MIMO Self-Encoded Spread Spectrum with Iterative Detection over Rayleigh Fading Channels

Abstract

Self-encoded spread spectrum (SESS) is a novel communication technique that derives its spreading code from the randomness of the source stream rather than using conventional pseudorandom noise (PN) code. In this paper, we propose to incorporate SESS in multiple-input multiple-output (MIMO) systems as a means to combat against fading effects in wireless channels. Orthogonal space-time block-coded MIMO technique is employed to achieve spatial diversity, and the inherent temporal diversity in SESS modulation is exploited with iterative detection. Simulation results demonstrate that MIMO-SESS can effectively mitigate the channel fading effect such that the system can achieve a bit error rate of 10⁻⁴ with very low signal-to-noise ratio, from 3.3 dB for a 2×2 antenna configuration to just less than 0 dB for a 4×2 configuration under Rayleigh fading. The performance improvement for the 2×2 case is as much as 6.7 dB when compared to an MIMO PN-coded spread spectrum system.

1. Introduction

Self-encoded spread spectrum (SESS) and multiple access communications have been proposed and shown to have a number of unique features [1, 2]. By deriving its spreading sequences from the randomness of the source stream, SESS provides a feasible implementation of random-coded spread spectrum and potentially enhances the transmission security. Since the use of PN code generators is obviated, SESS can simplify multirate transmissions with variable processing gains in multimedia applications [3]. In previous works, we have shown that the modulation memory in SESS can yield signal gain in AWGN channels [4]. We have also demonstrated that the inherent temporal diversity in SESS modulation can be exploited with iterative detection to significantly improve the system performance over time-varying fading channels [5].

It is also well known that PN-coded spread spectrum can be incorporated with multiple-input multiple-output (MIMO) techniques to achieve multipath and spatial diversities [6–8]. In delay-spread or frequency-selective fading, the system performance can be improved with a Rake receiver that can resolve and combine the multi-path signal components by matching the Rake fingers with the delayed PN codes to achieve multi-path diversity. By deploying multiple antennas at both transmitter and receiver, MIMO architectures are capable of mitigating channel fading by taking advantage of the spatial diversity and enhancing system capacity by employing spatial multiplexing [9–11].

In this paper, we propose a novel approach to combat against fading in wireless channels by incorporating SESS in MIMO system. Our motivation is based on the observation that SESS not only can achieve multi-path diversity like PN-coded spread spectrum, but it can also provide both inherent temporal diversity and signal gain. Our work combines orthogonal space-time block-coded MIMO technique and iterative detection to obtain spatial and temporal diversities, respectively. We determine the bit error rate (BER) performance of MIMO-SESS system over Rayleigh fading channels and show that the fading effects can be completely mitigated by exploiting diversities in both space and time domains. The proposed scheme in this paper employs only one spreading sequence which we refer to as the single-code scheme. Recently, we have reported a multicode Alamouti scheme that can enhance the system throughput [12] with multiple spreading sequences, but at the expense of a BER degradation. The orthogonal space-time block codes and multiple iterations in this paper thus could be incorporated into the multi-code scheme in order to improve the BER performance.

The rest of this paper is organized as follows. In Section 2, we briefly describe the original SESS system. The proposed MIMO-SESS with iterative detection is described in Section 3. Section 4 presents the simulation results and compares the proposed system to an MIMO PN-coded spread spectrum (MIMO-PNSS) system. Finally, Section 5 concludes this paper.

Notations
The superscripts ∗ and 𝑇 represent the conjugate and the transpose operations, respectively. |⋅|, 𝐸{⋅}, and diag(⋅) denote the absolute value of a scalar, the expectation operation, and the diagonal vector of a matrix, respectively. Matrix (vector) is represented by capital (small) bold letter, with 𝑎(𝑘,𝑛) representing the 𝑛th element of vector 𝐚(𝑘).

2. Background

In this section, we briefly review the concept of self-encoding spread spectrum. The readers are referred to the original description in [1] for more details.

2.1. Transmitter

A block diagram of SESS system is shown in Figure 1, where each rounded corner block represents one delay register, 𝑁 denotes the length of spreading sequence, and 𝑇_𝑏 is the bit duration.

(a) Transmitter

(b) Receiver

Figure 1 

Block diagram of SESS system.

The source information 𝑏 is assumed to be bipolar values of ±1. The bits are first spread by the self-encoded spreading sequence 𝐝(𝑘) of length 𝑁 at a chip rate of 𝑁/𝑇_𝑏. This sequence is constructed from the source information stored in the delay registers that are updated every 𝑇_𝑏. For example, the spreading sequence for the 𝑘th bit, 𝑏(𝑘), is given as𝐝(𝑘)=[𝑏(𝑘−1)𝑏(𝑘−2)⋯𝑏(𝑘−𝑁)]^𝑇. (1) Thus, with a random input data stream, the sequence is also random and time varying from one bit to another. We assume that the delay registers have been seeded by a randomly selected sequence (which has also been acquired at the receiver to initialize the despreading process).

The spread spectrum chips are then transmitted as𝐜(𝑘)=𝑏(𝑘)𝐝(𝑘). (2) Clearly, SESS signal has a modulation memory depth equal to 𝑁:𝑐(𝑘,𝑛)=𝑏(𝑘)𝑑(𝑘,𝑛)=𝑏(𝑘)𝑏(𝑘−𝑛). (3)

2.2. Receiver

We assume that the chips are transmitted over a wireless channel and each bit experiences independent Rayleigh fading that is constant over the bit duration. The channel coefficient for the 𝑘th bit interval is represented by 𝛼(𝑘). The received signal 𝑦(𝑘,𝑛) can be expressed as𝑦(𝑘,𝑛)=𝛼(𝑘)𝑐(𝑘,𝑛)+𝑒(𝑘,𝑛), (4) where the additive Gaussian noise 𝑒(𝑘,𝑛) has zero mean and variance 𝑁𝑁₀/2. Note that the noise is sampled at the chip rate and is broadband; its variance is thus the narrow-band noise variance 𝑁₀/2 multiplied by the spreading factor 𝑁.

The soft correlation estimate of the 𝑘th bit is computed from𝑏̂(𝑘)=1𝑁^𝑁_𝑛=1=1𝑦(𝑘,𝑛)𝑑(𝑘,𝑛)𝑁^𝑁_𝑛=1𝛼(𝑘)𝑐(𝑘,𝑛)𝑑(𝑘,𝑛)+𝑢(𝑘), (5) where1𝑢(𝑘)=𝑁^𝑁_𝑛=1𝑒(𝑘,𝑛)𝑑(𝑘,𝑛) (6) is the narrow-band Gaussian noise component of the correlation output and 𝑑(𝑘,𝑛) is the 𝑛th chip of the de-spreading sequence 𝐝(𝑘) generated by the delay registers. We assume that the receiver delay registers have been initialized by the same seed sequence at the transmitter. The content of the delay registers, 𝐝(𝑘), are then updated by the hard decision of the correlation estimate, 𝑏̃(𝑘)=sgn[𝑏̂(𝑘)]. Since the hard decision may be erroneous, the reconstructed de-spreading sequence could be different from the self-encoded spreading sequence at the transmitter. A bit error therefore will propagate through the delay registers for the next 𝑁 bits and cause self-interference that attenuates the strength of the despread signal 𝑏̂(𝑘). The correlation estimate given in (5) can be written as𝑏̂(𝑘)=1𝑁^𝑁_𝑛=1𝛼(𝑘)𝑐(𝑘,𝑛)𝑏̃(𝑘−𝑛)+𝑢(𝑘). (7) Substituting (3) into (7), we have𝑏̂(𝑘)=𝛼(𝑘)[1𝑁^𝑁_𝑛=1𝑏(𝑘−𝑛)𝑏̃(𝑘−𝑛)]𝑏(𝑘)+𝑢(𝑘). (8) Thus, the hard decision is performed on the soft estimate from the correlator output, a noisy and fading signal that is also subjected to self interference (expressed by the term inside the square bracket in (8)):̃̂̂𝑏(𝑘)=sgn[𝑏(𝑘)]=̂1,𝑏(𝑘)>0,−1,𝑏(𝑘)<0. (9)

3. MIMO-SESS with Iterative Detection

In this section, we seek to improve the performance of SESS system in fading channels by employing multiple antennas to take advantage of spatial diversity and by utilizing iterative detection to achieve temporal diversity. Figure 2 shows a block diagram of the proposed MIMO-SESS system.

(a) Transmitter

(b) Receiver

Figure 2 

Block diagram of the proposed MIMO-SESS system.

At the transmitter, the information bits first undergo spread spectrum modulation according to the self encoding procedure in Section 2.1. The spread spectrum chips are further encoded using orthogonal space-time block codes (OSTBCs) and then transmitted over an 𝑁_𝑡×𝑁_𝑟 MIMO fading channel, where 𝑁_𝑡 antennas and 𝑁_𝑟 antennas are deployed at the transmitter and receiver, respectively. We assume that the channel between each transmit/receive antenna pair is subjected to Rayleigh fading that remains constant over 𝑇_𝑏 but is independent from bit to bit [5, 13]. Furthermore, the channels for different transmit/receive antenna pairs experience independent fading. The signals from the received antennas are combined linearly to generate the output that is then detected with an iterative algorithm for bit recovery.

3.1. MIMO Transmission

In the above MIMO-SESS system, the (inner) MIMO encoding is independent from (outer) self-encoding as can be seen from Figure 2. The decoupling nature of this approach simplifies decoding at the receiver and lets us employ, in principle, any MIMO techniques for signal transmission, such as beamforming for spatial filtering [14] and for signal quality improvement [15–17], space-time block coding for spatial diversity [18–20], and layered space-time architecture for spatial multiplexing [21–23]. In this paper, we adopt OSTBC which provides full diversity gain and allows linear maximum-likelihood (ML) decoding [24].

Numerous OSTBCs have been reported in the literature. In this work, we employ the full-rate code 𝒢₂ for the two-transmit-antenna case, the half-rate code 𝒢₃ for the three-transmit-antenna case, and the half-rate code 𝒢₄ for the four-transmit-antenna case. The full-rate code 𝒢₂ was proposed in [25] and is known as the Alamouti scheme. The half-rate codes 𝒢₃ and 𝒢₄ were developed in [24]. The code matrices have been reproduced as follows:𝒢₂=𝑠₁𝑠₂−𝑠^∗₂𝑠^∗₁𝒢₃,=𝑠₁𝑠₂𝑠₃−𝑠₂𝑠₁−𝑠₄−𝑠₃𝑠₄𝑠₁−𝑠₄−𝑠₃𝑠₂𝑠^∗₁𝑠^∗₂𝑠^∗₃−𝑠^∗₂𝑠^∗₁−𝑠^∗₄−𝑠^∗₃𝑠^∗₄𝑠^∗₁−𝑠^∗₄−𝑠^∗₃𝑠^∗₂𝒢₄,=𝑠₁𝑠₂𝑠₃𝑠₄−𝑠₂𝑠₁−𝑠₄𝑠₃−𝑠₃𝑠₄𝑠₁−𝑠₂−𝑠₄−𝑠₃𝑠₂𝑠₁𝑠^∗₁𝑠^∗₂𝑠^∗₃𝑠^∗₄−𝑠^∗₂𝑠^∗₁−𝑠^∗₄𝑠^∗₃−𝑠^∗₃𝑠^∗₄𝑠^∗₁−𝑠^∗₂−𝑠^∗₄−𝑠^∗₃𝑠^∗₂𝑠^∗₁. (10)(11)(12)

In (10), (11), and (12), each row includes the symbols to be transmitted in one time slot via 𝑁_𝑡 transmit antennas, and each column consists of the symbols in one block for each antenna. The block sizes here are 2, 8, and 8 for the two-, three-, and four-antenna cases, respectively.

In order to enable linear maximum likelihood decoding of OSTBC, the channel should be stable in one block. Since we have assumed that the channel fading remains constant over a bit duration, this implies that the spreading factor 𝑁 should be a multiple of the block size. Notice that the total transmit power of MIMO system should be the same as single-antenna system. This means that the equivalent MIMO power per transmit antenna must be normalized by 𝑁_𝑡.

3.2. Space-Time Processing Example

In the following, we will use the Alamouti scheme for two transmit antennas as an example to describe space-time block coding and decoding procedures.

At the transmitter, the spread spectrum chips are divided into two streams according to 𝒢₂. A block of two consecutive chips, 𝑐(𝑘,2𝑖) and 𝑐(𝑘,2𝑖+1), are transmitted by sending 𝑐(𝑘,2𝑖) and −𝑐^∗(𝑘,2𝑖+1) to the first antenna, and 𝑐(𝑘,2𝑖+1) and 𝑐^∗(𝑘,2𝑖) to the second antenna. Here 𝑖∈{0,1,…,𝑁/2−1} is the block index for the chips of the 𝑘th bit. We note that in this case the power per antenna is one-half of the total power. The signals are then transmitted over an MIMO fading channel.

At the receiver, the signals from the antennas are first combined to achieve spatial diversity. For the 𝑚th receive antenna, the signals within the 𝑖th block are given as𝑥_𝑚(𝑘,2𝑖)=ℎ_1,𝑚(𝑘)𝑐(𝑘,2𝑖)+ℎ_2,𝑚(𝑘)𝑐(𝑘,2𝑖+1)+𝑒_𝑚𝑥(𝑘,2𝑖),_𝑚(𝑘,2𝑖+1)=−ℎ_1,𝑚(𝑘)𝑐^∗(𝑘,2𝑖+1)+ℎ_2,𝑚(𝑘)𝑐^∗(𝑘,2𝑖)+𝑒_𝑚(𝑘,2𝑖+1), (13)(14) where ℎ_1,𝑚(𝑘) and ℎ_2,𝑚(𝑘) denote the complex channel impulse response coefficients (for the 𝑘th bit) between the 𝑚th receive antenna and the first—and second—transmit antennas, respectively. 𝑒_𝑚 is the Gaussian noise with zero mean and variance 𝑁𝑁₀/2.

Again we assume that the delay registers in the receiver have been initially synchronized with the transmitter [26], and that perfect channel knowledge is available. Under these assumptions, diversity combining is carried out over two consecutive chip intervals per [11, 25]𝑦(𝑘,2𝑖)=^𝑁𝑟_𝑚=1(ℎ^∗_1,𝑚(𝑘)𝑥_𝑚(𝑘,2𝑖)+ℎ_2,𝑚(𝑘)𝑥^∗_𝑚(𝑘,2𝑖+1))+𝑤(𝑘,2𝑖)=𝛼(𝑘)𝑐(𝑘,2𝑖)+𝑤(𝑘,2𝑖),𝑦(𝑘,2𝑖+1)=^𝑁𝑟_𝑚=1(ℎ^∗_2,𝑚(𝑘)𝑥_𝑚(𝑘,2𝑖)−ℎ_1,𝑚(𝑘)𝑥^∗_𝑚(𝑘,2𝑖+1))+𝑤(𝑘,2𝑖+1)=𝛼(𝑘)𝑐(𝑘,2i+1)+𝑤(𝑘,2𝑖+1), (15)(16) where𝛼(𝑘)=^𝑁𝑟_𝑚=1(|ℎ_1,𝑚(𝑘)|²+|ℎ_2,𝑚(𝑘)|²),𝑤(𝑘,𝑛)=^{𝑁𝑟𝑚=1}(ℎ^∗_1,𝑚(𝑘)𝑒_𝑚(𝑘,𝑛)+ℎ_2,𝑚(𝑘)𝑒^∗_𝑚(𝑘,𝑛+1)),if𝑛iseven,_{𝑁𝑟𝑚=1}(ℎ^∗_2,𝑚(𝑘)𝑒_𝑚(𝑘,𝑛−1)−ℎ_1,𝑚(𝑘)𝑒^∗_𝑚(𝑘,𝑛)),if𝑛isodd. (17)(18)

The combined signals given in (15) and (16) can be expressed as𝑦(𝑘,𝑛)=𝛼(𝑘)𝑐(𝑘,𝑛)+𝑤(𝑘,𝑛), (19) and further in vector form𝐲(𝑘)=𝛼(𝑘)𝐜(𝑘)+𝐰(𝑘). (20)

Similar signal models and analysis can be developed for the three- and four-transmit-antenna cases. In general for an 𝑁_𝑡×𝑁_𝑟 MIMO system, (19) and (20) remain valid with𝛼(𝑘)=^𝑁𝑡𝑁_𝑛=1^𝑟_𝑚=1|ℎ_𝑛,𝑚(𝑘)|², (21) albeit with a more complicated noise term 𝐰(𝑘).

3.3. Iterative Detection

Since SESS signal has modulation memory, its optimum detection is the maximum likelihood sequence detector or the Viterbi detector. However, the complexity of the optimum detection grows exponentially as 2^𝑁, making it impractical to implement unless 𝑁 is rather small. On the other hand, iterative detection is suboptimum but linear in 𝑁, and can converge iteratively toward the optimum performance.

The iterative detection for SESS is described by the detailed block diagram illustrated in Figure 3. Notice that correlation detection provides the initial estimate 𝑏̃₀(𝑘) for the iterative detection. For simplicity we will ignore self interference in the following signal analysis, so that ∑^𝑁_𝑛=1𝑏(𝑘−𝑛)𝑏̃(𝑘−𝑛)=𝑁. This in fact is justified if the BER is sufficiently low. According to the description in Section 2, the soft correlation output or estimate of the 𝑘th bit is then given as𝑏̂₀(𝑘)=𝛼(𝑘)𝑏(𝑘)+𝑣₁(𝑘), (22) where 𝑣₁(𝑘) is the noise term.

Figure 3 

Block diagram of iterative detection.

The hard decision of the correlation output 𝑏̃₀(𝑘)=sgn[𝑏̂₀(𝑘)] is then used to construct the despreading sequences.

Now assuming that we have obtained the correlation detection of 𝑏(𝑘), then based on (3) and (19) the next 𝑁² chips of bits 𝑏(𝑘+1),𝑏(𝑘+2),…,𝑏(𝑘+𝑁) can be written in a square matrix 𝐏(𝑘) as shown by𝐏(𝑘)=.𝛼(𝑘+1)𝑏(𝑘+1)𝑏(𝑘)𝛼(𝑘+1)𝑏(𝑘+1)𝑏(𝑘−1)⋯𝛼(𝑘+1)𝑏(𝑘+1)𝑏𝛼(𝑘+2)𝑏(𝑘+2)𝑏(𝑘+1)𝛼(𝑘+2)𝑏(𝑘+2)𝑏(𝑘)⋯𝛼(𝑘+2)𝑏(𝑘+2)𝑏(𝑘−𝑁)⋮⋮⋱⋮𝛼(𝑘+𝑁)𝑏(𝑘+𝑁)𝑏(𝑘+𝑁−1)𝛼(𝑘+𝑁)𝑏(𝑘+𝑁)𝑏(𝑘+𝑁−2)⋯𝛼(𝑘+𝑁)𝑏(𝑘+𝑁)𝑏(𝑘) (𝑘−𝑁+1)(23)

Each element of this matrix represents a chip as given in (19) and the 𝑗th row includes the chips of bit 𝑏(𝑘+𝑗). Note that the noise term has been omitted due to the page limitation.

Observe that the 𝑘th bit, 𝑏(𝑘), are also present in the diagonal elements of the matrix 𝐏(𝑘). This means that the information of the 𝑘th bit is also included in the next 𝑁 bits. Thus, the modulation memory in SESS signals could be utilized to enhance the detection of the 𝑘th bit. The iterative detection employs the hard decisions of these next 𝑁 bits, [𝑏̃₀(𝑘+1)𝑏̃₀(𝑘+2)⋯𝑏̃₀(𝑘+𝑁)], in order to accumulate the time-diversity soft estimate of 𝑏(𝑘) as𝑏̂₁(𝑘)=1𝑁diag(𝐏(𝑘))[𝑏̃₀(𝑘+1)𝑏̃₀(𝑘+2)⋯𝑏̃₀(𝑘+𝑁)]^𝑇+𝑣₂=1(𝑘)𝑁^𝑁_𝑛=1𝛼(𝑘+𝑛)𝑏(𝑘)+𝑣₂(𝑘). (24) where 𝑣₂(𝑘) is a noise term.

The summation of the correlation estimate, 𝑏̂₀(𝑘), delayed by 𝑁𝑇_𝑏, and the time-diversity estimate, 𝑏̂₁(𝑘), provides the soft estimate for the hard decision of the first iteration. That is,𝑏̆₁(𝑘)=𝑏̂₀(𝑘)+𝑏̂₁1(𝑘)=(𝛼(𝑘)+𝑁^𝑁_𝑛=1𝛼(𝑘+𝑛))𝑏(𝑘)+𝑣(𝑘), (25) where 𝑣(𝑘)=𝑣₁(𝑘)+𝑣₂(𝑘).

The output of the first iteration 𝑏̃₁(𝑘) is then obtained by the hard decision of 𝑏̆₁(𝑘).

This first iterative bit decision 𝑏̃₁(𝑘) is an improvement over the initial estimate 𝑏̃₀(𝑘) because it is based on the combined soft estimates which yield not only a signal (to noise) gain of 3 dB but also the time diversity gain from SESS modulation memory. The improved bit decision can be further fed back again to reconstruct the de-spreading sequences for the correlation detection and also to reaccumulate the soft bit estimate from temporal diversity combining in the manner described by (24).

The block diagram for this second iteration is illustrated in the lower part in Figure 3. It is clear from the block diagram that procedure can be repeated iteratively. We note in addition that each iteration introduces a time delay of 𝑁𝑇_𝑏. Furthermore, the detector structure is partly similar to a Rake receiver that employs 𝑁 fingers to exploit the 𝑁-fold temporal diversity in SESS signals.

4. Simulation Results

In this section, the performance of the proposed MIMO-SESS system has been determined by simulations and is compared with a conventional MIMO-PNSS system. The length of the spreading sequence is set to 𝑁=64 unless noted otherwise. For each scenario and bit signal-to-noise ratio 𝐸_𝑏/𝑁_𝑜 (SNR), 100 runs of 100,000 bits have been simulated to obtain the average bit error rate.

Figure 4 compares two-transmit-antenna MIMO-SESS system without iterative detection to MIMO-PNSS system. Because iterative detection is not applied in this scenario, no temporal diversity gain is achieved. This means that the same performance can be expected with or without self-encoding. However, the MIMO-SESS system degrades at low SNR due to the effects of error propagation [1]. By employing MIMO technique, error propagation can be efficiently alleviated, as shown by the 2×2 scenario.

Figure 4 

Comparison between an MIMO-PNSS system and the proposed MIMO-SESS system without iterative detection, two transmit antennas using full-rate code 𝒢2, N = 64.

The results for the two-transmit-antenna MIMO-SESS system with iterative detection are plotted in Figure 5, which clearly shows that the BER performance has been significantly improved with iterative detection. With only one iteration, there is about 6.7 dB gain for the 2×2 scenario at 10⁻⁴ BER. This performance improvement can be attributed to the temporal diversity introduced by self-encoding, in addition to the signal gain from combining the soft estimates. Moreover, the improvement with the second iteration over the first iteration is quite marginal (less than 0.1 dB at 10⁻⁴ BER). This demonstrates that the algorithm is very efficient and converges extremely fast (to within one iteration in this case). Thus, in practice, the second iteration may not be necessary nor desirable given the diminishingly small gain and the increased computational complexity and delays.

Figure 5 

Performance of the proposed MIMO-SESS system with iterative detection, two transmit antennas using full-rate code 𝒢2, N = 64.

Figures 7 and 6 show the performance of the MIMO-SESS system with three- and four-transmit-antenna configurations, respectively. Clearly, the spatial diversity gains have increased compared to the two-transmit-antenna case as evident from the steeper slopes of the BER curves. This can also be quantified from the plots by observing, for example, that the overall gain at 10⁻⁴ BER has decreased from 6.7 dB for the 2×2 scenario to about 5.3 dB and 4.5 dB for the 3×2 and 4×2 configurations, respectively. Again, the performance gain with the iterative detection is significant for all antenna configurations. In particular, the 4×2 configuration requires an 𝐸_𝑏/𝑁_𝑜 of just less than 0 dB at 10⁻⁴ BER. It should be noted that these results also reflect additional coding gains (compared to the results in Figure 5) since the space-time codes are half rates.

Figure 6 

Performance of the proposed MIMO-SESS system with iterative detection, four transmit antennas using half-rate code 𝒢4, N = 64.

Figure 7 

Performance of the proposed MIMO-SESS system with iterative detection, three transmit antennas using half-rate code 𝒢3, N = 64.

Finally, Figure 8 shows the effects of varying the spreading lengths on the example 2×2 MIMO-SESS system. The performance of a BPSK system under AWGN and a 2×2 MIMO-PNSS system under Rayleigh fading has also been plotted for comparison. The plots show that the performance improves with the spreading length (especially at low SNR) because the temporal diversity increases with 𝑁 (more Rake fingers) as would be expected. When compared to the BPSK system under AWGN, the performance gain is nearly 5 dB at 10⁻⁴ BER, demonstrating that the fading effect on the BER has been completely mitigated with MIMO-SESS. The results also show that this performance can be achieved with a relatively small value of 𝑁 = 64.

Figure 8 

Performance of the proposed 2×2 MIMO-SESS system with iterative detection, N = 16, 32, 64, 128.

5. Conclusion

In this paper, we have described a novel MIMO-SESS technique with iterative detection as a means to provide temporal and spatial diversities for wireless communications over fading channels. The proposed scheme combines SESS with MIMO space-time block coding in a decoupling manner such that the number of antennas is not constrained by SESS structure. Our work has examined example OSTBC with two to four transmit antennas and one to two receive antennas in fast Rayleigh fading channels. The results show that SESS with iterative detection can significantly improve the overall BER performance, by as much as 6.7 dB gain over MIMO-PNSS for a 2×2 antenna configuration. The linear iterative detection is based on soft estimates and has been found to converge quickly after only one iteration, making it very efficient for a practical implementation.

Our work has shown that significant diversity gain can be exploited from SESS-modulated signals such that the proposed system can completely mitigate the fading effect and achieve a BER performance of only 3.3 dB SNR at 10⁻⁴ BER for the 2×2, to just less than 0 dB for the 4×2. The performance will be poorer if the channel fading is slow and the symbols experience block fading because the temporal diversity gain with SESS will be reduced, especially if the fading block length exceeds the spreading length. In general, the diversity loss due to slow or block fading could be compensated for and recovered with interleaving.

The approach in this paper could be extended to multi-code MIMO-SESS for BER improvement while enhancing the system throughput. Similarly, we anticipate that the spectral efficiency of the system can be further improved by incorporating QAM with SESS and by performing self encoding across multiple QAM symbols. We whould also like to point out that the decoupling nature of the proposed MIMO-SESS technique suggests the intriguing possibility of a coupled approach that may be advantageous. The design of such an encoding and decoding scheme (possibly together with error correction capability) for MIMO-SESS and its performance in fading channels could be an interesting and challenging problem.

Acknowledgments

The authors would like to express their appreciation for the valuable comments from the editor as well as the anonymous reviewers of their manuscript. This work was funded in part by Contract award FA9550-08-1-0393 from the U.S. Air Force Office of Scientific Research. They wish to thank Dr. J. Sjogren for his support of this study.