We therefore refer to 1. This allows investigation of the relative variations in received signal strength over an ensemble of measurements or simulations. A variety of such measurements have been reported [2—11], and results obtained include channel capacity, signal correlation structure in space, frequency, and time , channel matrix rank, path loss, delay spread, and a variety of other quantities. True array systems, where all antennas operate simultaneously, most closely model real-world MIMO communication, and can accommodate channels that vary in time.
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Other measurement systems are based on either switched array or virtual array archi- tectures. Switched array designs use a single transmitter and single receiver, sequentially connecting all array elements to the electronics using high-speed switches [12, 13]. Although this method has the advantage of eliminating mutual coupling, a complete channel matrix measurement often takes several In this section, we will provide details regarding a true array system for direct trans- fer matrix measurement and illustrate MIMO performance for representative propagation environments.
Following the discussion of the system, we will discuss additional processing that can be accomplished using MIMO system measurements to allow estimation of the physical multipath characteristics associated with the propagation channel. The sys- tem is calibrated before each measurement to remove the effects of unequal complex gains in the electronics. Once the IF data is collected, postprocessing is used to perform carrier and symbol timing recovery and code synchronization.
To locate the start of the codes, the signal from one of the NR receive antennas is correlated with a baseband representation of one of the transmit codes. A Fast Fourier Transform FFT of this result produces a peak at the IF when the selected transmit code is aligned with the same code in the receive signal.
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Subsequently, the waveform generated by moving a window along the despread signal is correlated against a complex sinusoid at the IF, and the phase of the result is taken as the carrier phase at the center of the recovery window. With the carrier and timing recovery performed, the channel transfer matrix can be extracted from the data using a maximum likelihood ML channel inversion technique. Data were collected using bit binary codes at a chip rate of Because channel changes occur on the timescales of relatively slow physical motion people moving, doors closing, etc.
The three different measurement scenarios considered here are listed in the table below. Multiple second data records — were taken for each scenario. These histograms are computed by treating each combination of matrix sample, transmit antenna, and receive antenna as an observation. Figure 1. The agreement between the analytical and empirical PDFs is excellent. The timescale of channel variation is an important consideration since this indicates the frequency with which channel estimation and perhaps channel feedback must occur to maintain reliable communication.
To assess this temporal variability, we can examine the temporal autocorrelation function for each element of the transfer matrix.
For all measurements, the correlation remains relatively high, indi- cating that the channel remains relatively stationary in time. The correlation between the signals on the antennas is another important indicator of performance since lower signal correlation tends to produce higher average channel capacity. We assume that the correlation functions at transmit and receive antennas are shift-invariant, and we therefore treat all pairs of antennas with the same spacing as independent observa- tions.
For the Finally, we examine the channel capacities associated with the measured channel matri- ces. Also, Monte Carlo simulations were performed to obtain capacity CCDFs for channel matrices whose elements are independent, identically distributed i. The array length is 2. However, as the array size increases, the simulated capacity continues to grow while the measured capacity per antenna decreases because of higher correlation between adjacent elements.
The capacity results of Figure 1. The second number in italics represents the capacity obtained when the normalization is applied over all H matrices considered in the study. Arrows are drawn from transmit to receive positions. The top and bottom number in each circle give capacity without and with propagation loss, respectively.
Conceptually, the simplest method for measuring the physical channel response is to use two steerable manually or electronically high-gain antennas. Typically, broad probing bandwidths are used to allow resolution of the multipath plane waves in time as well as angle. The resolution of this system is proportional to the antenna aperture for directional estimation and the bandwidth for delay estimation. Unfortunately, because of the long time required to rotate the antennas, the use of such a measurement arrangement is limited to channels that are highly stationary. However, attempting to extract more detailed information about the propagation environment requires a much higher level of postprocessing.
Theoretically, this information could be obtained by applying an optimal ML estimator. However, since many practical scenarios will have tens to hundreds of multipath components, this method quickly becomes computationally intractable. The resolution of such methods is not limited by the size of the array aperture, as long as the number of antenna elements is larger than the number of multipaths. While this double-directional channel characterization is very powerful, there are several basic problems inherent to this type of channel estimation.
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First, the narrowband assumption in the signal space model precludes the direct use of wideband data. Thus, angles and times of arrival must be estimated independently, possibly leading to suboptimal parameter estimation. This problem can be overcome by either applying alternating conventional beamforming and parametric estimation  or by applying advanced joint diagonalization methods [27, 28].
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However, in cases where these problems are not severe, the wealth of information obtained from these estimation procedures gives deep insight into the behavior of MIMO channels and allows development of powerful channel models that accurately capture the physics of the propagation environment. However, owing to the cost and complexity of conducting such measurement campaigns, it is important to have channel models available that accu- rately capture the key behaviors observed in the experimental data [30—33, 1].
When accurate, these models facilitate performance assessment of potential space-time coding approaches in realistic propagation environments. There are a variety of different approaches used for modeling the MIMO wireless channel. This section outlines several of the most widely used techniques and discusses their relative complexity and accuracy tradeoffs. Throughout this section, the superscript k will be dropped for simplicity. The simple linear relationship allows for convenient closed-form analysis, at the expense of possibly reduced modeling accuracy.
The multivariate complex normal distribution The multivariate complex normal MCN distribution has been used extensively in early as well as recent MIMO channel modeling efforts, because of simplicity and compatibility with single-antenna Rayleigh fading.
In fact, the measured data in Figure 1. From a modeling perspective, the elements of the channel matrix H are stacked into a vector h, whose statistics are governed by the MCN distribution. Covariance matrices and simplifying assumptions The zero mean MCN distribution is completely characterized by the covariance matrix R in 1.
Therefore, the simplifying assumptions of separa- bility and shift invariance can be applied to reduce the number of parameters in the MCN model. The separable Kronecker model appeared in early MIMO modeling work [31, 35, 7] and has demonstrated good agreement for systems with relatively few antennas 2 or 3. Shift invariance assumes that the covariance matrix is only a function of antenna sepa- ration and not absolute antenna location .
Complex and power envelope correlation The complex correlation in 1.
Space-Time Wireless Systems: From Array Processing to MIMO Communications
However, for cases in which only power information is available, a power envelope correlation may be constructed. Thus, for a given power correlation RP , we usually have a family of compatible complex envelope correlations. Although this method is convenient, for many scenarios it can lead to very high modeling error since only power correlations are required .
Early MIMO studies assumed an i. However, for most realistic scenarios the i. Although only approximate, closed-form expressions for covariance are most convenient for analysis. Alternatively, let rT and rR repre- sent the real transmit and receive correlation, respectively, for signals on antennas that are immediately adjacent to each other.
The exponential correlation model has also been proposed for urban measure- ments . Other methods for computing covariance involve the use of the path-based models in Section 1.
Even when the statistics of the path gains are not complex normal, the statistics of the channel matrix may tend to the MCN from the central limit theorem if there are enough paths. This approach potentially reduces a large set of channel matrix measurements into a smaller set When the Kronecker assumption holds, the number of parameters may be further reduced.
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