2.2 Co-located MIMO Elements
2.2.2 Spatial Multiplexing
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
0 1
2
3
Figure 2.3: Trellis diagram for a 4-QAM, four-state trellis code
which are mapped to the modulated symbols of −1 and 1. If the incoming symbols are 23, the trellis state will change to state 3.
At the receiver, ML sequence estimation using the Viterbi algorithm [74] may be employed for decoding the received signal. The studies of [48, 73] showed that increasing the number of states is capable of increasing the coding gain, while increasing the number of receive antennas may increase both the coding gain and the diversity gain.
2.2.2 Spatial Multiplexing
The family of space-time codes offers an improved BER performance for MIMO systems owing to possessing a diversity gain. However, they are unable to approach a tight lower bound on the MIMO channel capacity. Foschini [46] proposed a Layered Space-Time (LST) architecture, also known as a spatial multiplexing, which exploits the MIMO channels for the sake of improving the transmission rate rather than the BER. The throughput performance improvement due to employing spatial multiplexing is often referred to as having a multiplexing gain. In practice, there is a trade-off between the diversity gain and the multiplexing gain, when designing MIMO systems.
2.2.2.1 Layered Space-Time Transmitters Horizontal Layered Space-Time Coding
The schematic of the Horizontal Layer Space-Time (HLST) coding scheme is shown in Fig. 2.4. In HLSTs, the bit stream is first demultiplexed into nT separate streams by a serial-parallel converter.
Subsequently, each stream is independently encoded, interleaved and modulated before being trans-mitted from a specific antenna, hence, the spatial throughput becomes rs = nT. When the receiver employs nR receive antennas, the HLST may achieve a diversity order of nR, since each symbol is only transmitted from a single antenna and received by nR antennas. Therefore, the simple HLST constitutes a sup-optimal architecture. In order to increase the attainable diversity order as close to the expected value of (nT × nR), two modified arrangements, namely the Diagonal LST (DLST) [46]
2.2.2. Spatial Multiplexing 30
Demultiplex
Encoder Interleaver Modulator
Encoder Interleaver Modulator Input
bits
1
nT
Figure 2.4: The HLST architecture using separate channel codes in each layer.
and the threaded LST (TLST) [75] arrangements, were proposed.
• DLST: The DLST scheme, which is often known as D-BLAST, was proposed by Foschini [46].
Observe by comparing Figs. 2.4 and 2.5 that the initial signal processing in this scheme is similar to that of HLST. However, the demultiplexed streams of Fig 2.4 are passed through a ‘stream rotator’, which rotates the frame in a round-robin fashion before passing it to the transmit antenna. The codewords have to be of appropriate length to ensure that they are mapped to and transmitted over all nT transmit antennas. Again, the D-BLAST encoding scheme is illustrated in Fig. 2.5. According to the figure, there is an unexploited space-time area in the concept of D-BLAST, which is beneficial from a different perspective, since it facilitates optimal decoding at the receiver, as demonstrated in Chapter 11 of [73].
Demultiplex
Encoder Interleaver Modulator
Encoder Interleaver Modulator Input
bits
nT
Stream Rotator
1
Figure 2.5: The DLST architecture using separate channel codes in each layer. In contrast to Fig. 2.4, a stream rotator is inserted between the modulators and transmit antennas.
• TLST: The TLST scheme of Fig. 2.6 constitutes another version of HLST, where a spatial interleaver is inserted after the modulators. Owing to the spatial interleaver, the resultant codeword extends beyond the diagonal of the space-time stripe and wraps around, whilst creating multiple stripes, as shown in Fig. 2.6. This LST type offers an improved temporal diversity.
However, the joint decoding of multiple threads is required, resulting in a higher implementation complexity than both the general HLST scheme of Fig 2.4 and the D-BLAST scheme of Fig. 2.5.
2.2.2. Spatial Multiplexing 31
Demultiplex
Modulator
Modulator Input
bits
nT
Spatial Interleaver
1
Encoder
Figure 2.6: The TLST architecture using a single joint channel code for multiple layers. In contrast to the DLST architecture of Fig. 2.5, in TLST the encoder is incorporated in front of the demultiplexer, while the spatial-domain interleaver replaces the stream rotator.
Vertical Layered Space-Time Coding
Encoder Interleaver Modulator Input
bits
1
nT
Demultiplex
Figure 2.7: The VLST architecture using a single joint channel code for multiple layers, where a time-domain interleaver is arranged between the encoder and the modulator, instead of having a spatial-domain interleaver between the modulators and transmit antennas, as in Fig. 2.6.
The Vertical Layered Space-Time (VLST) philosophy, also known as V-BLAST, was proposed by Wolniansky et al. [58]. In the VLST architecture shown in Fig. 2.7 the bit stream is channel encoded along the temporal domain, interleaved and modulated before being demultiplexed into nT sub-streams. This type of LST has the benefit of spreading its information bits across all antennas.
However, the VLST’s joint decoding of the bit streams significantly increases the receiver’s complexity.
Again, the scheme is capable of achieving a spatial rate of rs = nT and a diversity order higher than nR, since the information symbols are spread over more than one antenna. The attainable coding gain depends on the choice of the temporal-domain channel codes employed and an array gain of nR is achievable.
2.2.2. Spatial Multiplexing 32 2.2.2.2 Layered Space-Time Receivers
The spatial multiplexing imposes spatial interference on the receive antennas at the receiver side. The interference-contaminated signal at the receiver may be represented by the following matrix operation
y = Hs + n, (2.8)
where y represents the received (nR× 1)-element vector, H is the channel matrix of size (nR× nT), s denotes the (nT × 1)-element transmit vector and n is the AWGN. Numerous decoding algorithms may be used at the receiver, which are briefly described below.
Maximum Likelihood Receiver
In the Maximum Likelihood (ML) receiver, the estimated symbol vector ˆs is decided by solving the following problem:
ˆ
s = arg min||y − Hs||2. (2.9)
The ML receiver searches for the entire space of legitimate transmit vectors. Therefore, the search complexity is proportional to MnT, where M is the number of the modulation levels. This type of receiver is referred to as the optimal ML receiver. In order to reduce its decoding complexity, the sphere decoding technique detailed in [76, 77] may be employed.
Zero-Forcing Receiver
The Zero-Forcing (ZF) receiver may be viewed as a linear filter, which separates the signal streams and decodes each stream independently, hence it is referred to as a linear receiver. The estimated symbol vector ˆs may be obtained as [46]
ˆ
s = (HHH)−1y = (HHH)−1(Hs + n) = H‡s + (HHH)−1n, (2.10) whereH denotes the complex conjugate transpose and ‡indicates the pseudoinverse of a matrix [78].
As shown in [73], this scheme attains a diversity order of (nR− nT + 1) for each stream and hence it is suboptimal.
Minimum-Mean-Square-Error Receiver
The Minimum-Mean-Square-Error (MMSE) receiver concept is based on minimizing the square of the estimation error caused by fading, noise and interference amongst cochannel signals. The estimated symbol vector ˆs is formulated as [79]
ˆ s =
µ1
γInR+ HHH
¶−1
HHy, (2.11)
where γ is the SNR.
Successive-Cancellation Receiver
Instead of jointly decoding the stream, the Successive-Cancellation (SuC) algorithm generally detects signals on a row-by-row basis. The effect of each detected and remodulated row is cancelled from the received signal, in order to reduce the interference. If the information of each layer is detected correctly, there is no error propagation. However, when the weakest stream’s signal is detected first, the errors might be propagated to the other stream during the decoding process, resulting in a degraded
2.2.3. Tradeoff between Spatial Coding and Spatial Multiplexing 33