Method and device for exploiting transmit diversity in time varying wireless communication systems

[0002] The present invention generally relates to the field of communication systems and more particularly, to achieving spatial diversity in a communication link having at least two transmitting antennas and at least one receive antenna while mitigating Doppler induced channel variations and multipath interference.

[0003] In a wireless communication system, a major design challenge is to maximize system capacity and performance in the presence of interference, and a time-varying multipath channel. A multipath channel is caused by the transmitted signal reflecting off objects near the transmitter and receiver and arriving at the receiver over multiple paths. Interference in a communication system can come from a variety of sources depending on the particular system deployment. If the system is in motion, then Doppler induced channel variations becomes a problem. Interference and multipath are major factors that limit the achievable performance and capacity of a communication system because both effects interfere with the ability of a communication receiver to properly decode the transmitted data.

[0004] In a multipath propagation channel, the transmitted signal propagates to the receiver over a finite number L_pathsof propagation paths, where each path has an associated time delay and complex gain. In such a channel, the communication receiver receives the superposition of L_pathsdelayed, attenuated, and phase-shifted copies of the transmitted signal. The number of paths L_pathsand their time delays and phase shifts depends on the physical location of the various scattering objects (such as buildings, automobiles, and trees) in the immediate vicinity of the transmitter and receiver. The complex attenuation (magnitude and phase) of each path depends on the length of each path, as well as the material composition of any scatterers or reflectors encountered along the path.

[0005] The presence of multipath can severely distort the received signal. In a multipath environment, the multiple copies of the transmitted signal can interfere constructively in some portions of the occupied bandwidth. In other portions of the occupied bandwidth, the multiple copies can interfere destructively at the receiver. This signal duplication causes unwanted variations in the received signal strength over the bandwidth occupied by the signal. Furthermore, if the difference in the path delays of the various propagation paths is significantly greater than the duration of a transmitted information symbol, then intersymbol interference is present at the receiver. When intersymbol interference is present, the received signal is corrupted by prior transmitted symbols propagating over paths having delays relative to the shortest path that are longer than the duration of an information symbol. The demodulation process (the process of determining which information symbol was transmitted) becomes difficult in the presence of intersymbol interference.

[0006] In mobile communication systems, the presence of multipath causes rapid variations, or fading of the received signal strength at a receive device. The fading of the received signal can cause the received signal level to drop significantly to levels sometimes reaching 20 dB below the average received signal level. The rate of variation in the transmitted signal is related to the relative velocity of the transmitter and receiver, and to the scattering objects in the environment. These rapid variations in the signal strength can severely interfere with the ability of the receiver to decode the transmitted information.

[0007] When more than one antenna is employed at a receiver, the presence of multipath that arrives at the receive array from multiple different directions can cause the signal variations, or fading, on each of the receive antennas to be uncorrelated. In other words, the presence of angular multipath scattering can cause the channel responses between a transmitter and each of the receive antennas to be independent of each other. The result is that the signal level on one receive antenna might be totally unrelated to the signal level on another receive antenna. This means that when one antenna is in a fade (i.e., the signal level has dropped to a very low level), the signal level on another antenna might not be in a fade. Not only does multipath cause the channel response to vary in time (due to motion) and frequency (due to intersymbol interference, as discussed earlier), but multipath can also cause the channel response to vary in space (across the elements of an antenna array).

[0008] A well-known practice for combating fading is to use a technique called “diversity.” In a generic sense, diversity is the practice of transmitting and/or receiving multiple copies of the signal and combining these copies in some optimal fashion. There are several methods of implementing diversity. One of the more popular forms of diversity is “receive spatial diversity,” in which multiple copies of the transmitted signal are received over multiple receive antennas and the outputs of the multiple antennas are combined in some optimal fashion, such as max-ratio combining. (Max ratio combining is the technique where the signals on the receive array are combined to maximize the signal to noise ratio at the output of the receive combiner.) The reason why exploiting receive spatial diversity improves receiver performance can be understood from the following line of reasoning. In a fading environment in which the fading processes are uncorrelated on the receive antennas, it is highly unlikely that all antennas of the receive array will simultaneously experience a severe drop in signal level. As a result, it is highly likely that at least one antenna element of the receive array is not in a fade, which means that at least one receive antenna is receiving a high-powered copy of the transmitted signal. The receiver will have difficulty when all receive antennas simultaneously go into a deep fade because no receive antenna can provide a high-powered version of the transmitted signal, and decoding errors will be highly likely. If the fading processes are all highly correlated, then all receive antennas will tend to go in and out of fades at the same time. When fades do occur in this case; the receiver will have difficulty decoding the signal because all antennas have faded simultaneously.

[0009] Another form of diversity similar to receive spatial diversity is “transmit spatial diversity” also called “transmit diversity.” Transmit spatial diversity, like receive spatial diversity, can be implemented in many ways. A primitive transmit diversity technique is to use multiple transmit antennas, where each transmit antenna simultaneously transmits the same signal. The receiver (either having a single receive antenna, or even multiple receive antennas) automatically receives the superposition of the two transmitted signals. When the channel responses between each of the transmit antennas and the receive antenna are uncorrelated, then a level of protection against fading can be achieved (for similar reasons as with receive diversity, as described above).

[0010] A properly designed transmit diversity scheme can be particularly valuable in a cellular system, especially when the subscriber device has only one receive antenna. In such a case, receive diversity is not an option on the downlink (base transmitting to the subscriber device) due to the single antenna design constraint that is often imposed by cost, size, and marketing requirements. Furthermore, many existing transmit diversity schemes suffer a performance loss when the propagation channel is rapidly varying. A transmit diversity scheme whose performance is not seriously degraded in a rapidly varying channel can also be particularly valuable in a mobile cellular system.

[0011] Thus, there is a significant need for a method and device for improved transmit diversity schemes so that the potential benefits of spatial diversity can be realized even when the receive device only has one receive antenna and even when the channel is rapidly varying.

[0012] FIG. 1 is an overview diagram of a preferred embodiment of a communication system in accordance with the present invention;

[0013] FIG. 2 is a block diagram illustrating a preferred embodiment of a receiver device in accordance with the present invention;

[0014] FIG. 3 is a block diagram illustrating a preferred embodiment of a transmitting device in accordance with the present invention;

[0015] FIG. 4 is a representation of an information burst for an Orthogonal Frequency Division Multiplexing (OFDM) system being grouped into coding intervals, which can be used by the communication system in which the receiving device in FIG. 2 and the transmitting device in FIG. 3 operate;

[0016] FIG. 5 is a representation of two adjacent information bursts being grouped into coding intervals, which can be used by the communication system in which the receiving device in FIG. 2 and the transmitting device in FIG. 3 operate;

[0017] FIG. 6 is a representation of an information burst for a single carrier system employing cyclic prefixes, which can be used by the communication system in which the receiving device in FIG. 2 and the transmitting device in FIG. 3 operate;

[0018] FIG. 7 is a representation of an information burst for a single-carrier system employing null prefixes, which can be used by the communication system in which the receiving device in FIG. 2 and the transmitting device in FIG. 3 operate;

[0019] FIG. 8 is a representation of a time invariant channel model for a single-input single-output time varying channel;

[0020] FIG. 9 is a flow chart representation of one embodiment of the method performed by the device of FIG. 2 for determining a symbol estimate for at least one transmitter in accordance with the present invention; and

[0021] FIG. 10 is a flow chart representation of one embodiment of the method performed by the communication system of FIG. 1 for transmitting at least two data symbol sequences; and a second embodiment of the method performed by the device of FIG. 2 for determining a symbol estimate for at least two transmitted data symbol sequences in accordance with the present invention.

[0022] In most scattering environments, antenna diversity is a practical, effective and widely applied technique for reducing the effect of multipath fading. One approach is to use multiple antennas at the receiver and perform combining or selection and switching in order to improve the quality of the received signal. An adaptive antenna array is an array of antennas connected to a communications receiver and operates by combining the signals received by the antennas so as to optimize in an adaptive fashion the receive characteristics of the array. By weighting and then summing the multiple antenna signals, the adaptive antenna array can adapt its angular response, sometimes called the array pattern, as well as its frequency response while it operates in response to changes in the propagation environment. While operating, the adaptive antenna attempts to maximize the reception of the signal received from a desired transmitting device, as it simultaneously minimizes the effects of all other interfering signals and noise. With a properly designed adaptive array combining algorithm, the act of maximizing the reception of the desired signal permits receive spatial diversity to be exploited. In a communication system, an adaptive array's ability to suppress interference and its ability to exploit receive diversity offers the potential to reduce co-channel interference, compensate for Doppler-induced inter-carrier interference, improve coverage quality, and increase overall system capacity.

[0023] A problem associated with employing both an equalizer and an adaptive antenna in a wireless communication system lies in the design of an algorithm and device having adequate ability to adapt to changes in the signal environment. For best performance, adaptive antennas that operate in a fast-fading multipath environment must adapt to the rapidly varying channel as well as to any changes in the nature of the desired and interfering signals. In a broadband system, a frequency-selective multipath channel will cause significant variations in the channel across the occupied bandwidth of the received signal. Equalization and interference-suppression algorithms, which cannot track these channel variations in both time and frequency, will suffer significant degradation in performance as measured by the Bit-Error Rate (BER) or Signal-to-Interference-plus-Noise Ratio (SINR).

[0024] A common ingredient in many equalizers and adaptive antenna arrays is an algorithm and device that estimates the characteristics of the multipath propagation environment, channel transfer function, or channel frequency response, between the desired transmitting device and each of the at least one receiving antenna at the communication receiver. When transmit spatial diversity is being employed in a broadband system operating in a frequency-selective environment, then an adaptive antenna array generally requires an estimate of the channel frequency response between each receiving antenna and each of the transmitting devices that are simultaneously sending information to the array. The channel estimation algorithm in this case should operate to simultaneously solve for the channel responses of the multiple transmitting devices. Performing a simultaneous estimate of the channel responses of multiple transmitting devices is a difficult operation in a mobile broadband communication system operating in a multipath-rich environment with high frequency selectivity. Strong interference caused by a spatial diversity deployment, Doppler induced Inter-Carrier Interference (ICI), or a low reuse factor causes even more difficulty to the channel estimation algorithms.

[0025] Alternatively, another approach for reducing the effect of multipath fading is to use multiple transmit antennas and utilize transmit diversity techniques to improve the quality of the received signal. One such procedure is the Alamouti technique, which with two transmit antennas and one receive antenna achieves a diversity order equal to the case where one transmit antenna is used and the receiver applies maximum-ratio receiver combining (MRRC) with two antennas. The technique may easily be generalized to two transmit antennas and M receive antennas to provide a diversity order of 2M. This is done without any feedback from the receiver to the transmitter and with small computation complexity. The scheme requires no bandwidth expansion, as redundancy is applied in space across multiple antennas, not in time or frequency.

[0026] When transmit diversity techniques, such as the Alamouti technique, are employed, the rapidly varying channel poses many of the same difficulties that were described above for receiver processing techniques. It can be shown that when the channel is rapidly varying, the benefits of employing transmit diversity are often not fully realized at the receiver. One of the advantages of the present invention is its ability to track and account for variations in the channel response so that the transmit diversity benefits can be realized even in a rapidly varying channel.

[0027] An overview of the Alamouti technique involves two antennas at the transmitter and one or two antennas at the receiver, and is described by equation (1) below. The derivation here is applicable to any time-varying channel, which may include the frequency domain of an OFDM system, or a flat-faded single carrier system, or the frequency domain of a single carrier, TDMA, or CDMA or other similar systems. The information symbol stream to be transmitted is broken up into disjoint pairs of symbols to be transmitted over a coding interval consisting of two symbol intervals. Each pair of symbols is then multiplexed onto two transmit antennas over the two symbol intervals of the coding interval according to the following encoding scheme. During the first symbol interval, the first transmit antenna transmits the first information symbol of the pair, and the second transmit antenna transmits the second information symbol of the pair. During the second symbol interval, the first transmit antenna transmits the negative conjugate of the second information symbol of the pair, and the second transmit antenna transmits the conjugate of the first information symbol of the pair. If x_i(k) is the symbol transmitted out of antenna i at symbol interval k, and s(k) is the (unit-average-power) k^thinformation symbol to be transmitted, the encoding scheme is as follows: $\begin{array}{cc}\begin{array}{cc}{x}_1\left(k\right)=1/\sqrt{2}s\left(k\right)& x_2\left(k\right)=1/\sqrt{2}s\left(k+1\right)\\ x_1\left(k+1\right)=1/\sqrt{2}{s}^{*}\left(k+1\right)& x_2\left(k+1\right)=1/\sqrt{2}{s}^{*}\left(k\right)\end{array}& \left(1\right)\end{array}$

[0028] In this encoding scheme, two information symbols are transmitted and received over two symbol intervals. Therefore, no additional bandwidth is used by this scheme. Although not necessary for the proper functioning of the technique, the 1/√{square root over (2)} in this equation is included so that the average transmit power is the same as when no transmit diversity technique is employed.

[0029] To decode the transmitted data stream, the receiver requires knowledge of the channel gain between each transmit antenna and the receive antenna(s). One possible decoding scheme assumes no variation in the channel response over a pair of symbol intervals. With a single antenna at the receiver, the decoding scheme is as follows: Let A_ijbe the complex channel gain between the j^thtransmit antenna and the i^threceive antenna. Let y_i(k) be the sample received on the i^threceive antenna during the k^thsymbol interval. According to this decoding scheme, a single-antenna receiver computes z(k) and z(k+1) to be an estimate of s(k) and s(k+1), respectively, as follows: $\begin{array}{cc}\left[\begin{array}{c}z\left(k\right)\\ z\left(k+1\right)\end{array}\right]=\left[\begin{array}{cc}{A}_{11}^{*}& A_{12}\\ A_{12}^{*}& -A_{11}\end{array}\right]\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\end{array}\right]& \left(2\right)\end{array}$

[0030] A two-antenna receiver computes z(k) and z(k+1) as follows: $\begin{array}{cc}\left[\begin{array}{c}z\left(k\right)\\ z\left(k+1\right)\end{array}\right]=\left[\begin{array}{cccc}{A}_{11}^{*}& A_{12}& A_{21}^{*}& A_{22}\\ A_{12}^{*}& -A_{11}& A_{22}^{*}& -A_{21}\end{array}\right]\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\\ y_2\left(k\right)\\ y_2^{*}\left(k+1\right)\end{array}\right]& \left(3\right)\end{array}$

[0031] Extending this scheme to more than two receive antennas is straightforward. A 2-transmit antenna, 1-receive antenna system (2Tx,1Rx) using this scheme has the same diversity characteristics as a one transmit antenna, two receive antenna system (1Tx,2Rx) that uses maximal ratio combining. However, the (2Tx,1Rx) Alamouti scheme has 3 dB performance degradation relative to a (1Tx,2Rx) Max Ratio system due to the power back off that must occur with two transmit antennas to make an equitable (i.e., power-equivalent) comparison.

[0032] An important aspect of the scheme presented above is that the receive combining scheme assumes a channel that is constant over the two successive symbol intervals of a coding interval. In a highly mobile channel, especially at the higher carrier frequencies, this assumption becomes difficult to maintain. Therefore, the performance of the above decoding scheme in highly mobile channels is diminished.

[0033] Briefly described, the present invention is a method and device for calculating the symbol estimates of the signals transmitted by at least two transmitting devices and received on at least one receive antenna. The method and device of the present invention also provides for a simple way of implementing an Alamouti style transmit diversity scheme so that spatial diversity can be achieved via frequency domain receive processing in high speed single carrier systems. The receive processing techniques account for time variations in the channel response and provide much better performance than existing methods which do not account for channel variations. The method of the present invention as is derived from the Alamouti Scheme is as follows:

[0034] The following discussion begins by first reviewing the input/output equations for a system in which a transmit array communicates with a receive array. Then system equations are modified to capture the space-time-block coding technique when the channel is varying over the encoding/decoding interval. Given the resulting formulation, a method is presented for decoding the space-time block coding technique in a way that tracks and accounts for the channel variations.

[0035] First, consider a system where an M_tx-element transmit array communicates with an M_rx-element receive array over a flat time-varying Rayleigh channel. The following equation relates the symbols transmitted by the transmit array to the symbols received at the receive array during symbol interval k:

[0036] y(k)=A(k)x(k)+n(k)  (4)

[0037] In this equation, x(k) is the M_tx×1 vector of symbols transmitted by the transmit array during symbol interval k. A(k) is the M_rx×M_txmatrix of complex channel gains between the transmit elements and the receive elements. y(k) is the M_rx×1 vector of symbols received by the receive array during symbol interval k. Finally, n(k) is the M_rx×1 vector of noise samples on the receive array. These quantities are given by: $\begin{array}{cc}y\left(k\right)=\left[\begin{array}{c}{y}_1\left(k\right)\\ y_2\left(k\right)\\ ⋮\\ y_{M_{\mathrm{RX}}}\left(k\right)\end{array}\right]A\left(k\right)=\left[\begin{array}{cccc}{A}_{11}\left(k\right)& A_{12}\left(k\right)& \cdots & A_{1M_{\mathrm{TX}}}\left(k\right)\\ A_{21}\left(k\right)& A_{22}\left(k\right)& \cdots & A_{2M_{\mathrm{TX}}}\left(k\right)\\ ⋮& ⋮& ⋰& ⋮\\ A_{M_{\mathrm{RX}}1}\left(k\right)& A_{M_{\mathrm{RX}}2}\left(k\right)& \cdots & A_{M_{\mathrm{RX}}{M}_{\mathrm{TX}}}\left(k\right)\end{array}\right]& \left(5\right)\\ x\left(k\right)=\left[\begin{array}{c}{x}_1\left(k\right)\\ x_2\left(k\right)\\ ⋮\\ x_{M_{\mathrm{TX}}}\left(k\right)\end{array}\right]n\left(k\right)=\left[\begin{array}{c}{n}_1\left(k\right)\\ n_2\left(k\right)\\ ⋮\\ n_{M_{\mathrm{RX}}}\left(k\right)\end{array}\right].& \left(6\right)\end{array}$

[0038] Note that only the spatial domain is captured by the above equations for a transmit and receive array. All matrices and vectors in these equations encompass the space dimension at a specific point in time.

[0039] In an Alamouti-style scheme, Equation (4) is still valid, but an overall system equation needs to be derived to capture the combined space-time processing that is inherent in space-time block coding. Consider one coding interval that consists of two symbol intervals k and k+1. In one embodiment, these symbol intervals are in the time domain of a flat faded single carrier system. In another embodiment, these symbol intervals are in the frequency domain of an OFDM or multicarrier system. In yet another embodiment, these symbols are in the frequency domain of a single carrier system in which the receiver is executing frequency-domain receive processing. In yet another embodiment, these symbols are in the frequency domain of a CDMA system where the received data is processed in the frequency-domain. If Equation (1) is combined with Equation (4), the following relationship is obtained, which captures Alamouti's encoding scheme for a single antenna receiver: $\begin{array}{cc}\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{A}_{11}\left(k\right)& A_{12}\left(k\right)\\ A_{12}^{*}\left(k+1\right)& -A_{11}^{*}\left(k+1\right)\end{array}\right]\left[\begin{array}{c}s\left(k\right)\\ s\left(k+1\right)\end{array}\right]+\left[\begin{array}{c}{n}_1\left(k\right)\\ n_1^{*}\left(k+1\right)\end{array}\right]& \left(7\right)\end{array}$

[0040] For a two-element receive array, the following is obtained: $\begin{array}{cc}\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\\ y_2\left(k\right)\\ y_2^{*}\left(k+1\right)\end{array}\right]=\frac{1}{\sqrt{2}}[\begin{array}{cc}{A}_{11}\left(k\right)& A_{12}\left(k\right)\\ A_{12}^{*}\left(k+1\right)& -A_{11}^{*}\left(k+1\right)\\ A_{21}\left(k\right)& A_{22}\left(k\right)\\ A_{22}^{*}\left(k+1\right)& -A_{21}^{*}\left(k+1\right)\end{array}][\begin{array}{c}s\left(k\right)\\ s\left(k+1\right)\end{array}]+[\begin{array}{c}{n}_1\left(k\right)\\ n_1^{*}\left(k+1\right)\\ n_2\left(k\right)\\ n_2^{*}\left(k+1\right)\end{array}]& \left(8\right)\end{array}$

[0041] As mentioned before, the 1/√{square root over (2)} in these equations is not necessary for the proper functioning of the technique, but is included so that the average power transmitted is the same as other diversity and non-diversity techniques.

[0042] Equations (7) and (8) can be simplified as follows: $\begin{array}{cc}\hat{y}\left(k\right)=\frac{1}{\sqrt{2}}H\left(k\right)\hat{s}\left(k\right)+\hat{n}\left(k\right)& \left(9\right)\end{array}$

[0043] where for a single element receiver: $\begin{array}{cc}\hat{y}\left(k\right)=\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\end{array}\right]H\left(k\right)=\left[\begin{array}{cc}{A}_{11}\left(k\right)& A_{12}\left(k\right)\\ A_{12}^{*}\left(k+1\right)& -A_{11}^{*}\left(k+1\right)\end{array}\right]\hat{s}\left(k\right)=\left[\begin{array}{c}s\left(k\right)\\ s\left(k+1\right)\end{array}\right]\hat{n}\left(k\right)=\left[\begin{array}{c}{n}_1\left(k\right)\\ n_1^{*}\left(k+1\right)\end{array}\right]& \left(10\right)\end{array}$

[0044] For a two element receiver: $\begin{array}{cc}\hat{y}\left(k\right)=\left[\begin{array}{c}{y}_1\left(k\right)\\ y_1^{*}\left(k+1\right)\\ y_2\left(k\right)\\ y_2^{*}\left(k+1\right)\end{array}\right]H\left(k\right)=\left[\begin{array}{cc}{A}_{11}\left(k\right)& A_{12}\left(k\right)\\ A_{12}^{*}\left(k+1\right)& -A_{11}^{*}\left(k+1\right)\\ A_{21}\left(k\right)& A_{22}\left(k\right)\\ A_{22}^{*}\left(k+1\right)& -A_{21}^{*}\left(k+1\right)\end{array}\right]\hat{s}\left(k\right)=\left[\begin{array}{c}s\left(k\right)\\ s\left(k+1\right)\end{array}\right]\hat{n}\left(k\right)=\left[\begin{array}{c}{n}_1\left(k\right)\\ n_1^{*}\left(k+1\right)\\ n_2\left(k\right)\\ n_2^{*}\left(k+1\right)\end{array}\right]& \left(11\right)\end{array}$

[0045] Equation (11) can be rewritten another way: $\begin{array}{cc}\hat{y}\left(k\right)=\frac{1}{\sqrt{2}}{H}_1\left(k\right)s\left(k\right)+\frac{1}{\sqrt{2}}{H}_2\left(k\right)s\left(k+1\right)+\hat{n}\left(k\right)& \left(12\right)\end{array}$

[0046] where H_i(k) is the i^thcolumn of the matrix H(k). Equations (9) through (12) provide a concise expression that captures the structure of Alamouti's rate one space-time block coding scheme.

[0047] Recovering the transmitted symbols s(k) and s(k+1) involves the following linear combining scheme: $\begin{array}{cc}z\left(k\right)=W^H\left(k\right)\hat{y}\left(k\right),\mathrm{where}W\left(k\right)=\left[\begin{array}{cc}{w}_1\left(k\right)& w_2\left(k\right)\end{array}\right],\mathrm{and}z\left(k\right)=\left[\begin{array}{c}z\left(k\right)\\ z\left(k+1\right)\end{array}\right]& \left(13\right)\end{array}$

[0048] where z(k) and z(k+1) are estimates for s(k) and s(k+1), respectively. The receive weight matrix W(k) is dimensioned 2M_rx×2.

[0049] Combining Equations (12) and (13), results in: $\begin{array}{cc}z\left(k\right)=\frac{1}{\sqrt{2}}{w}_1^H\left(k\right)H_1\left(k\right)s\left(k\right)+\frac{1}{\sqrt{2}}{w}_1^H\left(k\right)H_2\left(k\right)s\left(k+1\right)+w_1^H\hat{n}\left(k\right)& \left(14\right)\\ z\left(k+1\right)=\frac{1}{\sqrt{2}}{w}_2^H\left(k\right)H_1\left(k\right)s\left(k\right)+\frac{1}{\sqrt{2}}{w}_2^H\left(k\right)H_2\left(k\right)s\left(k+1\right)+w_2^H\hat{n}\left(k\right)& \left(15\right)\end{array}$

[0050] Some comments are in order concerning the structure of Equations (9) through (15). First, Equation (12) has a structure similar to a two-user receive SDMA system. In other words, H_i(k) of (12) is analogous to the vector channel response for user i at time k in a receive SDMA system. ŷ(k) is analogous to the vector of received samples on the receive array at time k, and {circumflex over (n)}(k) is analogous to the vector of noise samples on the receive elements at time k. The quantities s(k) and s(k+1) in (12) are analogous to the user signals transmitted by the two SDMA users, respectively.

[0051] Second, if the channel is constant across the two symbol intervals of the coding interval, then the columns of the channel matrix H(k) are orthogonal. A constant channel means A_ij(k)=A_ij(k+1) for i,j={1,2}. The resulting orthogonality permits an easy receive combining scheme whereby w_i(k) is simply chosen to be a scalar multiple of H_i(k) for i=1,2. (The scalar multiple is chosen according to the particular detection criterion: i.e., MMSE, ML, etc.). With an orthogonal channel matrix, no component of s(k+1) will be present in the estimate for s(k), and vice versa.

[0052] Third, if the channel is not constant across the two symbol intervals of the coding interval, then the columns of the channel matrix H(k) are not orthogonal. The combining scheme where w_i(k) is simply chosen to be a scalar multiple of H_i(k) will result in a component of s(k+1) being left over in the estimate for s(k), and vice versa. As is shown below, this cross-talk results in a bit error floor.

[0053] Fourth, the symbol intervals of a coding interval do not necessarily have to be adjacent in time. A coding interval can be made up of symbol intervals that are not adjacent in time or in frequency (when the system is a multicarrier or OFDM system).

[0054] The formulation presented above provides the basis for the method of the present invention, which provides a receiver processing algorithm, which handles time-varying channels. The goal of the receiver processing algorithm of the present invention is to recover the information symbols s(k) and s(k+1) by processing the vector of received samples ŷ(k). The preferred embodiment of the present invention employs the minimum mean square error criteria to compute a combining weight vector that better handles time varying channels. Other criteria can also be used to determine the frequency-domain information symbols, s(k) and s(k+1), such as joint detection of the frequency-domain information symbols (i.e., the maximum likelihood criteria) and successive cancellation. It can be shown that the receive weight matrix in Equation (13) that minimizes the mean square error between the transmitted symbol vector ŝ(k) and the combined output vector z(k) is given by: $\begin{array}{cc}W\left(k\right)={\frac{1}{\sqrt{2}}\left[\frac{1}{2}H\left(k\right)H^H\left(k\right)+{\sigma }^{2}I\right]}^{-1}H\left(k\right)& \left(16\right)\end{array}$

[0055] where I is an 2M_rx×2M_rxidentity matrix and σ²is the noise power at the receive antenna(s). In another preferred embodiment, σ²I in Equation (16) is replaced by a 2M_rx×2M_rxmatrix, R_n(k), that is an estimate of the space-time covariance matrix of the interference plus noise at subcarrier k. Equation (16) provides much better performance in mobile channels than when w_i(k)=αH_i(k), (where α is a scalar constant).

[0056] It is noted that with a single receive antenna and no channel variations; Equation (16) reduces to the following: $\begin{array}{cc}W\left(k\right)=\frac{\sqrt{2}}{{\left|A_{11}\right|}^2+{\left|A_{12}\right|}^2+2{\sigma }^2}H\left(k\right)& \left(17\right)\end{array}$

[0057] With two receive antennas and no channel variations; Equation (16) does not reduce to Alamouti's combining scheme. With two receive antennas, however, the method of setting W(k)=αH(k), where α is a scalar constant, is a zero-forcing (i.e., maximum likelihood) solution, while Equation (16) is an MMSE solution. At high SNRs and constant channels (H(k) having orthogonal columns), the zero forcing and MMSE solutions will have virtually identical performance.

[0058] The method outlined above for two transmit elements can be extended to a three and four element rate one-half space-time block code as follows. Begin by reviewing the input/output equations for a system where an M_tx-element transmit array communicates with an M_rx-element receive array over a flat Rayleigh channel. The following equation relates the symbols transmitted by the transmit array to the symbols received at the receive array during symbol interval k:

[0059] y(k)=A(k)x(k)+n(k)  (18)

[0060] In this equation, x(k) is the M_tx×1 vector of symbols transmitted by the transmit array during symbol interval k. A(k) is the M_rx×M_txmatrix of complex channel gains between the transmit elements and the receive elements at symbol interval k. y(k) is the M_rx×1 vector of symbols received by the receive array during symbol interval k. Finally, n(k) is the M_rx×1 vector of noise samples on the receive array. These components are written as follows: $\begin{array}{cc}y\left(k\right)=\left[\begin{array}{c}{y}_1\left(k\right)\\ y_2\left(k\right)\\ ⋮\\ y_{M_{\mathrm{RX}}}\left(k\right)\end{array}\right]A\left(k\right)=\left[\begin{array}{cccc}{A}_{11}\left(k\right)& A_{12}\left(k\right)& \cdots & A_{1M_{\mathrm{TX}}}\left(k\right)\\ A_{21}\left(k\right)& A_{22}\left(k\right)& \cdots & A_{2M_{\mathrm{TX}}}\left(k\right)\\ ⋮& ⋮& ⋰& ⋮\\ A_{M_{\mathrm{RX}}1}\left(k\right)& A_{M_{\mathrm{RX}}2}\left(k\right)& \cdots & A_{M_{\mathrm{RX}}{M}_{\mathrm{TX}}}\left(k\right)\end{array}\right]& \left(19\right)\\ x\left(k\right)=\left[\begin{array}{c}{x}_1\left(k\right)\\ x_2\left(k\right)\\ ⋮\\ x_{M_{\mathrm{TX}}}\left(k\right)\end{array}\right]n\left(k\right)=\left[\begin{array}{c}{n}_1\left(k\right)\\ n_2\left(k\right)\\ ⋮\\ n_{M_{\mathrm{RX}}}\left(k\right)\end{array}\right]& \left(20\right)\end{array}$

[0061] In space-time block coding, the complex symbols to be transmitted on each transmit element are some function of a set of K data symbols s₁through s_Kthat are to be transmitted over P symbol intervals. (The symbols s₁through s_Kare typically ordinary QAM or PSK symbols that may or may not already be coded according to some existing (i.e., non-spatial) coding scheme.) Since K data symbols are transmitted over P symbol intervals, the rate of the space-time block code is defined to be K/P. The space-time block code is defined by the P×M_txcode matrix, which specifies how the uncoded data symbols are transformed into the complex symbols to be transmitted out the array. The entries of the code matrix are linear combinations of the K data symbols s₁through s_Kand their conjugates. The (i,j)^thentry of the code matrix provides an expression that determines the complex symbol that will be transmitted out the j^thantenna at symbol interval i within the encoding/decoding interval. For example, the Alamouti 2-Transmitter, Rate-1 space-time block code utilized earlier in this document has a code matrix given by: $\begin{array}{cc}{G}_2=\left[\begin{array}{cc}{s}_1& s_2\\ -s_2^{*}& s_1^{*}\end{array}\right]& \left(21\right)\end{array}$

[0062] Half rate codes for three and four transmit elements have code matrices that are given respectively by: $\begin{array}{cc}{G}_3=\left[\begin{array}{ccc}{s}_1& s_2& s_3\\ -s_2& s_1& -s_4\\ -s_3& s_4& s_1\\ -s_4& -s_3& s_2\\ s_1^{*}& s_2^{*}& s_3^{*}\\ -s_2^{*}& s_1^{*}& -s_4^{*}\\ -s_3^{*}& s_4^{*}& s_1^{*}\\ -s_4^{*}& -s_3^{*}& s_2^{*}\end{array}\right]G_4=\left[\begin{array}{cccc}{s}_1& s_2& s_3& s_4\\ -s_2& s_1& -s_4& s_3\\ -s_3& s_4& s_1& -s_2\\ -s_4& -s_3& s_2& s_1\\ s_1^{*}& s_2^{*}& s_3^{*}& s_4^{*}\\ -s_2^{*}& s_1^{*}& -s_4^{*}& s_3^{*}\\ -s_3^{*}& s_4^{*}& s_1^{*}& -s_2^{*}\\ -s_4^{*}& -s_3^{*}& s_2^{*}& s_1^{*}\end{array}\right]& \left(22\right)\end{array}$

[0063] In both of these two codes, K=4 symbols are transmitted over P=8 symbol intervals, so both codes have a rate equal to one-half.

[0064] When the M_rx×M_txmatrix of complex channel gains A(k) is changing significantly across the encoding/decoding interval (symbols k=1,2, . . . ,K), decoders that do not track the channel will suffer a significant loss in BER performance.

[0065] The following procedure enables the decoder in the present invention to handle channel variation. For a given code matrix, equations must first be set up for the signals received at the receive array in terms of the transmitted data symbols s_ithrough s_K. It turns out that codes G₂, G₃, and G₄have a special structure that greatly simplifies the expressions for the received signal vectors when the channel is not constant over the P symbol intervals of the encoding/decoding interval. For the G₂, G₃, and G₄codes, the received signal vector at time k can be shown to be: $\begin{array}{cc}y\left(k\right)=\frac{1}{\sqrt{M_{\mathrm{zx}}}}A\left(k\right)C\left(k\right)s+n\left(k\right)k=1,\dots ,\frac{P}{2}y\left(k\right)=\frac{1}{\sqrt{M_{\mathrm{zx}}}}A\left(k\right)C\left(k\right)s^{*}+n\left(k\right)k=\frac{P}{2}+1,\dots ,P& \left(23\right)\end{array}$

[0066] where

[0067] s=[s₁s₂. . . s_K]^T  (24)

[0068] is the length K vector of unit power uncoded data symbols (e.g., QAM or PSK) to be delivered to the receiver over the P symbol intervals. The inverse square root of the number of transmit antennas is included in (23) because the total power transmitted by the array is normalized to unity. Note that the conjugation of the s vector for k=P/2+1, . . . , P is a result of the lower P/2 rows in the coding matrices being composed only of conjugate values of the data symbols. (The analysis for other codes not having this structure would need modification.) In (23), the P matrices C(k), where k=1. . . P, are of dimension M_tx×K and are determined by inspecting the code matrix of the space-time block code being used. For example, for the G₂code in Equation (21), P=2, and C(1) and C(2) are given by: $\begin{array}{cc}C\left(1\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]C\left(2\right)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]& \left(25\right)\end{array}$

[0069] For the G₃code of Equation (22), P=8, and the C(k) matrices are defined as follows: $\begin{array}{cc}C\left(1\right)=C\left(5\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]& \left(26\right)\\ C\left(2\right)=C\left(6\right)=\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right]& \\ C\left(3\right)=C\left(7\right)=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right]& \\ C\left(4\right)=C\left(8\right)=\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\end{array}\right]& \end{array}$

[0070] For the G₄code of Equation (22), P=8, and the C(k) matrices are defined as follows: $\begin{array}{cc}C\left(1\right)=C\left(5\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(27\right)\\ C\left(2\right)=C\left(6\right)=\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right]& \\ C\left(3\right)=C\left(7\right)=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right]& \\ C\left(4\right)=C\left(8\right)=\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right]& \end{array}$

[0071] The encoding process for these codes can be summarized as:

[0072] $\begin{array}{cc}\begin{array}{cc}x\left(k\right)=C\left(k\right)s& k=1,\dots ,\frac{P}{2}\\ x\left(k\right)=C\left(k\right)s^{*}& k=\frac{P}{2}+1,\dots ,P\end{array}& \left(28\right)\end{array}$

[0073] where x(k), and s are vectors defined in Equations (20) and (24) respectively.

[0074] In order to recover the data symbols, a space-time receive vector is created to collect the received array data over the encoding/decoding interval as follows: $\begin{array}{cc}\hat{y}=\frac{1}{\sqrt{M_{\mathrm{rx}}}}\mathrm{Hs}+\hat{n}& \left(29\right)\end{array}$

[0075] where: $\begin{array}{cc}\begin{array}{c}\hat{y}=\left[\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ ⋮\\ y\left(\frac{P}{2}\right)\\ y^{*}\left(\frac{P}{2}+1\right)\\ y^{*}\left(\frac{P}{2}+2\right)\\ ⋮\\ y^{*}\left(P\right)\end{array}\right]H=\left[\begin{array}{c}A\left(1\right)C\left(1\right)\\ A\left(2\right)C\left(2\right)\\ ⋮\\ A\left(\frac{P}{2}\right)C\left(\frac{P}{2}\right)\\ A^{*}\left(\frac{P}{2}+1\right)C\left(\frac{P}{2}+1\right)\\ A^{*}\left(\frac{P}{2}+2\right)C\left(\frac{P}{2}+2\right)\\ \\ A^{*}\left(P\right)C\left(P\right)\end{array}\right]\\ \hat{n}=\left[\begin{array}{c}n\left(1\right)\\ n\left(2\right)\\ ⋮\\ n\left(\frac{P}{2}\right)\\ n^{*}\left(\frac{P}{2}+1\right)\\ n^{*}\left(\frac{P}{2}+2\right)\\ \\ n^{*}\left(P\right)\end{array}\right]\end{array}& \left(30\right)\end{array}$

[0076] Recovering the transmitted symbols s(k) and s(k+1) involves the following linear combining scheme:

[0077] z=W^Hŷ, where W=[w₁w₂. . . w_K], and z=[z₁z₂. . . z_K]^T  (31)

[0078] where Z_kis an estimate for s_k, k=1, . . . ,K. The receive weight matrix W is dimensioned PM_rx×K. The MMSE solution for the weight matrix is given by: $\begin{array}{cc}W={\frac{1}{\sqrt{M_{\mathrm{rx}}}}\left[\frac{1}{M_{\mathrm{rx}}}{\mathrm{HH}}^H+{\sigma }^{2}I\right]}^{-1}H& \left(32\right)\end{array}$

[0079] where I is an PM_rx×PM_rxidentity matrix and σ²is the noise power at the receive antenna(s). In another preferred embodiment, σ²I in Equation (32) is replaced by a PM_rx×PM_rxmatrix, R_n(k), that is an estimate of the space-time covariance matrix of the interference plus noise at subcarrier k.

[0080] Note that the preferred embodiment of the combining weights of Equation (32) of the present invention is found using the MMSE criteria. Other criteria can also be used to determine the frequency-domain information symbols such as joint detection of the frequency-domain information symbols (i.e., the maximum likelihood criteria) and successive cancellation.

[0081] Once these weights are applied to the received data, the resulting combined data would be sent to a detection device for determining the transmitted information. In the case of an OFDM system, the received data is in the frequency domain, and the combined output is also in the frequency domain and is sent to a detection device. However, in the case of a single carrier system using frequency domain processing, the combined frequency domain data would be converted to the time domain with a Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT) operation, and the resulting time-domain data would be sent to the detection device.

[0082] Another embodiment of the present invention involves a single carrier system in which at least two antennas are employed at the transmitter and at least one antenna is employed at the receiver. The time domain of this single carrier system is divided up into successive “bauds,” where each baud consists of L_pcyclic prefix symbols followed by N data symbols. The L_pcyclic prefix symbols are a repetition of the last L_psymbols of the N data symbol sequence and are used in a manner analogous to how the cyclic prefix is used in a typical OFDM system. It is assumed that the length of the cyclic prefix is greater than or equal to the length of the effective impulse response of the channel between the transmitter and receiver, which is equal to the length of the RF channel plus twice the number of significant “tails” of the pulse waveform.

[0083] Two types of bauds are transmitted in this system: a “pilot baud” or a “data baud.” In a pilot baud, the transmitted N-symbol data sequence is known at the receiver and is typically used for channel estimation and other purposes. The N-symbol data sequence transmitted in a data baud carries the information being conveyed between the transmitter and the receiver.

[0084] A part of the present invention is the following encoding process in which the transmitter must use two antennas to deliver two independent length N data sequences to the receiver over a two-baud time interval. These two sequences are denoted s[m,b₁] and s[m,b₂], where m is the time-index variable of the sequence (m=0 . . . N−1), and b₁and b₂refer to the first and second sequence, respectively.

[0085] In the preferred embodiment, the encoding process of the present invention is as follows. In the data portion of the first baud interval, the first antenna transmits an unmodified copy of the first sequence s[m,b₁]. The second antenna transmits an unmodified copy of the second sequence s[m,b₂] in the data portion of the first baud interval. In the data portion of the second baud interval, the first antenna transmits a sequence that is a negative, conjugated, time-reversed and circularly shifted (i.e., a modulo N shift) copy of the second information sequence s[m,b₂]. In addition, in the data portion of the second baud interval, the second antenna transmits a sequence that is a conjugated, time-reversed and circularly shifted copy of the first information sequence s[m,b₁]. Defining x_j[m,b_d] to be the information sequence transmitted by the j^thantenna in data portion of the d^thbaud, then:

[0086] x₁[m,b₁]=1/√{square root over (2)}s[m,b₁] x₂[m,b₁]=1/√{square root over (2)}s[m,b₂]

[0087] x₁[m,b₂]=−1/√{square root over (2)}s*[D−m,b₂] x₂[m,b₂]=1/√{square root over (2)}s*[D−m,b₁]  (33)

[0088] where D (0≦D≦N−1) is an integer that causes an arbitrary circular shift of the sequence by D symbols. For example, for D=0, the contents of the data portion within the second baud interval on the first antenna are given as:

[0089] −1/√{square root over (2)}[s*[0,b₁] s*[N−1,b₁] s*[N−2,b₁] . . . s*[1,b₁]]  (34)

[0090] For example, for D=N−1, the contents of the data portion within the second baud interval on the first antenna are given as:

[0091] −1/√{square root over (2)}[s*[N−1,b₁] s*[N−2,b₁] s*[N−3,b₁] . . . s*[0,b₁]]  (35)

[0092] For other values of D (i.e., 0<D<N−1), the contents of the data portion within the second baud interval on the first antenna are given as:

[0093] −1/√{square root over (2)}[s*[D,b₁] s*[D−1,b₁] . . . s*[0,b₁] s*[N−1,b₁] s*[N−2,b₁] . . . s*[D+1,b₁]]  (36)

[0094] Equations (34) through (36) can be rewritten to show the contents of the second baud interval on the second antenna in accordance with (33). It will be shown below that the value of D does not affect performance as long as the receiver knows what value of D is being used by the transmitter.

[0095] Once the data portions of the two baud intervals are determined according to (33), the last L_psymbols of the data portion are copied to the beginning of the baud interval to form the cyclic prefix. Note that the inverse of the square root of 2 is included in (33) to make the total power transmitted by the two element array equivalent to a single antenna that transmits the two symbol sequences without any transmit diversity scheme.

[0096] At the receiver, N samples are collected and sent to an FFT for the frequency domain processing. Let y_i[m,b] be the length N data sequence (m=0,1, . . . ,N−1) received by the i^threceive antenna during the b^thbaud interval. It is assumed that the time sample corresponding to sample m=0 corresponds to t=−N_pT_s, where T_sis the symbol duration, N_pis the number of significant non-causal “tails” of the pulse shaping waveform, and t=0 is defined to be the time sample of the first symbol of the data portion of the baud of the earliest multipath component. Choosing the N sampling times in this manner causes the effective impulse response between the transmitter and receiver to be causal even though the transmitted waveform is technically non-causal due to the non-causal tails of the pulse waveform. Given the use of a cyclic prefix greater than or equal to the length of the effective impulse response, y_i[m,b] can be shown to be: $\begin{array}{cc}{y}_i\left[m,b\right]=\sum _{j=1}^J\sum _{n=0}^{L_p-1}{\hat{h}}_{\mathrm{ij}}\left[n\right]x_j\left[m-n,b\right]& \left(37\right)\end{array}$

[0097] where J denotes the number of transmit antennas (two in this case) and

[0098] ĥ_ij[n]=h((n−N_p)T_s)  (38)

[0099] and h(t) is the convolution of the transmit and receive pulse shaping filters and the RF channel. In the frequency domain, the received data has the following form: $\begin{array}{cc}{Y}_i\left[k,b\right]=\sum _{m=0}^{N-1}{y}_i\left[m,b\right]e^{-j2\pi km/N}=\sum _{j=1}^J{H}_{\mathrm{ij}}\left[k,b\right]X_j\left[k,b\right]& \left(39\right)\end{array}$

[0100] where $\begin{array}{cc}{X}_j\left[k,b\right]=\sum _{m=0}^{N-1}{x}_j\left[m,b\right]e^{-j2\pi km/N}& \left(40\right)\end{array}$

[0101] and $\begin{array}{cc}{H}_{\mathrm{ij}}\left[k,b\right]=\sum _{m=0}^{L_p-1}{\hat{h}}_{\mathrm{ij}}\left[m,b\right]e^{-j2\pi km/N}& \left(41\right)\end{array}$

[0102] Given the encoding scheme in Equation (33), Equation (40) becomes $\begin{array}{cc}\begin{array}{cc}{X}_1\left[k,b_1\right]=\frac{1}{\sqrt{2}}S\left[k,b_1\right]& X_2\left[k,b_1\right]=\frac{1}{\sqrt{2}}S\left[k,b_2\right]\\ \begin{array}{c}{X}_1\left[k,b_2\right]=\\ -\frac{1}{\sqrt{2}}{e}^{-\mathrm{j2\pi }Dk/N}{S}^{*}\left[k,b_2\right]\end{array}& \begin{array}{c}{X}_2\left[k,b_2\right]=\\ \frac{1}{\sqrt{2}}{e}^{-\mathrm{j2\pi }\mathrm{Dk}/N}{S}^{*}\left[k,b_1\right]\end{array}\end{array}& \left(42\right)\end{array}$

[0103] where $\begin{array}{cc}S\left[k,b\right]=\sum _{m=0}^{N-1}s\left[m,b\right]e^{-j2\pi km/N}& \left(43\right)\end{array}$

[0104] When M_rxreceive antennas are used with the encoding scheme of (33), the received data can be expressed as follows: $\begin{array}{cc}\hat{Y}=\frac{1}{\sqrt{2}}\hat{H}S+\hat{N}& \left(44\right)\end{array}$

[0105] where the following definitions are used: $\begin{array}{cc}\begin{array}{c}\hat{Y}=\left[\begin{array}{c}Y\left[k,b_1\right]\\ Y^{*}\left[k,b_2\right]\end{array}\right],\\ \hat{H}=\left[\begin{array}{cc}{H}_1\left[k,b_1\right]& H_2\left[k,b_1\right]\\ H_2^{*}\left[k,b_2\right]e^{j2\pi \mathrm{Dk}/N}& -H_1^{*}\left[k,b_2\right]e^{j2\pi \mathrm{Dk}/N}\end{array}\right],\\ S=\left[\begin{array}{c}S\left[k,b_1\right]\\ S\left[k,b_2\right]\end{array}\right],\hat{N}=\left[\begin{array}{c}N\left[k,b_1\right]\\ N^{*}\left[k,b_2\right]\end{array}\right]\end{array}& \left(45\right)\\ Y\left[k,b\right]=\left[\begin{array}{c}{Y}_1\left[k,b\right]\\ Y_2\left[k,b\right]\\ ⋮\\ Y_{M_{\mathrm{rx}}}\left[k,b\right]\end{array}\right],H_j\left[k,b\right]=\left[\begin{array}{c}{H}_{1j}\left[k,b\right]\\ H_{2j}\left[k,b\right]\\ ⋮\\ H_{M_{\mathrm{rx}}j}\left[k,b\right]\end{array}\right],N\left[k,b\right]=\left[\begin{array}{c}{N}_1\left[k,b\right]\\ N_2\left[k,b\right]\\ ⋮\\ N_{M_{\mathrm{rx}}}\left[k,b\right]\end{array}\right]& \left(46\right)\end{array}$

[0106] The frequency-domain transmitted symbol sequences are estimated by the receiver as: $\begin{array}{cc}Z\left[k,b\right]=W^H\hat{Y},\mathrm{where}W=\left[\begin{array}{cc}{w}_1& w_2\end{array}\right],\mathrm{and}Z\left[k,b\right]=\left[\begin{array}{c}Z\left(k,b_1\right)\\ Z\left(k,b_2\right)\end{array}\right]& \left(47\right)\end{array}$

[0107] where the MMSE-based decoder for this system is to choose the W matrix according to: $\begin{array}{cc}W={\frac{1}{\sqrt{2}}\left[\frac{1}{2}\hat{H}{\hat{H}}^H+{\sigma }^{2}I\right]}^{-1}\hat{H}& \left(48\right)\end{array}$

[0108] where I is an 2M_rx×2M_rxidentity matrix and σ²is the frequency-domain noise power on a frequency bin. In another preferred embodiment, σ²I in Equation (48) is replaced by a 2M_rx×2M_rxmatrix, R_n(k), that is an estimate of the space-time covariance matrix of the interference plus noise at subcarrier k.

[0109] With the exception of the exponential terms e^j2πDk/Nin (45), the encoding scheme of the present invention embodied by Equation (33) has caused Equations (44) through (48) to be identical to the equations described earlier for a generic flat-faded transmit diversity system. Note that if D is zero, the exponential terms in (45) will equal one and drop out of the equation. Also, note that if H_j[k,b₁]=H_j[k,b₂] for j=1,2, then the two columns of Ĥ are orthogonal regardless of the value of D. The value of D, therefore, will not affect the attainable diversity benefits of the proposed scheme.

[0110] If the integer shift factor D is not equal to 0, then the receiver has two choices for accounting for the nonzero value of D. The first choice is to incorporate the exponential terms e^j2πDk/Nin (45). The second choice is based on the observation that an exponential phase shift in the frequency domain is equivalent to a circular shift in the time-domain. Therefore, the second choice is to circularly shift the received data vector (37) by D before carrying out the FFT. D can then be set to zero in the frequency domain equations of (45).

[0111] If there are three transmitters, then an encoding strategy for an embodiment of the present invention in a single carrier system having cyclic prefixes is given as follows: $\begin{array}{cc}\begin{array}{ccc}{x}_1\left[m,b_1\right]=s\left[m,b_1\right]& x_2\left[m,b_1\right]=s\left[m,b_2\right]& x_3\left[m,b_1\right]=s\left[m,b_3\right]\\ x_1\left[m,b_2\right]=-s\left[m,b_2\right]& x_2\left[m,b_2\right]=s\left[m,b_1\right]& x_3\left[m,b_2\right]=-s\left[m,b_4\right]\\ x_1\left[m,b_3\right]=-s\left[m,b_3\right]& x_2\left[m,b_3\right]=s\left[m,b_4\right]& x_3\left[m,b_3\right]=s\left[m,b_1\right]\\ x_1\left[m,b_4\right]=-s\left[m,b_4\right]& x_2\left[m,b_4\right]=-s\left[m,b_3\right]& x_3\left[m,b_4\right]=s\left[m,b_2\right]\\ x_1\left[m,b_5\right]=s^{*}\left[D-m,b_1\right]& x_2\left[m,b_5\right]=s^{*}\left[D-m,b_2\right]& x_3\left[m,b_5\right]=s^{*}\left[D-m,b_3\right]\\ x_1\left[m,b_6\right]=-s^{*}\left[D-m,b_2\right]& x_2\left[m,b_6\right]=s^{*}\left[D-m,b_1\right]& x_3\left[m,b_6\right]=-s^{*}\left[D-m,b_4\right]\\ x_1\left[m,b_7\right]=-s^{*}\left[D-m,b_3\right]& x_2\left[m,b_7\right]=s^{*}\left[D-m,b_4\right]& x_3\left[m,b_7\right]=s^{*}\left[D-m,b_1\right]\\ x_1\left[m,b_8\right]=-s^{*}\left[D-m,b_4\right]& x_2\left[m,b_8\right]=-s^{*}\left[D-m,b_3\right]& x_3\left[m,b_8\right]=s^{*}\left[D-m,b_2\right]\end{array}& \left(49\right)\end{array}$

[0112] If there are four transmitters, then an encoding strategy for an embodiment of the present invention in a single carrier system having cyclic prefixes is given as follows: $\begin{array}{cc}\begin{array}{cccc}{x}_1\left[m,b_1\right]=s\left[m,b_1\right]& x_2\left[m,b_1\right]=s\left[m,b_2\right]& x_3\left[m,b_1\right]=s\left[m,b_3\right]& x_4\left[m,b_1\right]=s\left[m,b_4\right]\\ x_1\left[m,b_2\right]=-s\left[m,b_2\right]& x_2\left[m,b_2\right]=s\left[m,b_1\right]& x_3\left[m,b_2\right]=-s\left[m,b_4\right]& x_4\left[m,b_2\right]=s\left[m,b_3\right]\\ x_1\left[m,b_3\right]=-s\left[m,b_3\right]& x_2\left[m,b_3\right]=s\left[m,b_4\right]& x_3\left[m,b_3\right]=s\left[m,b_1\right]& x_4\left[m,b_3\right]=-s\left[m,b_2\right]\\ x_1\left[m,b_4\right]=-s\left[m,b_4\right]& x_2\left[m,b_4\right]=-s\left[m,b_3\right]& x_3\left[m,b_4\right]=s\left[m,b_2\right]& x_4\left[m,b_4\right]=s\left[m,b_1\right]\\ x_1\left[m,b_5\right]=s^{*}\left[D-m,b_1\right]& x_2\left[m,b_5\right]=s^{*}\left[D-m,b_2\right]& x_3\left[m,b_5\right]=s^{*}\left[D-m,b_3\right]& x_4\left[m,b_5\right]=s^{*}\left[D-m,b_4\right]\\ x_1\left[m,b_6\right]=-s^{*}\left[D-m,b_2\right]& x_2\left[m,b_6\right]=s^{*}\left[D-m,b_1\right]& x_3\left[m,b_6\right]=-s^{*}\left[D-m,b_4\right]& x_4\left[m,b_6\right]=s^{*}\left[D-m,b_3\right]\\ x_1\left[m,b_7\right]=-s^{*}\left[D-m,b_3\right]& x_2\left[m,b_7\right]=s^{*}\left[D-m,b_4\right]& x_3\left[m,b_7\right]=s^{*}\left[D-m,b_1\right]& x_4\left[m,b_7\right]=-s^{*}\left[D-m,b_2\right]\\ x_1\left[m,b_8\right]=-s^{*}\left[D-m,b_4\right]& x_2\left[m,b_8\right]=-s^{*}\left[D-m,b_3\right]& x_3\left[m,b_8\right]=s^{*}\left[D-m,b_2\right]& x_4\left[m,b_8\right]=s^{*}\left[D-m,b_1\right]\end{array}& \left(50\right)\end{array}$