Spatial multiplexing MIMO systems

We assume a MIMO OFDM system with M_T transmit antennas and M_R ≥ M_T receive antennas. We denote the K subcarriers by the subscript k = 1, …, K throughout this chapter. The input-output model is captured by:

(1)

where is the complex-valued M_R × M_T channel matrix, is the M_T -dimensional transmitted vector, where each , j = 1, …, M_T, is chosen from a complex-valued constellation Ω_j of the order w_j,k = |Ω_j,k|, is the circularly symmetric complex additive white Gaussian noise vector of size M_R, and is the M_R-element received vector. Note that we do not restrict all the parallel M_T streams to use the same modulation order; rather, each stream, which corresponds to one of the antennas of one of the users, may be using either the 4-, 16-, or 64-QAM modulation. Also, note that even though we focus on one transmitter with multiple antennas, the results in this section can be extended to multiple users such that the sum of the number of antennas of all of them equals M_T.

We assume that the complete channel information, i.e., the channel matrix coefficients for each subcarrier, are known in the receiver. Moreover, since the detection procedure for each subcarrier is performed independently of other subcarriers, we will drop the k subscript unless it is needed.

Therefore, the original MIMO model can be simplified to .

The preceding MIMO equation can be further decomposed into real-valued numbers as follows [7]:

(2)

corresponding to:

(3)

with M = 2M_T and N = 2M_R presenting the dimensions of the new model. We call the ordering in (2) the conventional ordering. Using the conventional ordering, all the computations can be performed in real values, which would simplify the implementation complexity. Note that after real-valued decomposition, each s_i, i = 1, …, M, in s is chosen from a set of real numbers, Ω_i′, with p′_i = √p_i elements. For instance, for a 64-QAM modulation, each s_i can take any of the values in the set Ω′ = {±7, ±5, ±3, ±1}. The general optimum detector for such a system is the maximum-likelihood (ML) detector which minimizes y−Hs 2 over all the possible combinations of the s vector. Notice that for high order modulations and large number of antennas, this detection scheme incurs an exhaustive exponentially growing search among all the candidates, and is not practically feasible in a MIMO receiver. However, it has been shown that using the QR decomposition of the channel matrix, the distance norm can be simplified [8, 9, 10] as follows:

(4)

where H = QR, QQ_H = I, and y′ = Q^Hy. Note that the transition in (4) is possible through the fact that R is an upper triangular matrix.

The norm in (4) can be computed in M iterations starting with i = M. When i = M, i.e., the first iteration, the initial partial norm is set to zero, T_M+1(s_(M+1)) = 0. Using the notation of [11], at each iteration the Partial Euclidean Distances (PEDs) at the next levels are given by

(5)

With s_(i) = [s_i,s_i+1,…,s_M]_T, and i = M,M−1,…,1,where

(6)

(7)

One can envision this iterative algorithm as a tree traversal with each level of the tree corresponding to one i value, and each node having p′i children.

The tree traversal can be performed in either a breadth-first or a depth-first manner. In the depth-first tree search [12, 11, 13], only one node is expanded at any time and once the end of the tree is reached, the traversal continues by visiting the new nodes in the higher levels of the tree. Therefore, each level can be visited more than one time.

In the breadth-first tree search [14, 15], however, each level is only visited once, and more than one node per level is expanded. Once the end of the tree, or the leaves level, is reached, the minimum candidate is chosen. A typical breadth-first tree search is the K-best detector. In the K-best detector, at each level, only the best K nodes, i.e. the K nodes with the smallest T_i, are chosen for expansion. Note that such a detector requires sorting a list of size K × p′ to find the best K candidates. For instance, for a 16-QAM system with K = 10, this requires sorting a list of size K × p′ = 10 × 4 = 40 at most of the tree levels. This introduces a long delay for the next processing block in the detector unless a highly parallel sorter is used. Highly parallel sorters, on the other hand, consist of a large number of compare-select blocks, and result in dramatic area increase.

Flex-Sphere detector

In order to simplify the sorting step, which significantly reduces the delay of the detector, a sort-free strategy can be utilized. Moreover, we discuss how using a new modified real-valued decomposition ordering (M-RVD) scheme can help in designing a flexible architecture. Finally, we discuss the design and implementation of the flexible sphere detector, Flex-Sphere [16, 17], that can support a range of modulation orders and number of antennas.

Modified real-valued decomposition (M-RVD) ordering

For the sort-free detector described in the preceding section, we can use a modified real-valued decomposition (M-RVD) ordering which improves the BER performance compared to the ordering given in equation (2). The new decomposition is summarized as [20]:

(8)

or,

(9)

where is the permuted channel matrix of equation (3) whose columns are reordered to match the other vectors of the new decomposition ordering in equation (8). It is worth noting that since the difference between RVD and M-RVD is the grouping of the signals, there is no extra computational cost associated with this modified ordering. We will see in the following sections how this ordering could reduce the latency for Flex-Sphere.

FPGA design of the configurable detector for SDR handsets

In this section, the main features of the architecture and the FPGA implementation of the SDR handset detector are presented. We use the Xilinx System Generator [21] to implement the proposed architecture. In order to support all the different numbers of antenna/user and modulation orders, the detector is designed for the maximal case, i.e., M_T × M_R, 64-QAM case, and configurability elements are introduced in the design to support different configurations.

Modified real-valued decomposition (M-RVD)

Using the real-valued decomposition, the two extra adders that are required for each complex multiplication can be avoided, thus avoiding the unnecessary FPGA slices on the addition operations. Moreover, while using the complex-valued operations requires the SE ordering of [11], which would be a demanding task given the configurable nature of the detector, with the real-valued decomposition, the SE ordering can be implemented more efficiently and simply for the proposed configurable architecture as described earlier. Also, note that even though some of the multiplications can be replaced with shift-adds in an area-optimized ASIC design, for an FPGA implementation, the appropriate design choice is to use the available embedded multipliers, commonly known as XtremeDSP and DSP48E in Virtex-4 and Virtex-5 devices.

It is noteworthy that if the conventional real-valued decomposition of (3) were employed, then the results for a 2 × 2 system would have been ready only after going through all the in-phase tree levels and the first two quadrature levels. However, with the modified real-valued decomposition (M-RVD), each antenna is isolated from other antennas in two consecutive levels of the tree. Therefore, there is no need to go through the latency of the unnecessary levels. Thus, using the M-RVD technique offers a latency reduction compared to the conventional real-valued decomposition.

Xilinx FPGA implementation results for M_T=3

Table 5-2 presents the System Generator implementation results of the Flex-Sphere on a Xilinx Virtex-4 FPGA, xc4vfx100-10ff1517 [21] for 16-bits precision. The maximum number of detectable streams is set to M_T = 3. The maximum achievable clock frequency is 250 MHz. Since the design folding factor is set to F = 8, the maximum achievable data rate, i.e., M_T = 3 and p_i = 64, is:

(12)

Table 5-2: FPGA resource utilization summary of the proposed Flex-Sphere for the Xilinx Virtex-4, xc4vfx100-10ff1517 device.

Xilinx FPGA implementation results for M_T=4

Table 5-3 presents the System Generator implementation results of the Flex-Sphere on a Xilinx Virtex-5 FPGA, xc5vsx95t-3ff1136 [21] for 16-bits precision and M_T = 4. The maximum achievable clock frequency is 285.71 MHz. Since the design folding factor is set to F = 8, the maximum achievable data rate, i.e., M_T = 4 and p_i = 64, is:

(13)

Table 5-3: FPGA resource utilization of the proposed Flex-Sphere.

The Flex-Sphere can support different numbers of antennas and modulation orders, and achieves high data rate requirements of various wireless standards. Table 5-4 summarizes the data rates for all of the different scenarios of the M_T = 4, Virtex-5 implementation.

Table 5-4: Data rate for different configurations of the 4 × 4.

Simulation results

In this section, we present the simulation results for the Flex-Sphere, and compare the performance of the FPGA fixed-point implementation with that of the optimum floating-point maximum-likelihood (ML) results. The Xilinx System Generator implementation of the Flex-Sphere detector is shown in Figure 5-5. Prior to the M-RVD, introduced earlier, we employ the channel ordering of [22] to further close the gap to ML. Also, we make the assumption that all the streams are using the same modulation scheme. We assume a Rayleigh fading channel model, i.e., complex-valued channel matrices with the real and imaginary parts of each element drawn from the normal distribution.

In order to ensure that all the antennas in the receiver have similar average received SNR, and none of the users’ messages are suppressed with other messages, a power control scheme is employed. Figure 5-6 shows the simulation results for the maximal 4 × 4 configuration. As can be seen, the proposed hardware architecture implementation performs within, at most, 1 dB of the optimum maximum-likelihood detection.

Beamforming in wideband systems

The physical layer of the WiMAX standard achieves wideband communication via Orthogonal Frequency Division Multiplexing (OFDM). OFDM is a multicarrier scheme in which subcarriers occupy orthogonal narrowband channels. Beamforming is a narrowband scheme that can be easily extended to wideband OFDM systems by applying beamforming techniques per subcarrier.

Figure 5-7 shows a beamforming OFDM system where channel state information is sent from the receiver to the transmitter via a feedback channel. The system has M_T transmitter antennas, M_R receiver antennas, and K data subcarriers. is used to represent the complex symbol transmitted on subcarrier k, 1 ≤ k ≤ K. The time span of the channel impulse response is assumed to be shorter than the time span of the cyclic prefix (this is true for any well designed OFDM system); the baseband relationship between the complex symbol transmitted on subcarrier k and the corresponding received signal y_k is given by:

(14)

The M_T × 1 vector used for beamforming subcarrier k is represented by w_bk. The K beamforming vectors w_b1,…, w_bK are chosen from a beamforming codebook W of cardinality |W| = 2_B, hence, W = {w₁,w₂,…,w_|W|}, and index b_k specifies the index of the beamforming vector chosen to beamform subcarrier k, 1 ≤ b_k ≤ |W|. The channel matrix corresponding to subcarrier k is represented by the M_R × M_T matrix , the entries of are i.i.d., and each entry is a circularly symmetric complex Gaussian random variable with zero mean and unit variance per complex dimension. To simplify the analysis, it is assumed that the receiver has perfect knowledge of the channel matrix . The M_R × 1 vector z_k is the Maximum Ratio Combining (MRC) vector for subcarrier k. The M_R × 1 noise vector has i.i.d. entries where each entry is assumed to be circularly symmetric complex Gaussian with zero mean and variance N₀ per complex dimension. Each subcarrier is transmitted with the same average power E_s; this constraint is met by setting E[] = E_s and w_bk  = 1. The WiMAX standard specifies beamforming codebooks for different numbers of transmitter antennas and codebook cardinalities; all the beamforming vectors in the WiMAX codebooks satisfy w_bk  = 1. (Note that · is used to denote the vector two-norm and (·)_H denotes the conjugate transposition of a matrix.)

The SNR for subcarrier k is given by:

(15)

the MRC vector and the index of the beamforming vector that maximize SNR_k are [5]

(16)

and

(17)

respectively.

In a beamforming FDD system like the one considered in the WiMAX standard, the beamforming codebook W is known to both the transmitter and the receiver. The K indices b₁, b₂,…,b_K computed by the channel quantizer at the receiver are fedback to the transmitter. Based on these indices, the transmitter beamforms subcarriers 1, 2,…, K using vectors w_b1, w_b2,…,w_bK respectively. The K indices b₁, b₂,…,b_K computed by the channel quantizer are also used to compute the MRC vectors as shown in (16). Since the codebook has cardinality |W| = 2^B, each index b_k is represented using B bits. The total amount of feedback bits is equal to KB, which corresponds to B bits of feedback information per subcarrier. The WiMAX standard specifies different possible mechanisms to send the KB bits of feedback information from the receiver to the transmitter; in some configurations (e.g. adjacent subcarrier permutations [26]) the amount of feedback bits can be reduced by clustering subcarriers.

The process of quantizing each subcarrier channel into B bits takes place at the channel quantizer. Since each subcarrier goes through an independent channel, each subcarrier channel must be quantized independently. In order to reutilize hardware resources the quantizer processes one subcarrier at a time, as shown in Figure 5-7. The input-output relationship of the channel quantizer is given by equation (17) which is computed by exhaustive searching over the elements in the codebook W.

The system depicted in Figure 5-7 achieves full diversity if |W| ≥ M_T and the beamforming codebook is designed based on the Grassmannian beamforming criterion [5]. The WiMAX standard specifies codebooks for 2, 3, and 4 transmit antennas. For 2 transmit antennas the WiMAX standard specifies a codebook of size |W| = 8; for 3 and 4 transmitter antennas the standard specifies one codebook of size |W| = 8 and one codebook of size |W| = 64. The WiMAX codebooks seem to be designed based on the Grassmannian criterion [27] and it can be verified via Monte Carlo simulation that WiMAX codebooks achieve full diversity.

Computational requirements and performance of a beamforming system

Implementation of the beamforming and MRC blocks in Figure 5-7 is straightforward; the most intensive computation in each of the blocks is a vector multiplication. Also, the K beamforming and K MRC blocks can be reduced to one beamforming block and one MRC block by processing one subcarrier at a time.

Implementation of the channel quantizer in Figure 5-7 can require a large amount of resources depending on the number of transmitter antennas and codebook size. The entries of are complex numbers and the WiMAX standard specifies that the entries of w_i are complex numbers rounded to four decimal places. Hence, computing for all the |W| codewords requires |W|M_T M_R complex multiplications, |W|M_T M_R −|W|M_R complex additions, 2|W|M_R real multiplications, and 2|W|M_R − |W| real additions. After computing for all |W| codewords the channel quantizer searches for the codeword with the largest . Implementing a tree search requires |W|−1 relational blocks that compare two inputs and then output the greater of the two.

The total amount of resources required for channel quantization for one subcarrier in different WiMAX configurations is shown in Table 5-5. Notice that as the number of antennas and codebook cardinality increases, the number of resources required increases dramatically. The maximum number of embedded multipliers in an FPGA is 512 for the Xilinx Virtex-4 family [28] and 1056 for the Xilinx Virtex-5 family [29]. Hence, implementing the channel quantizer can become a bottleneck for the implementation of WiMAX systems.

Table 5-5: Resources required for channel quantization.

The amount of resources required for channel quantization can be reduced by reutilizing resources, but resource reutilization would increase the latency of the channel quantizer. Reutilization is possible as long as the timing constraints are met. In a WiMAX system the timing constraints can be quite tight, specially in high mobility scenarios and in frequency division duplexing (FDD) mode. In a high mobility scenario the channel coherence interval decreases and feedback information must be sent before it becomes stale and in FDD mode feedback information can potentially be sent as soon as it is available. In both FDD mode and high mobility the faster the feedback information is sent the more throughput the system will have, since more payload data will be sent during a coherence interval.

Another alternative to reduce the amount of resources required for channel quantization is to use a mixed codebook scheme as proposed in [30]. In a mixed codebook scheme, the WiMAX codebook is used at the transmitter for beamforming and at the receiver for MRC, and a mapped version of the WiMAX codebook is used at the receiver’s channel quantizer. The entries in the mapped version of the WiMAX codebook belong to {0,+1,−1,+j,−j}; hence, all the complex multiplications required for channel quantization can be implemented via simple changes and swapping of the real and imaginary parts of the channel matrix entries. Table 5-6 compares the amount of resources required for channel quantization when using a mapped WiMAX codebook with the amount of resources required when using a WiMAX codebook. The mapped version of the WiMAX codebook can be obtained via the vector mapping proposed in [30]. Using a mixed codebook scheme will affect performance because the codebook used for beamforming at the transmitter and MRC at the receiver is different from the codebook used for channel quantization at the receiver. However, the performance loss is very small, since by construction, the mapped WiMAX codebook has quantization regions similar to the quantization regions of the WiMAX codebook from which it is obtained [30]. Notice that the mixed codebook scheme remains WiMAX compliant because the mapped WiMAX codebook is only used for channel quantization.

Table 5-6: Resources required for channel quantization using a WiMAX codebook and a mapped WiMAX codebook. Results correspond to a system with 4 transmitter and 4 receiver antennas and a codebook of 64 codewords.

Observe from Table 5-6 that when using a mapped WiMAX codebook for channel quantization, the number of multiplications required reduces to zero. This reduction is obtained by increasing the number of five input multiplexers and negators (a negator block is a very simple block that computes the arithmetic negation or two’s complement of its input). For both ASIC and FPGA implementations, 2048 five input multiplexers, 2048 negators, and 64 real multipliers require fewer resources than implementing 1024 complex multipliers.

Figure 5-8 shows the performance of different beamforming WiMAX configurations. All simulations were run for a one subcarrier model, and the results obtained represent the per subcarrier behavior in an OFDM system. The results in Figure 5-8 correspond to the best case scenario of perfect channel estimate at the receiver, noiseless and zero delay feedback, and floating-point processing. The results in Figure 5-8 show that in a feedback-based beamforming system, such as FDD WiMAX, few bits of feedback can achieve performance close to the ideal case of infinite feedback. Also, observe that using a mixed codebook scheme (labeled MC in the figure) results in small performance degradation, and as shown in Table 5-6, the amount of resources can be significantly reduced by using a mixed codebook scheme. Figure 5-8 also shows the performance of the Alamouti STC [31], which is an open loop scheme. Observe that the closed loop beamforming scheme outperforms the open loop scheme.

Beamforming experiments using WARPLab

WARPLab is a framework for rapid prototyping of physical layer algorithms. The WARPLab framework combines the ease of MATLAB with the capabilities of the Wireless Open Access Research Platform (WARP) developed at Rice University [32]. This section describes in detail the WARPLab framework and presents experiment results that show the performance gains that can be obtained with beamforming in a WiMAX system.

Experiment setup and results

This section shows experiment results for a beamforming system with two transmitter antennas and one receiver antenna. The system was designed and tested using the WARPLab framework. Experiments were performed over a wireless channel emulated using Spirent’s SR-5500 wireless channel emulator. The experiment setup is shown in Figure 5-11. Two radios at the transmitter node are connected to the two RF inputs of the channel emulator which emulates two independent RF wireless channels. The two RF outputs of the channel emulator are added to emulate a 2 × 1 system; the output of the RF adder is connected to one radio at the receiver node. The channel emulator is connected to the host computer via Ethernet; the host computer controls the two WARP nodes and the channel emulator via the Ethernet links.

The system implemented is a single subcarrier system, the results representing the per subcarrier behavior in an OFDM system. Table 5-7 summarizes the experiment conditions; the two emulated RF channels were both set using exactly the same parameters. Since the delay spread is much smaller than the symbol period, the transmitted signal goes through a flat fading channel.

Table 5-7: Experimental conditions.

Transmission of the payload data was done in packets of 110 symbols. The number of symbols per packet was limited to 110 due to the characteristics of the transmitted signal (symbol time of 3.2 μs and sampling frequency of 40 MHz) and the maximum number of samples that can be stored per receiver radio in a node, which is limited to 2₁₄ samples [34]. As shown in Figure 5-12, two pilot sequences were sent before transmission of payload. The first pilot sequence was used for computation of the channel estimate used for channel quantization, and the total pilot energy was set equal to twice the energy per symbol. The second pilot sequence was a beamformed pilot sequence, which was used to obtain an estimate of the channel, times the beamforming vector. This estimate was used for MRC at the receiver and the total pilot energy was equal to the total energy per symbol. The error-free feedback channel was implemented in the host PC (the host PC is connected to both transmitter and receiver). The feedback delay was approximately 60 ms, which is the time it took to send training, estimate the channel, quantize the channel estimate, and start transmitting the beamformed signal.

In order to compare the performance of a closed loop scheme like beamforming and an open loop scheme like Alamouti, a 2 × 1 Alamouti scheme was also implemented and tested using the WARPLab framework. In the Alamouti scheme, transmission of the payload data was also done in packets of 110 symbols. For the Alamouti implementation, only one pilot sequence was sent and payload was sent immediately after the pilot sequence. The total pilot energy was equal to the total energy per symbol.

Simulation results in Figure 5-8 showed that the beamforming scheme has better performance than the Alamouti scheme and that the mixed codebook scheme for efficient implementation results in small performance degradation. These two observations are verified by the experiment results shown in Figure 5-13. The results also show that, as expected, few feedback bits have performance close to the performance of infinite feedback. Observe that the plots of the experiment results are not as smooth as the plots of the simulation results. This is very likely due to hardware non-linearities and the fact that simulations were run more for total number of bits than were experiments because simulations ran faster.