is given by: (39)p1|0=8116p(4,3γb8)+116p(4,γb8)-818p(3,3γb8)+38p(3,γb8)+40532p(2,3γb8)+40532p(1,3γb8)+4532p(2,γb8)-13532p(1,γb8).
As is the case for Nr=1, the probability of bit error is given by:(40)Pb<p15∣0+4p1∣0+3p3∣0.


## 4.3. BER Performance of MIMO-CDMA Using Parity-Bit-Selected Spreading

The upper bounds on the BER performance of MIMO-CDMA employing parity-bit-selected spreading that were derived in the previous subsections are shown in Figure3. We have also simulated the BER performance using the same assumptions that were made in the derivation of the equations.Figure 3
Comparison of BER upper bound to simulated performance of MIMO-CDMA employing parity-bit-selected spreading forNt=4 and Nr=1,2,3, and 4.We see from Figure3 that the upper bound is tight as Eb/No increases as we would expect. This is due to the probability of the intersection of events (the distance between the received vector and desired vector is greater than two or more incorrect vectors) tends towards zero as Eb/No increases.We also noticed that asEb/No increases, Pb of (25) and (40) tends towards p15∣0 of (14) and (29), respectively. This means that as Eb/No increases, the message being inverted is the most likely detection error. In other words, the most likely error is to detect the other message from the same coset. Let us consider the coset {0000,1111}. If we were to place 1111 in a different coset, for Nr=1, the distance squared d15,02 would decrease to 2Eb|α11+α21+α31+α41|2, thus increasing the probability of this error. If we were to include other messages in this coset, for example, 0011, then the distance squared between 0000 and 0011 would be 4Eb|α31+α41|2 which is less than d15,02, making it the dominant error. Thus, the pairing of a message with its one's complement is the optimum coset pairing with respect to the BER for this scheme.It can be shown that the BER of a conventional single-user MIMO-CDMA system operating in frequency nonselective Rayleigh fading has a probability of bit error given by:(41)Pb=p(Nr,1NrEbNo),
when the fading on the different links is independent and the receiver has perfect knowledge of the channel gains. The conventional system assigns each antenna a unique spreading waveform, and the set of Nt waveforms are mutually orthogonal. From the above discussion, we know that as Eb/No tends towards infinity, the probability of bit error for a single-user MIMO-CDMA system using the parity-bit-selected spreading technique has a bit error rate of(42)Pb=p(Nr,NtNrEbNo).
Therefore, the MIMO-CDMA system employing the parity-bit-selected spreading technique provides an asymptotic gain of 10log(Nt) over the conventional MIMO-CDMA system under the conditions discussed in this paper. As Eb/No increases, the actual gain approaches but is less than 10log(Nt). This gain can be traded off against additional users accessing the common channel.


## 5. Conclusion

In this paper, we derived an analytical expression for the BER of a single user MIMO-CDMA system employing the parity-bit-selected spreading technique in frequency nonselective Rayleigh fading under the assumptions of independently fading links and known channel gains. The results obtained from the analytical expression were compared against the simulated BER performance of the same system. The comparison shows that the derived upper bound is very tight to the simulated performance for moderate to high levels of signal to noise ratio. The expression also allows us to determine that for MIMO-CDMA employing parity-bit-selected spreading, the most likely error results in an inversion of the message. This then further allowed us to determine that parity-bit-selected spreading provides an asymptotic gain of10log(Nt) compared to conventional MIMO-CDMA. This gain can be traded off against additional users, thus increasing the spectral efficiency of MIMO-CDMA systems at an expense of increased transmitter and receiver complexity.



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*Source: 516929-2011-08-29.xml*