# Patent

## Title
Non-line of sight wireless communication system and method

## Abstract

A non-line of sight backhaul system and method are described that provides self-alignment of the antennas beams of the wireless radios of the system, that provides robust operation in licensed and unlicensed frequency bands, that facilitates the use of a reduced number of frequency channels from M to 1 and that enables operation in a non-line of sight (NLOS) propagation environment.

## Background
FIELD
The disclosure relates a wireless backhaul system that uses non-line of sight wireless communications and the non-line of sight wireless communications system and method.
BACKGROUND
A backhaul system is a communication system that is used to communicate certain data from a cellular network, for example, back to the central system in a communications system. Various different backhaul systems are well known that are both wireless communication systems and wired communication systems. Most of the current wireless backhaul systems are point to point (P2P) systems that operate in the licensed FDD (frequency division multiplexed or a frequency division protocol) microwave bands from 6 GHz to 80 GHz. These systems use high gain parabolic dishes which must be manually pointed and also rely on the low sidelobe performance of the dish to reduce (but not eliminate) co-channel interference. Moreover, these products must be used where line-of-sight is available.
These existing backhaul systems do not provide self alignment and realignment of the antenna beams or any increase in effective link spectral efficiency by using interference cancellation. These existing systems also do not operate in a non-line-of sight propagation environment, are not able to double the spectral efficiency by using two polarizations in this environment and have reduced link reliability due to fading. These existing systems do not cancel radio interference to optimize signal to interference and noise ratio (SINR) and do not cancel interference from other self-generated co-channel interference. These existing systems also do not have multi-target beam-forming that enhances spectral efficiency and can provide exceptional data concentration in small amounts of spectrum.
Furthermore, given the shortage of spectrum for broadband wireless and the need to increase both link rate and network capacity, wireless carriers are migrating from a traditional macro-cellular network topology 10 to a micro-cellular and pico-cellular topologies 20 as shown in FIG. 1. That is, instead of using a few high-high powered base-stations to cover large areas, they supplement these with many outdoor micro-cells and pico-cells as shown in FIG. 1.
The base station capacity (measured in Mbps) has been increasing slowly as operators migrate from 2.5G and 3G technologies (HSDPA, HSPA, CDMA 2000, CDMA EVO, etc) to 4G technologies (WiMax and LTE). However, 4G technology increases spectral efficiency only 50% relative to the previous generation. Yet, 4G will not solve the 10-fold increase needed to maintain a macro-cell topology while increasing user capacity. Hence, emerging wireless architectures solve the throughput problem by increasing “capacity density” (measured in Mbps per km2), and not by increasing capacity alone.
Increasing capacity density 10-fold can be solved by a dense deployment of micro- and pico-cells. Although these cells have limited range (100 to 500 meters), they retain the capacity similar to macro-cells, for the LTE standard 15 Mbps in 10 MHz of bandwidth and 39 Mbps in 10 MHz of bandwidth with 3-sectored implementations. Thus, a macro-cell with a range of 1 km (urban propagation) and capacity density of 15 Mbps/km2, evolves into a micro-cellular topology featuring a capacity density of 135 Mbps/km2 using 9 micro-cells.
Thus, it is desirable to provide a non-line of sight backhaul system and method that overcomes the above limitations of existing backhaul systems and it is to this end that the disclosure is directed.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates a typical backhaul system migration from macro-cells;
FIG. 2 illustrates an example of a non-line of sight system such as a non-line of sight, point to multipoint backhaul star architecture system;
FIG. 3 illustrates an example of an implementation of a wireless non-line of sight backhaul system;
FIG. 4 illustrates details of the operation of the wireless non-line of sight backhaul system;
FIG. 5 illustrates a method for multiplicative spectral efficiency in the non-line of sight system;
FIG. 6 illustrates an example of an implementation of both concentrating backhaul radio and the terminating backhaul radios of the system;
FIG. 7 illustrates an example of a 2-dimensional beamforming network subsystem;
FIG. 8 illustrates an implementation of the communication protocol between the CBR and TBR of the system;
FIG. 9 illustrates a set of uplink and downlink RF channels on the system; and
FIG. 10 illustrates the RD channels.



DETAILED DESCRIPTION OF ONE OR MORE EMBODIMENTS
The disclosure is particularly applicable to a wireless non-line of sight backhaul system and method described below and it is in this context that the disclosure will be described. It will be appreciated, however, that the system and method has greater utility since the wireless non-line of sight backhaul system and method can be implemented in other manners that are within the scope of the disclosure.
The system is a wireless, non-line of sight backhaul system that enables self alignment and realignment of the antennas beams of the wireless radios of the system, that provides robust operation in licensed and unlicensed frequency bands, that facilitates the use of a reduced number of frequency channels from M to 1 and that enables operation in a non-line of sight (NLOS) propagation environment. The system may provide an increase in effective link spectral efficiency by using Extreme Interference Cancellation (EIC) and up to double the spectral efficiency by using two polarizations. The system also provides an improvement in link reliability by reducing fading. The system also cancels the radio interference to optimize signal to interference and noise ratio (SINR) and cancels interference from other self-generated co-channel interference. The system can be operated in a line of sight, near line of sight and non-line of sight radio frequency (RF) propagation environments. The system also provides multi-target beam-forming that enhances spectral efficiency and can provided exceptional data concentration in small amounts of spectrum. The non-line of sight system described below may be use a single radio frequency channel to support the wireless backhaul requirements of a carrier. Now, an implementation of the system is described in more detail.
FIG. 2 illustrates an example of a non-line of sight system 30 such as a non-line of sight, point to multipoint backhaul star architecture system. In one implementation, the non-line of sight system has a star topology with a point-to-concentrating multipoint (P2CMP) connectivity that has two types of wireless nodes: concentrator backhaul radios (CBRs) and terminating backhaul radios (TBRs). The non-line of sight system 30 may also have a point to point architecture, a multiple point to point architecture and a point to multipoint architecture. Each node in this wireless architecture may be referenced simply as concentrator nodes (CN) and end nodes (EN) evocative of their positions in the network. Each concentrator node aggregates traffic from L full bandwidth TBRs with capacity Q for a total of L*Q bps. Note that the P2CMP also embodies the notion of L*Lsb TBRs links with capacity Q/Lsb connected to the CBR where Lsb is the number of subbands using either frequency division multiplexing (FDM) or time division multiplexing (TDM). Each TBR is attached to the micro or pico base station thus providing wireless backhaul connectivity. The physical connection with the TBR is via the native output of the base station.
FIG. 3 illustrates an example of an implementation of a wireless non-line of sight backhaul system 50. The system has one or more links (microwave or RF links) terminated at a central backhaul radio 54 (CBR). In one example, the system has M links where M is a number greater than 1. Each link is comprised of a termination backhaul radio 52 (TBR) and 1/Mth of the CBR with a data capacity of Q bps per link where Q is a predetermined number that ranges from 1 to 15 bps. The data capacity measured at the CBR 54 is M*Q bps. Each link is comprised of a shared adaptive array 58 at the CBR 54 and one or more directional antennas 60 at each TBR. Optionally, the antenna at each TBR can be realized with an adaptive array. If each TBR has an adaptive array, then the TBR will self align its antenna pattern to the CBR. This is accomplished by computing the array's beamforming weights using the frame start preamble (FSP) of the CBR as a reference. The weights may be computed from the Weiner equation using the array covariance matrix and the cross-correlation of the data with the FSP. Alternately, known reference symbols in the CBR may be used instead of the FSP as the desired signal. For all TBR antenna types, the CBR will self align its antenna pattern to all of the TBRs. When adaptive arrays are used for both the CBR and TBRs, the CBR and TBR adaptive arrays seek to maximize the signal-to-interference and noise ratio (SINR) by pointing an antenna beam toward the other end of the link, and by reducing interference by directing spatial nulls of the array toward these sources of link degradation.
In more detail, the wireless backhaul system 50 may have an architecture as shown in FIG. 3 which there is a central/concentrating backhaul radio (CBR) 54 that is communicating with up to 10 terminating backhaul radios (TBRs) 52 and each of the TBRs may be located adjacent a picocell and provide a connection between the picocell and the network as a backhaul network. In more detail, a wireless backhaul is used to connect wireless base stations to the core network and/or the operator's point-of-presence and facilitates the backhaul connection of all types of base stations, including femto, pico, micro, mini, and macro base stations. Moreover, this same technology is effective in wireless broadband bridging and last mile extensions of copper, cable and fiber plant. The wireless backhaul system described herein is able to handle the capacity requirements of 3G systems, 4G systems and future wireless protocols (including wireless data protocols.)
The CBR 54 may further comprise an adaptive antenna array 58 (that permit multiple simultaneous beams of the same channel) and a piece of CBR shared equipment 56. The CBR 54 can handle multiple simultaneous beams using the same channel because the system is able to perform extreme interference cancellation to eliminate interference between the various TBR signals. In this invention, this is accomplished by directing spatial nulls in the array antenna pattern in the directions of all interfering TBR while forming a beam peak in the direction of the desired TBR. Moreover, this process is replicated for each desired TBR connected to the CBR thus forming multiple beams and mutual spatial nulling that cancels interference. In conventional PtMP systems, the CBR-like hub can usually handle multiple beams, but those beams each use separate RF channels which wastes bandwidth.
While the maximum SINR optimization criterion is described above, other criteria also may be used (e.g. max SIR beamforming or non-linear data direction beamforming). Of particular importance is an extreme interference cancellation (EIC) feature at each TBR 52 if a high order (6 to 256 elements) adaptive array is employed at the TBR 52 using the equations (E1) through (E23) and equation G1 described below. This feature enables a boost in the SINR from 10 dB to 25 dB nominally in typical system deployments. In addition, depending on the coding rate, EIC is projected to boost link capacity from 1-2 bps/Hz to over 5-6 bps/Hz. Furthermore, if dual polarizations are used, then the capacity can be increased by a factor of 2 (maximum).
Multi-target beamforming introduces a unique beam and interference nulling solution for each TBR 52. Thus, the CBR 54 issues M beams, one each to M TBRs. Each of these beams may use one of M separate frequency channels, or one of M separate subchannels within the overall channel. Alternately, the beams may use the same frequency channel. In the case in which the beams use the same frequency channel, the adaptive array eliminates the interference from the M-1 other beams using spatial nulling techniques. Alternately, the beams may use a combination of M/K channels or subchannels where K is integer sub-multiple of M. In this case, the adaptive array eliminates the interference from the M/K-1 other beams using spatial nulling techniques.
The non-line-of-sight backhaul operation involves angle and delay spread array processing to remove the effects of frequency selective channel responses due to multipath. This process is described using the channel model described in Equation (B1) through (B2) Moreover, it deals with multiple copies of the signal arriving from Q disparate angles of arrival. Conceptually, this involves creating a separate beamforming/null steering solution for each of Q signal paths at each delay spread value, then adaptive combining the Q outputs of the individual paths to optimize the SINR of the link. Two-dimensional beamforming in space for each time-delayed multipath is used. This may be implemented as a tapped delay line beamformer. Alternately, the beamforming operations may be realized efficiently by transforming the array signals between the frequency and time domains.
For the case of dual polarization, the 2 dimensional beamformer can operate on 2M antennas where M in the number of antennas with one polarization. Many algorithms as described above yield an optimal solution to this problem if all antennas/polarizations are used in the formulation.
FIG. 4 illustrates details of the operation of the wireless non-line of sight backhaul system. As shown, a NLOS signal path (that may be diffracted or reflected) is used. Each TBR 52, using its adaptive antenna array (to receive multiple signals) and a beamformer (BFN), can perform spatial and temporal equalization of the various signals received by the antenna array (A1, . . . , An) to extract the signal from the multipath signals caused by the NLOS signal paths. This allows the wireless backhaul system to work without line of sight between the radios.
Beamforming and Interference Cancellation
Adaptive beamforming is used at both the CBR 54 and TBRs 52 to increase array gain and reduce interference. In adaptive beamforming, N1 and N2 antennas, respectively, are adaptively combined to yield the optimum signal-to-interference and noise (SINR) ratio at both ends of the link. Physically, a CBR adaptive beam is pointed in the direction of the TBR and the TBR points its adaptive beam back to the CBR as shown in FIG. 3. Moreover, N1-1 spatial nulls (zeros) are directed at interference caused by other TBRs (managed interference) in the network while N2-1 spatial nulls (zeros) are directed at interference caused by other CBRs. In practical systems, the number of antennas N1 and N2 can range from 2 to 256. By cancelling all interference, the desired link may achieve its highest capacity since it will be essentially noise limited rather that interference limited. Moreover, since a range of a link is now not interference limited but is dependent only on path loss to the TBR's location, the EIRP and the receiver's sensitivity are enhanced by the 2-dimensional beamforming gain at the TBR and CBR.
Automatic Beamsteering, in Changing Multipath Propagation and SINR Optimization
No special knowledge of the TBR's direction of arrival, location, or the interferers' directions of arrival is required to automatically point the optimized antenna beams between the TBR and CBR. Moreover, the adaptive beamforming solution optimizes the receiver SINR and may be updated rapidly to follow any temporal changes in the signal's angle of arrival, power levels, phase, time delay or other changes in the vector signature received on the array due to multipath or time-varying multipath. Special reference symbols embedded within the uplink and downlink transmission provide the signal structure so that algorithms can generate adaptive weights which cancel interference and direct beams even in time varying propagation channels without operator intervention. For example, those processes are described in Equations (C1) through (C23) for Space Time Adaptive Processing (STAP) and Equations (E1) through (E15d) through Space Frequency Adaptive Processing (SFAP).
Multi-Beam and Multiplicative Spectral Efficiency
FIG. 5 illustrates a method for multiplicative spectral efficiency in the non-line of sight system. In the system, Multi-beam array processing extends single beam beamforming by providing L simultaneous links to L TBRs 52 within the footprint of the CBR 54 using the same RF channel at full bandwidth. It accomplishes this by using at least N-1 spatial nulls per link to remove co-channel interference caused by other L-1 TBRs in attached to the CBR cell as shown in FIG. 5. In the system, each link optimizes SINR in the presence of the co-channel self-interference to maximize throughput.
If the system uses time-division duplexing (TDD or time division duplexing), the system described here will preserve the “null directions” in the downlink transmission by using a retro-directive computation of the array processing solution. As a result, TBRs experience no undesired interference from the other in-cell links and other out-of-cell CBRs. Thus, the downlink capacity is maximized for all in-cell links and out-of-cell CBRs.
Universal Frequency Reuse
TDD multi-beam array processing described above and used by the system has remarkable implications. For example, the capacity of the CBR has increased L-fold because the CBR now serves L times as many TBRs in any time epoch. Furthermore, the spectral efficiency is L times greater. This enables broadband backhaul to be deployable in modest amounts of spectrum. The system and method described herein supports an entire metro-area backhaul or BWA network with a single frequency channel, thus minimizing the need for large amounts of spectrum.
FIG. 6 illustrates an example of an implementation of both concentrating backhaul radio 54 and the terminating backhaul radios 52 of the system. In particular, both the CBR 54 and each TBR 52 has four subsystems including an antenna subsystem 100, a transceiver subsystem 102, a beamforming subsystem 104, such as a 2-dimensional beamforming system and a baseband radio subsystem 106.
The antenna subsystem 100 may comprise a plurality of antennas 101, such as N1 as described above wherein N1 is a predetermined number of antennas in the adaptive array for each TBR or N2 antennas wherein N2 is a predetermined number of antennas in the adaptive array for the CBR. Each antenna has 2 feed points that are orthogonal (or quasi-orthogonal). For example, the orthogonal (or quasi-orthogonal) feed points may be vertical/horizontal, left-hand-circular/right-hand-circular, or slant left/slant right as examples. The vertical/horizontal (V/H) feed points for each antenna are shown in FIG. 6. Each antenna may be passive or active. The antennas can be arrayed in a linear array, a two dimensional array or a 3-dimensional array. Example geometries of the antennas include flat panel, circular, semi-circular, square or cubic implementations. The antennas maybe placed in the array in an arbitrary fashion as well.
The transceiver subsystem 102 may further comprise a plurality of transceivers 102a, such as N1 transceivers for each TBR and N2 for the CBR, that provide one channel for each of the 2 polarization feeds from the associated antenna as shown in FIG. 6. Each transceiver channel provides a radio frequency (RF) receiver and an RF transmitter. Each transceiver 102a provides coherent or quasi-coherent down-conversion and up-conversion between RF and a complex baseband.
Each RF receiver in each transceiver may include a preselection filter, a low noise amplifier, a mixer, low pass filter and local oscillator to convert the RF signal down to the complex baseband in a well known manner. The complex baseband may be converted to digital in-phase and quadrature signals using two analog-to-digital converters in a converter unit 102b. 
Each RF transmitter in each transceiver may include two digital-to-analog converters in the converter unit 102b, two low pass filters, a mixer and a local oscillator (LO) to convert the baseband signal to RF in a well known manner. For a TDD system, the LO can be shared between the receiver and the transmitter. The output of the mixer drives an RF preamplifier, transmit (Tx) filter and power amplifier completing the transmitter. The transmitter and receiver are connected to the antenna feed via a TR switch (TDD) or a diplexing filter (FDD).
The beamforming subsystem 104 may receive the signals from each of the transceivers as shown in FIG. 6. The beamforming subsystem may have a weigh processor 104a and a beamforming network (BFN) 104b that is implemented using a processor to perform the beamforming processes described below. In one implementation, a two dimensional (2D) space-time adaptive processing (STAP) or space-frequency adaptive processing (SFAP) bidirectional beam forming network may be used. The beamforming subsystem 104 is also connected to the baseband radio subsystem 106 that has a plurality of well known baseband radios (1 to K in FIG. 6) that are coupled to a hub or switch to route the data traffic.
FIG. 7 illustrates an example of a 2-dimensional beamforming network subsystem 104. The subsystem is realized in the digital domain as a tapped-delayline for each antenna polarization. The output of each tapped delayline is summed with the outputs of the other delaylines for each antenna and polarization to form the output. Two 2D-BFNs are implemented, one for each polarization where the dimension of the 2D-BFNs are 2N1×K where K is the number of taps. Note that FIG. 7 shows signal flow a receiver described above. For the transmitter, the signal flow is reversed through the beamformer 104.
The computation of the weights for the 2D-BFN is mechanized by the weight control processor 104c. The receiver signal vectors at the delay taps are made available to the processor in order to compute the complex weights at each tap. A number of optimal and near optimal algorithms are detailed in subsequent sections. The 2D-BFN may be implement in hardware (discrete logic, ASICs, SOCs), software (on a DSP, CPU or GPU), or firmware (e.g. FPGA). Note the 2D-BFN may be implemented in the analog domain at baseband, IF or RF.
At the CBR, two 2D-BFNs are implemented for each of the L TBRs attached to the CBR. In this case, N2 antennas are processed. The dimension of the 2D-BRNs is 2N2×K where K is the number of taps. In a more efficient CBR implementation, a pair of 2D-BRN of dimension N2×K may be used if the channel is reciprocal and the TBR uses a pair of 2N2×K 2D-BRN.
The subsystem 104 may contain an optional data direction feedback circuit 104d. This circuit and its advantages will be described later in the description of the system.
Returning in FIG. 6, the baseband subsystem 106 may further comprise a dual channel baseband radio processer (BRP). The BRP is comprised of a physical layer, media access controller (MAC) and a network layer. The physical (PHY) layer implements modulation/demodulation, coding/decoding, encryption/decryption, and other PHY functions. The MAC provides frame building, scheduling, queuing, flow control, and layer 2 signaling. The network layer provides the interface between the core network and MAC implementing encapsulation, packet inspection, shaping, policing, and QoS mechanism. The BRP can be implemented with industry-wide radio standards such as GSM, W-CMDA, IEEE-802.16 and LTE or be implemented with a custom PHY and MAC.
Prorata Cost Advantage of the Point-to-Concentrating Multipoint Architecture
Traditional MW backhaul uses a point-to-point backhaul radio architecture. In the disclosed system, the point-to-concentrating multipoint (P2CMP) architecture shares the cost of all common CBR equipment such as antennas, transceivers, LNAs, PAs, filters, local oscillators, system control, backplanes, powering, and cabling cost over L TBRs with capacity Q per link, thus lowering the overall cost of each link. Traditional point to multipoint (P2MP) cannot make the same claim since the capacity per link is only Q/M. When P2MP is normalized to the capacity of Q bps per link, the cost per link is often higher that the P2P equivalent, or the spectral efficiency is degraded by a factor of L such that spectrum cost become prohibitive.
Simultaneous/Sequential Adaptive Array Processing
The system described herein implements simultaneous array adaption of all TBRs receiving the desired downlink CBR signals and sequential adaption of each end of the link. Typically two independent data streams are transmitted on the vertical and horizontal arrays from the CBR. All TBRs compute receiver weight vectors {right arrow over (w)} of the downlink that are applied to the TBR beamformers to estimate the downlink signals. The TBRs also estimate the SINR, {right arrow over (γ)} for the data stream(s) and sends it to the other end of the link as payload or signaling. The TBR transmit weights g are formed from the scaled conjugate of received weight vectors noting again the decoration denotes uplink. The transmit weights are scaled by the power control variable and sent to the transmit beamformer where the uplink data is processed before sending the weighted data streams to the antenna arrays.
The CBR then receives all uplink signals from L TBRs and computes receiver weight vectors wj where j is the TBR link index. The uplink SINR γj is also computed for each of the links and will be sent to the TBRs on the downlink. Next, the transmit weights {right arrow over (g)}j are computed from the scaled conjugate of the receive weights wj and applied to the downlink data.
Note weight adaption ping-pongs in a sequential fashion between the CBR and TBR, improving the network's performance at each iteration as shown in FIG. 8. Since the CBR computes j simultaneous weights in the presence of the self interference of all TBRs, the method may be known as simultaneous/sequential weight adaptation. This approach will lead to an optimal solution across the network provided the transmitter powers are also adaptively controlled according to a power control criteria described earlier.
Channel Model
The channel model for the system and method can be decomposed into a narrowband representation where the received signal yi(n) is a result of receive beamforming using N1 antennas and transmission beamforming using N2 antennas and propagation through an RF channel matrix connecting transmit antennas to receiver antennas pair wise:

yi(n)=wiHHijgj√{square root over (pj)}dj(n)+wiεi(n)

where wi is the N1×1 weight vector at the end node for link i, Hij is the N1×N2 complex channel matrix for the CBR associated with link j to the end node associated with link i, gj is the N2×1 normalized downlink weight vector, √pj is the complex transmit voltage on the downlink j, dj(n) is the transmit signal for downlink j at time sample n and εi(n) is a complex white Gaussian noise process seen at TBR i. The uplink and downlink RF channels are depicted in FIG. 9. The tilde decoration denotes uplink in FIG. 9. From FIG. 9, the following equation for the expected output signal-to-interference and noise ratio (SINR) seen be downlink i as,

γi≡|wiHHiigi|2pi/Σj≠i|wiHHijgj|2pj+σε2wiHwi  (A1)

γ≡Tiipi/Σj≠iTijpj+σε2  (A2)

T↓ij≡(w{right arrow over ( )}↓i^HH↓ijw↓j^*p↓i)/(∥w{right arrow over ( )}↓i^∥^2∥w↓j^∥^2)  A3)

g↓j≡(w↓j^*)/∥w↓j^∥^2  (A4)

p=[p0,p1 . . . pN−1]T  (A5)

P=1Tp  (A6)

Optimal Power Control
If the channel between TBRs and CBRs is reciprocal or quasi-reciprocal within the time span of the adaptive loop, then it can be shown that the optimal transmitter weights are the scaled conjugate of the receiver weights. This is known as retro-directive beamforming. Moreover, if the transmit power is adaptively adjusted using an algorithm driven by the receive SINR and a target SINR γj, then the network of CBRs and TBRs will converge to their target SINRs using minimum total network power P as described in the equations above. If the total network power can be minimized, then interference into adjacent cells is minimized thus enhancing the capacity of these cells and the overall network performance in general. The target SINR is determined by the bit error rate requirements of the communication link as determined by the modulation and coding scheme.
One simple method of power control computes the new power level from last power level and ratio of the target SINR to the received SINR. The equation can be expressed as:

p(l+1)=p(l)* γj/γj  (A7)

where γj is the target SINR to close the link at acceptable BER and γ is the SINR as defined above
In this embodiment, the system has equalization gains (to be defined later) since the system needs the per-subcarrier equalization to be reflected in the power control on the transmit side of the equation. The effect is cumulative over all the nodes in the network. Note this algorithm estimates the SINR of link continuously. If it is too high with respect to the target SINR, then the next Tx power is reduced by ratio between the target and current SINR.
Each receiver will optimize its beamformer weights by maximizing the SINRs at both ends of the link. Normally we place no constraints on the weights. However it is possible to place structural constraints on the weights to simplify computations. If our optimization process increases the SINR and if our SINR targets remain constant; we will necessarily obtain a new solution that decreases all of the transmit powers and thus reducing P according to the equation above.
NLOS, STAP, SFAP, Polarization, Multiuser Processing, Equalization & Subbanding
Non Line-of-Sight Propagation Model
For the system and method, the model of the received data x(n) can be expressed as the sum of multipath signals arriving on p distinguishable paths each with steering vector ap. The received data also includes spatial white Gaussian interference i(n) generated. Hence the model may be expressed as:

x(n)=Ad(n)+i(n)  (B1)

A≡[a1,a2, . . . ap]  (B2)

d(n)≡[√{square root over (a1)}e−jθ1d(n−τ1),√{square root over (a2)}e−jθ1d(n−τ2) . . . √{square root over (ap)}e−jθ1d(n−τp)]T 

where √{square root over (ap)}e−jθp is the amplitude and phase of path p, ap is the steering vector for path p and τp is the time delay for path p. d(n) is the transmitted data and i(n) is the interference vector modeled as Gaussian white noise. The dimension of matrix A is N antenna sensors by P multipaths.
The Space-Time Adaptive Processor
An estimate {circumflex over (d)}(n) of the original signal d(n) can be realized by processing the received data with a 2-dimensional filter in the dimensions of space and time, hence a space-time adaptive processor (STAP). This filter may be written as the linear convolution of receive vector x(n) at time n with the K1+K2+1 time taps of the filter where each time tap has coefficients wH(k) for −K1≦k≦K2:

wH(n)*x(n)≡{circumflex over (d)}(n)  (C1)





  
    
      
        
          
            
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w(n)≡[wH(K1),wH(K1−1), . . . ,wH(−K2)]H  (C3)

The error between the output of the STAP filter {circumflex over (d)}(n) and the desired signal d(n) can be expressed as ε(n). We seek to minimize the expected value of the error power μ where we replace the expectation with a time average over a suitably large interval over the time index n.

ε(n)={circumflex over (d)}(n)−d(n)  (C4)

μ=<|ε(n)|2>n  (C5)





  
    
      
        
          
            
              
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                  ⁢
                  
                    
                      
                        x
                        H
                      
                      ⁡
                      
                        (
                        
                          n
                          -
                          
                            k
                            ′
                          
                        
                        )
                      
                    
                    ⁢
                    
                      w
                      ⁡
                      
                        (
                        
                          k
                          ′
                        
                        )
                      
                    
                  
                
                -
                
                  
                    
                      
                        d
                        ^
                      
                      *
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      7
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
      
    
  




where the estimate of the signal and the signal conjugate is written as follows:




  
    
      
        
          
            
              
                
                  
                    d
                    ^
                  
                  *
                
                ⁡
                
                  (
                  n
                  )
                
              
              ≡
              
              ⁢
              
                
                  ∑
                  
                    
                      k
                      ′
                    
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  
                    
                      x
                      H
                    
                    ⁡
                    
                      (
                      
                        n
                        -
                        
                          k
                          ′
                        
                      
                      )
                    
                  
                  ⁢
                  
                    w
                    ⁡
                    
                      (
                      
                        k
                        ′
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      8
                    
                    )
                  
                
              
            
          
        
        
          
            
              
                
                  d
                  ^
                
                ⁡
                
                  (
                  n
                  )
                
              
              ≡
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  
                    
                      w
                      H
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  ⁢
                  
                    x
                    ⁡
                    
                      (
                      
                        n
                        -
                        k
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      9
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
      
    
  




The time averaged error power can be written as follows:




  
    
      
        μ
        =
        
          〈
          
            
              
                ∑
                
                  k
                  =
                  
                    -
                    
                      K
                      1
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  
                    w
                    H
                  
                  ⁡
                  
                    (
                    k
                    )
                  
                
                ⁢
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      k
                    
                    )
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        k
                        ′
                      
                      =
                      
                        -
                        
                          K
                          1
                        
                      
                    
                    
                      K
                      2
                    
                  
                  ⁢
                  
                    
                      
                        x
                        H
                      
                      ⁡
                      
                        (
                        
                          n
                          -
                          
                            k
                            ′
                          
                        
                        )
                      
                    
                    ⁢
                    
                      w
                      ⁡
                      
                        (
                        
                          k
                          ′
                        
                        )
                      
                    
                  
                
              
            
            -
            
              
                
                  
                    d
                    ^
                  
                  *
                
                ⁡
                
                  (
                  n
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  
                    
                      w
                      H
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  ⁢
                  
                    x
                    ⁡
                    
                      (
                      
                        n
                        -
                        k
                      
                      )
                    
                  
                
              
            
            +
            
              
                
                  d
                  ^
                
                2
              
              ⁡
              
                (
                n
                )
              
            
            -
            
              
                
                  d
                  ^
                
                ⁡
                
                  (
                  n
                  )
                
              
              ⁢
              
                
                  ∑
                  
                    
                      k
                      ′
                    
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  
                    x
                    H
                  
                  ⁢
                  
                    (
                    
                      n
                      -
                      
                        k
                        ′
                      
                    
                    )
                  
                  ⁢
                  
                    w
                    ⁡
                    
                      (
                      
                        k
                        ′
                      
                      )
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        (
        C10
        )
      
    
  




Since the system would like to minimize the error power as a function of the tap weights to minimize the mean squared error, the partial differentials with respect to the tap weights can be taken as follows:




  
    
      
        
          
            
              x
              ⁡
              
                (
                
                  n
                  -
                  k
                
                )
              
            
            ⁢
            
              
                x
                H
              
              ⁡
              
                (
                
                  n
                  -
                  
                    k
                    1
                  
                
                )
              
            
            ⁢
            
              (
              
                w
                ⁡
                
                  (
                  
                    k
                    ′
                  
                  )
                
              
              )
            
          
          =
          
            
              2
              ⁢
              
                
                  ∑
                  
                    
                      k
                      1
                    
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  R
                  ⁢
                  
                    
                  
                  ⁢
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  
                    x
                    ⁡
                    
                      (
                      
                        
                          -
                          k
                        
                        ,
                        
                          k
                          ′
                        
                      
                      )
                    
                  
                  ⁢
                  
                    w
                    ⁡
                    
                      (
                      
                        k
                        ′
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  f
                  ⁢
                  
                    
                  
                  ⁢
                  or
                  ⁢
                  
                    
                  
                  ⁢
                  j
                
              
            
            =
            k
          
        
        ,
        
          otherwise
          ⁢
          
            
          
          ⁢
          0
        
      
    
    
      
        (
        C11
        )
      
    
  
  
    
      
        
          
            δμ
            
              δ
              ⁢
              
                
              
              ⁢
              
                
                  w
                  H
                
                ⁡
                
                  (
                  j
                  )
                
              
            
          
          =
          
            
              
                
                  
                    d
                    ^
                  
                  *
                
                ⁡
                
                  (
                  n
                  )
                
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    
                      -
                      
                        K
                        1
                      
                    
                  
                  
                    K
                    2
                  
                
                ⁢
                
                  
                    
                      w
                      H
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  ⁢
                  
                    x
                    ⁡
                    
                      (
                      
                        n
                        -
                        k
                      
                      )
                    
                  
                
              
            
            =
            
              
                2
                ⁢
                
                  
                    
                      d
                      ^
                    
                    *
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      
                        -
                        
                          K
                          1
                        
                      
                    
                    
                      K
                      2
                    
                  
                  ⁢
                  
                    
                      x
                      ⁡
                      
                        (
                        
                          n
                          -
                          k
                        
                        )
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    f
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    j
                  
                
              
              =
              k
            
          
        
        ,
        
          otherwise
          ⁢
          
            
          
          ⁢
          0
        
      
    
    
      
        (
        
          C
          ⁢
          
            
          
          ⁢
          12
        
        )
      
    
  
  
    
      
        
          δμ
          
            δ
            ⁢
            
              
            
            ⁢
            
              
                w
                H
              
              ⁡
              
                (
                j
                )
              
            
          
        
        =
        
          
            
              
                d
                ^
              
              ⁡
              
                (
                n
                )
              
            
            ⁢
            
              
                ∑
                
                  
                    k
                    ′
                  
                  =
                  
                    -
                    
                      K
                      1
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  
                    x
                    H
                  
                  ⁡
                  
                    (
                    
                      n
                      -
                      
                        k
                        ′
                      
                    
                    )
                  
                
                ⁢
                
                  w
                  ⁡
                  
                    (
                    
                      k
                      ′
                    
                    )
                  
                
              
            
          
          =
          0
        
      
    
    
      
        (
        
          C
          ⁢
          
            
          
          ⁢
          13
        
        )
      
    
  
  
    
      
        
          δμ
          
            δ
            ⁢
            
              
            
            ⁢
            
              
                w
                H
              
              ⁡
              
                (
                j
                )
              
            
          
        
        =
        
          
            
              
                d
                ^
              
              2
            
            ⁡
            
              (
              n
              )
            
          
          =
          0
        
      
    
    
      
        (
        
          C
          ⁢
          
            
          
          ⁢
          14
        
        )
      
    
  




Hence, the equations can be rewritten as follows:




  
    
      
        
          δμ
          
            δ
            ⁢
            
              
            
            ⁢
            
              
                w
                H
              
              ⁡
              
                (
                j
                )
              
            
          
        
        =
        
          
            2
            ⁢
            
              
                ∑
                
                  
                    k
                    ′
                  
                  =
                  
                    -
                    
                      K
                      1
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  
                    R
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ⁡
                  
                    (
                    
                      
                        k
                        ′
                      
                      ,
                      
                        -
                        k
                      
                    
                    )
                  
                
                ⁢
                
                  w
                  ⁡
                  
                    (
                    
                      k
                      ′
                    
                    )
                  
                
              
            
          
          -
          
            2
            ⁢
            
              
            
            ⁢
            
              
                r
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  d
                
              
              ⁡
              
                (
                
                  -
                  k
                
                )
              
            
          
        
      
    
    
      
        (
        C15
        )
      
    
  




where the following expressions are defined:

rxd(−k)≡<x(n−k)d*(n)>n  (C16)

Rxx(k′,−k)≡Rxx(k′−k)≡<x(n−k)xH(n−k′)>n  (C17)

Setting the partial derivatives to zero for each weight vector, the essential equations can be rewritten as:

Σk′=−K1K2Rxx(k′,−k)w(k′)=rxd(−k) for −K1≦k≦K2.  (C18)

The above equation can be rewritten in matrix form as:

RxxW=Rxd  (C19)

where

W≡[wH(−K1),wH(−K1+1), . . . ,wH(K2)]H  (C20)

Rxd(n)≡[rxdH(K1),rxdH(K1−1), . . . ,rxdH(−K2)]H  (C21)





  
    
      
        
          R
          XX
        
        =
        
          [
          
            
              
                
                  
                    R
                    xx
                  
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                ...
              
              
                
                  
                    R
                    xx
                  
                  ⁡
                  
                    (
                    
                      
                        -
                        
                          K
                          1
                        
                      
                      -
                      
                        K
                        2
                      
                    
                    )
                  
                
              
            
            
              
                ⋮
              
              
                ⋱
              
              
                ⋮
              
            
            
              
                
                  
                    R
                    xx
                  
                  ⁡
                  
                    (
                    
                      
                        K
                        1
                      
                      +
                      
                        K
                        2
                      
                    
                    )
                  
                
              
              
                ...
              
              
                
                  
                    R
                    xx
                  
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        C22
        )
      
    
  




If the above matrix equation is rewritten in summation form, the equation becomes:




  
    
      
        
          
            ∑
            
              k
              =
              
                -
                
                  K
                  1
                
              
            
            
              K
              2
            
          
          ⁢
          
            
              
                R
                xx
              
              ⁡
              
                (
                
                  
                    k
                    ′
                  
                  +
                  k
                
                )
              
            
            ⁢
            
              W
              ⁡
              
                (
                
                  k
                  ′
                
                )
              
            
          
        
        =
        
          
            R
            xd
          
          ⁡
          
            (
            k
            )
          
        
      
    
    
      
        (
        C23
        )
      
    
  




Fast Transform Methods for Solving the STAP Problem
The above equations are formulated as a correlation. Thus, a linear transform via the well known Discrete Fourier Transform (DFT) to further simplify the computations are further simplified by replacing the correlation with multiplications in the transformed domain using a fast convolution/correlation theorem efficiently implemented via the well know Discrete Fourier Transform (DFT). First, the DFT F is defined as the K×K matrix transformation as follows:

F≡[ωmk] For 0≦m,k≦K−1 and ω=e−j2π/K  (D1)

Note the property of the inverse DFT is as follows:

F−1=FH  (D2)

Given this definition, the correlation expression above in the transformed domain can be rewritten as follows:




  
    
      
        
          F
          ⁢
          
            
              ∑
              
                
                  k
                  ′
                
                =
                
                  -
                  
                    K
                    1
                  
                
              
              
                K
                2
              
            
            ⁢
            
              
                
                  R
                  xx
                
                ⁡
                
                  (
                  
                    k
                    ,
                    
                      k
                      ′
                    
                  
                  )
                
              
              ⁢
              
                W
                ⁡
                
                  (
                  
                    k
                    ′
                  
                  )
                
              
            
          
        
        =
        
          
            FR
            xd
          
          ⁡
          
            (
            k
            )
          
        
      
    
    
      
        (
        D3
        )
      
    
  
  
    
      
        
          
            ∑
            
              k
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              ∑
              
                
                  k
                  ′
                
                =
                
                  -
                  
                    K
                    1
                  
                
              
              
                K
                2
              
            
            ⁢
            
              
                
                  R
                  xx
                
                ⁡
                
                  (
                  
                    k
                    +
                    
                      k
                      ′
                    
                  
                  )
                
              
              ⁢
              
                W
                ⁡
                
                  (
                  
                    k
                    ′
                  
                  )
                
              
              ⁢
              
                ω
                mk
              
            
          
        
        =
        
          
            ∑
            
              k
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xd
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              ω
              mk
            
          
        
      
    
    
      
        (
        
          D
          ⁢
          4
        
        )
      
    
  




For 0≦m,k≦K−1 and ω=e−j2π/K where K≧K1+K2+1. Let k+k′=q and k=q−k′




  
    
      
        
          
            ∑
            
              q
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              ∑
              
                
                  k
                  ′
                
                =
                
                  -
                  
                    K
                    1
                  
                
              
              
                K
                2
              
            
            ⁢
            
              
                
                  R
                  xx
                
                ⁡
                
                  (
                  q
                  )
                
              
              ⁢
              
                ω
                mq
              
              ⁢
              
                W
                ⁡
                
                  (
                  
                    k
                    ′
                  
                  )
                
              
              ⁢
              
                ω
                
                  mk
                  ′
                
              
            
          
        
        =
        
          
            ∑
            
              k
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xd
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              ω
              mk
            
          
        
      
    
    
      
        (
        D5
        )
      
    
  
  
    
      
        
          
            ∑
            
              q
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xx
              
              ⁡
              
                (
                q
                )
              
            
            ⁢
            
              ω
              mq
            
            ⁢
            
              
                ∑
                
                  
                    k
                    ′
                  
                  =
                  
                    -
                    
                      K
                      1
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  W
                  ⁡
                  
                    (
                    
                      k
                      ′
                    
                    )
                  
                
                ⁢
                
                  ω
                  
                    -
                    
                      mk
                      ′
                    
                  
                
              
            
          
        
        =
        
          
            ∑
            
              k
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xd
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              ω
              mk
            
          
        
      
    
    
      
        (
        
          D
          ⁢
          
            
          
          ⁢
          6
        
        )
      
    
  




 Rxx(−m) W(m)= Rxd(−m)  (D7)

where the over bar indicates the Discrete Fourier Transform as follows




  
    
      
        
          
            W
            _
          
          ⁡
          
            (
            m
            )
          
        
        =
        
        ⁢
        
          
            ∑
            
              
                k
                ′
              
              =
              
                -
                
                  K
                  1
                
              
            
            
              K
              2
            
          
          ⁢
          
            
              W
              ⁡
              
                (
                
                  k
                  ′
                
                )
              
            
            ⁢
            
              ω
              
                -
                
                  mk
                  ′
                
              
            
            ⁢
            
              
            
            ⁢
            
              (
              D8
              )
            
          
        
      
    
  
  
    
      
        
          
            
              R
              _
            
            xd
          
          ⁡
          
            (
            
              -
              m
            
            )
          
        
        =
        
        ⁢
        
          
            ∑
            
              k
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xd
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              ω
              mk
            
            ⁢
            
              
            
            ⁢
            
              (
              D9
              )
            
          
        
      
    
  
  
    
      
        
          
            
              R
              _
            
            xx
          
          ⁡
          
            (
            
              -
              m
            
            )
          
        
        =
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            
              K
              -
              1
            
          
          ⁢
          
            
              
                R
                xx
              
              ⁡
              
                (
                q
                )
              
            
            ⁢
            
              ω
              mq
            
            ⁢
            
              
            
            ⁢
            
              (
              D10
              )
            
          
        
      
    
  




Finally, the space-time filter solution can be expressed as:

W(k′)=F−1 W(m)=F−1 Rxx−1(−m) Rxd(−m)  (D11)

For each of taps on the space-time filter −K1≦k′≦K2.
Improving Computational Efficiency
Note that for efficiency, an FFT and IFFT can be used to replace the DFT in the above equations. In this case, replace 3K2 complex multiply/accumulates to realize the matrix multiplications associated with the DFTs and IDFT with 3log2 K complex multiply/accumulates using the fast Fourier transform method. This results in one to two orders of magnitude savings depending on the length of the filter in the time domain.
Furthermore, note that the weight solution for the space-time filter requires the formation of K covariance matrices of dimension N×N and K cross-correlation vectors of length N. This is a considerable reduction in computation complexity compared to the original problem.
Improving Numerical Accuracy
Now express Rxx(−m) in terms of its QR decomposition which factors the covariance matrix into an orthonormal basis matrix Q and an upper triangular matrix R. of the underlying data X:

 Rxx(−m)= XH X  (D12)

 X= QRx  (D13)

where Rx is the Cholesky factor of the covariance matrix Rxx. The Cholesky factor is an upper triangular matrix and where Q is a orthonormal matrix:

 Rxx(−m,m′)= RxH QH QRx  (D14)

 QH Q=I  (D15)

And finally, substituting the Cholesky factor for the covariance matrix leads to the following expression. Note that the equation may be solved for W efficiently using two back substitutions:

 RxH(−m,m′) Rx(−m,m′) W(m′)= Rxd(−m)  (D16)

Space Frequency Adaptive Processing
The 2-dimensional STAP beamformer may be realized in the frequency domain by exploiting the Fourier Transform of the baseband signals from the array. In this case, the signal is “channelized” by the transform into multiple frequency subbands such that the array response is constant or nearly constant across the subchannel. The implication is that the subchannel frequency support should be a fraction of the inverse of the RMS delay dispersion of the signal's multi-path components. Hence, narrowband beamforming may be performed on each subchannel. This is known as Space Frequency Adaptive Processing (SFAP).
Note that the number of subchannels is approximately equal to the number of STAP taps in the delayline. SFAP may be the preferred embodiment for many signal types including OFDM, OFDMA and SC-FDMA. These signals are naturally constructed in the frequency domain using data subchannels, pilots for demodulation and preambles as can be observed by examining the specifications for LTE and 802.16. The formation the SFAP equations begin with a model of the beamforming symbols:

Xw=d+e  (E1)

where X is the received signal matrix of M rows of time samples by N antennas, w is the CBR receive beamforming weight vector of length K, d is the desired symbol vector of length M and e is the error in this model due to noise plus interference. Pre-multiply (D1) by XH to get:

XHXw=XHd+XHe  (E2)

The minimum mean squared error solution is obtained by choosing the weight vector so that the received signal matrix is orthogonal to the error vector:

XHe=0  (E3)

Equation (E2) can be written as:

Rxxw=Rxd  (E4)

where the auto-correlation matrix and the cross-correlation vector are respectively:

Rxx=XHX  (E5)

Rxd=XHd  (E6)

Solving (E4) for the weight vector yields:

w=Rxx−1Rxd  (E7)

Substituting (E7) into (E1) and solving for the error vector yields (E8) and the error power per symbol is as follows and expanding (E9) by (E8) yields (E10):

e=X(XHX)−1XHd−d  (E8)

σe2=eHe/N  (E9)

Nσe2=dHd−dHX(XHX)−1XHd  (E10)

Comparing (E10) and (E7), the error power can be expressed as a function of the weight vector:

Nσe2=dHd−Re(RxdHw)  (E11)

The QR decomposition of X is:

X=QRx  (E11a)

where Q is an orthonormal matrix the same size as X (M rows of subcarriers by N antennas):

QHQ=I  (E11b)

and Rx the Cholesky factor matrix of the auto-correlation matrix and is a N by N upper-triangular matrix. Substituting (E11a) into (E5), we obtain:

Rxx=RxHQHQRx  (E11c)

But because Q is orthonormal (D11b), the auto-correlation matrix is simply the product of two Cholesky factors:

Rxx=RxHRx  (E12)

If the Cholesky factors of (E12) is substituted into the error power of (E10):

Nσe2=dHd−RxdH(RxHRx)−1Rxd  (E13)

which can be written as:

Nσe2=dHd−uHu  (E14)

where the biased estimate of the desired symbol vector is:

u=Rx−HRxd  (E15)

Note that u is the solution to the equation:

RxHu=Rxd  (E15a)

which can be solved by back substitution of the upper triangular matrix RxH into Rxd. Substituting (E12) into (A7) yields the weight vector:

w=Rx−1Rx−HRxd  (E15b)

which can also be written as:

Rxw=Rx−HRxd  (E15c)

By substituting (E15) into (E15c), the following equation is obtained:

Rxw=u  (E15d)

which can be solved for w by back substitution of the upper triangular matrix Rx into u.
Equalization and Subbanding
The formulation of the (A1) through (A15) assumes that the matrix X contains M subcarriers in the estimation of the first and second order statistics. For each subcarrier, this may be satisfied by collecting M subcarriers over the time interval of JTs where Ts is the symbol time of the OFDM(A) baseband. J is selected to satisfy certain constraints dictated by the mis-adjustment of the linear combiner with respect to the desired signal. A nominal figure of merit for M is 4 times the degrees of freedom (4×DOF) for a mis-adjustment loss of <1 dB.
Alternately, the time-bandwidth product constraint maybe satisfied by forming the first and second or statistics over Msc×Msym=M subcarriers where Msym is the number of subcarrier per subband and Nsym is the number of symbols required to satisfy the M subcarrier time-bandwidth product (TBP) constraint.
This formulation is valid provided that the phase and amplitude of the steering vector is relatively constant over the subband of Msc subcarriers. In fact, this method is advantageous in lowering the number of training symbols to meet the TBR required for link adaption.
In the system, we note that while maximizing Msc subcarriers subject to the SINR constraints is advantageous, it ultimately fails as the received SINR declines as Msc subcarriers gets too large subject to the dispersion of the multipath. However, the method may add a final processing step to improve performance for a constrained number of subcarriers and model the collection of data over the subband as a constant steering vector perturbed by small variation in phase and amplitude. Often, this is a low order model in both amplitude and phase. Hence, is a small phase and amplitude ramp across the subband can be estimated accurately by pilot subcarriers present/insert in the baseband of the signal (e.g. Wimax and LTE).
So the final processing in the frequency domain is to collect the first and second order statistics and to estimate the post beamforming phase and amplitude tilt across the subband for receiver processing. Then upon transmit, conjugate the phase and apply the amplitude corrections while applying them to the transmitter weight derived from the receiver weight vector. Note that the phase and amplitude tilt is unique for each subcarrier within in the subband. Thus each subcarrier has a unique linear combiner Rx weight and Tx weight conjugate.
This process is defined as subband equalization and yield superior network performance. It is an additional component in achieving EIC.
Equalization and Power Control
For SFAP, it may not be intuitively obvious how transmit equalization helps, there is still a relatively simple mathematical reason why it improves performance. If the method looks at the sum total transmit power on the uplink or on the downlink, reciprocity says that these two quantities are equal and are used as a network wide performance metric which may be known as quantity P. On the downlink this as a function of all the receiver weights at the TBRs called W, and as a function of all the transmit weights at the concentrators G. Thus P(W,G) becomes the network performance metric to be optimized. Technically this metric is a function of the weights on each subcarrier. Optimizing the receiver weights at the TBRs yields, minw P(W,G). Equalization, in this context, optimizes components of W. At the TBRs, if the system does not transmit with the conjugate of the equalization gains, then the method is refusing to optimize over all the components of G. In terms of the degrees of freedom associated with just the equalization components, that's half the degrees of freedom in the network and this performance suffers. Better performance is achieved optimizing both transmit and receive weights according to minw minG P(W,G).
Reciprocity proves that the optimal G are the conjugate of the concentrator receiver weights and thus are included the equalization gains as well. The effect is cumulative over all the nodes in the network. The fact that P(W,G) is reciprocal means that optimizing over equalization gains and transmitting with them necessarily improves all nodes in the network.
In this system and method, it is more intuitive to think in terms of what equalization does to the weights qualitatively. If a minimum complexity rank 1 beam is all that is required in the direction of the intended node, more degrees of freedom are then devoted to interference cancellation. Thus equalization will improve the channel in the direction of the intended node and that's all it has to do to improve overall network performance.
Capacity Matching, MCS Downshifting
In this system and method, the real-time capacity requirements of equipment attached (i.e. a base station) to and from the TBR can be sensed and the modulation and coding scheme (MCS) can be adaptively adjusted to meet these requirement. This is known as MCS downshifting and upshifting. In the downshifting scheme, a moving average of the uplink and downlink capacity is computed by subtracting the “filler” symbols from the total symbols in order to estimate the number of “traffic” symbols. Next, a new and lower MCS level is computed that would essentially pack all symbols and remove the filler. Headroom is added to this computation so as not to remove all filler. This is now the safe MCS level and the downshift can be commanded to the new level via PHY signaling (e.g. MAP). In the upshifting scheme, the data queues for the uplink and downlink are monitored. If these are not emptied on each transmission interval, a higher MCS is warranted and the safe MCS level can be computed from a queue statistics. The upshift can be commanded to the new level via PHY signaling (e.g. MAP). Since the SINR requirements are lower for a MCS downshift, the TX power of the link can be lowered reducing interference power. Thus, downshifting increases the overall capacity of the network by allowing additional links (>L) or higher MCS levels for other links.
Best Serving CBR, Network Resiliency and Handover.
In this system and method, the TBRs may establish connections to multiple CBRs for the purpose of selecting the best serving CBR. The best serving CBR may be sensed by computing the SINR of each link, estimating the quality of the multipath channel, determining the loading of the CBR, estimating the potential SINR degradation to other links currently connected to the CBR, or any combinations of these metrics. The TBR will then request connection to the best serving CBR and generate a neighbor list of other CBRs. The list will include the required timing parameters and power levels to each CBR. Using the neighbor list, a TBR can quickly switch over to another CBR if either the CBR fails or the propagation channel becomes impaired. This provides a level of network resiliency and reliability.
Subbanding and Subzoning
In this system and method, the full capacity Q bps of each link can be subdivided into Lsb sublinks with each with capacity Q/Lsb. This maybe realized by using Lsb subbands in the frequency domain. Alternately, a frame of data symbols may be divided into Lsb subzones in the time domain. Note, that various linear combinations of capacity may be realized by aggregating integer numbers of subbands and/or subzones. A combination of subbanding and subzoning is also supported.
Dual Polarization, Optimal Polarization Separation
For this system, the system may transmit with two independent data streams d1(n) and d2(n) on two different polarization with transmit weight g1 and g2, respectively. This effectively doubles the data rate. Unfortunately, there is cross-coupling between the polarizations due to propagation through an RF channel that mixes the polarization as shown in FIG. 10.
The receiver sees a mixture of the desired data d1(n) via the principle channel matrix H11 and an interference term d2(n) via the cross coupling matrix of the other polarization H12 according the first equation below. In a similar fashion, the receiver for the orthogonal polarization sees a mixture the desired data d2(n) via the principle channel matrix H22 and an interference term d1(n) via the cross coupling term of the other polarization H21 according the second equation below.

x1(n)=H11g1√{square root over (p1)}d1(n)+H12g2√{square root over (p2)}d2(n)+i1(n)  (F1a)

x2(n)=H21g1√{square root over (p1)}d1(n)+H22g2√{square root over (p2)}d2(n)+i2(n)  (F1b)

The first equation maybe recast as the desired signal being received on aperture a11(n) in the presence of cross-coupled interference on aperture a12(n) and generalized interference aperture i1(n) where the apertures are defined here. The same is true for the second equation.

a11(n)=H11g1  (F2a)

a12(n)=H12g2  (F2b)

Given this formulation, the optimal weights for the down link (CBR to TBR) can be computed by forming the 2N1×1 receiver data vector and computing the relevant first and second order statistics below. This is the optimal MMSE solution for narrowband signals.




  
    
      
        
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{right arrow over (w)}1=Rxx−1Rxd1  (F4a)

{right arrow over (w)}2=Rxx−1Rxd2  (F4b)

The equations above maybe extended in a straightforward fashion for wideband data receive in multipath using the STAP and SFAP solutions previously above.
For the uplink (TBR to CBR), the weight solutions can be computed as above with 2N2 degrees of freedom, or with substantially lower complexity via the equations below. In this case, this invention computes weights using x1(n) and x2(n) instead of x(n). The polarization wavefronts arrive without cross-coupling since the TBR transmit weights were derived from the conjugate of the downlink TBR receive weights which are known to removed the cross-coupling components.

 w1=Rx1x1−1Rx1d1  (F5a)

 w2=Rx2x2−1Rx2d2  (F5b)

Note, any orthogonally polarized antennas sets may be substituted with the same result such as vertical and horizontal, slant left and right, or RHCP and LHCP.
The Reference Symbols and Data Directed Feedback
The reference signals for link adaption may be realized from special training symbols, embedded pilots (e.g OFDM pilots for data demodulation), preambles, statistical significant data replications, baseband signal replications (e.g. cyclic prefix) in either the time or frequency domain, the signal's known constellation (known modulus), or by data directed feedback techniques using payload data or by combinations of above.
In one embodiment of this system and method, it can be illustrated how the reference signals may be utilized. In this scheme, the CBR sends the desired vertical and horizontal reference signals dv(k) and dh(k) on the vertical and horizontal array respectively simultaneously. The reference signals are generated by modulating OFMDA subcarriers in the frequency domain using elements of codes derived from CAZAC sequences. Alternately, the reference signals can be the modulation on the data symbols of a single carrier in the time domain. CAZAC codes are a set of orthogonal codes with constant amplitude and zero circular auto-correlation and low circular cross-correlation. Hence dv(k) and dh(k) have low or zero cross-correlation. The CAZAC codes can be expressed as



  
    
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  for




  
    
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f or k=0, 1, . . . , L−1 and r relatively prime to L,q

CAZAC codes from the same family are assigned to each TBR within the footprint of the CBR. Each TBR is assigned two CAZAC codes, one for each polarization. Because each code has low cross-correlation with the others, high quality estimates of cross correlation vectors are formed. This is important to minimize array mis-adjustment of the linear combiner due to noise weight vectors computed from the normal equations.

w=Rxx−1Rxd  (F6)

For the SFAP embodiment, Rxx−1 and Rxd are estimated over a block of P*L/B subcarriers containing the reference signals where P is the number of OFMDA symbols modulated by the reference signal and B is the number of subbands. A subband is defined as a group of K/B adjacent subcarriers where the relative differences between steering vectors measured on each member subcarrier as small. Thus, the linear combiner weights for the receiver are updated for each block of reference signal subcarriers. The SINR of the received signal maybe also estimated over this block. The SINR is useful for transmission power control when relayed to the other end of the link.
Reference Symbols Derived from Data Directed Feedback
Often the 2D-BFN weight accuracy derived from above can be enhanced by computing the first and second order statistics over a longer time interval not limited to the reference symbols. In this case, the data payload part of the frame may be used. One embodiment correlates a delayed copy of the received data at multiple tap delays with an estimate of the data re-modulated using the same modulation and coding scheme (MCS) of the link. The data covariance is computed over this same time interval. This technique is known as STAP/SFAP data directed feedback (DDF) from and is illustrated in the figure above.
By using DDF, the embodiment achieves higher efficiency and link capacity since fewer reference symbols are required for link convergence when supplemented with a payload-derived “reference symbols” extracted from the payload. Moreover, DDF may extract reference symbols from the entire receive interval. This enables robust interference cancellation of unmanaged interference occurring in the data payload part of the subframe and but not available in the reference symbol part of the subframe. In this case, this is the preference embodiment for unlicensed frequency bands or licensed bands with unmanaged interference.
Multiuser STAP
In the system, the CBR performs a link “concentration” function by simultaneously connecting to L TBRs using L (single polarization) or 2L (dual polarization) sets of STAP weights. The governing L equations are given below:




  
    
      
        
          
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where the equation is the index over L links connected to the CBR and 1≦l≦L. Note for best performance, d1 should have low or zero cross-correlation among all dj where j≠l.
Multiuser SFAP
In the system, the CBR concentration function for L simultaneous links to L TBRs may realized in the frequency domain yields the multiuser SFAP solution. The governing L (or 2L for dual polarization) equations are given below:

u1=Rx−1Rxd1  (G2)

Rxw1=u1  (G3)

where u1 is the index over L links connected to the CBR and 1≦l≦L. Note for best performance, d1 should have low or zero cross-correlation among all dj where j≠l.
Dynamic Multiuser Power Allocation
The CBR antenna power delivered by each power amplifier (PA) is the sum of all powers for L links as weighted by the complex element gm of the transmit weight vector g where m is the antenna/PA index.
The power P of all links in given below

p=[p0,p1 . . . pL−1]T  (H1)

p=1Tp  (H2)

Power must be allocated to each link to maintain the target SINR and the total power to each PA cannot exceed the PPA. One technique is to allocate equal maximum power PPA/L for each link. This is a very conservative method and suboptimal since some links require more power and some require less due to distance variation from the CBR. This method causes the power required at the PA to be over specified.
In the system, the power can be allocated differently according to the following equations:

p→pmax  (H3)

ppa=Gpmax  (H4)

f[pmax pa,ppa]→pmax  (H5)

where the columns of G contain the power in each element of the transmit weight vectors for the L links. In this implementation, p is the initial estimate of the link powers derived from initial ranging. The power ppa to each PA can then be evaluated using the power scaling factors from each of L transmit weights contained in G post multiplied by the initial estimate of pmax. Based on the computed PA power, and a vector of maximum permissible powers pmax,pa, a new pmax can be computed by a variety of functions, methods and/or iterations. If there is a reasonable spread between power requirements of the links due to “near-far” distance variation, then this method yields significantly better performance.
This system may thus be called dynamic multiuser power allocation and can improve the power available to TBRs at the edge of coverage by 2-6 dB depending on L, N1, N2 and spatial distribution of the end points.
While the foregoing has been with reference to a particular embodiment of the invention, it will be appreciated by those skilled in the art that changes in this embodiment may be made without departing from the principles and spirit of the disclosure, the scope of which is defined by the appended claims.

## Claims

1. A wireless broadcast radio for backhaul applications, wireless Ethernet replacement applications and broadband wireless drop applications, the radio comprising:
an adaptive antenna array including at least two adaptive antennas;
a beamforming network connected to the adaptive antenna array;
a weigh processor connected to the beamforming network; and
one or more baseband radios connected to the beamforming network that deliver a stream of data to an external output of the wireless broadcast radio;
wherein the adaptive antenna array permits non-line of sight communications with another wireless broadcast radio; and
a channel from each adaptive antenna in the adaptive antenna array, wherein the beamforming network uses an extreme interference cancellation having a space-time filter with a quarter bandwidth resolution that inverts the channel from each adaptive antenna in the adaptive antenna array and eliminates interference between the channels from each adaptive antenna in the adaptive antenna array, by directing spatial nulls in the array antenna pattern in the directions of all interfering terminating backhaul radio (TBR) while forming a beam peak in the direction of the desired TBR.

2. The radio of claim 1, wherein the weight processor computes space-time weights using a fast fourier transform and a fast correlation process.

3. The radio of claim 1, wherein the weight processor computes space-time weights wherein the signal is transformed into a frequency domain.

4. The radio of claim 1, wherein the beamforming network performs space-time adaptive processing and extreme interference cancellation that a time-bandwidth required for a predetermine signal to noise ratio and adaptive processing loss.

5. The radio of claim 1, wherein the beamforming network performs one of space-time adaptive processing and space-frequency adaptive processing and performs low order receiver equalization of an output from one of the space-time adaptive processing and space-frequency adaptive processing.

6. The radio of claim 1, wherein the adaptive antenna array is adjustable in one or more of azimuth, polarization, time and frequency.

7. The radio of claim 1, wherein the adaptive antenna array has adaptive down-tilt by adapting an elevation pattern of the adaptive antenna array in an outer adaptive loop independent of a faster inner loop for polarization and azimuth control.

8. The radio of claim 1, wherein the beamforming network receives a reference signal for link adaption, wherein the reference signal is one of a set of special training symbols, an embedded pilot, a preamble, a statistical significant data replication, a baseband signal replication, a known constellation of a signal and a data directed feedback.

9. The radio of claim 8, wherein the reference signal is CAZAC code.

10. The radio of claim 8, wherein the data directed feedback uses a payload to derive the reference signals.

11. The radio of claim 1, wherein the beamforming network calculates a signal covariance matrix using a voltage domain representation.

12. The radio of claim 1, wherein the weight processor generates a unique weight for each subcarrier in the baseband radios in one of an orthogonal frequency division multiple access baseband signal and an orthogonal frequency-division multiplexing baseband signal.

13. The radio of claim 1, wherein the beamforming network performs dynamic multiuser power allocation.

14. The radio of claim 1, wherein the beamforming network performs modulation and coding downshifting to enhance network capacity.

15. A wireless broadcast radio for backhaul applications, wireless Ethernet replacement applications and broadband wireless drop applications, the radio comprising:
an adaptive antenna array including at least two adaptive antennas;
a beamforming network connected to the adaptive antenna array;
a weigh processor connected to the beamforming network; and
one or more baseband radios connected to the beamforming network that deliver a stream of data to an external output of the wireless broadcast radio;
wherein the adaptive antenna array permits non-line of sight communications with another wireless broadcast radio; and
a channel from each adaptive antenna in the adaptive antenna array, wherein each channel has at least two polarizations and each channel carries a stream of independent data,
wherein the beamforming network performs processing of the two polarizations by doubling a degree of freedom in one of space-time adaptive processing and space-frequency adaptive processing and performs extreme interference cancellation having a space-time filter with a quarter bandwidth resolution that inverts the channel from each adaptive antenna in the adaptive antenna array and eliminates interference between the channels from each adaptive antenna in the adaptive antenna array, by directing spatial nulls in the array antenna pattern in the directions of all interfering terminating backhaul radio (TBR) while forming a beam peak in the direction of the desired TBR.