A Physical Layer Abstraction for Maximum Likelihood Demodulation of MIMO Signals

Transcription

1 A Physical Layer Abstraction for Maximum Likelihood Demodulation of MIMO Signals R. Ramésh, Havish Koorapaty, Thomas Cheng and Kumar Balachandran Ericsson Research, RTP P.O. Box 139, Research Triangle Park, NC 2779, USA Abstract We develop a physical layer abstraction to model the performance of MIMO with maximum likelihood demodulation (MLD) in a system simulation. MLD is inherently a non-linear process, and thus it is difficult to find a linear abstraction to its performance. The abstraction is based on bounds on performance obtained from the QR & QL decompositions of the channel. The accuracy of the abstraction is demonstrated using link simulations. In addition, the abstraction has low complexity. The model is used in a system simulation to evaluate the gains obtained with MLD. I. INTRODUCTION Multi-Input Multi-Output (MIMO) transmission with several transmit and receive antennas has been proposed as a means to improve the throughput of wireless communication systems by transmitting multiple streams over the air interface. Two possible MIMO encodings exist: Horizontal encoding, wherein separate codewords are transmitted over the multiple streams, and Vertical encoding, wherein a single codeword is multiplexed over the multiple streams. MIMO transmissions using Orthogonal Frequency Division Multiplexing (OFDM) may be received using a Minimum Mean-Squared Error (MMSE) receiver on a per-tone basis. The various codewords in a horizontally encoded format may be decoded successively, by decoding one code word with an MMSE receiver that treats the effect of all other codewords as colored noise, subtracting the effect of that code word from the received signal waveform, and subsequently decoding the other codewords in the same manner. Such MMSE reception with successive interference cancellation (MMSE-SIC), with proper link adaptation on both streams, is known to achieve the open-loop capacity of the MIMO channel [1]. A similar MMSE-SIC receiver is not possible with vertically encoded MIMO signals. However, an MLD-based demodulator has been suggested as an improvement over the MMSE demodulator [4] without SIC. The MLD demodulator calculates the squared-error metric between the received signal and the product of the matrix channel and all hypotheses of the MIMO symbols. The hypothesis that minimizes the squared error gives the maximum likelihood estimate of the transmitted signal. Soft bit information can be calculated by taking the difference of squared-distance metrics between the maximum likelihood MIMO symbol estimates and MIMO symbols with bits flipped from the ML symbol estimates. The complexity of MLD is quite high, and approaches to lower the complexity have been studied [2]. Radio network system simulations typically use an abstraction of the physical layer to model link performance. Many simulation methodologies utilize a block fading channel model that uses an instantaneous snapshot of the desired signal s and the interferers channels. Along with knowledge of the transmit power, pathlosses, shadowing and noise variance, the instantaneous Block Error rate (BLER) can then be calculated from its dependency on the Signal-to-Interferenceand-Noise Ratio (SINR) to determine whether an error event has occurred. Calculation of the SINR for linear receivers such as MMSE and MMSE-SIC receivers is straightforward; expressions can be found in [13] and [14]. However, it is difficult to arrive at a good abstraction for non-linear receivers such as the MLD. Such an abstraction should attempt to model the performance of the receiver based on the knowledge of the channel of the desired and interfering signals and knowledge of the signal, interference and noise powers. A desirable output of the abstraction is an effective SINR (typically represented as an equivalent SINR for an AWGN channel), which is then used in a characterization of SINR versus BLER to obtain an instantaneous BLER, which could then be converted to an error event indication using a coin tossing experiment. Measures other than the SINR can be used as well. For example, the Received Block Information Rate (RBIR) metric can be used [6]. The SINR can be mapped to an RBIR value, or alternatively, the parameters of the simulation can be used to directly calculate an RBIR value. A mapping of RBIR to BLER is used to predict the performance. In [], the Mean Mutual Information per bit (MMIB) metric is also used as an alternative to the RBIR, with simulation parameters used to calculate the MMIB directly. The primary focus of this paper is to derive a simple and accurate abstraction of the performance of MLD. The chosen approach bounds the performance on each stream from above and below, and uses a mean of these bounds to characterize the performance achievable. Link simulations are used to demonstrate the accuracy of the model. We also use the abstraction in a system simulation, and compare the results to the system performance obtained using the MMSE receiver. The rest of the paper is organized as follows: in Section II, we provide some background, and describe previous approaches to address the problem. In Section III, we derive the proposed method. In Section IV, we refine the param-

2 eters for the method, and evaluate the performance of the method via link simulations. In Section V, we evaluate system performance with a system simulation using the abstraction derived in this paper for MLD. We conclude in Section VI. This paper provides detailed analysis of a 2x2 MIMO system, but it appears possible to generalize the method for MIMO systems with more antennas. II. BACKGROUND For a 2x2 MIMO system, the received signal is given by [ ] h11 h t = Hs + n, H = 12 (1) h 21 h 22 where H is the MIMO channel on a per tone basis, t = [t 1,t 2 ] T is the received vector signal, s =[s 1,s 2 ] T are the transmitted symbols on the two streams, and n =[n 1,n 2 ] T is AWGN. The maximum likelihood demodulator estimates the transmitted symbols as those that minimize the squared error, i.e., E =min t Hŝ 2 (2) ŝ This requires a search over all possible postulates of s, which can be prohibitively complex with higher level modulations such as 64-QAM. The search space is then 496 sets of 64-QAM symbols over each OFDM tone. In addition, the demodulator will have to generate soft information for the different bits to be used by a decoder that follows; this typically requires finding the metrics for many symbols with a bit flipped. The generation of soft information can be done after the most likely symbols have been detected. As stated before, there has been prior work on reducing the complexity of this demodulator. Decomposition of the matrix channel can help in reducing the complexity of the maximum likelihood demodulation process and in the formulation of a physical layer abstraction. One such decomposition is the Singular Value Decomposition (SVD), given by the factorization H = UΛV H (3) where U and V are unitary matrices, and Λ is a diagonal matrix with non-negative diagonal entries known as the singular values, whose ratio is related to the condition number of H. If the matrix V is applied at the transmitter and the matrix U H is applied at the receiver (essentially, a form of precoding), then the equivalent channel is Λ, and we have two independent streams, whose performance can be characterized using the squared singular values as the symbol energy. Note that the noise variance is not changed due to multiplication by a unitary matrix. Thus, the SINR on each stream is obtained from knowledge of the singular values, and knowledge of the noise or impairment variance. If the matrix V is not applied at the transmitter, the matrix U H can still be applied at the receiver without changing the noise variance to give us an equivalent channel of ΛV H. This equivalent channel represents independent streams working on a signal constellation skewed by the unitary matrix V H. While multiplication by a unitary matrix does not change the total signal energy, it does skew the constellation, thereby affecting demodulation properties. It has been suggested in [7] that the squares of the singular values be used as the signal energy values for the two streams. This is indeed an accurate measure of signal power, but we find that the approximation to actual demodulation performance is poor due to the skewing of the constellation. Therefore, alternative methods are needed. An alternative abstraction, suggested in [], models the RBIR or the MMIB directly using knowledge of Λ and V. The RBIR or MMIB is modeled as a polynomial or exponential function of parameters that depend on the condition number of the channel, the actual singular values themselves, and the skew of the unitary matrix V. A significant amount of fine tuning is done to refine the abstraction for the many classifications of the above parameters, giving rise to a reasonably good, albeit complex, model. In another approach [8] [9], it is suggested that the capacity expression for the instantaneous MIMO channel be used to generate an abstraction for the performance of MLD. A receiver that merely uses Maximum Likelihood Demodulation, and not an exhaustive (and prohibitively complex) Maximum Likelihood Detection that also decodes the codeword, is not expected to achieve capacity. In such a case, it is likely that an abstraction based on capacity will overestimate performance. In this paper, we use an alternative decomposition that does not have the skew problem, but has to deal with interference between the streams. Upper and lower bounds are then derived to characterize the performance of the streams given the parameters of the decomposition, and empirical means of these bounds are used to arrive at the abstraction. III. PHYSICAL LAYER ABSTRACTION BASED ON QL AND QR FACTORIZATIONS A suggested method to reduce the complexity of MLD starts with QR factorization of the channel matrix [2]. The search for the ML estimate of the symbols is then done hierarchically. The QR decomposition of the channel is [ ] r11 r H = QR, R = 12 (4) r 22 where Q is a unitary matrix, and R is an upper triangular matrix as shown above. The demodulation process first multiplies the received signal by the inverse of the unitary matrix Q to get x 1 = r 11 s 1 + r 12 s 2 + m 1 () x 2 = r 22 s 2 + m 2 (6) where x =[x 1 x 2 ] T = Q H r and m =[m 1 m 2 ] T = Q H n. The symbol s 2 can be detected using only the second equation above, and can then be applied to the first equation to detect s 1. In most cases, this will give the maximum likelihood estimate of s 1 ands 2. However, this will not work in many cases. For example, when the matrix H is ill-conditioned, the value of r 22 will be low, and the values of s 2 detected from the second equation alone will not be accurate. Thus, a more complicated search will be needed to obtain the maximum likelihood estimate. Nevertheless, it can be seen that a signal power of r 22 2 gives a lower bound on the performance of the

3 detection of s 2, since we can always do better by using the information in the first equation also. Also, the best detection for s 1 is achieved when all its energy can be used to combat noise rather than to combat the interference due to s 2 as per the first equation above. Thus, a signal power of r 11 2 gives an upper bound to the performance of s 1. Another interpretation of this fact is that this is the performance than can be achieved for s 1 if s 2 were perfectly known. It can be shown that r 11 is the maximum absolute value achievable for the first element of any decomposition of H into a product of a unitary matrix followed by another matrix, thus justifying the claim of an upper bound. It can be also shown that r 11 r 22 for any matrix H, thus the above bounds are consistent. The QR factorization of a matrix is well known, but the similar QL factorization, which factorizes a matrix into the product of a unitary matrix and a lower triangular matrix is not as popular 1. Nevertheless, it is easy to derive the QL factorization if a method for the QR factorization is available by flipping the order of the rows and columns of the matrix, calculating the QR factorization, and flipping the rows and columns of the resultant matrices. Using the QL factorization, we obtain H = Q 1 L, where Q 1 is unitary, and L is lower triangular. Thus, we have y 1 = l 11 s 1 + w 1 (7) y 2 = l 21 s 1 + l 22 s 2 + w 2 (8) Again, we can see that using a signal power of l 11 2 gives a lower bound on the performance of the detection of s 1, and using a signal power of l 22 2 gives an upper bound to the performance of s 2. From the above discussion, it is clear that the performance of the detection of s 1 bounded between signal powers of l 11 2 and r 11 2, and the performance of the detection of s 2 is bounded between signal powers of r 22 2 and l Thus, it is reasonable to expect that a physical layer abstraction that uses some kind of an empirical mean value of the above quantities might give reasonable performance. For two quantities a and b, the generalized mean [1], in terms of a parameter k, is defined as: ( a k + b k ) 1 k G(a, b, k) = (9) 2 It can be shown that the arithmetic mean,the harmonic mean, and the geometric mean are special cases of the generalized mean for values of k equal to 1,-1 and (as a limit). In our evaluation, we have attempted to evaluate the parameter k that provides a good match to the observed performance in link simulations for different modulation and coding schemes. IV. PERFORMANCE EVALUATION In this section, we refine the accuracy of the proposed abstraction by tuning the parameter k, and evaluate the performance of the different abstractions proposed. In the first set of simulations, we evaluate the performance of the different abstractions for a non-dispersive 2x2 MIMO channel. The 1 We note that the usage of the QR & QL factorization for MIMO decoding has been suggested in [3]. Performance of PHY Abstractions for QPSK Rate 1/2 Coding: Non Dispersive Channel 1 SNR Error at 1% BLER (db) Fig. 1. Performance of PHY Abstractions for QPSK with Rate 1/2 Coding Performance of PHY Abstractions for 16 QAM Rate 1/2 Coding: Non Dispersive Channel 1 SNR Error at 1% BLER (db) Fig. 2. Performance of PHY Abstractions for 16-QAM with Rate 1/2 Coding modulation and coding schemes evaluated are from the IEEE e system [11], also known as WiMAX: QPSK and 16- QAM with coding rates 1/2 and 3/4, and 64-QAM with coding rates 1/2, 2/3, 3/4 and /6. The performance is simulated for a number of randomly generated channel realizations. The predicted BLER as per the abstraction is calculated for each channel realization, and a simulated BLER is obtained using multiple signal and noise realizations. The simulated BLER versus SINR results are compared to the performance predicted by the abstraction. The difference in SINR for 1% BLER is plotted in Figures 1 through 4 for a few representative modulation and coding schemes. Each point on these plots corresponds to a single channel realization. The figures show that most abstractions do quite well for low condition numbers, but are inaccurate at high condition numbers. In general, the SVD-based abstraction appears to be quite inaccurate with a high spread of SINR error, and typically seems to underestimate the performance. The modified SVD based abstraction, as per the MMIB approach in [], is able to achieve a relatively low spread of SINR error by adapting the parameters of the abstraction with the condition number of the channel and the structure of the unitary matrix involved in the SVD. The proposed abstraction, based on the generalized mean of the diagonal elements of the QR and QL

4 Performance of PHY Abstractions for 64 QAM Rate 1/2 Coding: Non Dispersive Channel 1 1 SNR Error at 1% BLER (db) Fig. 3. Performance of PHY Abstractions for 64-QAM with Rate 1/2 Coding Performance of PHY Abstractions for 64 QAM Rate /6 Coding: Non Dispersive Channel SNR Error at 1% BLER (db) Fig. 4. Performance of PHY Abstractions for 64-QAM with Rate /6 Coding decompositions of the channel matrix appears to achieve a relatively low spread of SINR error. However, this requires a suitable choice of the parameter k that depends on the particular modulation and coding scheme. The performance achieved using the more common arithmetic and geometric means is also shown for comparison purposes. It is desirable and possible to lower the complexity of the abstraction by choosing a single value of k for each modulation in use. This requires a certain accuracy of the abstraction for this single value of k for all coding schemes used with the modulation in a practical environment. The ITU Pedestrian B channel model was chosen as an appropriate dispersive channel to expect in practice [12]. We simulated the performance of the allowable Modulation and coding schemes in WiMAX and obtained an ensemble average of the frame error rate. We also calculated the BLER as per the abstraction over each channel realization and found the average. These results are plotted in Figures through 7 for the different modulation schemes. The abstraction is quite accurate for QPSK and 16-QAM for the corresponding chosen values of k. A single value of k does not seem to provide the same level of accuracy for 64-QAM. Hence, we chose to use a single value of k for 64- QAM that is more accurate for coding schemes with higher Fig.. Fig. 6. BLER BLER Performance of MLD Abstraction for QPSK in PedB Channel Performance of PHY Abstractions for QPSK in a Ped. B Channel Performance of MLD Abstraction for 16 QAM in PedB Channel Performance of PHY Abstractions for 16-QAM in Ped. B Channel rate, and somewhat optimistic in regard to the performance of coding schemes with lower rate. However, it is to be noted that these lower coding rates are expected to be used with lower probability, particularly since a scheme of equivalent rate is available with a more robust modulation. It is indeed possible to overcome this slight inaccuracy with a choice of a separate k for each modulation and coding scheme. We have chosen to use a single k per modulation for reasons of complexity. V. SYSTEM SIMULATION PERFORMANCE The abstraction derived in earlier sections was used in a system simulation to compare the performance of different transmission schemes and receivers for MIMO. The system simulation uses a network of 19 3-sector cells, and models shadow fading with correlation. between base stations. Fast dispersive fading using the 3GPP SCM Urban Macro channel model with a base station antenna spacing of 1 wavelengths, and mobile station antenna spacing of. wavelengths, with an angle spread of 2 degrees at the BS. The inter-site distance is m, and the transmit power is 2W. Full buffer traffic is simulated over the WiMAX air interface with the PUSC permutation. Only the downlink is simulated. TDD is assumed with a DL/UL ratio of 29:18, with 7 symbols being used for signaling overhead on the downlink subframe. Link adaptation is performed with the allowable modulation and coding schemes in WiMAX. Coding rates lower than 1/2

5 BLER Performance of MLD Abstraction for 64 QAM in PedB Channel Rate 2/3 Simulated Rate /6 Simulated Rate 2/3 Abstraction Rate /6 Abstraction TABLE I SPECTRAL EFFICIENCY ACHIEVED FOR VARIOUS MIMO CONFIGURATIONS.ALL THROUGHPUTS AND DATA RATES ARE NORMALIZED IN UNITS OF BITS/SECOND/HZ/SECTOR.NOTE THAT THE USER RATES TAKE INTO ACCOUNT THE FACT THAT 1 USERS ARE PRESENT PER SECTOR. MIMO Receiver Sector 1% User 9% User Encoding Algorithm Throughput Rate Rate Vertical MMSE Vertical MLD Vertical S-MLD Horizontal MMSE-SIC Fig Performance of PHY Abstractions for 64-QAM in Ped. B Channel are achieved using repetition in WiMAX, but are treated as true coding rates in our simulation, which leads to slightly optimistic performance. A BLER target of 1% is used for link adaptation. Chase combining is used for HARQ. An aggressive CQI delay of ms, which is equal to the frame duration in WiMAX, is used for scheduling and link adaptation. The proportional fair scheduler is used. An average of ten users per sector is assumed. The PHY abstraction derived for MLD operated under the assumption of spatially white noise. In a practical system, the interference is certainly not white, and thus we emulate the applcation of a whitening filter to the received signal before calculation of the abstraction. In a simplified abstraction, we assume that the noise white; we term this Simplified MLD, or S-MLD. The SINR s achieved with the various receiver models are calculated using the abstractions, and used to calculate the RBIR. Using the RBIR and the knowledge of the coding rate used, the BLER can be calculated. Link adaptation is achieved by choosing the modulation and coding scheme that maximizes throughput while achieving the BLER target. We compare the system and user throughputs achieved of MIMO with Vertical encoding using the MMSE receiver, the MLD receiver and the S-MLD receiver in Table V. We also compare their performance with that of the MMSE- SIC receiver, which has been used with Horizontal MIMO encoding. The results for sector throughput of the system, and user throughputs at 1 percentile (bad users) and 9 percentile (good users) are shown. The table shows that the MLD receiver provides about a 6% improvement in sector throughput over the MMSE receiver. However, the S-MLD receiver provides a lower sector throughput than the MMSE receiver. This result seems to underscore the importance of noise whitening in an interference environment. As expected, the MMSE-SIC receiver with horizontal encoding outperforms the ML demodulation based receiver with vertical encoding, showing about 7% improvement. VI. CONCLUSIONS We have derived a simple and effective abstraction for maximum likelihood demodulation of MIMO signals. The performance of the abstraction was evaluated via link simulations and found to deliver accurate results. The use of the abstraction in a system simulation gave results for MLD that show an improvement over the MMSE receiver provided noise is whitened before the channel is decomposed. In addition, we find that the improved performance of vertical MIMO encoding, achieved using MLD, still lags the performance of horizontal MIMO encoding with an MMSE-SIC receiver in a system environment. REFERENCES [1] M. K. Varanasi and T. Guess, Optimum decision feedback multiuser equalization with successive decoding achieves the total capacity of the Gaussian multiple-access channel, in Proc. Asilomar Conf. on Signals, Systems, and Computers, Monterey, CA, Nov. 1997, pp [2] Kyeong Jin Kim, Jiang Yue, Iltis, R.A. and Gibson, J.D., A QRD- M/Kalman filter-based detection and channel estimation algorithm for MIMO-OFDM systems, IEEE Transactions on Wireless Communications, Volume 4, Issue 2, March 2, pp [3] R. van Nee, A. van Zelst and G. 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