Multiple-Input Multiple-Output Measurements and Modeling in Manhattan

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL Multiple-Input Multiple-Output Measurements and Modeling in Manhattan Dmitry Chizhik, Jonathan Ling, Peter W. Wolniansky, Reinaldo A. Valenzuela, Fellow, IEEE, Nelson Costa, and Kris Huber Abstract Narrowband multiple- input multiple- output (MIMO) measurements using 16 transmitters and 16 receivers at 2.11 GHz were carried out in Manhattan. High capacities were found for full, as well as smaller array configurations, all within 80% of the fully scattering channel capacity. Correlation model parameters are derived from data. Spatial MIMO channel capacity statistics are found to be well represented by the separate transmitter and receiver correlation matrices, with a median relative error in capacity of 3%, in contrast with the 18% median relative error observed by assuming the antennas to be uncorrelated. A reduced parameter model, consisting of 4 parameters, has been developed to statistically represent the channel correlation matrices. These correlation matrices are, in turn, used to generate matrices with capacities that are consistent within a few percent of those measured in New York. The spatial channel model reported allows simulations of matrices for arbitrary antenna configurations. These channel matrices may be used to test receiver algorithms in system performance studies. These results may also be used for antenna array design, as the decay of mobile antenna correlation with antenna separation has been reported here. An important finding for the base transmitter array was that the antennas were largely uncorrelated even at antenna separations as small as two wavelengths. Index Terms Bell Labs Layered Space Time (BLAST), capacity, correlation, multiple-input multiple-output (MIMO), spatial channel model. I. INTRODUCTION INFORMATION theory research has shown that enormous capacity gains over single-antenna systems can be achieved using multiple transmit and multiple receive arrays (MIMO) by exploiting multipath in the rich scattering wireless channel [1], [2]. Capacity grows linearly with the number of transmit antennas, with fixed radiated power and bandwidth. The capacity gain for MIMO systems assumes a rich scattering environment where the channel between transmit and receive antennas is uncorrelated and entries of the channel transfer matrix are assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables [1]. Practical signaling schemes have been proposed which achieve a large fraction of the Shannon capacity, and one such scheme is called Bell Labs Layered Space Time or BLAST [1], [3]. Given a suitable Manuscript received December 7, 2001; revised October 29, D. Chizhik, J. Ling, P. W. Wolniansky, and R. A. Valenzuela are with Lucent Technologies, Holmdel, NJ USA ( chizhik@lucent.com; jonling@lucent.com; pww@lucent.com; rav@lucent.com). N. Costa and K. Huber are with the Communications Research Laboratory, McMaster University, Hamilton, ON L8S 4L8, Canada ( costasoma.mcmaster.ca; huber@soma.mcmaster.ca). Digital Object Identifier /JSAC wireless channel, BLAST and other MIMO signaling schemes may be the solution for the capacity bottleneck in cellular systems. In [4], the 4 4 median system capacity in suburban New Jersey was reported to be 15 b/s/hz at 20 db system signal-tonoise ratio (SNR), which is 68% of the corresponding capacity of the complex Gaussian i.i.d. channel. Other measurements [5] [7] have also shown high capacity in suburban areas. In this paper, we present the results of a survey of narrowband MIMO channel measurements in midtown Manhattan. The base array was placed at a height of 100 m at the corner of 35th Street and 8th Avenue, and the terminal array was mounted on a van. Many other buildings in the area were above the base transmit array. We evaluate the Shannon capacity of the MIMO channel for a plausible configuration of four antennas for handheld, and sixteen antennas for laptop terminal devices. To observe the relative merit of a given MIMO channel, the measured capacity is compared to the capacity of the i.i.d. complex Gaussian channel. We also compare the measured capacity to the capacity of a single transmitter and single receiver, and of conventional receiver diversity for the same number of receive antennas. We then present a theoretical model to describe the measured data, and show comparisons between model predictions and data. In Section II, we describe our measurement apparatus, experimental design, calibration, and impact of measurement noise on experimental design. As validation of equipment we also show the measurements made in an open parking lot and compare to theoretical capacity for free space and over ground plane propagation. In Section III, we present the measured capacity results. In Sections IV VII, the theoretical model used to describe the data is presented, Sections VIII and IX show the model to data comparison, and Section X contains the conclusions. II. EXPERIMENT DESIGN A narrowband channel sounder was built to measure the complex channel coefficients between a 16-element transmit array and a 16-element receive array. We now describe the equipment, data processing, and calibration. A. Radio Chain The transmitter array radiated a unique continuous wave (CW) tone from each of its antennas. The tones were separated by 2 khz, thus occupying 32 khz. This bandwidth is regarded as narrow, so that any differences between the antennas are due to spatial displacements of the antennas, and not due to frequency selective effects. The receiver consisted of 16 identical radio chains, and a digitizer, which can simultaneously /03$ IEEE

2 322 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003 Fig. 1. Photo of base antenna array and configuration showing polarization. Fig. 2. array. Physical size, spacing, and polarization arrangement of the terminal elements with alternating polarizations and oriented in the vertical plane. Mutual coupling between adjacent terminal array elements was measured and found to be less than 30 db, which has negligible effect on capacity. The mobile array was mounted 1.5 meters above ground. Each element was a slot antenna with half power beam width of 70, and gain of 4 dbi. The terminal array had the same type of elements as the base array. Sufficient spacing between array elements is important for low correlation and high capacity. If the base array is mounted on a high building or tower, little scattering may be available at the base, thus base elements must be widely spaced to resolve a narrow angular spread. In [8], it is shown for a high base array with 2 root-mean-square (rms) angle of arrival spread that a spacing of four wavelengths, with dual polarized antennas, is sufficient to achieve 80% of the median capacity of an uncorrelated Gaussian channel. For the terminal array, we expect significant local scattering, to produce wide angular spread, so one-half wavelength was judged sufficient for low correlation. C. Impact of Measurement Noise on Measured Capacity The capacity of the MIMO system with transmitters and receivers has been derived in [1] as sample all receivers. To identify the tones at the receiver, the fast Fourier transform (FFT) was used during post processing. Channel matrices were taken every 1.5 ms which at speeds of 20 mi/h corresponds to an matrix measurement every 0.2 of a wavelength. Ordinarily, it is considered that the channel decorrelation distance is 0.5, derived for omnidirectional power spectrum at the array. As these measurements used directional antennas, it is expected that the narrower angular spread leads to larger coherence distances. Indeed, it was found that the channel remained unchanged over the duration of the measurement period. Both transmitter and receiver were frequency locked to GPS. Noise measurements were taken continuously outside the transmit band. The carrier frequency was 2.11 GHz, and transmit power was 23 dbm per element. The uncompensated gain variations of transmit and receive amplifiers of 0.8 db may be modeled as multiplicative noise, which produces less than 1% variation in measured capacity. B. Antenna Arrays The base array is a horizontal linear array of eight pairs of antennas, shown in Fig. 1. Each pair consists of a vertical and horizontally polarized radiating slot element. The length of the entire array is 3 m, which is twenty wavelengths at 2 GHz, corresponding to typical spacing of diversity antennas of cellular base stations. The terminal array is rectangular, Fig. 2, with approximately one-half wavelength between neighboring b/s/hz (1) We define the average SNR, also called the system SNR, as, where is the average received power per receive element. Note that is an arbitrary parameter chosen to be typical of the intended system conditions and unrelated to the measurement SNR value which is maintained at a high level to ensure accurate results. is the normalized channel matrix, whose entries have unit average power, is the power of additive white Gaussian noise (AWGN), is an identity matrix of dimension, and denotes the complex conjugate transpose. Each measured is entered into (1). is set to 10 db for every as if the transmitter array was power controlled to maintain this target SNR value. Measured s were taken at sufficiently high measurement SNR, denoted as SNR, to guarantee capacity error of less than 10%. It is useful to compare the capacities of a MIMO system and an optimum combining receive diversity system in a Rayleigh i.i.d. channel. For a system, capacity is achieved by using an optimum combining system and is given by [2], where is a chi-squared random variable with 2 degrees of freedom. When is doubled, the median capacity improves by 1 b/s/hz, and the tails of distribution become a smaller fraction of the median. To verify system accuracy, a keyhole connection was made, by combining all the outputs from the transmitter to a combiner, through a single cable and attenuator, and then through a

3 CHIZHIK et al.: MIMO MEASUREMENTS AND MODELING IN MANHATTAN 323 Fig. 3. Measured and predicted capacity for a horizontal transmit array in a parking lot. In the predictions, the receiver element locations were adjusted slightly to include edge diffraction. splitter, which connects to all receivers. The measurement signal to noise ratio was 40 db. At 20 db, the theoretical keyhole capacity [11] is given by b/s/hz, with one effective degree of freedom (EDOF) [9] and 16-fold receiver diversity. The measured capacity was b/s/hz and the number of EDOF was measured to be 1.06, computed as in [9]. D. Outdoor Calibration Measurements were first collected in a very simple environment, where the results may be compared to predictions. Propagation path loss and capacity are used to compare measured and predicted results. The transmitter and receiver were taken to an empty parking lot to obtain measurements in an environment for which a theoretical model is expected to be accurate. In this case, propagation in free space over a ground plane was used to predict path loss and capacity, as in [18]. The area where the measurements were taken had no obstructions in the first Fresnel zone. The mobile receive antenna array was mounted either on a tripod or on a side of a van to assess the impact of the vehicle. The base transmit array was mounted both vertically and horizontally. The midpoint of both arrays was always at a height of about 1.5 m above the ground. When base transmit array was mounted vertically, the edge of it was resting on the ground. Only the results for the horizontally oriented array are reported here, with similar results obtained for the vertically mounted transmit array. The capacity was computed at a high value of 20-dB system SNR to highlight possible deviations of the system due to spurious effects, such as noise. The capacity of actual field measurements were computed at 10 db system SNR. The measured and theoretically predicted capacities are shown in Fig. 3. The theoretical prediction was done using ray tracing that included both direct and ground reflected rays [10]. In the theoretical calculation, the locations of mobile antenna elements were adjusted slightly to account for diffraction from the edges of the antenna back plane. Both in free space and in the presence of a ground place, capacity is very high when arrays are close, and then drops off rapidly with distance, which may be attributed to flattening of the wave fronts. The capacity is computed at a fixed system SNR, thus increased path loss does not affect the capacity here. It may be observed that the measured and predicted capacities agree within 6%. It was also found that, not accounting for antenna back plane diffraction, results in 15% error in capacity in free space conditions. This error is expected to decrease in scattering environments, where the relative impact of weak spurious scattering is smaller. It is useful to consider some limiting values of capacity. The expected range of capacities for various channel conditions at the system SNR of 20 db falls between two limiting cases. First, the maximum capacity correspond to the case where the singular values of the matrix are all equal. For this case the capacity is given by b/s/hz (55.4 b/s/hz at 10 db ), where it is assumed that. At the lower end of the range, we consider the environment where the elements of the matrix corresponding to antennas with the same polarization degenerate to a single channel. The entire matrix then has only two communication channels corresponding to the two states of polarization. Such a channel arises for example in free space at large distances between the transmitter and receiver arrays. The matrix of this channel may be represented by a sum of two eight-element dyads, as there are eight receivers and eight transmitters for each state of polarization and there is no cross-polarization coupling. The capacity is now given by b/s/hz (12.7 b/s/hz at 10 db ). If the environment is such that even the two states of polarization are collapsed into a single keyhole (dyad) channel, the minimum capacity becomes b/s/hz, (7.3 b/s/hz at 10 db ). Both the single and the double keyhole cases above did not include possible beam forming at the transmitter, as this would require additional calibration and knowledge of the channel at the transmitter. In the case of a fully scattering channel where the matrix entries are distributed as complex Gaussian variables, the median capacity is 88 b/s/hz for, with a very small spread around the median. Note that all outdoor measurements and theory remained within the limiting cases of 16 equal eigenvalues and 2 equal nonzero eigenvalues. III. MEASURED CAPACITY RESULTS The base array was placed at a height of 100 m facing east at the corner of 35th Street and 8th Avenue in the dense urban environment of midtown Manhattan, and the terminal array was mounted on the side of a van, at a height of 1.5 m. The mobile array antennas were, therefore, oriented perpendicular to the street. Some matrices were collected. Fig. 4 gives an overview of the measurements by showing drive locations, and the gray scale capacity at 10 db using all 16 antennas of both terminal and base arrays. The lines from the base denote the 6-dB beam width of the base antenna elements. Only locations where the measured SNR was above 10 db are shown. The measured capacity is seen to be high for most locations within the main lobe of the base antenna, except for the

4 324 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003 Fig. 4. Measured capacity of a system at =10dB. two radial streets, which may be considered nearly line of sight streets. In this study, capacity of sub arrays of 2 2 and 4 4 were also considered, as handheld devices of current interest may be limited to a small number of antennas. It was found that capacity in b/s/hz is correlated to (distance) with a correlation coefficient of The small positive value of the correlation coefficient implies that the capacity increases slightly with larger distance between the base and the terminal. Note that here the capacity is computed at a fixed system SNR, as discussed later. The scattering ring model [12] predicts decreasing capacity with distance. The capacities from all measurement locations that fall within the antenna element pattern are plotted in Fig. 5 as complementary cumulative distribution functions (CCDF) for 2 2, 4 4, and systems, at of 10 db. In computing the capacity of a 4 4 system, four elements of the mobile terminal that were closest to each other were chosen, which included two elements of each polarization. At the base array two pairs of antennas separated by two wavelengths were chosen. Each pair included both polarizations. To compute the capacity of a 2 2 system, a pair of closest spaced dual polarized antennas are selected from both the base and terminal Fig. 5. are theoretical predictions for full scattering channel. CCDF for all measurements 2 2 2, 4 2 4, and Dashed lines arrays. The dashed lines are reference theoretical capacities for the complex Gaussian i.i.d. (full scattering) channel for

5 CHIZHIK et al.: MIMO MEASUREMENTS AND MODELING IN MANHATTAN 325 Fig. 6. CCDF of capacity for different transmit element spacing, the same array size and. It may be observed that for a system, 90% of capacities are above 23 b/s/hz. The median capacities of the 4 4 and systems were found to be 90% and 78%, respectively, of the corresponding capacities of complex Gaussian i.i.d. channels. For reference, median capacities in Rayleigh i.i.d. channels of 1 1, 1 2, 1 4, and 1 16 systems are 3.5, 4.4, 5.4, and 7.3 b/s/hz, respectively, at 10-dB system SNR. Note that median capacities include the effect of the power gain obtained from multiple receive antennas, but not the diversity gain of using multiple antennas. We now investigate for a 4 4 system, whether capacity will increase by increasing the antenna element spacing at the base. Fig. 6 shows the CCDF of capacity for a pair of co-polarized base antennas separated by 2, 6, and 20 wavelengths. Only a small increase in capacity with spacing is observed, and the base antennas are nearly uncorrelated at two wavelengths. Future measurements may determine if it is possible to reduce the spacing of base antennas further without significant loss in capacity. For system level modeling, the relationship between capacity and shadow fading is important. One may speculate that in a faded location the dominant paths are attenuated and scattering is richer leading to higher capacity. To examine this relationship, a linear least mean squares fit is done to average path loss with distance: (distance). The local average power is computed over about 4 m. The average received power predicted by this relationship is 10-dB higher at 1 km than the prediction given by the COST-231 Hata model [13]. Excess path loss is defined as loss, in decibels, relative to this linear relationship. The excess path loss is often identified as shadow fading loss. In Fig. 7, the measured capacity at db and excess path loss are plotted for each location as a scatter plot. We note that channel tends to become more rich with increasing excess path loss, with a correlation coefficient of 0.7 between capacity and excess path loss. The apparent improvement in channel richness at shadowed locations does come at the expense of reduced Fig. 7. Capacity versus excess path loss. SNR available at those locations. In a real system, the operating point SNR is usually heavily influenced by interference. To predict the capacity in a real system, matrices may be generated as described below and normalized to provide SNRs consistent with cell layout, scheduling, etc., as well as path loss. IV. MODEL DESCRIPTION In [8], a model has been proposed that describes the MIMO channel matrix as a Gaussian matrix with correlated entries. To test this model the correlation matrices are computed from our measured data and used to spatially filter a synthetic complex Gaussian i.i.d. entries matrices. The resulting CDF of channel capacity may be compared to the CDF of measured capacity to assess the adequacy of the correlated Gaussian model. We note that the most general correlation relations may be impossible to get for a large (i.e., 16 16) matrix, as the required time and space required by our measurement equipment to collect the requisite data exceeds the expected coherence times and space over which the channel statistics remain constant. So, a simpler model is formulated, again based on [8]. In this model, it is assumed that receivers are only correlated with receivers and transmitters only correlated with transmitters. In line with this simpler hypothesis, the data is processed to obtain these correlations. As in the more general case, the synthetic data set with the same correlations will be generated. The CDF of synthetic data set capacity will be compared to the measured capacity CDF to assess model accuracy. The most general type of correlation between the various elements of may be represented as a four-dimensional (4-D) tensor, so that a general matrix with correlated entries may be obtained from the matrix which has uncorrelated complex Gaussian entries using the operator expression where each entry is (2) (3)

6 326 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003 In this paper, a special case of the above relationship is considered, i.e., modeling the correlation among receiver and transmitter array elements independently from one another. The underlying justification for this approach is to assume that only immediate surroundings of the antenna array impose the correlation between array elements and have no impact on correlations observed between the elements of the array at the other end of the link [8], [14]. One way to effect this type of antenna signal correlation is to pre- and post-multiply the channel transfer matrix by the appropriate transformation matrices The above limit is valid if the complex spectrum is an ordinary function, which suffices for physical sources. Similar results may be obtained if contains distributions, such as a discrete plane wave. In that case, the spatial average (7) is still zero provided the averaging is done over an area. Correlation between received signals on the two antennas may be similarly computed as (4) where and are the correlation matrices of transmit antennas and receive antenna arrays, respectively, and the square roots are matrix square roots. For example, the element of the receive antenna covariance matrix is the covariance between receive antennas and. If (2) holds, the general correlation matrix used in (2) is related [19] to the transmit and receive correlation matrices through, where is the Kronecker product. In the special case of no correlation between antennas of orthogonal polarization, the covariance matrix, thus, takes on a block diagonal form The above allows for the spatial correlations to be different for the two states of polarization. Given antennas in the array, the total covariance matrix has the dimension, while the submatrices have the dimension. It is the goal of this work to measure the correlation matrices described above and compare the resulting capacity predictions with measured capacities. V. SPATIAL STATISTICS AND ANGULAR SPECTRA The local spatial statistics of the received (or transmitted) signal field may be expressed in terms of the angular spectra. The received signals and on two antennas located at and, respectively, may be expanded in terms of plane waves where the wave vector is assumed to lie in the horizontal plane, and are the complex amplitudes of the plane waves arriving at the receive antennas. The mean value of the signal field may be defined as the spatial average of the received signal, computed over a spatial aperture of length (5) (6) (7) where is the power angular spectrum as a function of angular coordinate, is the antenna separation, is the average spatial coordinate for the two antennas, and is the average wave vector for the two angular spectra in (8). The limits of the integral over the angular coordinate are set from 0 to 2, thus excluding inhomogeneous waves, which are assumed to decay away from the source so as to be negligible. VI. DATA PROCESSING FOR ANTENNA CORRELATIONS The MIMO channel measurement data were processed to get the separate antenna correlations, over a small spatial scale, say 4 m (28.2 wavelengths), where such statistics are assumed to remain constant. At 10 m/s the van would have moved 4 m in 0.4 s. Collecting this data at a rate of 1.56 ms per matrix, results in 256 matrices that will be processed as a single block. Assuming the data becomes essentially independent for a spatial separation of 1.5 wavelength, this would produce on the order of 20 independent measurements of instantaneous correlation. VII. MEASURED POLARIZATION PROPERTIES A. Measured Cross-Polarized Coupling Cross-polarized coupling was computed using (8) Xpol (9) where the order of the subscripts of the elements of corresponds to receivers and transmitters, respectively. The subscripts and refer to vertical and horizontal polarizations, respectively. The matrix elements used in (9) corresponded to two transmitters with different polarizations, with averaging carried out over all receivers fitting the indicated subscripts as well as over time. The median cross-polarization is found to be 3 db, which may be compared to 6 db reported in Lee and Yeh [15]. The

7 CHIZHIK et al.: MIMO MEASUREMENTS AND MODELING IN MANHATTAN 327 higher value of cross polarization reported here may be a consequence of pointing the mobile antenna perpendicular to the street, thus encouraging more scattering, as compared to receiving signals that propagate along the street. B. Measured Cross-Polarized Correlation Normalized correlation coefficient is computed for two transmitters with orthogonal polarizations, using signals received by vertically polarized receivers. A separate calculation is done using horizontally polarized receivers. Only one sense of polarization for receivers is used at a time so as to average quantities from presumably the same statistical distribution (10) The averaging indicated by the brackets above is carried out both over all receivers of a particular polarization, as well as over time. As there are eight receivers for each polarization and 256 matrices are taken sequentially in time and used for processing, there are measured quantities averaged in (10). Similarly, the normalized correlation coefficient for two cross-polarized receivers is computed using signals from the vertically polarized transmitters. A separate calculation is done using horizontally polarized transmitters. As it was found the correlation coefficient between two crosspolarized signals is predominantly below 0.5 (and mostly below 0.2), the cross-polarized signals may be assumed to be uncorrelated. This is justified by consulting [8], where 80% of complex Gaussian i.i.d. capacity was found for covariance values of less than 0.55, at 4 separation for a Gaussian shaped spectrum with 2 rms angular spread. The covariance will be further reduced owing to the fact that cross-pol variances are used to scale the normalized correlation coefficients to obtain the covariance that is used to compute the correlated matrices. Note that as the cross-polarized antennas are separated spatially, the correlation coefficient is affected by both polarization and spatial statistics. VIII. MODELING USING MEASURED CORRELATIONS AT THE TRANSMITTER AND AT THE RECEIVER We now compute covariance matrices separately between co-polarized transmitters and receivers and use them to spatially filter synthetically generated complex Gaussian i.i.d. matrices, as suggested in (4). The capacity of measured and synthetically generated matrices is shown in Fig. 8, for a particular data run along Park Avenue North. While the agreement between the measured and modeled capacities may be judged to be generally good, the remaining disagreement may be attributed to a number of factors, such as errors in estimating the correlations from the data, as well as limitations of the model itself, which represents the matrix as a zero-mean Gaussian process with separable correlations. As before, the predictions are obtained using measured correlation matrices. While the two capacities generally agree within a few percent, there is a deep drop in measured capacity at a location corresponding to 43 on the axis. This data was collected while passing through a tunnel on Park Avenue. The point is remarkable in that the measured correlations have remained sufficiently low, as indicated by the Fig. 8. Capacities of measured and synthetic H matrices, generated using measured correlation matrices at the transmitter and receiver arrays. Correlations between cross-polarized antennas are assumed to be zero. Fig. 9. Cumulative distribution of the relative error between the capacities of measured H matrices and synthetic H matrices created from measured correlations at the transmitter and receiver arrays. Cross-polarized antennas are assumed to be uncorrelated. high values of predicted capacity. That location is, therefore, akin to a keyhole [11], in that the correlations do not capture the relevant behavior of the channel. Fortunately, such points have been found in this work to be rather rare, as evidenced by the good agreement between the measurements and predictions based on correlation matrices. We now quantify the data-model agreement by plotting the cumulative distribution of the relative error between capacities computed from measured matrices and synthetic matrices generated using the measured correlation matrices, as in (4). The relative error is computed at every location as Relative Error (11) The cumulative distribution of the relative error is plotted in Fig. 9, as a curve labeled exact correlation. We note that the median relative error is about 3%, while the 90th percentile error is 18%. We, therefore, conclude that correlation matrices

8 328 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003 Fig. 10. Measured and fitted correlation coefficient for vertically polarized mobile receivers at one location. adequately represent the spatial statistics of the channel and that the procedure of generating spatially correlated Gaussian channel matrices gives capacities in close agreement with measurements. For comparison, the relative error between measured capacities and those generated using uncorrelated Rayleigh i.i.d. matrices is also shown in Fig. 9. We see that in the case of independent antenna assumption, the median error is 18%, while the 90th percentile error is 93%. IX. REDUCED CHANNEL MODEL In the previous sections, it was found that the correlation matrices at the base transmitter and at the mobile receiver are representative of the spatial characteristics of the channels. For a MIMO system, the transmitter and receiver correlation matrices are 256 elements each. It would be very useful to reduce the number of parameters needed to describe the correlation matrices. For example, in [8], the correlation matrices were derived using a particular angular spectrum assumption, with angular spectrum being described by a single rms angular spread. Generally, the correlation between antennas is expected to decrease as the antennas are separated. In Fig. 10, the measured correlation coefficients between vertically polarized mobile receive antennas are plotted as a function of antenna separation at one particular location. The total separation between antennas was used without regard to whether the antennas were separated vertically or horizontally. Thus, all co-polarized pairs shown in Fig. 2 were used. Also plotted is a model correlation coefficient, which was chosen as an exponentially decaying function, with the decay rate fitted to the correlation data using the least mean square deviation criteria. The correlation coefficient between two spatially separated (and uncoupled) antennas and is related to the power angular spectrum by a Fourier transform relationship (12) Fig. 11. Measured and fitted correlation coefficient for vertically polarized base transmitters at one location. Using the exponential form of the correlation coefficient used to represent the data (13) we find that the corresponding angular spectrum is, where is the wave vector component along the axis joining the antennas. The 3 db point of the above angular spectrum is given by, where is the angle of incidence at the array, measured from the normal to the array. The above relationship can be used to determine the angular spread at the remote antenna. The above discussion on the modeling of the correlation and angular spectrum should be treated as an attempt to represent the data in a compact form. At any particular location the angular spectrum may not be generally expected to be monotonic. It should also be noted here that as the remote antenna array was mounted on the side of the van during measurements, the remote antenna elements had their main lobe pointed perpendicular to the street. The angular spread thus obtained would, therefore, be appropriate to the specific condition of the remote antenna pointing across the street. Similar findings have been obtained in [17], where the angular spectra at the mobile were computed from measured Doppler spectra of the received signals. Measured correlation coefficients for base transmit antennas at one location are shown in Fig. 11. As we note that there is no clear decrease of the base transmitter correlation coefficient with antenna separation, it was decided (based on many such plots) to fit instead a constant value to represent the transmitter correlation. Lack of a clear trend of the base antenna correlation coefficient with antenna separation may be attributed to multiple bundles of rays arriving from different directions. Similar findings have been reported in wideband measurements collected in Newark [16]. The value of the constant transmitter correlation coefficient was chosen according to the least mean square deviation criteria. The resulting correlation coefficient between two co-polarized transmit antennas and is given by (14)

9 CHIZHIK et al.: MIMO MEASUREMENTS AND MODELING IN MANHATTAN 329 Both transmitter and receiver correlation parameters vary from location to location, so that these correlation parameters form a distribution. These parameters may be used to generate transmitter and receiver correlation matrices. The resulting model has the advantage over the actual measured correlation matrices in that only four parameters are needed to describe it. We now quantify the error incurred when using the reduced parameter correlation model to spatially correlate the matrices. Note that the entries of the covariance matrices computed from data are, in general, complex. The above parametric fit was done only to the amplitude of the measured correlation parameters. The approximate covariance matrices computed from these parameters are therefore strictly real. The approximation, therefore, consists of both omitting the phase of the covariance matrices and of representing the amplitude of the covariance in terms of a single parameter. In Fig. 9, a cumulative distribution of the relative error is plotted between the capacities computed from measured matrices and capacities computed from synthetic matrices, with correlations generated by the four-parameter model. Note that in this comparison, the parameters used for each location were those computed from the measured data at that location. This is done solely to test the adequacy of the reduced parameter model to represent the data. The median error is 3% and that at the 90th percentile error is 30%, which is comparable to what was found when using measured correlation matrices. Now, we proceed to compute capacity using synthetically generated matrices. The spatial statistics of each matrix, defined by the four correlation parameters, vary from location to location. This would correspond, for example, to having different angle spreads at different locations. It was decided to take the measured distributions of the correlation parameters and view them as a correlated set of Gaussian variables, defined by a 4 1 vector mean and a 4 4 covariance matrix (15) where is a real Gaussian random variable with zero mean and unit variance,, and the square root is the matrix square root. The values of the mean set and the covariance matrix obtained from the data are tabulated below (16) The correlation parameters and are the decay parameters (per wavelength) for the vertically and horizontally polarized mobile antennas, respectively, while and are the correlation for the vertically and horizontally polarized base antennas. Note that the model above shows no dependence of base antenna correlation on the antenna separation. The measurements were collected with the smallest separation between base antennas of two wavelengths. The model does not imply, therefore, that base antennas have low correlation when the antennas are brought closer than two wavelengths. Equations (15) and (16) now give the recipe for generating the correlation parameters from a 4-D vector distribu- Fig. 12. Cumulative distributions of measured and predicted capacities. tion. For the receive arrays, each entry of the receive correlation matrix is given by (13), corresponding to the correlation between two co-polarized receive antennas and separated by the distance. For transmit antennas, the correlation coefficient between two co-polarized antennas is given by (14). The resulting ensemble of matrices, generated using (4), has two associated statistical spatial scales: small scale variations which are generated for particular correlation matrices and choosing different realizations of, and large scale variations which result from generating the correlation matrices, defined by the correlation parameters, given by (16). The cumulative distribution of capacities, computed from synthetically generated matrices, are compared in Fig. 12 to the distribution of measured capacities. The plotted distribution represents the large-scale variation in scattering richness. Each point in the CDF of measured capacity is a median capacity measured from 0.4 seconds of data collection. As the van speed varied from 6 m/s to 12 m/s, this corresponds to median capacity reported every 2.4 to 4.8 m. Also shown is the CDF of the capacity generated from the actual set of correlation parameters deduced from data, which gives somewhat closer agreement to data than capacities generated using (15). Both predictions uniformly predict higher capacity than measurements, corresponding to the fact that the nonseparable correlations, as well as non-gaussian statistics are not modeled. The reduced parameter model overpredicts the median measured capacity by about 10%. Similar accuracy is achieved at most other outage levels. X. CONCLUSION Narrowband MIMO measurements using 16 transmitters and 16 receivers were carried out in Manhattan. Measured median capacities of the full array were found to be 35 b/s/hz at 10 db system SNR, which is within 80% of the Rayleigh i.i.d. channel capacity. Subarray capacities for a 2 2 and 4 4 systems were found to be 5.5 and 10 b/s/hz, respectively, which is within 90% of the corresponding Rayleigh i.i.d. capacities. Correlation model parameters are derived from data and used

10 330 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003 to predict capacities, which are then compared to measured capacities. It was found that the spatial MIMO channel statistics are well represented by the separate transmitter and receiver correlation matrices, with a median relative error in capacity of 3%, in contrast to the 18% median relative error observed by assuming the antennas to be uncorrelated. A reduced parameter model, consisting of four parameters, has been developed to generate correlation matrices. When these correlation matrices are used to impose spatial correlation on the synthetic matrices, the median relative error in capacity between measured and synthetic channels was found to be 3%. Furthermore, the set of four covariance parameters are, in turn, modeled as a correlated Gaussian random vector with four elements. Its statistics are completely described by a four-element vector mean and a 4 4 covariance matrix. The resulting set of 20 parameters is used to generate different realizations of the covariance matrices. These covariance matrices are, in turn, used to generate correlated matrices with capacities that are consistent with measurements collected in Manhattan. The spatial channel model reported allows simulations of matrices for arbitrary antenna configurations, consistent with the NYC measurements. These channel matrices may be used to test receiver algorithms in system performance studies. These results may also be used for antenna array design, as the decay of remote antenna correlation with antenna separation has been reported here. An important finding for the base transmitter array was that the antennas were largely uncorrelated even at antenna separation as small as two wavelengths. [8] D. Chizhik, F. Rashid-Farrokhi, J. Ling, and A. Lozano, Effect of antenna separation on the capacity of BLAST in correlated channels, IEEE Commun. Lett., vol. 4, pp , Nov [9] N. Amitay, M. J. Gans, H. Xu, and R. A. Valenzuela, The effects of thermal noise on the accuracy of measured BLAST capacities, Electron. Lett., vol. 37, no. 9, pp , Apr. 26, [10] P. F. Driessen and G. J. Foschini, On the capacity formula for multiple input-multiple output wireless channels: A geometric interpretation, IEEE Trans. Commun., vol. 47, pp , Feb [11] D. Chizhik, G. J. Foschini, and R. A. Valenzuela, Capacities of multi-element transmit and receive antennas: Correlations and keyholes, Electron. Lett., vol. 36, no. 13, pp , June 22, [12] W. C. Jakes, Ed., Microwave Mobile Communications. New York: Wiley, [13] EURO-COST231, Urban Transmission Loss Models for Mobile Radio in the 900 and 1800 MHz Bands: The Hague, Revision 2, Sept [14] K. I. Pedersen, J. B. Andersen, J. P. Kermoal, and P. Mogensen, A stochastic multiple-input multiple-output radio channel model for evaluation of space-time coding algorithms, in Proc. 52nd IEEE Vehicular Technology Conf., VTS Fall VTC 2000, vol. 2, 2000, pp [15] W. C.-Y. Lee and Y.-S. Yeh, Polarization diversity system for mobile radio, IEEE Trans. Commun., vol. COM-20, pp , Oct [16] A. L. Moustakas, S. Arunachalam, K. H. Wu, and H. Heller, Wideband spatio-temporal characterization of the urban PCS channel, presented at the Int. Symp. 3G Infrastructure and Services, Athens, Greece, July [17] H. Xu, M. J. Gans, D. Chizhik, J. Ling, P. Wolniansky, and R. A. Valenzuela, Spatial and temporal variations of MIMO channels and impacts on capacity, in Proc. IEEE Int. Conf. Communications (ICC), vol. 1, New York, May 2002, pp [18] P. Kyritsi and D. Chizhik, Capacity of multiple antenna systems in free space and above perfect ground, IEEE Commun. Lett., vol. 6, pp , Aug [19] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems, IEEE Trans. Commun., vol. 48, pp , Mar ACKNOWLEDGMENT The authors gratefully acknowledge many insightful discussions they have had with M. Gans, A. Yeh, N. Amitay, H. Xu, I. Korisch, D. Samardzija, J. Aaron, T. Sizer, D. Taylor, M. MacDonald, R. Storz, and G. Rittenhouse. REFERENCES [1] G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multiple antennas, Bell Labs Tech. J., vol. 1, no. 2, pp , Autumn [2] G. J. Foschini and M. Gans, On the limits of wireless communications in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, no. 3, p. 311, Mar [3] G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolniansky, Simplified processing for wireless communication at high spectral efficiency, IEEE J. Select. Areas Commun., vol. 17, pp , Nov [4] T. Sizer, D. Taylor, W. MacDonald, R. Storz, C. Tran, D. Mumma, M. Gans, N. Amitay, H. Xu, R. Valenzuela, and G. Rittenhouse, Measurement of system capacity using BLAST for mobile applications, Lucent, Holmdel, NJ, Internal Tech. Memorandum. [5] C. Martin, J. Winters, and N. Sollenberger, Multiple-input multipleoutput (MIMO) radio channel measurements, in Proc. IEEE Vehicular Technology Conf., Fall 2000, pp [6] M. Gans, N. Amitay, Y. S. Yeh, H. Xu, T. C. Damen, R. A. Valenzuela, T. Sizer, R. Storz, D. Taylor, W. M. MacDonald, C. Tran, and A. Adamiecki, Outdoor BLAST measurement system at 2.44 GHz: Calibration and initial results, IEEE J. Select. Areas Commun., vol. 20, pp , Apr [7] D. Chizhik, J. Ling, P. Wolniansky, R. Valenzuela, N. Costa, and K. Huber, BLAST radio propagation measurements with 16 element arrays in suburban New Jersey, Lucent, Holmdel, NJ, Internal Tech. Memorandum. Dmitry Chizhik received the Ph.D. degree in electrophysics from the Polytechnic University, Brooklyn, NY. His thesis work was in ultrasonics and nondestructive evaluation. In 1991, he joined the Naval Undersea Warfare Center, New London, CT, where he did research in scattering from ocean floor, geoacoustic modeling and shallow water acoustic propagation. In 1996, he joined Bell Laboratories, Lucent Technologies, Holmdel, NJ, working on radio-propagation modeling and measurements, using deterministic and statistical techniques. His recent work has been in measurement, modeling, and channel estimation of MIMO channels. The results are used both for determination of channel-imposed bounds on channel capacity, as well as for optimal antenna array design. His research interests are in acoustic and electromagnetic wave propagation, signal processing, and communications. Jonathan Ling received the B.Sc. degree in electrical engineering from Rutgers University, Piscataway, NJ, in 1992, and the M.Sc. degree in computer science in 1999 from Stevens Institute of Technology, Hoboken, NJ, where he is currently working toward the Ph.D. degree in electrical engineering. He joined the Wireless Communications Research Department of Bell Laboratories, Holmdel, NJ, in He conducted radio propagation measurements and modeling in microcellular environments and expanded the WiSE ray-tracing model to outdoor scenarios. Recently, he completed a 16 antenna by 16 antenna MIMO measurement in Manhattan. He has also been involved in evaluating wireless system performance through simulation.

11 CHIZHIK et al.: MIMO MEASUREMENTS AND MODELING IN MANHATTAN 331 Peter W. Wolniansky received the B.S.E.E. and M.S.E.E. degrees from Boston University, Boston, MA, in 1983 and 1986, respectively. He performed system tests on the HAWK missile system for Raytheon Corporation, Bedford, MA, from 1983 to 1984 and studied optical data storage for Sony Corporation, Tokyo, Japan, from 1986 to Since 1988, he has been with Lucent Technologies, Bell Labs Innovations, Holmdel, NJ, as a Member of Technical Staff. His duties have included system engineering of modem networks, millimeter radio design, and propagation studies. His recent activities include multichannel radio design and field studies using antenna arrays. Nelson Costa received the B.A.Sc degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. He is currently working toward the Ph.D. degree at McMaster University, Hamilton, ON, Canada. He is constructing a wideband MIMO software defined radio for use in the characterization of the wideband MIMO channel. Reinaldo A. Valenzuela (M 85 SM 89 F 99) received the B.S. degree from the University of Chile, Santiago, and the Ph.D. degree from the Imperial College of Science and Technology of the University of London, London, U.K. At Bell Laboratories, Holmdel, NJ, he studied indoor microwave propagation and modeling, packet reservation multiple access for wireless systems and optical wavelength division multiplexing (WDM) networks. He became Manager of the Voice Research Department at Motorola Codex, Boston, MA, where he was involved in the implementation integrated voice and data packet systems. On returning to Bell Laboratories, he led a multidisciplinary team to create a software tool for wireless system engineering (WiSE), now in widespread use in Lucent Technologies. He is interested in microwave propagation measurements and models, intelligent antennas, third-generation wireless system and space time systems achieving high capacities using transmit and receive antenna arrays. He has published over 80 papers and has 12 patents. Dr. Valenzuela is the Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS and the IEEE TRANSACTIONS ON WIRELESS. He received the Distinguished Member of Technical Staff award and is Director of the Wireless Communications Research Department. Kris Huber received the B.Sc. degree from the Department of Physics, and the M.Eng. degree from the Electrical and Computer Engineering Department, in 1997 and 2000, respectively, McMaster University, Hamilton, ON, Canada, where he is currently working toward the Ph.D. degree in the Electrical and Computer Engineering Department. He currently works in the Adaptive Systems Laboratory at McMaster University under Dr. Simon Haykin. His research interests include Bayesian learning, tracking, multiple antenna systems, and signal processing; all applied with a particular emphasis toward wireless communication systems.