First, fundamentals of both OFDM and MIMO are studied in chapter 2, and so are the techniques for MIMO data detection. Secondly, the combination of OFDM and MIMO is described in chapter 3, followed by some existed simplification algorithms for data detection. Chapter 4 introduces the new simplified MIMO OFDM data detection algorithms from some major existed algorithms and other detection techniques as well, while chapter 5 includes simulations. Finally, we conclude with some remarks of the work in chapter 6.
Chapter 2
OFDM and MIMO Fundamentals
2.1 OFDM System Models
The principle of multicarrier system is to seperate the data stream into several parallel ones, each modulated by a specific subcarrier and discrete Fourier Transform (DFT) is used in the baseband modulation and demodulation. Through this approach, only a pair of oscillator (for I-part and Q-part) is needed instead of multiple oscillators to modulate different signals at different carriers.
When signals pass through a time-dispersive channel, inter-symbol interference (ISI) and inter-carrier interference (ICI) usually occur in the receiver and cyclic prefix (CP) was introduced to combat ISI and ICI. Cyclic prefix, shown in Figure 2.1, is a copy of the tail part of a symbol, which is inserted in between the symbol to be transmitted and its preceding symbol. As long as the cyclic prefix length is longer than its experiencing time-dispersive channel length, ISI can be avoided. At the same time, the cyclic prefix along with its symbol makes a periodic signal and maintains the properties of circular convolution and subcarrier orthogonality that prevents the ICI effect.
Figure 2.1 Cyclic prefix of an OFDM symbol
2.1.1 Continuous Time Model
In this section, a continuous-time model is used to introduce the whole OFDM baseband system including the transmitter and receiver. In the transmitter, the transmitted data is split into multiple subchannels with overlapping frequency bands.
The spectrum of OFDM signal is shown in Figure 2.2. It is clear that the spectrum of each subchannel is spreading to all the others, but is zero at all other subcarrier frequencies, because of the sinc function property, which is the key feature of the orthogonality.
Figure 2.3 (a) shows a typical continuous-time OFDM baseband modulator, in which the transmitted data is split into multiple parallel streams which are modulated by different subcarriers and then transmitted simultaneously.. At the receiver, the received signal is demodulated simultaneously by multiple matched filters and then the data on each subchannel is obtained by sampling the outputs of matched filters, as shown in Figure 2.3 (b).
Cyclic prefix
Ts Tg
Figure 2.2 Spectrum of an OFDM signal
Figure 2.3 (a) Continuous-time OFDM baseband modulator
Figure 2.3 (b) Continuous-time OFDM baseband demodulator
2.1.2 Discrete Time Model
As mentioned previously, to simultaneously transmit multiple data, the transmitter must modulate data with multiple subcarriers and the receiver must demodulate with multiple matched filters. In fact, the modulation and demodulation can be implemented efficiently by using digital IDFT/DFT operations, because they can be respectively represented as
which are the same as IDFT operation of the transmitted data x(k) and DFT operation of the received data r(k), respectively.
Figure 2.4 shows the discrete-time baseband OFDM model. The IDFT transforms the frequency-domain data into time-domain data which is delivered over the air and experiences multi-path channel, denoted as h( mn, ). At the receiver, to recover the signal in frequency domain, DFT is adopted in the demodulator as a matched filter. Then the frequency domain signal of each subchannel is obtained from its DFT output.
Figure 2.4 Discrete-time OFDM system model
2.1.3 Effect of Cyclic Prefix
Assume the given channel is quasi-static, i.e., constant during the transmission of an OFDM symbol and variable symbol wise, where the quasi-static impulse response is
) , (tτ
h , t is the time index and τ is the channel path delay. The received signal r(t)can be expressed as
) multipath channels, orthogonality as shown in Figure 2.2 will be destroyed by ISI and ICI.
However, as long as the cyclic prefix length is longer than that of h( mn, ), ISI effect can be avoided. At the same time, linear convolution of s(n) and h( mn, ) turns out to be circular convolution. It is known that circular convolution in time domain results in multiplication in frequency domain when the channel is stationary so that the received signal y(k) in frequency domain is the product of transmitted data x(k) and subcarrier channel response H(k). Thus, the orthogonality is maintained (if h(n,m) is fixed within the symbol length) and data can be easily recovered by one-tap channel equalizer, i.e., dividing y(k) by the correspondingH(k).
2.2 Concept of Multiple-Input Multiple-Output (MIMO) Systems
MIMO system architectures provide better spectral efficiency than conventional systems because of the benefit of multiple antenna or space diversity both at the transmitter and receiver.
MIMO systems provide the ability to turn multipath propagation, which is traditionally a drawback of wireless transmission, into a benefit. Since MIMO systems effectively take advantage of random fading and multipath delay spread, the signals transmitted from each transmit antenna appear highly uncorrelated at each receive antenna and the signals travel through different spatial channels. Then the receiver can exploit these different spatial channels and separate the signals transmitted from different antennas at the same frequency band simultaneously.
2.2.1 MIMO System Model
MIMO is a promising technology suited for high-speed broadband wireless communications. Through space division multiplexing, the MIMO technology can transmit multiple data streams in independent parallel spatial channels, thereby increasing the total transmission rate of the system.
MIMO systems can be simply defined. Considering an arbitrary wireless communication system, a link is considered for which the transmitter is equipped with Nt
1. The channel is constant during the transmission of a packet. It means the communication is carried out in packets that are of shorter time-span than the coherence time of the channel.
2. The channel is memoryless. It means that an independent realization of channel is drawn for each use of the channel.
3. The channel is frequency-flat fading. It means that constant fading over the bandwidth is desired in the case of narrowband transmissions. It also indicates that the channel gain can be represented by a complex number.
4. Only a single user transmits signals at any given time, so the received signal is corrupted by AWGN only.
With these assumptions, it is common to represent the input/output relations of a narrowband, single-user MIMO link by the complex baseband vector notation
n Hx
y= + (2.5)
where x is the Nt×1 transmit vector, y is the Nr×1 receive vector, H is the Nr×Nt
channel matrix, and n is the Nr×1 additive white Gaussian noise (AWGN) vector at some instant in time. All of the coefficients hij comprise the channel matrix H and represent the complex gain of the channel between the jth transmit antenna and the ith receive antenna.
The channel matrix can be written as
⎟⎟
Coefficients {hij} reflect unknown channnel properties of the medium, usually Rayleigh distributed in a rich scattering environment without line-of-sight (LOS) path. If α and β are independent and Gaussian distributed random variables, then |hij| is a Rayleigh distributed random variable. Actually, coefficients {hij} are likely to be subject to varying degrees of fading and change over time. Therefore, the determination of the channel matrix is an important and necessary aspect of MIMO processing. If all these coefficients are known, there will be sufficient information for the receiver to eliminate interference from other transmitters operating at the same frequency band.
Figure 2.5 Wireless MIMO transmission model [17]
2.3 Signal Detection Algorithms for MIMO Systems
In this section zero-forcing (ZF) criterion is considered due to consideration of lower complexity. Zero-forcing techniques receive an input vector y and send it to a filter bank which eliminates the mutual interference without caring about noise.
2.3.1 Maxima-Likelihood (ML) Detection
Since modern transmission systems are digital, each element of transmitted vector x is chosen from a finite set, which is denoted as A, such as BPSK, QPSK and 16-QAM.
Hence, a transmitted signal x belongs to ANt. The optimum maximum-likelihood (ML) detector searches over the whole set of transmit signals x ∈ ANt, and decides in favor of the transmit signal xML that minimizes the Euclidian distance to the receive vector x, i.e.
min 2
arg y Hx
xML = x∈ANt − (2.8)
The computational effort is of order MNt, where M denotes size of finite set A.
When using high modulation scheme or many transmitting antennas, ML detection is impractical.
2.3.2 Suboptimal Algorithms
Although ML detection reaches optimal performance, it is not feasible for large numbers of transmit antennas or high modulation schemes. In the sequel, some suboptimal algorithms are investigated. The target is to find algorithms that have performance near ML and low complexities.
2.3.2.1 The Linear Detection Method
For linear detection method, the received vector y is linearly multiplied with a matrix G, and then a parallel decision follows. Zero-forcing criterion means that mutual
interferences between the transmitted signals shall be perfectly suppressed without concerning noise. As a result, matrix G is generated by the Moore-Penrose pseudo-inverse (denoted by (·)+) of the channel matrix H[18].
H H
ZF H H H H
G = +=( )−1 (2.9)
where H is assumed to have full column rank. The parallel decision maps each element of the filtered output vector
n
onto an element of the symbol alphabet through a minimum distance quantization. The estimation errors correspond to the main diagonal elements of the error covariance matrix
( )( )
[
~ − ~ −]
= 2( )
−1=E xZF x xZF x H n HHH
ZF σ
φ (2.11)
that equals the covariance matrix of the filtered noise. Small eigenvalues of HHH will result in large estimation errors owing to noise amplification, which is especially severe in systems with equal number of transmit and receive antennas. Using random matrix theory [19], it can be shown that asymptotically for NT = NR → ∞ the noise amplification tends to infinity.
2.3.2.2 The V-BLAST Detection Method
Vertical – Bell Laboratories Layered Space-Time (V-BLAST) [9] is proposed to improve performance of the linear detection method by utilizing successive interference cancellation based on zero-forcing criterion. It suggests that transmitted signals are
( )
i iwhere i means the order index a signal is detected.
The x~ is quantized to i x , and the interference of this signal is removed by i subtracting it from the received signal y and nulling the i-th column of the channel matrix.
Nulling and canceling are repeated until all signal is detected, as summarized in the following algorithm steps:
Begin
The main computational bottleneck is the computation of pseudo inverse for Nt
times. For saving computation, channel response is assumed to be invariant within a packet, as assumption 1 suggests. The optimal detection order and each nulling vector gZF
is validly used to detect received signals in the whole packet since they share the same channel matrix. That is, pseudo inverse of the channel is not updated
2.3.2.3 The SQRD Detection Method
Although V-BLAST algorithm achieves good performance, it still costs high computation power. The sorted QR decomposition (SQRD) detection algorithm [10,11] is introduced in the sequel based on QR decomposition. SQRD algorithm costs rather lower computation power with small performance degradation. The method’s operation is listed below.
The SQRD algorithm [10,11] is an extension to the modified Gram-Schmidt procedure by reordering the columns of the channel matrix before each orthogonalization step. The basic idea is to minimize |rk,k| in the order it is computed (from 1 to Nt ) instead of to maximize in the order of detection (from Nt to 1). |rk,k| denotes absolute value of the element at k-th row and k-th column in R matrix. This is motivated because the signal detected last has effects on only few other layers through error propagation and therefore has small SNR’s. Since r1,1 is simply the norm of the column vector h1, the first step in the SQRD algorithm is simply to permute the column of H with minimum norm to this position. During the following orthogonalization of the vectors h2, . . . , hNt with respect to the normalized vector h1, the first row of R is obtained. In the second step, r2,2 is determined in a similar fashion from the remaining Nt-1 orthogonalized vectors, et cetera.
BLAST, but in many cases of interest the performance degradation is small compared to the reduced complexity [10,11]
Begin
exchange columns in the first Nr+i-1 rows of Q (2.14.h)
i
Chapter 3
MIMO Application to OFDM
Algorithms introduced for MIMO transmission require frequency-flat fading channels, and it limits its application to narrowband transmissions. For real broadband transmission systems, channel conditions are often frequency-selective fading. A technique alleviating severe effect of frequency-selective fading is demanded. OFDM technique is a good solution for this purpose in wireless transmission owing to its advantages [3,4,5].
3.1 MIMO-OFDM Architecture
According to Section 2.1, OFDM turns frequency-selective fading channel into several frequency-flat fading subchannels, solving the major problem in wideband transmission systems. Boubaker et. al [12] proposed MIMO-OFDM using V-BLAST algorithm to detect transmitted signals on each subcarrier. MIMO-OFDM transceiver architectures are also proposed, as shown in Figure 3.1 and 3.2.
Subchannels are orthogonal to each other in OFDM systems. Hence, in single-input-single-output (SISO) OFDM systems, the received signals are product of channel response and transmitted signal, as shown in equation 2.4. In MIMO systems, signals transmitted from different antenna on a subcarrier simultaneously interfere each other, but signals at different subcarriers are independent. At the receiver antennas, a linear combination of the transmitted signal and channel response on each subcarrier is observed, as equation 2.5 shows. That corresponds to assumptions of MIMO systems. On each subchannel, a space division multiplexing (SDM) like V-BLAST is applied. That is, the task is to recover x from the received signal y and channel state information (CSI) H on each subcarrier.
Figure 3.2 System achitecture of an OFDM and V-BLAST receiver [12]
3.2 Simplified Data Detection Algorithms for MIMO OFDM Systems
On each subchannel, a V-BLAST or SQRD detector is used since we can formulate the problem as equation 2.5. For most practical OFDM systems, there are too many subcarriers to implement a detector for each subcarrier. Fortunately, channel responses of successive subcarriers are often similar. Some simplified algorithms are proposed based on this observation.
3.2.1 Boubaker’s Algorithm [13]
In [13], Boubaker et. al proposed an algorithm based on V-BLAST which divide used subcarriers into groups. Since data detection and channel inversion can be separated to two parts, the same nulling vectors corresponding to a subcarrier of a group is used to
detect data of all the group. That is, the algorithm assumes that the channel conditions on all subcarriers in this group are identical. Therefore, a dramatically reduction of computation cost is obtained. In addition, a new detection algorithm is proposed. A sub-optimal ordering is obtained simply from the first pseudo-inverse by sorting ||(Gl)j ||2 in an ascending order, for j = 1,. . . , M, where G1 = HH. Then, the series of calculations of pseudo-inverse can be replaced by applying Gram-Schmitt Orthogonalization (GSO) once to H, provided that the decoding ordering is known in advance.
3.2.2 Liu’s Algorithm [20]
For good channel conditions, [13] provide a good way for both good performance and low complexity, while for slightly poor channels, this may not be enough. Liu and Yang [20] proposed an algorithm for this situation. They use V-BLAST to determine pseudo inverse matrixes at some subcarrier of a group and then use equation 3.1 to approximate pseudo inverse matrixes at other subcarriers.
(
A−ε)
+≈ A+ +A+εA+ (3.1)With equation 3.1, pseudo inverse at (k+1)-th subchannel is calculated by using pseudo inverse of the k-th subchannel.
+ +
+
+ ≈ + − +
+1) ( ) ( ) ( ( ) ( 1)) ( )
(k H k H k H k H k H k
H (3.2)
where H(k) is channel response of k-th subchannel.
Other pseudo inverse is also calculated.
3.3 Introduction to WWiSE 802.11n Proposal
To examine validity of proposed algorithms in next chapter, WWiSE proposal, a proposal for next generation 802.11n standard, is adopted for simulation. 802.11n standard utilize both MIMO and OFDM techniques for efficient wireless LAN. In the following, brief introduction is described.
WWiSE proposal [21] emphasize backward compatibility with existing installed base, building on experience with interoperability in 802.11g and previous 802.11 amendments which are mainly designed for indoor wireless internet applications. Hence, we review the physical layer of wireless LAN 802.11a [22] system which is based on OFDM technology. The main system parameters of IEEE 802.11a Wireless LAN standard are listed in Table 3.1.
Table 3.1 Parameters and specifications of 802.11a system [22]
Signal bandwidth 20MHz Sample duration 50ns
FFT length 64
Used subcarriers 52 Data subcarriers 48 Pilot subcarriers 4
Symbol period 3.2us (64 samples) Cyclic prefix 0.8us (16 samples) Subcarrier spacing 312.5 kHz
Modulation BPSK, QPSK, 16QAM, 64QAM Channel coding 1/2 convolutional, constrain length 7,
Optional puncturing
Data rate 6, 9, 12, 18, 24, 36, 48, 54 Mbps
In 802.11a standard, a frame is composed of three fields. Figure 3.3 shows the packet format which facilitates synchronization and channel estimation of the receiver. In the preamble field, the preambles are composed of ten repeated short symbols and two repeated long symbols. The total duration of short symbols is 8us and so is that of long symbols. Since the SIGNAL field contains the most important information of the packet, such as frame length and modulation, synchronization and channel estimation must be finished before decoding of the SIGNAL.
Figure 3.3 Frame structure of 802.11a [22]
For the purpose of compatibility with the 802.11 legacy devices, the legacy part
Short Training Symbol Field
• Frame Timing Sync.
• Coarse CFO Sync.
• Symbol Timing Sync.
Long Training Symbol Field
• Fine CFO Sync.
• Channel Estimation.
cyclical delay is adopted in WWiSE proposal, whose format is used for simulation.
Figure 3.4 shows the cyclical delay format in WWiSE. The maximum number of the spatial data streams is four.
Figure 3.4 Cyclical delay format of the preamble in WWiSE [21]
STRN stands for the short training sequence. LTRN represents the long training symbol. GI2 is the guard interval of the long training symbol.
STRN 400 ns cs
STRN
STRN 600 ns cs
GI21 GI2 LTRN
GI23
LTRN 100 ns cs
LTRN 1700 ns cs
GI22
LTRN 1600 ns cs STRN
200 ns cs
Chapter 4
The Proposed Data Detection Algorithms
Since the algorithms to be proposed are for MIMO-OFDM systems to reduce complexity of V-BLAST detection, the technique are also useful to simplify the linear detection and SQRD detection algorithms. In the sequel, some simplification algorithms are introduced based on the designs of [13,20]
4.1 The Simplified Linear Detection Methods
In the linear detection method, a received vector is simply multiplied with pseudo inverse of channel. Direct computation of pseudo inverse is costly. Although pseudo inverse is calculated only once for each subcarrier, an approximation with good performance and low cost is highly required, because it is calculated N times for a symbol.
Based on the concept of [13], we propose two simplified algorithms. The simplified algorithm 1, for each two suncarriers, compute inverse of one subcarrier, and directly
applies it to the data detection of the other subcarrier. Based on the approximation to pseudo inverse in [20], obviously it is also useful in linear detection. Our proposed linear detection algorithm 2 is that for each two suncarriers, we compute the inverse of one subcarrier, while approximate the other using the computed one.
Figure 4.1 Scenario of the proposed linear detection algorithm 1
Begin
4.2 The Simplified V-BLAST Detection Method
Simplified algorithms for V-BLAST detection method are already proposed in [13,20]. [13] assumed successive subcarriers share a channel response so that vectors of a subcarrier is applied to detect signal of other subcarriers.
Figure 4.3 The simplified V-BLAST detection algorithm in [13]
In [20], the simplification, however, miss an important assumption. The subcarrier which is decoded by approximation is assumed to have the same decode order with that of the subcarrier which provides the pseudo inverse. At each step of V-BLAST, a column
of channel matrix is set to zero in order to ignore the effect of the corresponding transmitting antenna. If the pseudo inverse is used to estimate inverse of other subcarriers, the subcarriers have to recognize the nulled columns, which stand for detected signals. In other words, the optimal decoding order is assumed not to change.
Under this assumption, there are some opportunities for further simplification.
The simplified algorithm has identical performance but with lower complexities. Owing to unchanged decoding order, what we need is merely a row of the approximated pseudo inverse which represents the transmitted signal to be detected, rather than the inverse, as
The simplified algorithm has identical performance but with lower complexities. Owing to unchanged decoding order, what we need is merely a row of the approximated pseudo inverse which represents the transmitted signal to be detected, rather than the inverse, as