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Multiple-Input Multiple-Output Fixed Wireless Radio Channel Measurements and Modeling Using Dual-Polarized Antennas at 2.5 GHz Vinko Erceg, Senior Member, IEEE, Pitchaiah Soma, Daniel S. Baum, and Severine Catreux
Abstract—This paper presents outdoor propagation measurements together with derivative analysis, modeling, and simulation of the 2 2 fixed wireless multiple-input multiple-output (MIMO) channel. Experimental data were collected in the suburban residential areas of San Jose, CA, at 2.48 GHz by using dual-polarized antennas. Measurement results include the estimation of path loss, Rician -factor, cross-polarization discrimination (CPD), correlation coefficients, and the MIMO channel capacity. An elaborate -factor model that assumes variation over location, time, and frequency is developed. Distance-dependent CPD models of the variable and constant signal components are proposed. A generalized 2 2 MIMO channel model is then derived based on the correlation among the path loss, the copolarized -factor, and the CPD’s distribution of the constant and scattered signal components. Finally, the MIMO channel response is simulated using the newly developed model, and results are found to be well in agreement with measurements.
data and simplistic channel models exist for a MIMO type configuration, where the channel coefficients are arranged in a matrix [2]–[7]. Although certain assumptions can be made on the properties of this channel type, measurements and modeling are important to determine the actual capacity of MIMO systems.
Index Terms—Capacity, correlation coefficient, , measurements, multiple-input multiple-output (MIMO), path loss, propagation modeling, Rician-factor, XPD.
It has been shown that the use of polarization for spatial multiplexing-based MIMO systems can lead to significant performance improvements [7]. A thorough understanding and accurate modeling of the cross-polarization discrimination (CPD) and -factors is necessary because these parameters have a significant influence on the MIMO channel matrix condition number, and consequently capacity. For locations with high -factors, the capacity of a 2 2 MIMO system (two transmit and two receive antennas) was found to be low when using the single polarization configuration, but high when using the dual polarization configuration because in this case, the CPD yields a full rank matrix.
I. INTRODUCTION
C
OMMUNICATION systems that use multiple antennas at both the base station and subscriber ends have drawn considerable attention in recent years as they were shown to greatly improve coverage reliability, data rates, or combination thereof, thereby enhancing the overall system performance. For example, it is well known that the information-theoretic capacity of spatial multiplexing-based multiple-input multiple-output (MIMO) systems grows linearly with the number of antenna array elements for fixed power and bandwidth [1]. This source of greater spectrum efficiency solves the capacity needs of future communication systems aiming at providing high-speed data, such as fixed broadband wireless access (BWA) networks, wireless local loop (WLL) networks, and high-speed mobile networks. Most of the available literature on the attainable capacity by MIMO systems is of a theoretical nature. Only limited empirical Manuscript received May 2002; revised January 2003 and June 2003; accepted September 2003. The editor coordinating the review of this paper and approving it for publication is D. Gesbert. V. Erceg and S. Catreux were with Iospan Wireless Inc., San Jose, CA 95134 USA. They are now with are with Broadcom Corporation, San Diego, CA 92128 USA (e-mail:
[email protected]). P. Soma was with Iospan Wireless Inc., San Jose, CA 95134 USA. He is now with Polaris Wireless Inc., San Jose, CA 95054 USA. D. S. Baum was with Iospan Wireless Inc., San Jose, CA 95134 USA. He is now with ETH University, Zurich 8092, Switzerland. Digital Object Identifier 10.1109/TWC.2004.837298
The objective of propagation modeling is to provide a set of parameters, which describe the signal behavior at a desired frequency band and geographic location using statistical or deterministic methods. In the open literature, numerous models can be found for the more conventional single-input single-output (SISO) channel. Path loss, -factor, gain reduction factor, delay spread, and tapped delay line impulse response models were summarized in [8] for the fixed wireless SISO channel at 1.9 GHz.
In this paper we present the results from a measurement campaign conducted in the ISM band (2.4–2.5 GHz) in the San Francisco Bay area, using a coherent 2 2 MIMO channel measurement system spanning a bandwidth of 4 MHz. Based on these results, we derive a space, time, frequency -factor model and a distance-dependent CPD model. These are combined into a new generalized 2 2 channel model that predicts the channel matrix at any given distance as a function of channel parameters such as path loss, -factor, CPD values of the constant and scattered signal components, and correlation coefficients. Note that a simplified structure of the model was presented in an earlier contribution [6]. Later, we use the newly developed model to generate a MIMO channel response. The corresponding capacity simulation results are then compared to the estimates obtained directly from the measurements and are found to be well in agreement. We also show the benefits of our model over the simpler Rayleigh model, where the 2 2 channel matrix elements are independent identically distributed (i.i.d.), zero-mean unit-variance complex Gaussian random variables.
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The remaining sections of this paper are organized as follows: Section II gives a brief description of the measurement system and the field measurement campaign. The estimation and modeling of channel parameters such as the Ricean -factor, the CPD, and the correlation coefficient are presented in Section III. Section IV presents a new generalized 2 2 MIMO channel model. Section V introduces the simulation procedures and validates the model by using metrics such as capacity. Section VI concludes this paper. II. MEASUREMENT SYSTEM AND DATA COLLECTION The coherent measurement system was designed based on the swept frequency-domain sounding technique to measure the channel frequency response. Narrow-band test signals were swept over a band of 4 MHz in steps of 200 KHz every 84 ms. Transmitted signals at different antennas were separated by few kilohertz, well within the coherence bandwidth of the channel. The narrow-band receiver was also swept synchronously with the transmitter, with timing references derived from rubidium clocks. The transmitter consisted of two directive antennas, each having a gain of 17 dBi and azimuthal beamwidth of 90 . The two transmit antennas were separated by ten wavelengths, one with 45 and the other with 45 polarization orientation. At the receiver side, two colocated directive antennas with slanted polarization orientation of 45 were used, each having a gain of 12 dBi and an azimuthal beamwidth of 90 . The receiving antennas were mounted on a retractable mast at a height of 3 m and the transmitter was installed on the rooftop of an office building at a height of 20 m. The outdoor measurement campaign was conducted in the ISM band (2.4–2.5 GHz) at 60 locations in San Jose, CA, for a medium range macrocell (0.2–7 km), typical to broadband fixed wireless systems. The environment can be characterized as suburban with residential blocks (one to two floors), commercial buildings (two to three floors), trees (10–20 m high), and lightly hilly terrain. Most of the data were collected during winter season with reduced foliage. At each test location two fixed measurements 1 m apart were taken to obtain a better statistical average over local space. The signals were collected at 20 different frequency values (or tones) separated by 200 kHz. The measurement data were recorded over a duration of 5 min in the direction of the strongest signal, which turned out to be the direct transmitter–receiver path in most cases. The recorded data were later streamed to a computer hard disc for later processing. The lowest average signal-to noise ratio (SNR) measured at a particular location was approximately 25 dB. At most locations, SNR was significantly larger than 25 dB so that the accurate estimation of channel parameters was possible.
Fig. 1. Path loss versus distance scatter plot.
into two parts (a fixed and variable part, both distance-dependent) and combine it together with the -factor model. In the -factor and complex envelope correlation coefficient estimation, we use the signal power rather than the complex signal envelopes because the measurement system introduces a slow time-varying phase roll due to an imperfect synchronization of the rubidium standards. The phase roll was slow enough not to corrupt the relative complex signal phases in each of the MIMO channel snapshots for all the frequencies measured. A. Path Loss Fig. 1 shows the scatter plot of the path loss, obtained from the average of four copolarized received signals1 as a function of distance for all measurement locations. The median path loss can be obtained by overlying a least mean square error PL (LMSE) regression fit curve on the measured data as shown in Fig. 1 and is given by PL
dB
(1)
is 1 km. where The standard deviation about median path loss is measured at 9.03 dB, a typical finding for this type of environment [8]. The reader is referred to Fig. 6(a) to find the cumulative distribution function (cdf) of the path loss variation about the distance-dependent mean (median), or, in other words, the cdf of the shadow fading component, plotted on a probability scale. The straight line indicates a Gaussian distribution (of the shadow fading in dB). For comparison, Fig. 1 also shows COST-231 Hata model [8] and Path Loss Model B [8] median path loss curves. B. Rician K-Factor
III. MODELING VARIOUS CHANNEL PARAMETERS In this section we present models for the path loss, -factor, CPD, and correlation coefficients. These are important since they are a part of the proposed MIMO model in Section IV. We use a previously published -factor model [10] and extend it to the frequency-selective time-varying MIMO case. The modeling of the CPD is novel in the fact that we separate the CPD
The narrow-band -factors of the 2 2 channel snapshots were computed at each frequency tone (20 tones), 5-min intervals, and for all measurement locations (distributed up to 7 km distance) using the moment method described in [9]. Fig. 2 1At each location, the transmit and receive antennas with the same polarization yield two copolarized received signals, and the adjacent measurement 1 m away yields another two.
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Fig. 2. CDFs of copolarized, cross-polarized, and combined cell area.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER 2004
K -factors over a
shows the cdf curves of narrow-band -factors for the copolarand cross-polarized signals . Recall ized signals and provides valuthat the knowledge of both able insights on the MIMO channel matrix condition number and hence on the MIMO capacity. and versus Fig. 3 shows the scatter plots of distance, together with the median narrow-band -factor model (copolarized) reported in [10]. The input parameters used in the model from [10] are consistent with our measurement setup, i.e., 90 receive antenna beamwidth, 3 m receive antenna height, and 2.5 seasonal factor (no leaves). Additionally, the distance-dependent median -factors for the copolarized and cross-polarized signals signals were obtained using the LMSE regression fit method on our measurement data and are given by dB dB
(2) (3)
where, again, km, consistent with (1) in the case of the path loss modeling. We find that the cross-polarized -factor, on the average, is lower than the copolarized -factor. This result is intuitive because the cross-polarized antenna receives mostly scattered multipath components that often couple into the cross-polarization as a direct result of scattering. Moreover, the median copolarized -factor value predicted by (2) closely matches the results predicted by the model in [10]. For example, the predicted 1 km intercept and slope of the model are 9.5 dB and 0.5, respectively, while (2) yields 8.53 dB intercept and 0.439 slope (4.39 dB per decade). In addition to predicting the median -factor, studies in [10] suggest that the narrow-band -factor follows a log-normal distribution over time, frequency, and user location, with the median being a simple function of season, antenna height, antenna beamwidth, and distance. We proceed with analyzing the -factors distributions obtained from our experimental data, extending the results from [10] to the frequency-selective time-varying MIMO case.
Fig. 3.
K -factor versus distance scatter plot.
1) Log-Normality of -Factor Over Time, Frequency, and User Location: We first characterize the composite distribution of the -factor, i.e., we observe its variations over the three combined domains composed by time, frequency, and user location. To that end, we normalize the copolarized and cross-polarized -factor values with the distance-dependent median values given by (2) and (3), respectively. Fig. 4(a) and (b) shows the cdf curves of the median-normalized copolarized and cross-polarized -factors on a probability scale. Evidence of log-normality is demonstrated by the straight line. The corresponding standard deviations are 7.87 and 6.39 dB for the copolarized and cross-polarized -factors, respectively. These are the standard deviations of the composite distribution of the -factors. Next, we isolate the time, frequency, and location variations of the -factors from the composite distribution, so we may characterize them individually. As will be seen below, we resort to both our experimental data analysis results as well as earlier results from [10] to show the log-normality property of each of these components and to estimate their corresponding standard deviation. 2) Log-Normality of -Factor Over Frequency: To show the log-normal characteristic of the -factor over frequency, we isolate the frequency-dependent component of the -factor, , for each of the eight subchannels2 (two sets of four called subchannels sampled 1 m apart) and for all locations. In all cases, we subtract its mean over frequency (dB values) and finally combine all the data together in a common group. Fig. 4(c) shows the corresponding cdf of the frequency-dependent -factor. In the range of 1–99% the distribution follows well a straight line, confirming the log-normal distribution for -factor. The corresponding standard deviation is 2.93 dB. 3) Log-Normality Over Time: We now show the log-normal characteristic of the -factor over time. The study is not as straightforward as in the frequency case because our experimental data at hand offer only a one-time 5-min short-term measurement per location, which is insufficient to estimate the 2A subchannel refers to the channel between a pair of transmit and receive antenna element.
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Fig. 4. CDFs of
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K -factors about mean values.
long-term variations of the -factor over time. However, we can use the results reported in [10] since they are based on 24-h measurements, where the time traces recorded at each frequency were divided into 15-min segments and the -factor was computed for each segment. Since the results from [10] assume a single polarization, we limit our study to the copolarized -factor. We feel comfortable with using the results from [10] because the copolarized -factor statistics are in excellent agreement (Fig. 3). Note that the results in [10] characterize the variations of the -factor over the user locations on the one hand, and over the combined time/frequency domains on the other hand. To arrive to our conclusions on the variations of the -factor over the time domain only, we use the results from the combined time/frequency study in [10] together with our results on the variations over frequency only. In [10] it was found that temporal/frequency and user location variation of the -factor both follow a log-normal distribution. This, together with our finding on the frequency-only variation, suggests that the time-only variation is also log-normally distributed. The temporal/frequency standard deviation of the -factor and location standard deviation of the -factor were both found to be 5.7 dB. The root mean square (rms) sum of the two standard deviations yields a standard deviation of 8 dB corresponding to the composite distribution. From our measurements, we found that the standard deviation of the comdB, posite distribution of the copolarized -factor is which we may now decompose into the rms sum of two identical components, as suggested by [10]
easily calculated using the following equation (where all three standard deviations add on an rms basis, assuming statistical independence): (5) dB. We conclude that the variations The result is of the copolarized -factor over the time domain follow a lognormal distribution, with a standard deviation of 4.74 dB. 4) Log-Normality Over Time/User Location: Studies from [10] already proved that the variations of the -factor over the user locations are log-normally distributed with a standard deviation of 5.7 dB. With our measurements, we confirm this finding by plotting on Fig. 4(d) the cdf of the location/time-dependent component of the -factor. Fig. 4(d) shows that the composite location/time distribution of the -factor is again log-normal with a composite standard deviation of 7.3 dB. We end this section by formulating a narrow-band -factor model, valid for any branch of a MIMO fixed wireless channel, that consists of a and three standistance-dependent median -factor dard deviations that describe the time, frequency, and location variations
dB
(6)
, and are zero mean, unit variance Gaussian random where variables and are standard deviations with values of is the copolarized 5.56, 4.74, and 2.93 dB, respectively. -factor that is location (distance) dependent. We further define
(4) The result is dB. Since we found earlier from our experimental data analysis that the frequency standard is 2.93 dB, the time standard deviation , can be deviation
dB
(7)
These -values are variations about the medians. Since the generalized channel matrix model of Section IV uses the channel
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dB
(10)
dB
(11)
where , and are zero-mean unit variance Gaussian random , and are the corresponding standard devariables and viations with the values of 4.90, 5.37, and 2.62 dB, respectively. , on the average, is smaller than the The result that the is intuitive since the scattered signal components often couple into the cross-polarization as a direct result of scattering. and have This also explains why the results for similar median values. If the model is to be used at distances beyond a distance where the median CPD values reach 0 dB or below (approximately 10 km; see Fig. 5), the median values should be limited to 0 dB. D. Complex Envelope Correlation Coefficient
Fig. 5. CPD versus distance scatter plot.
parameters in linear values, we define the linear counterparts of . Following the various -factor components as this notation, we rewrite (6) in the linear domain as (8)
C. Cross-Polarization Discrimination (CPD) The CPD is defined as the ratio of the copolarized average received power to the cross-polarized average received power. Fig. 5 shows the CPD variation with distance of the total, con, and stant, and time-variable signal components , respectively (in the text, we regard CPD as an acronym s as values). Again, the knowledge of each compoand nent is relevant to the modeling of the MIMO channel matrix, as it influences the matrix condition number. At each location, four CPD values were averaged (the transmit and receive antennas with different polarization yield one pair of CDP values, and the adjacent measurement 1 m away yields another pair). The four CPD values were calculated by first averaging the received power over time and frequency and then taking the ratios. The straight lines, obtained from an LMSE fit, represent the distance-dependent mean (median) of the CPD values. To look further into the characteristics of the CDP variations about their mean, we plot on Fig. 6(b), (c), and (d) the cdf curves , and on a probability of the variables scale, after normalizing each of them by their respective distance-dependent mean. One can immediately recognize that the variations (dB values) closely follow the Gaussian distribution. These observations enable us to propose a model for the time, frequency, and local space average CPD. Similarly to the path loss and -factor models, we express the CPD as the function and a Gaussian variof a distance-dependent median, ation about the median
The complex envelope correlation coefficients (worst case) between various branches of the 2 2 MIMO channel matrix were computed at each frequency tone and for all locations using power correlation coefficient and -factor as described in [11]. The worst case complex envelope correlation is real and bipolar and has a distribution very close to that of the power correlation coefficient, and the distribution is roughly Gaussian [11]. Fig. 7(a) shows the scatter plot versus distance of the worst case transmit and receive complex envelope correlation coefficients (magnitudes) averaged over frequency, subchannels, and two locations 1 m apart. No significant distance dependency of the complex envelope correlation coefficient was found. Fig. 7(b) shows the corresponding cdfs. Transmit correlation is defined as correlation between transmit antennas for a given receiving antenna, and receive correlation is defined as correlation between receive antennas for a given transmitter antenna. As expected, transmit complex envelope correlation coefficients are found to be on average less than receive complex envelope correlation coefficients, since the combination of both polarization and space diversity at the transmitter randomizes the signals more than the polarization diversity at the receiver alone. The average transmit and receive complex envelope correlation coefficients are found to be 0.16 and 0.19 and the standard deviation is 0.25 (for both), respectively. For simplicity, we assume a fixed mean value for both cases, which is a mean of the two (0.175). , In this section we have shown that the values of PL (in dB) can be modeled as Gaussian random variand ables with distance-dependent medians. To complete the model, we determine the correlation properties (covariance matrix) of about their medians, simithe variations of PL larly to [11]. The resulting correlation coefficients are presented in Table I. In Appendix A we present a method on how to deand other termine the correlation coefficients between is available from the measurements. variables if only IV. GENERALIZED 2
dB
(9)
2 MIMO CHANNEL MODEL
In this section lower case variables point to linear values while upper case variables indicate dB values (except for matrix ).
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Fig. 6. (a) CDF of path loss normalized by distance-dependent mean. (b)–(d) CPD variations about their distance-dependent mean values.
Fig. 7. (a) Complex envelope correlation coefficients (magnitude) versus distance scatter plot. (b) Corresponding cdfs. TABLE I CORRELATION MATRIX C
pressed as follows: (12)
Assuming that the total received signal follows a Rician distribution that is composed of a constant signal with mean path gain and a Rayleigh-distributed time-varying signal with mean path (where both mean path gains are expressed in linear gain units), the instantaneous MIMO channel matrix can be ex-
and are the normalized [explained in (17)] conIn (12), , stant and variable channel MIMO matrices of size where and denote the number of receive and transmit antennas, respectively. We now make use of the channel parameter models developed in the previous section to provide a complete, elaborate model for the matrix . First, we write the expression for the received complex signal gain of a narrow-band fixed wireless SISO channel [11] (13)
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where is the mean path gain, is zero-mean complex Gaussian of unit variance, and is the phase of constant signal component. Assuming the SISO channel uses a single polarization, (13) may be rewritten as
2)
(14) is given in (8). where If the transmit and receive antennas use a different polarization, (13) becomes
(15)
3)
and represent the CPD values of the constant and where variable signal components modeled by (10) and (11), respecand tively. It is interesting to note from (15) that are related by (16) Each subchannel of the MIMO channel matrix follows the model given by (14) and (15) depending on the antenna polarizations. In the case of a 2 2 system, the channel matrix can be rewritten as follows:
(17)
correspond to subDiagonal (cross-diagonal) elements of channels in which the transmit and receive antennas have the is the local average path gain same (different) polarization. and is the phase of the subchannel between th receiving and th transmitting antennas in the constant channel matrix. is a mean-to-median ratio of the log-normal Constant and is given by variable (18) where is the standard deviation of (2.93 dB) and s are correlated, using complex envelope correlation coefficient, complex Gaussian random variables with zero mean, and unit variance. Underlying assumptions to (17) are the following. , and correspond to local mean 1) values that are variable over frequency, time, and local space. The CPD values were calculated by first averaging the receive signal power over time, frequency, and local space and then taking the ratios. The average
4)
path gain was calculated by averaging the path gain over time, frequency, and local space. is the same for all four subchannels and for all user locations over a cell area in a given block of time (note that the time block duration depends on wind conditions and is rather variable: it ranges from minutes up to hours, or even longer, depending on the environment, trees, foliage, season, and wind conditions). When there is no wind, all locations in a given time block experience high -factors (i.e., most signal components are constant in time). During windy conditions, moving trees and leaves cause fading, and therefore, lower -factors (some or, in the extreme case, all signal components become variable). The values of are assumed to change from time block to time block, i.e., a different value of can be drawn from its corresponding log-normal distribution. The variation of the -factor in frequency, due to multipath arriving at different delays, is mainly characterized by a variation of the constant signal power, while the variable signal power is constant versus frequency. The frequency-independent aspect of the variable signal power is easy to show by writing the as a summation of independent complex signal delayed echoes, then computing its Fourier transform and finally calculating its corresponding mean-square value as a function of frequency. Based on our measurements, the standard deviation of the variable signal power over frequency was found to be less than 1 dB for all locations. This explains why the frequency , appears only in the component of , denoted by constant channel matrix in (17). The frequency variations of the -factor are independent from one subchannel to another, due to phase differences in each subchannel (0.012 correlation was found from the experimental data analysis). The frequency pattern of the -factor, or equivalently the frequency pattern of the constant signal power, changes over time depending on the environment, trees, foliage, season, and wind conditions. Different can be drawn from its corresponding values of log-normal distribution at each block of time, location, and frequency. A channel characteristic that we do not model in detail in the in this paper is the distribution of the phases constant channel matrix. Depending on their distribution (correlation), the performance of a MIMO communication system can be quite different for Rician channels [7]. It is desirable to have a full rank fixed matrix in order to obtain the multiplexing gain (increased data rate in MIMO systems). When a dual-polarization antenna system is used, as is the case in our measurements, the rank of the constant signal matrix is often full thanks to the effect of the CPD (orthogonal matrix) [7]. Under these conditions, the correlation of is irrelevant, and we choose to model the phases for a 2 2 MIMO system with dual-polarized antennas as uniformly distributed between 0 and 2 .
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TABLE II SUMMARY OF VARIOUS CHANNEL MODEL PARAMETERS
We summarize the following channel parameter properties. , and are functions of (change over) • distance only. is a function of (changes over) blocks of time only. • and are functions of (change over) location, • blocks of time, frequency, and subchannels. and are mutually independent and are also • , and . independent of In Section III we have shown that the dB values of , and can be modeled as Gaussian random variables, where the first four variables are correlated and have distance-dependent medians. In summary, the model consist of: 1) a vector of four Gaussian processes (dB values of , and ) varying about their median and correlated by the covariance matrix given in Table I; and parameters modeled in dB as mutually 2) uncorrelated Gaussian random variables that describe time and frequency variation, respectively, of the -factor, independent of all other variables; 3) mutually correlated complex Gaussian random vari, which are independent of all other variables ables and have a common correlation coefficient. Table II summarizes all model parameters and their values. In the next section we show how to use the model in simulations.
3)
(19) 4)
Generate the four correlated location (distance)-depen) by scalar dent channel parameters (4 1 vector multiplying each of the four elements of vector with its corresponding standard deviation organized and by adding the corresponding into a 4 1 vector (see Table II) mean value vector dB
5)
6) 7)
8) 9)
V. SIMULATION PROCEDURE AND MODEL VALIDATION A. Simulation Procedure We describe below a list of steps to follow in order to simulate the 2 2 MIMO channel (note that the model can be extended MIMO channel case). to the more general 1) Generate a set of distances (usually uniformly distributed over a cell area). At each distance, follow steps ii)–xii). 2) Generate the median dB values, organized into a 4 1 , of the following channel model paramevector , and ters: PL using (1), (2), (10), and (11), respectively.
of independent Gaussian Generate a 4 1 vector distributed random variables with zero mean and unit by mulvariance. Then obtain the correlated vector tiplying with a 4 4 correlation matrix given in Table I using the following equation [11]:
10)
11)
(20)
Generate the complex envelope correlation coefficient using mean and standard deviation (Gaussian distribution, see Table II). Iterate over time blocks (for example, 5–15 min time interval). At each time block follow steps vii)–xii). , obtained as a Gaussian Generate a realization of distributed random variable with zero mean and corresponding variance. Iterate over frequency. At each frequency follow steps ix)–xii). Generate an independent set of four realizations of , obtained as mutually independent Gaussian random variables with zero mean and corresponding variance. Generate a set of four independent random phases whose values are uniformly distributed in the range of [0,2 ). matrix of correlated, complex Generate a 2 2 with zero Gaussian distributed random variables mean and unit variance, where the correlation is obtained by using the complex envelope correlation coefficient and where represents the number of observation time samples at a given user location. To correlate these variables, either the method that uses transmit
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Mean capacity versus distance scatter plots.
Fig. 9. Mean capacity cdf plots.
12)
and receive correlation matrices [3], [4] or the method described in iii) can be used. An additional time correlation of samples can be achieved by using the Doppler spectrum of the fixed wireless channel [8]. Generate the 2 2 MIMO channel matrix by substituting the above simulation parameters, converted into linear units, into (17).
B. Validation of the Model To validate the model, the channel capacity metric is used to compare the results obtained directly from the measured data with the results obtained using the newly developed model. We also highlight the value of our model over the more simplistic i.i.d. Rayleigh model.
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Channel capacity is defined as the highest transfer rate of information that can be sent with low (zero) probability of error. reFor transmit antennas using equal transmit power and ceiving antennas, the generalized formula for the channel cais given by [1] pacity bps/Hz
(21)
In (21), denotes the channel matrix, is its , conjugate transpose, is a unit diagonal matrix of size and is the average SNR at the receiver. The channel capacity obtained from measurements was computed at a fixed SNR to the value of 15 dB 3 by normalizing the channel matrix local mean power obtained from the average of the two diagonal (copolarized) components of the channel matrix. The MIMO channel simulation was carried out to generate the channel using the methodology described in Section V for matrix various distances, 20 frequency tones, and 5-min time interval (block ), in the same way as in the measurements. The channel capacity was estimated using (21) and averaged over the time and frequency domain. Fig. 8 shows the mean capacity estimated directly from the measurement data as well as the simulated capacity as a function of distance. The simulation and measurements are well in agreement. Simulations indicate slightly lower capacity results; this is most likely due to pessimistic (worst case) complex envelope correlation coefficients. Fig. 8 also shows the mean capacity results using the simple Rayleigh model (2 2 channel matrix elements are i.i.d., zeromean unit-variance complex Gaussian random variables). This model does not produce any variation of the mean capacity since it does not include variations of -factor, CPD, and corresponding correlation coefficients. From Fig. 8, we can observe that the mean capacity obtained from experiments and by using our model can vary significantly, from 6.5 to 10 bps/Hz, while the simple Rayleigh model predicts a constant 8.25 bps/Hz. Fig. 9 shows the cumulative statistics of the mean capacity using the proposed model, measurements, and simple Rayleigh model. This figure highlights the advantage of our model over the simple Rayleigh model, since it predicts the cdf of capacity more accurately.
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copolarized -factor, and CPD distribution of the constant and scattered signal components. The considered channel parameters were found to follow the log-normal distribution. Using the developed model, channel simulations were carried out at different distances and used to simulate the corresponding capacity, which was then compared to the capacity results obtained directly from the measurements. The two estimates were found to be in a very good agreement. The MIMO channel model presented can be used to carry out end-to-end system simulations and estimate system performance under various channel conditions. APPENDIX CORRELATION BETWEEN SPLIT VARIABLES by and In this Appendix we denote by . We examine the correlation between PL and , where is uncorrelated with and PL . Our aim is to arrive at the correlation coefficient between PL and . We can write the correlation and , assuming zero mean coefficient between PL for all variables as PL PL PL PL PL PL
where
PL
, so that PL
PL
each location the average measured SNR was greater than 25 dB so that the capacity estimates at 15 dB SNR are accurate (uncorrelated noise does not artificially influence the capacity results).
(23)
PL
Our unknown is the correlation coefficient between PL and PL
VI. CONCLUSION
3At
(22)
PL
PL In this paper, we presented results from our 2 2 MIMO fixed outdoor channel measurement campaign conducted at 2.48 GHz in the suburban residential areas of San Jose, CA. The recorded data over local space, time, and frequency were analyzed to compute the path loss, Rician -factor, CPD, correlation coefficients, and capacity. The -factor results were found to be in excellent agreement with a previously reported model. Simple models were developed for the CPD values of the constant and scattered signal components. A generalized 2 2 MIMO channel model was presented based on the correlation properties between the path loss,
PL
Assuming PL arrive at PL
(24)
PL is known and combining (23) with (24), we
PL
PL (25)
In a similar way we determined other correlation coefficients that involve in Table I. ACKNOWLEDGMENT The authors would like to thank L. J. Greenstein, A. J. Paulraj, R. Krishnamoorthy, D. Gore, R. Nabar, D. Gesbert, J. Tellado,
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G. J. Foschini, C. Oestges, A. J. Rustako, D. Michelson, S. Ghassemzadeh, and reviewers for their valuable discussions, comments, and suggestions. The authors also gratefully acknowledge the assistance of M. Azam, K. R. Adams, and V. Lagmen in conducting this extensive propagation measurement campaign successfully. REFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “MIMO wireless channels: Capacity and performance prediction,” in Proc. IEEE Globecom Conf., San Francisco, CA, Nov. 2000, pp. 1083–1088. [3] 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. IEEE Veh. Technol. Conf., Boston, MA, Sept. 2000, pp. 893–897. [4] J. P. Kermoal, L. Schumaker, F. Frederiksen, and P. E. Mogensen, “Polarization diversity in MIMO radio channels: Experimental validation of a stochastic model and performance assessment,” in Proc. IEEE Veh. Technol. Conf., Atlantic City, NJ, Oct. 2001. [5] C. C. Martin, J. H. Winters, and N. R. Sollenberger, “Multiple-input multiple-output (MIMO) radio channel measurements,” in Proc. IEEE Veh. Technol. Conf., Fall 2000. [6] P. Soma, D. S. Baum, V. Erceg, R. Krishnamoorthy, and A. J. Paulraj, “Analysis and modeling of multiple-input multiple-output (MIMO) radio channel based on outdoor measurements conducted at 2.5 GHz for fixed BWA applications,” in Proc. IEEE ICC’2002 Conf., New York, Apr. 2002. [7] V. Erceg, D. Baum, S. Pitchaiah, and A. J. Paulraj, “Capacity obtained from multiple-input multiple-output channel measurements in fixed wireless environments at 2.5 GHz,” in Proc. IEEE ICC’2002 Conf., New York, Apr. 2002. [8] V. Erceg et al., Channel models for fixed wireless applications. IEEE 802.16 Broadband Wireless Access working group Doc. IEEE802.16.3c-01/29r4. [9] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment-method estimation of the Ricean -factor,” IEEE Commun. Lett., vol. 3, pp. 175–176, June 1999. [10] L. G. Greenstein, S. Ghassemzadeh, V. Erceg, and D. G. Michelson, “Ricean -factors in narrowband fixed wireless channels,” in Proc. WPMC Conf., Amsterdam, the Netherlands, Sept. 1999. [11] D. G. Michelson, V. Erceg, and L. J. Greenstein, “Modeling diversity reception over narrowband fixed wireless channels,” in IEEE MTT-S Symp. Technologies for Wireless Applications, 1999, pp. 95–100.
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Vinko Erceg (M’92–SM’98) received the B.Sc. and Ph.D. degrees in electrical engineering from the City University of New York in 1988 and 1992, respectively. From 1990 to 1992, he was a Lecturer in the Department of Electrical Engineering at the City College of New York. Concurrently he was a Research Scientist with SCS Mobilecom, Port Washington, NY, working on a code division multiple access (CDMA) system for mobile communications. In 1992, he joined AT&T Bell Laboratories and, in 1996, AT&T Laboratories—Research as a Principal Member of Technical Staff in the Wireless Communications Research Department, where he has been working on signal propagation as well as other projects related to the systems engineering and performance analysis of personal and mobile communication systems. From 2000 to 2002, he was Director and Principal Engineer for Iospan Wireless, where he managed the System Validation and Channel Modeling Group responsible for systems, propagation, deployment, and performance issues of a multiple-input multiple-output OFDM communication system. Afterwards, he was the Director of the Space-Time Processing (STP) division of Zyray Wireless Inc. He lead new technology development utilized in both WCDMA and WLAN (802.11) solutions. Currently, he is with Broadcom Corporation, San Diego, CA, working on WCDMA and HSDPA systems.
Pitchaiah Soma was born in Bapatla, India, in 1974. He received the B.Tech. degree in electronics and communications engineering from Nagarjuna University, India, and the M.Tech. degree in microwave signals and systems engineering from the Indian Institute of Technology, Kharagpur. His work experience includes propagation measurements and channel modeling for GSM, LMDS, and MIMO-based MMDS broadband wireless applications. He was also involved in the development of site-specific three-dimensional ray tracing channel models for PCS applications. He is currently developing a propagation tool based on GIS data such as terrain, clutter, and roads and prediction optimization algorithms using drive test data for mobile location estimation applications. His research interests include propagation measurements, channel modeling, and system performance studies.
Daniel S. Baum received the Dipl.-Ing. degree in electrical engineering/communications from Karlsruhe University of Technology, Karlsruhe, Germany, in 2000. He is currently pursuing the Ph.D. degree at the Communication Theory Group, Federal Institute of Technology (ETH), Zürich, Switzerland. From 2000 to 2001, he was a Research Assistant with the Smart Antenna Research Group, Information System Laboratory, Stanford University, Stanford, CA. He conducted research in channel measurements and modeling for broadband fixed wireless applications and was active in the IEEE 802.16 working group in developing NLOS channel models. He was also a Consultant to Sprint Corp. for G2 MMDS technology evaluation and test plan development. In 2001, he joined Iospan Wireless Inc., San Jose, CA, a startup company developing the first high-speed broadband fixed-wireless system based on MIMO-OFDM technology. There he was active in channel modeling, performance testing, and system integration. At ETH, he is supervising the hardware laboratory and the development of a real-time MIMO testbed. His current research interests are in the areas of signal processing for wireless communications and MIMO antenna systems including networking and real-world aspects.
Severine Catreux received the M.Sc. degree in electrical engineering and the Ph.D. degree from National Institute of Applied Sciences (INSA), Rennes, France, in 1996 and 2000, respectively. Between 1996 and 1999, she was at the University of Victoria, BC, Canada, where she studied array signal processing techniques for digital radio communications systems. From April 1999 to early 2000, she completed her doctoral studies in the Wireless Communications Research Department of AT&T Laboratories—Research, with emphasis on the data throughputs attainable by MIMO systems. From 2000 to 2002, she was with Iospan Wireless Inc., San Jose, CA, a startup company developing a high-speed broadband fixed wireless system based on the MIMO-OFDM technology. From 2002 to 2004, she was a Senior Systems Engineer with Zyray Wireless Inc., San Diego, CA, working in the areas of space–time processing for both wireless LANs and W-CDMA applications. Her research interests include adaptive signal processing for digital communications, wireless MIMO systems, and other cutting-edge technologies for future generations of wireless systems. Currently, she is with Broadcom Corporation, San Diego, working on WCDMA and HSDPA systems.
