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

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 of 10log(𝑁_𝑡) 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.

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 has 𝑁_𝑡 transmit and 𝑁_𝑟 receive antennas. The serial data, whose bit rate is 𝑅_𝑏, is converted into 𝑁_𝑡 parallel data streams, each with bit rate 𝑅_𝑏/𝑁_𝑡. The 𝑖th data stream of user 𝑚 is spread by spreading waveform 𝑤_𝑚𝑖(𝑡), which is an antipodal signal with chip rate 𝑅_𝑐 and is selected from a set of mutually orthogonal spreading waveforms 𝐶_𝑚={𝑐_𝑚1(𝑡),𝑐_𝑚2(𝑡),𝑐_𝑚𝑁(𝑡)}. In other words, 1𝑇^𝑇₀𝑐_𝑚𝑖(𝑡)𝑐_𝑚𝑗(𝑡)𝑑𝑡=1,𝑖=𝑗,0,𝑖≠𝑗, (1) where 𝑇=𝑁_𝑡/𝑅_𝑏 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 𝐶_𝑚 can be described mathematically by 𝑐_𝑚𝑖(𝑡)=^𝐿_𝑗=1𝑐^(𝑗)_𝑚𝑖𝑝𝑡−𝑗𝑇_𝑐, (2) where 𝑐^(𝑗)_𝑚𝑖=±1 is the 𝑗th bipolar chip of the 𝑚th user's 𝑖th spreading waveform, 𝑇_𝑐=1/𝑅_𝑐 is the chip interval, 𝐿 is the number of chips in the spreading waveform and is given by 𝐿=𝑇/𝑇_𝑐 = 𝑅_𝑐𝑁_𝑡/𝑅_𝑏, and 𝑝(𝑡) is the rectangular chip pulse shape given by 𝑝(𝑡)=1,0≤𝑡≤𝑇_𝑐,0,otherwise. (3)

Different users are assigned unique sets of spreading waveforms. Therefore, 𝐶_𝑚∩𝐶_𝑙=∅ for 𝑚≠𝑙.

The transmitter, receive antennas and link gains are shown in Figure 1. On interval (𝑛−1)𝑇≤𝑡≤𝑛𝑇 (signaling interval 𝑛), the data to be transmitted is 𝐦^(𝑛)=[𝑚₁^(𝑛),𝑚₂^(𝑛),…,𝑚_𝑁^(𝑛)_𝑡], where 𝑚_𝑘^(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),…,𝑏_𝑁^(𝑛)_𝑡], where 𝑏_𝑘^(𝑛) for our baseband model is ±1 with equal probability and 𝐸[𝑏_𝑘^(𝑛)𝑏_𝑔^(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, 𝑏_𝑘^(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 𝑤^(𝑛)_𝑚𝑘(𝑡). The complex channel gain between transmit antenna 𝑘 and receive antenna 𝑗 on the 𝑛th signaling interval is 𝛼^(𝑛)_𝑖𝑗.

Figure 1 

MIMO-CDMA transmitter and link gains.

The receiver for MIMO-CDMA is shown in Figure 2. 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 𝑢^(𝑛)_𝑗𝑘. 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),...,𝑝_𝑁^(𝑛)], where 𝑁 is the number of spreading codes used and is given by 𝑁=2^length(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)=𝐩_𝑥, then 𝑤^(𝑛)_𝑚𝑖(𝑡)=𝑐_𝑚𝑥(𝑡−𝑛𝑇)for𝑖=1,2,…,𝑁_𝑡. (4)

For the spreading strategy described by (4), the decision variables of Figure 2 are given by 𝑢^(𝑛)_𝑗𝑘=^𝑁𝑡_𝑖=1𝐸_𝑏𝑏_𝑖^(𝑛)𝛼^(𝑛)_𝑖𝑗+𝑛^(𝑛)_𝑗𝑘𝑛,𝑘=𝑥,_{(𝑛)𝑗𝑘},𝑘≠𝑥, (5) where 𝐸_𝑏 is the received energy per bit.

Let 𝐮^(𝑛)=[𝑢^(𝑛)₁₁,𝑢^(𝑛)₁₂,…,𝑢^(𝑛)_1𝑁,𝑢^(𝑛)₂₁,…,𝑢^(𝑛)_2𝑁,…,𝑢_𝑁^(𝑛)_𝑟𝑁] which is a 1×𝑁_𝑟𝑁 vector. The 𝑥th channel gain matrix, 𝐇_𝑥^(𝑛) is an 𝑁_𝑡×𝑁_𝑟𝑁 matrix, is defined as𝐇_𝑥^(𝑛)=𝟎_{𝑁𝑡,(𝑥−1)}𝐡₁^(𝑛)𝟎_{𝑁𝑡,(𝑁−1)}𝐡₂^(𝑛)𝟎_{𝑁𝑡,(𝑁−1)},𝐡₃^(𝑛)⋯𝐡_𝑁^(𝑛)_𝑟𝟎_{𝑁𝑡,(𝑁−𝑥)}, (6) where 𝟎_𝑎,𝑏 is an 𝑎×𝑏 all zero matrix and 𝐡_𝑖^(𝑛)=[𝛼^(𝑛)_1𝑖,𝛼^(𝑛)_2𝑖,…,𝛼_𝑁^(𝑛)_𝑡𝑖]^𝑇. The channel matrix used on signaling interval 𝑛 depends on the spreading waveform used during that interval. For example, if the transmitter employs 𝑐_𝑚1(𝑡−𝑛𝑇), then 𝐇₁^(𝑛) is the appropriate channel gain matrix to use.

We can now express 𝐮^(𝑛) as 𝐮^(𝑛)=𝐸_𝑏𝐛^(𝑛)𝐇_𝑏^(𝑛)+𝐧^(𝑛), (7) where 𝐇_𝑏^(𝑛) is the channel matrix associated with data vector 𝐛^(𝑛) and 𝐧^(𝑛)=[𝑛^(𝑛)₁₁,𝑛^(𝑛)₁₂,…,𝑛^(𝑛)₂₁,…,𝑛_𝑁^(𝑛)_𝑟𝑁] is a 1×𝑁_𝑟𝑁 noise vector. The elements of 𝐧^(𝑛) are uncorrelated zero mean complex Gaussian random variables with variance 𝜎²_𝑛.

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 𝐸_𝑏𝐛_𝑖𝐇_𝑏^(𝑛)_𝑖, where 𝐇_𝑏𝑖 is the channel matrix associated with transmitted message 𝐛_𝑖. The receiver selects 𝐛^(𝑛) as the message vector that minimizes the square of the Euclidean distance as shown in: 𝐛^(𝑛)=min_{𝐛𝑖∈𝐵}𝑢^(𝑛)−𝐸_𝑏𝐛_𝑖𝐇_𝑏^(𝑛)_𝑖², (8) 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 for 𝑁_𝑡=4.

From Table 1, 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 𝑐_𝑚1(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 between 𝐮^(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 order 𝐿, 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 (𝐸_𝑏/𝑁_𝑜). The pdf of 𝐸_𝑏/𝑁_𝑜 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:𝑝(𝐿,𝑥)=1−𝜇2^𝐿𝐿−1_𝑘=0𝑘𝐿−1+𝑘1+𝜇2^𝑘, (9) where 𝜇=𝑥/(𝑥+1).

4.1. Single Receive Antenna, 𝑁_𝑟=1

We start by assuming that there is only one receive antenna. Thus, the received vector is 𝐮^(𝑛)=𝑢^(𝑛)₁₁,𝑢^(𝑛)₁₂,…,𝑢^(𝑛)_1𝑁. (10)

In the absence of noise, the set of all received vectors form an 8 dimensional constellation. The points of the constellation are given in Table 2. The variable 𝐸_𝑏 is the average energy per bit.

Table 2 

The received vectors for the parity-bit-selected spreading technique for 𝑁_𝑡=4 and 𝑁_𝑟=1.

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, 𝑑_15,0. This distance is given by𝑑²_15,0=𝐮₁₁₁₁−𝐮₀₀₀₀²=4𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁². (11)

We let 𝑋=𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁ 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 by𝑓_𝑍(𝑧)=14𝑒^−𝑧/4,0<𝑧<∞. (12)

The probability of incorrectly detecting message 15 when message 0 is transmitted is𝑃()=115∣02erfc𝑑²_15,04𝑁_𝑜=12erfc𝐸_𝑏𝑍𝑁_𝑜, (13) where 𝑁_𝑜 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)]=𝑝_15∣0 is found by multiplying 𝑃(15∣0) by the pdf of 𝑍 and integrating from 0 to ∞. This is shown in [19] to be𝑝_15|0=^∞₀18erfc𝐸_𝑏𝑍𝑁_𝑜𝑒^−𝑧/4=1𝑑𝑧21−4𝛾_𝑏1+4𝛾_𝑏=𝑝1,4𝛾_𝑏, (14) where 𝛾_𝑏=𝐸_𝑏/𝑁_𝑜 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 distance 𝑑_1,0 is given by𝑑²_1,0=𝐮₀₀₀₁−𝐮₀₀₀₀²=𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁²+𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁−𝛼^(𝑛)₄₁². (15)

It can be shown that |𝐴+𝐵|²+|𝐴−𝐵|²=2|𝐴|²+2|𝐵|², thus 𝑑²_1,0 becomes𝑑²_1,0=2𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁²+𝛼^(𝑛)₄₁². (16)

Let 𝑌_𝐴=𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁ and 𝑌_𝐵=𝛼^(𝑛)₄₁. Also, let 𝑊_𝐴=|𝑌_𝐴|² and 𝑊_𝐵=|𝑌_𝐵|². Finally, let 𝑊=𝑊_𝐴+𝑊_𝐵. 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 𝑊_𝐴 and 𝑊_𝐵 are𝑓_𝑊𝐴(𝑤)=13𝑒^−𝑤/3𝑓,0<𝑤<∞,_𝑊𝐵(𝑤)=𝑒^−𝑤,0<𝑤<∞. (17) Since the channel gains are i.i.d., 𝑊_𝐴 and 𝑊_𝐵 are independent. Thus the pdf of 𝑊=𝑊_𝐴+𝑊_𝐵 is given by𝑓_𝑊(𝑤)=𝑓_𝑊𝐴(𝑤)∗𝑓_𝑊𝐵(𝑤)=12𝑒^−𝑤/3−12𝑒^−𝑤,0<𝑤<∞, (18) where ∗ denotes convolution. It can be seen that 𝑓_𝑊(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]𝑝_1∣0=^∞₀12erfc𝐸_𝑏𝑊2𝑁_𝑜𝑓_𝑊(𝑤)=3𝑑𝑤41−3𝛾_𝑏2+3𝛾_𝑏−141−𝛾_𝑏2+𝛾_𝑏=32𝑝31,2𝛾_𝑏−12𝑝11,2𝛾_𝑏. (19)

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. Therefore 𝑝_1∣0=𝑝_2∣0=𝑝_4∣0=𝑝_7∣0=𝑝_8∣0=𝑝_11∣0=𝑝_13∣0=𝑝_14∣0.

The distance between message 3 and message 0 is given by𝑑²_3,0=𝐮₀₀₁₁−𝐮₀₀₀₀²=𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁+𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁²+𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁−𝛼^(𝑛)₃₁−𝛼^(𝑛)₄₁²=2𝐸_𝑏𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁²+𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁². (20)

Here, we let 𝑉=𝑉_𝐴+𝑉_𝐵, where 𝑉_𝐴=|𝛼^(𝑛)₁₁+𝛼^(𝑛)₂₁|² and 𝑉_𝐵=|𝛼^(𝑛)₃₁+𝛼^(𝑛)₄₁|² are independent chi-square distributed random variables. The pdfs of 𝑉_𝐴 and 𝑉_𝐵 are𝑓_𝑉𝐴(𝑣)=𝑓_𝑉𝐵(𝑣)=12𝑒^−𝑣/2,0<𝑣<∞. (21) The pdf of 𝑉 is 𝑓_𝑉(𝑣)=𝑓_𝑉𝐴(𝑣)∗𝑓_𝑉𝐵(𝑣), which is𝑓_𝑉(𝑣)=14𝑣𝑒^−𝑣/2,0<𝑣<∞, (22) 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]𝑝_3∣0=^∞₀12erfc𝛾_𝑏𝑉2𝑓_𝑉(𝑣)𝑑𝑣=𝑝2,𝛾_𝑏. (23)

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, 𝑝_3∣0=𝑝_5∣0=𝑝_6∣0=𝑝_9∣0=𝑝_10∣0=𝑝_12∣0.

Using a union bound, we can estimate the probability of detection error when message 0 is transmitted as𝑃(𝐸∣0)<¹⁵_𝑖=1𝑝_𝑖∣0<𝑝1,4𝛾_𝑏3+8𝑝1,2𝛾_𝑏1−4𝑝1,2𝛾_𝑏+6𝑝2,𝛾_𝑏. (24)

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 by𝑃_𝑏<𝑝1,4𝛾_𝑏3+4𝑝1,2𝛾_𝑏1−2𝑝1,2𝛾_𝑏+3𝑝2,𝛾_𝑏. (25)

4.2. Multiple Receive Antennas, 𝑁_𝑟>1

In the single user case, the use of multiple receive antennas provides BER improvement through receive diversity. The received vector is a 1×8𝑁_𝑟 vector. As an example, 𝐮₀₀₀₀ is𝐮₀₀₀₀=𝐮₀₀₀₀₁∣𝐮₀₀₀₀₂∣⋯∣𝐮_0000𝑁𝑟, (26) where 𝐮_0000𝑖=[−𝐸_𝑏/𝑁_𝑟(𝛼^(𝑛)_1𝑖+𝛼^(𝑛)_2𝑖+𝛼^(𝑛)_3𝑖+𝛼^(𝑛)_4𝑖),0,…,0].

The probability that message 15 is detected when message 0 is sent is given by𝑝_15∣0=^∞₀12erfc𝛾_𝑏𝑍^(𝑁𝑟)𝑁_𝑟𝑓_{𝑍𝑟)(𝑁}(𝑧)𝑑𝑧, (27) where 𝑍^(𝑁𝑟)=𝑍₁+𝑍₂+⋯+𝑍_𝑁𝑟. Each of the 𝑍_𝑖's has a pdf that is given by (12). Assuming no spatial correlation, the 𝑍_𝑖s are mutually independent; therefore 𝑓_{𝑍𝑟)(𝑁}(𝑧) is the convolution of (12) with itself 𝑁_𝑟−1 times. Thus, as shown in [19], 𝑓^(𝑁𝑟)_𝑍(𝑧) is𝑓_{𝑍𝑟)(𝑁}(𝑧)=1𝑁_𝑟−1!4^𝑁𝑟𝑧^𝑁𝑟−1𝑒^−𝑧/4,0<𝑧<∞, (28) which is a chi-square distribution with 2𝑁_𝑟 degrees of freedom. Therefore, from [19], we can show that 𝑝_15∣0 is𝑝_15∣0=𝑝𝑁_𝑟,4𝛾_𝑏𝑁_𝑟. (29)

The probability of detecting message 3 when message 0 is transmitted is given by 𝑝_3|0=^∞₀12erfc𝛾_𝑏𝑉^(𝑁𝑟)𝑁_𝑟𝑓_{𝑉𝑟)(𝑁}(𝑣)𝑑𝑣, (30) where 𝑉^(𝑁𝑟)=𝑉₁+𝑉₂+⋯+𝑉_𝑁𝑟, and each 𝑉_𝑖 has distribution 𝑓_𝑉𝑖(𝑣) that is given by (22). Therefore, 𝑓_{𝑉𝑟)(𝑁}(𝑣) is the convolution of (22) with itself 𝑁_𝑟−1 times. We can show that 𝑓_{𝑉𝑟)(𝑁}(𝑣) is given by𝑓_{𝑉𝑟)(𝑁}(𝑧)=12𝑁_𝑟−1!2^2𝑁𝑟𝑣^{2𝑁𝑟−1}𝑒^−𝑣/2,0<𝑣<∞, (31) which is a chi-square distribution with 4𝑁_𝑟 degrees of freedom. Therefore, 𝑝_3|0 is𝑝_3∣0=𝑝2𝑁_𝑟,𝛾_𝑏𝑁_𝑟. (32)

Lastly, the probability 𝑝(1∣0) is given by𝑝_1∣0=^∞₀12𝛾erfc^𝑏𝑊^(𝑁𝑟)2𝑁_𝑟𝑓_{𝑊𝑟)(𝑁}(𝑤)𝑑𝑤, (33) where 𝑊^(𝑁𝑟)=𝑊₁+𝑊₂+⋯+𝑊_𝑁𝑟. Each 𝑊_𝑖 has distribution 𝑓_𝑊𝑖(𝑤) given by (18). To find the pdf of 𝑊^(𝑁𝑟) we must convolve (18) with itself 𝑁_𝑟−1 times. Therefore:𝑓_{𝑊𝑟)(𝑁}(𝑤)=14𝑤𝑒^−𝑤/3+14𝑤𝑒^−𝑤−34𝑒^−𝑤/3+34𝑒^−𝑤,0<𝑤<∞ (34) for 𝑁_𝑟=2. For 𝑁_𝑟=3 it is given by: 𝑓_{𝑊𝑟)(𝑁}(𝑤)=𝑒^−w/3−𝑒^−𝑤1𝑤16²+27−916𝑤𝑒16^−w/3+𝑒^−𝑤,0<𝑤<∞. (35)

For 𝑁_𝑟=4 the pdf is given by: 𝑓_{𝑊𝑟)(𝑁}(𝑤)=𝑒^−w/3+𝑒^−𝑤1𝑤96³+45𝑤−𝑒32^−w/3−𝑒^−𝑤×3𝑤16²+13532,0<𝑤<∞. (36)

Using these pdfs and the results in [19], we can find 𝑝_1∣0. For 𝑁_𝑟=2,𝑝_1∣0 is given by:𝑝_1∣0=94𝑝2,3𝛾_𝑏4+14𝑝𝛾2,^𝑏4−94𝑝1,3𝛾_𝑏4+34𝑝𝛾1,^𝑏4. (37)

For 𝑁_𝑟=3, 𝑝_1∣0 is given by: 𝑝_1|0=278𝑝𝛾3,^𝑏2−18𝑝𝛾3,^𝑏6−81𝑝𝛾162,^𝑏2−9𝑝𝛾162,^𝑏6+81𝑝𝛾161,^𝑏2−27𝑝𝛾161,^𝑏6. (38)

For 𝑁_𝑟=4, 𝑝_1∣0 is given by: 𝑝_1|0=81𝑝164,3𝛾_𝑏8+1𝑝𝛾164,^𝑏8−818𝑝3,3𝛾_𝑏8+38𝑝𝛾3,^𝑏8+405𝑝322,3𝛾_𝑏8+405𝑝321,3𝛾_𝑏8+45𝑝𝛾322,^𝑏8−135𝑝𝛾321,^𝑏8. (39) As is the case for 𝑁_𝑟=1, the probability of bit error is given by:𝑃_𝑏<𝑝_15∣0+4𝑝_1∣0+3𝑝_3∣0. (40)

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 Figure 3. 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 for 𝑁𝑡=4 and 𝑁𝑟=1,2,3, and 4.

We see from Figure 3 that the upper bound is tight as 𝐸_𝑏/𝑁_𝑜 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 𝐸_𝑏/𝑁_𝑜 increases.

We also noticed that as 𝐸_𝑏/𝑁_𝑜 increases, 𝑃_𝑏 of (25) and (40) tends towards 𝑝_15∣0 of (14) and (29), respectively. This means that as 𝐸_𝑏/𝑁_𝑜 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 𝑁_𝑟=1, the distance squared 𝑑²_15,0 would decrease to 2𝐸_𝑏|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 4𝐸_𝑏|𝛼₃₁+𝛼₄₁|² which is less than 𝑑²_15,0, 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:𝑃_𝑏=𝑝𝑁_𝑟,1𝑁_𝑟𝐸_𝑏𝑁_𝑜, (41) 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 𝐸_𝑏/𝑁_𝑜 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 of𝑃_𝑏=𝑝𝑁_𝑟,𝑁_𝑡𝑁_𝑟𝐸_𝑏𝑁_𝑜. (42) Therefore, the MIMO-CDMA system employing the parity-bit-selected spreading technique provides an asymptotic gain of 10log(𝑁_𝑡) over the conventional MIMO-CDMA system under the conditions discussed in this paper. As 𝐸_𝑏/𝑁_𝑜 increases, the actual gain approaches but is less than 10log(𝑁_𝑡). 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 of 10log(𝑁_𝑡) 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.