# Patent

## Title
Calibration of transmit and receive chains in a MIMO communication system

## Abstract

Techniques for calibrating the transmit and receive chains at a wireless entity are described. For a pre-calibration, N first overall gains for a receiver unit and N transmitter units in the transmit chain are obtained, where N>1. Each first overall gain is for a combined response for the receiver unit and the associated transmitter unit. N second overall gains for a transmitter unit and N receiver units in the receive chain are also obtained. Each second overall gain is for a combined response for the transmitter unit and the associated receiver unit. The gain of each transmitter unit and the gain of each receiver unit are determined based on the first and second overall gains. At least one correction matrix is then derived based on the gains of the transmitter and receiver units and is used to account for the responses of these units.

## Background
CLAIM OF PRIORITY
This application is a continuation of, and claims the benefit of priority from, U.S. patent application Ser. No. 10/816,999, filed Apr. 2, 2004 now U.S. Pat. No. 7,486,740 and entitled “Calibration of Transmit and Receive Chains in a MIMO Communication System,” which is fully incorporated herein by reference for all purposes.


FIELD
The present invention relates generally to data communication, and more specifically to techniques for calibrating transmit and receive chains at a wireless entity in a multiple-input multiple-output (MIMO) communication system.
BACKGROUND
A MIMO system employs multiple (NT) transmit antennas and multiple (NR) receive antennas for data transmission. A MIMO channel formed by the NT transmit and NR receive antennas may be decomposed into NS spatial channels, where NS≦min {NT, NR}. The NS spatial channels may be used to transmit data in parallel to achieve higher overall throughput or redundantly to achieve greater reliability.
To obtain high performance, it is often necessary to know the response of the entire transmission path from a transmitting entity to a receiving entity. This transmission path, which may be called an “effective” channel, typically includes a transmit chain at the transmitting entity, the MIMO channel, and a receive chain at the receiving entity. The transmit chain includes NT transmitter units, one transmitter unit for each transmit antenna. Each transmitter unit contains circuitry (e.g., digital-to-analog converter, filter, amplifier, mixer, and so on) that performs signal conditioning on a baseband signal to generate a radio frequency (RF) transmit signal suitable for transmission from the associated transmit antenna. The NT transmitter units may have different responses due to differences in the circuitry within these units. The receive chain includes NR receiver units, one receiver unit for each receive antenna. Each receiver unit contains circuitry (e.g., filter, amplifier, mixer, analog-to-digital converter, and so on) that performs signal conditioning on an RF receive signal from the associated receive antenna to obtain a received baseband signal. The NR receiver units may also have different responses due to differences in the circuitry within these units.
The effective channel response includes the responses of the transmit and receive chains as well as the response of the MIMO channel. Channel estimation may be simplified and performance may be improved if the responses of the transmit and receive chains can be determined and accounted for. The simplification in channel estimation is especially desirable for a MIMO system in which the downlink and uplink share a single frequency band in a time division duplex manner, as described below.
There is, therefore, a need in the art for techniques to calibrate the transmit and receive chains at the transmitting and receiving entities in a MIMO system.
SUMMARY
Techniques for calibrating the transmit and receive chains at a wireless entity are described herein. The wireless entity may be a user terminal or an access point. The responses of the transmit and receive chains may be determined and accounted for by performing a pre-calibration, a field calibration, and/or a follow-on calibration.
For pre-calibration, N first overall gains for a receiver unit and N transmitter units in the transmit chain are obtained, one first overall gain for each transmitter unit, where N>1. Each first overall gain is indicative of a combined response for the receiver unit and the associated transmitter unit. N second overall gains for a transmitter unit and N receiver units in the receive chain are also obtained, one second overall gain for each receiver unit. Each second overall gain is indicative of a combined response for the transmitter unit and the associated receiver unit. The overall gain for transmitter unit i and receiver unit j may be obtained by sending a test signal (e.g., a tone) via transmitter unit i, measuring the test tone received via receiver unit j, and computing the overall complex gain as the ratio of the received test signal level to the sent test signal level. The gain of each transmitter unit is determined based on the N first overall gains, and the gain of each receiver unit is determined based on the N second overall gains. At least one correction matrix is then derived based on the gains of the N transmitter units and the gains of the N receiver units. The at least one correction matrix is used to account for the responses of the transmitter and receiver units at the wireless entity.
For field calibration, an access point transmits a MIMO pilot (described below) on the downlink, and a user terminal transmits a MIMO pilot on the uplink. Estimates of the MIMO channel responses for the downlink and uplink are obtained based on the downlink and uplink MIMO pilots, respectively, and used to derive at least one updated correction matrix for each wireless entity, as described below. The updated correction matrices for both wireless entities may be used in place of the correction matrices obtained for these entities via pre-calibration.
For follow-on calibration, one wireless entity (e.g., the access point) transmits two different pilots, and the other wireless entity (e.g., the user terminal) estimates the errors in the correction matrices for the access point and the user terminal based on the pilots, as described below. The correction matrices for both wireless entities may then be updated based on the determined errors.
In general, pre-calibration, field calibration, and follow-on calibration may be performed at any time and in any order. Various aspects and embodiments of the invention are described in further detail below.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows an overall process for calibration and normal operation;
FIG. 2 shows a block diagram of a wireless entity;
FIG. 3 shows a process for performing pre-calibration;
FIG. 4 shows the transmit and receive chains at an access point and a user terminal;
FIG. 5 shows the use of a correction matrix for each transmit and receive chain;
FIG. 6 shows the use of correction matrices on the transmit paths;
FIG. 7 shows the use of correction matrices on the receive paths; and
FIG. 8 shows a block diagram of the access point and the user terminal.



DETAILED DESCRIPTION
The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments.
FIG. 1 shows a flow diagram of a process 100 performed by an access point and a user terminal for calibration and normal operation. Initially, pre-calibration may be performed separately for the access point (block 110a) and the user terminal (block 110b) to derive correction matrices that may be used to account for the responses of the transmit/receive chains at these entities. The pre-calibration may be performed during manufacturing, after deployment, or at some other time. Field calibration may be performed jointly by the access point and the user terminal in the field to obtain updated correction matrices for these entities (block 120).
For normal operation, the user terminal may transmit data on the uplink using a correction matrix, if any, for the transmit path of the user terminal (block 132). The access point may receive the uplink transmission using a correction matrix, if any, for the receive path of the access point (block 134). The access point may also transmit data on the downlink using a correction matrix, if any, for the transmit path of the access point (block 136). The user terminal may receive the downlink transmission using a correction matrix, if any, for the receive path of the user terminal (block 138).
Follow-on calibration may be performed jointly by the access point and the user terminal to estimate the errors in the correction matrices and to update the correction matrices for these entities (block 140). In general, pre-calibration, field calibration, follow-on calibration, or any combination thereof may be performed to obtain the correction matrices for the access point and user terminal. Furthermore, the different types of calibration may be performed at any time and in any order.
A MIMO system may utilize a frequency division duplex (FDD) or a time division duplex (TDD) channel structure. For an FDD MIMO system, the downlink and uplink are allocated separate frequency bands, and the MIMO channel response for one link may not correlate well with the MIMO channel response for the other link. In this case, the responses of the transmit and receive chains for each wireless entity may be determined (e.g., by performing pre-calibration), and each chain may be accounted for with a respective correction matrix, as described below.
For a TDD MIMO system, the downlink and uplink share the same frequency band, with the downlink being allocated a portion of the time and the uplink being allocated the remaining portion of the time. The MIMO channel response for one link may be highly correlated with the MIMO channel response for the other link and may even be assumed to be reciprocal of one another. That is, if H represents a channel response matrix from antenna array A to antenna array B, then a reciprocal channel implies that the coupling from array B to array A is given by HT, where “T” denotes a transpose. Channel estimation may be simplified for a reciprocal channel since the channel response for one link (e.g., the uplink) may be estimated based on a pilot received via the other link (e.g., the downlink). For a TDD MIMO system, calibration may be performed in a manner to take advantage of the correlation between the downlink and uplink channel responses, as described below.
Pre-calibration, field calibration, and/or follow-on calibration may be performed to derive correction matrices for the transmit path, the receive path, or both the transmit and receive paths at the access point and user terminal, as described below. For simplicity, the following description assumes a noise-free environment and channel estimation without errors. Thus, noise terms are not shown in the equations below. Also, the receiver units are assumed to have sufficient (e.g., 30 dB or more) isolation among one another.
Pre-Calibration
FIG. 2 shows a block diagram of a wireless entity 200 equipped with N antennas, where N>1. Wireless entity 200 may be a user terminal, which may also be called a wireless device, a mobile station, or some other terminology. Wireless entity 200 may also be an access point, which may also be called a base station or some other terminology.
On the transmit path, a data processor 210 receives and processes (e.g., encodes, interleaves, and modulates) data to obtain data symbols. As used herein, a “data symbol” is a modulation symbol for data, and a “pilot symbol” is a modulation symbol for pilot. Pilot symbols are known a priori by both the transmitting and receiving entities. Data processor 210 may further perform spatial processing on the data symbols and provides N streams of transmit symbols to N transmitter units (TMTR) 224a through 224n. As used herein, a “transmit symbol” is a symbol to be transmitted from an antenna. Each transmitter unit 224 processes its transmit symbol stream to generate an RF transmit signal, which is then routed through a circulator 226 and via an antenna connector 228 to an antenna (not shown in FIG. 2). The processing by each transmitter unit 224 typically includes digital-to-analog conversion, amplification, filtering, and frequency upconversion.
On the receive path, one or more RF transmit signals (e.g., from another wireless entity) are received by each of the N antennas (not shown in FIG. 2) at wireless entity 200. The RF receive signal from each antenna is provided via connector 228 and routed through circulator 226 to an associated receiver unit (RCVR) 234. Each receiver unit 234 processes its RF receive signal and provides a stream of received symbols to data processor 210. The processing by each receiver unit 234 typically includes frequency downconversion, amplification, filtering, and analog-to-digital conversion. Data processor 210 performs receiver spatial processing (or spatial matched filtering) on the received symbols from all N receiver units 234a through 234n to obtain detected symbols, which are estimates of the data symbols sent by the other wireless entity. Data processor 210 further processes (e.g., demodulates, deinterleaves, and decodes) the detected symbols to obtain decoded data.
The signals for the transmit path at wireless entity 200 may be expressed as:

ztx=Tx,  Eq (1)

where x=[x1 x2 . . . xN]T is a vector with N transmit (TX) baseband signals for the N antennas, where xi is the TX baseband signal for antenna i; T is a diagonal matrix with N complex gains for the N transmitter units; and ztx=[ztx,1 ztx,2 . . . ztx,N]T is a vector with N RF transmit signals for the N antennas, where ztx,i is the RF transmit signal for antenna i.

The signals for the receive path at wireless entity 200 may be expressed as:

y=Rzrx,  Eq (2)

where zrx=[zrx,1 zrx,2 . . . zrx,N]T is a vector with N RF receive signals for the N antennas, where zrx,i is the RF receive signal for antenna i; R is a diagonal matrix with N complex gains for the N receiver units; and y=[y1 y2 . . . yN]T is a vector with N receive (RX) baseband signals for the N antennas, where yi is the RX baseband signal for antenna i.

The RF and baseband signals are a function of time, but this is not indicated above for simplicity.
The matrices T and R are of dimension N×N and may be expressed as:




  
    
      
        
          
            
              T
              _
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          t
                          11
                        
                      
                      
                        0
                      
                      
                        ...
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          t
                          22
                        
                      
                      
                        ...
                      
                      
                        0
                      
                    
                    
                      
                        ⋮
                      
                      
                        ⋮
                      
                      
                        ⋱
                      
                      
                        ⋮
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        ...
                      
                      
                        
                          t
                          NN
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                and
                ⁢
                
                  
                
                ⁢
                
                  R
                  _
                
              
              =
              
                [
                
                  
                    
                      
                        r
                        11
                      
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        r
                        22
                      
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                    
                      ⋮
                    
                    
                      ⋱
                    
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      
                        r
                        NN
                      
                    
                  
                
                ]
              
            
          
          ,
        
        ⁢
        
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          3
          )
        
      
    
  




where tii is the complex gain for transmitter unit i and rii is the complex gain for receiver unit i, for i=1 . . . N. The responses of the transmitter and receiver units are typically a function of frequency. For simplicity, a flat frequency response is assumed for the transmitter and receiver units. In this case, the response of each transmitter unit is represented by a single complex gain tii, and the response of each receiver unit is also represented by a single complex gain rii.

FIG. 3 shows a flow diagram of a process 300 for performing pre-calibration for wireless entity 200. Process 200 is described below with reference to FIG. 2.
An overall gain r11·t11 of the transmitter unit and receiver unit for antenna 1 is first obtained (block 312). This may be achieved, for example, by short-circuiting connector 228a for antenna 1 with a termination connector having a center conductor connected to circuit ground. With connector 228a short circuited, the RF transmit signal ztx,1 and the RF receive signal zrx,1 at a point between circulator 226a and connector 228a may be expressed as:

zrx,1=−ztx,1,  Eq (4)

where the signal inversion is due to the short circuit at connector 228a. A TX baseband signal x1 (e.g., a single tone) is then applied to transmitter unit 224a and an RX baseband signal y1 at the output of receiver unit 234a is measured. The TX baseband signal level should be such that the reflected signal from the short-circuited antenna port should not cause any damage. The RF transmit signal ztx,1 and the RX baseband signal y1 may be expressed as:

ztx,1=t11·x1, and  Eq (5)

y1=r11·zrx,1=−r11·ztx,1,  Eq (6)

where the rightmost quantity in equation (6) is obtained using equation (4). Combining equations (5) and (6), the following is obtained:





  
    
      
        
          
            r
            11
          
          ·
          
            t
            11
          
        
        =
        
          -
          
            
              
                y
                1
              
              
                x
                
                  1
                  ⁢
                  
                    
                  
                
              
            
            .
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          7
          )
        
      
    
  




Equation (7) indicates that the overall gain r11·t11 may be obtained as the negative of the ratio of the TX baseband signal level to the RX baseband signal level for antenna 1, with connector 228a shorted.
In FIG. 3, an index i is then initialized to 2 (block 314). An overall gain rii·t11 of transmitter unit 1 for antenna 1 and receiver unit i for antenna i is obtained (block 316). An overall gain r11·tii of transmitter unit i for antenna i and receiver unit 1 for antenna 1 is also obtained (block 318). The overall gains rii·t11 and rii·t11 may be obtained as follows. The termination connector is removed from connector 228a and a test cable with a known characteristic is connected between connector 228a for antenna 1 and connector 228i for antenna i. A TX baseband signal x1 (e.g., a single tone) is applied to transmitter unit 224a and an RX baseband signal yi from receiver unit 234i for antenna i is measured. The RX baseband signal yi may be expressed as:

yi=rii·zrx,i=rii·αcable·ztx,1=rii·αcable·t11·x1,  Eq (8)

where αcable is a known complex value for the loss and phase shift of the test cable. The overall gain rii·t11 may then be computed as:





  
    
      
        
          
            r
            ii
          
          ·
          
            t
            11
          
        
        =
        
          
            1
            
              α
              cable
            
          
          ·
          
            
              
                y
                i
              
              
                x
                1
              
            
            .
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          9
          )
        
      
    
  




Similarly, a TX baseband signal xi (e.g., a single tone) is applied to transmitter unit 224i for antenna i and the RX baseband signal x1 from receiver unit 234a is measured. The RX baseband signal y1 may be expressed as:

y1=r11·zrx,1=r11·αcable·ztx,i=r11·αcable·tii·xi,  Eq (10)

The overall gain r11·tii may then be computed as:




  
    
      
        
          
            r
            11
          
          ·
          
            t
            ii
          
        
        =
        
          
            1
            
              α
              cable
            
          
          ·
          
            
              
                y
                1
              
              
                x
                i
              
            
            .
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          11
          )
        
      
    
  




Equations (9) and (11) indicate that an overall gain rii·tjj may be obtained as a scaled version of the ratio of the RX baseband signal level for antenna i to the TX baseband signal level for antenna j, where the scaling is by 1/αcable.
A determination is then made whether index i is equal to N (block 320). If the answer is ‘no’, then index i is incremented by one (block 322) and the process returns to block 316 to determine another pair of overall gains for another antenna. Otherwise, if the answer is ‘yes’ for block 320, then the process proceeds to block 330.
FIG. 2 shows the use of circulators 226 to route (1) the TX baseband signals from transmitter units 224 to the antennas and (2) the RF receive signals from the antennas to receiver units 234. Circulators are typically used for a TDD system in which the downlink and uplink share the same frequency band. Switches may also be used for the TDD system to route signal to and from the antennas. In this case, the overall gain r11·t11 for antenna 1 is not obtained by shorting connector 228a but may be obtained as:




  
    
      
        r
        11
      
      ·
      
        t
        11
      
    
    =
    
      
        
          (
          
            
              r
              11
            
            ·
            
              t
              ii
            
          
          )
        
        ·
        
          (
          
            
              r
              jj
            
            ·
            
              t
              11
            
          
          )
        
      
      
        
          r
          jj
        
        ·
        
          t
          ii
        
      
    
  
  ,




where rjj·tii is the overall gain for receiver unit j and transmitter unit i.

Block 312 provides the overall gain r11·t11 for antenna 1. The N−1 iterations of blocks 316 and 318 provide 2(N−1) overall gains, r11·t22 through r11·tNN and r22·t11 through rNN·t11, for antenna 1 and each of antennas 2 through N. A matrix {tilde over (T)} of gains for the N transmitter units may be obtained based on the N overall gains r11·t11 through r11·tNN for the N transmitter units and receiver unit 1 (block 330), as follows:




  
    
      
        
          
            T
            ~
          
          _
        
        =
        
          
            [
            
              
                
                  
                    
                      r
                      11
                    
                    ·
                    
                      t
                      11
                    
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      r
                      11
                    
                    ·
                    
                      t
                      22
                    
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  
                    
                      r
                      11
                    
                    ·
                    
                      t
                      NN
                    
                  
                
              
            
            ]
          
          ⁢
          
            

          
          ⁢
          
            
          
          =
          
            
              
                (
                
                  
                    r
                    11
                  
                  ·
                  
                    t
                    11
                  
                
                )
              
              ·
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        
                          t
                          22
                        
                        
                          t
                          11
                        
                      
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                    
                      ⋮
                    
                    
                      ⋱
                    
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      
                        
                          t
                          NN
                        
                        
                          t
                          11
                        
                      
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                r
                11
              
              ·
              
                
                  T
                  _
                
                .
              
            
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          12
          )
        
      
    
  




Equation (12) indicates that {tilde over (T)} is a scaled version of T, where the scaling is by r11.
Similarly, a matrix {tilde over (R)} of gains for the N receiver units may be obtained based on the N overall gains r11·t11 through rNN·t11 for transmitter unit 1 and the N receiver units (block 332), as follows:




  
    
      
        
          
            R
            ~
          
          _
        
        =
        
          
            [
            
              
                
                  
                    
                      r
                      11
                    
                    ·
                    
                      t
                      11
                    
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      r
                      22
                    
                    ·
                    
                      t
                      11
                    
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  
                    
                      r
                      NN
                    
                    ·
                    
                      t
                      11
                    
                  
                
              
            
            ]
          
          ⁢
          
            

          
          ⁢
          
            
          
          =
          
            
              
                (
                
                  
                    r
                    11
                  
                  ·
                  
                    t
                    11
                  
                
                )
              
              ·
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        
                          r
                          22
                        
                        
                          r
                          11
                        
                      
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                    
                      ⋮
                    
                    
                      ⋱
                    
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      
                        
                          r
                          NN
                        
                        
                          r
                          11
                        
                      
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                t
                11
              
              ·
              
                
                  R
                  _
                
                .
              
            
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          13
          )
        
      
    
  




Equation (13) indicates that {tilde over (R)} is a scaled version of R, where the scaling is by t11.
At least one correction matrix that can account for the responses of the transmitter and receiver units may be derived based on the matrices {tilde over (T)} and {tilde over (R)} (block 334), as described below. The process then terminates.
FIG. 4 shows a block diagram of the transmit and receive chains at an access point 410 and a user terminal 450 in a MIMO system 400. For the downlink, at access point 410, transmit symbols (denoted by a vector xdn) are processed by a transmit chain 424 and transmitted from Nap antennas 428 and over a wireless MIMO channel. At user terminal 450, Nap downlink signals are received by Nut antennas 452 and processed by a receive chain 454 to obtain received symbols (denoted by a vector ydn). For the uplink, at user terminal 450, transmit symbols (denoted by a vector xup) are processed by a transmit chain 464 and transmitted from Nut antennas 452 and over the MIMO channel. At access point 410, Nut uplink signals are received by Nap antennas 428 and processed by a receive chain 434 to obtain received symbols (denoted by a vector yup).
Transmit chain 424 includes Nap transmitter units for the Nap access point antennas and is characterized by a diagonal matrix Tap with Nap complex gains for the Nap transmitter units, or diag(Tap)={tap,11 tap,22 . . . tap,NapNap}. Receiver chain 434 includes Nap receiver units for the Nap access point antennas and is characterized by a diagonal matrix Rap with Nap complex gains for the Nap receiver units, or diag(Rap)={rap,11 rap,22 . . . rap,NapNap}. Pre-calibration may be performed for access point 410 to obtain matrices {tilde over (T)}ap and {tilde over (R)}ap, which are scaled versions of Tap and Rap, respectively, as described above for FIG. 3.
Similarly, transmit chain 464 includes Nut transmitter units for the Nut user terminal antennas and is characterized by a diagonal matrix Tut with Nut complex gains for the Nut transmitter units, or diag(Tut)={tut,11 tut,22 . . . tut,NutNut}. Receive chain 454 includes Nut receiver units for the Nut user terminal antennas and is characterized by a diagonal matrix Rut with Nut complex gains for the Nut receiver units, or diag(Rut)={rut,11 rut,22 . . . rut,NutNut}. Pre-calibration may also be performed for user terminal 450 to obtain matrices {tilde over (T)}ut and {tilde over (R)}ut, which are scaled versions of {tilde over (T)}ut and {tilde over (R)}ut, respectively.
The following relationships may be expressed based on equations (12) and (13):

{tilde over (T)}ap=rap,11·Tap,  Eq (14)

{tilde over (R)}ap=tap,11·Rap,  Eq (15)

{tilde over (T)}ut=rut,11·Tut,  Eq (16)

{tilde over (R)}ut=tut,11·Rut,  Eq (17)

where tap,11 is the gain of the transmitter unit for access point antenna 1; rap,11 is the gain of the receiver unit for access point antenna 1; tut,11 is the gain of the transmitter unit for user terminal antenna 1; and rut,11 is the gain of the receiver unit for user terminal antenna 1.

In an embodiment, the response of each transmit and receive chain is accounted for based on a correction matrix derived for that chain. The correction matrix for each chain may be computed as the inverse of the diagonal matrix for that chain. A correction matrix for a transmit chain is applied prior to the transmit chain, and a correction matrix for a receive chain is applied after the receive chain.
FIG. 5 shows the use of a separate correction matrix to account for the response of each transmit and receive chain at an access point 410a and a user terminal 450a. On the downlink, at access point 410a, the transmit vector xdn is first multiplied with a correction matrix {tilde over (T)}ap−1 by a unit 522, processed by transmit chain 424, and transmitted from Nap antennas 428. At user terminal 450a, the Nap downlink signals are received by Nut antennas 452, processed by receive chain 454, and further multiplied with a correction matrix {tilde over (R)}ut−1 by a unit 556 to obtain the received vector ydn.
On the uplink, at user terminal 450a, the transmit vector xup is first multiplied with a correction matrix {tilde over (T)}ut−1 by a unit 562, processed by transmit chain 464, and transmitted from Nut antennas 452. At access point 410a, the Nut uplink signals are received by Nap antennas 428, processed by receive chain 434, and further multiplied with a correction matrix {tilde over (R)}ap−1 by a unit 536 to obtain the received vector yup.
The combined gain of unit 522 and transmit chain 424 may be computed as (1/rap,11)·I, where I is the identity matrix with ones along the diagonal and zeros elsewhere. Similarly, the combined gain of receive chain 434 and unit 536 may be computed as (1/tap,11)·I, the combined gain of receive chain 454 and unit 556 may be computed as (1/tut,11)·I, and the combined gain of unit 562 and transmit chain 464 may be computed as (1/rut,11)·I. The use of a correction matrix for each transmit/receive chain results in an essentially flat response across the transmitter/receiver units in that chain. The scaling factor (e.g., 1/rap,11) may be accounted for by simply scaling the transmit symbols and/or the transmit power for all antennas by the same amount.
For a reciprocal channel (e.g., a TDD MIMO system), the channel response matrix for the downlink may be denoted as H, and the channel response matrix for the uplink may be denoted as HT. The received vectors for the downlink and uplink, without any correction matrices, may be expressed as:

ydn=RutHTapxdn, and  Eq (18)

yup=RapHTTutxup.  Eq (19)

From equations (18) and (19), the “effective” downlink and uplink channel responses, Hdn and Hup, which include the responses of the applicable transmit and receive chains, may be expressed as:

Hdn=RutHTap and Hup=RapHTTut.  Eq (20)

As shown in equation set (20), if the responses of the transmit and receive chains at the access point are not equal to the responses of the transmit and receive chains at the user terminal, then the effective downlink and uplink channel responses are not reciprocal of one another, or RutHTap≠(RapHTTut)T.
In another embodiment, the responses of the transmit and receive chains at each wireless entity are accounted for by a single correction matrix applied on the transmit path prior to the transmit chain. The two equations in equation set (20) may be combined to obtain the following:

HupT=TutRut−1HdnTap−1Rap=Kuttx−1HdnKaptx or Hup=(Kuttx−1HdnKaptx)T,  Eq (21)

where Kaptx=Tap−1Rap and Kuttx=Tut−1Rut. Kaptx is an Nap×Nap diagonal matrix for the access point and is equal to the ratio of the receive chain response Rap to the transmit chain response Tap, where the ratio is taken element by element. Similarly, Kuttx is an Nut×Nut diagonal matrix for the user terminal and is equal to the ratio of the receive chain response Rut to the transmit chain response Tut.

Equation (21) may also be expressed as:

Hcup=HupKuttx=(HdnKaptx)T=HcdnT,  Eq (22)

where {tilde over (K)}uttx is the calibrated channel response for the uplink; and Hcdn is the calibrated channel response for the downlink.

Pre-calibration may be performed for the access point to obtain matrices {tilde over (T)}ap and {tilde over (R)}ap Pre-calibration may also be performed for the user terminal to obtain matrices {tilde over (T)}ut and {tilde over (R)}ut. Correction matrices {tilde over (K)}aptx and {tilde over (K)}uttx for the access point and user terminal, respectively, may then be derived as:




  
    
      
        
          
            
              
                K
                ~
              
              _
            
            aptx
          
          =
          
            
              
                
                  
                    T
                    ~
                  
                  _
                
                ap
                
                  -
                  1
                
              
              ⁢
              
                
                  
                    R
                    _
                  
                  ~
                
                ap
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  
                    (
                    
                      
                        r
                        
                          ap
                          ,
                          11
                        
                      
                      ·
                      
                        
                          T
                          _
                        
                        ap
                      
                    
                    )
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  (
                  
                    
                      t
                      
                        ap
                        ,
                        11
                      
                    
                    ·
                    
                      
                        R
                        _
                      
                      ap
                    
                  
                  )
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              =
              
                
                  
                    
                      
                        t
                        
                          ap
                          ,
                          11
                        
                      
                      
                        r
                        
                          ap
                          ,
                          11
                        
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ap
                      
                        -
                        1
                      
                    
                  
                  ⁢
                  
                    
                      R
                      _
                    
                    ap
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                
                =
                
                  
                    k
                    aptx
                  
                  ·
                  
                    
                      K
                      _
                    
                    aptx
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          23
          )
        
      
    
  
  
    
      
        
          
            
              
                K
                _
              
              ~
            
            uttx
          
          =
          
            
              
                
                  
                    T
                    ~
                  
                  _
                
                ut
                
                  -
                  1
                
              
              ⁢
              
                
                  
                    R
                    ~
                  
                  _
                
                ut
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  
                    (
                    
                      
                        r
                        
                          ut
                          ,
                          11
                        
                      
                      ·
                      
                        
                          T
                          _
                        
                        ut
                      
                    
                    )
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  (
                  
                    
                      t
                      
                        ut
                        ,
                        11
                      
                    
                    ·
                    
                      
                        R
                        _
                      
                      ut
                    
                  
                  )
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              =
              
                
                  
                    
                      
                        t
                        
                          ut
                          ,
                          11
                        
                      
                      
                        r
                        
                          ut
                          ,
                          11
                        
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ut
                      
                        -
                        1
                      
                    
                  
                  ⁢
                  
                    
                      R
                      _
                    
                    ut
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                
                =
                
                  
                    k
                    uttx
                  
                  ·
                  
                    
                      K
                      _
                    
                    uttx
                  
                
              
            
          
        
        ,
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          24
          )
        
      
    
  




where kaptx kaptx and kuttx kuttx are two scalars defined as kaptx=tap,11/rap,11 and kuttx=tut,11/rut,11.

FIG. 6 shows the use of correction matrices on the transmit paths to account for the responses of the transmit and receive chains at an access point 410b and a user terminal 450b. On the downlink, at access point 410b, the transmit vector xdn is first multiplied with the correction matrix {tilde over (K)}aptx by a unit 622, processed by transmit chain 424, and transmitted from Nap antennas 428. At user terminal 450b, the Nap downlink signals are received by Nut antennas 452 and processed by receive chain 454 to obtain the received vector ydn.
On the uplink, at user terminal 450b, the transmit vector xup is first multiplied with the correction matrix {tilde over (K)}uttx by a unit 662, processed by transmit chain 464, and transmitted from Nut antennas 452. At access point 410b, the Nut uplink signals are received by Nap antennas 428 and processed by receive chain 434 to obtain the received vector yup.
The received vectors for the downlink and uplink, with the correction matrices {tilde over (K)}aptx and {tilde over (K)}uttx applied on the transmit path at the access point and user terminal, respectively, may be expressed as:

ydn=RutHTap{tilde over (K)}aptxxdn=kaptxRutHTapTap−1Rapxdn=kaptxRutHRapxdn, and  Eq (25)

yupRapHTTut{tilde over (K)}uttxxup=kuttxRapHTTutTut−1Rutxup=kuttxRapHTRutxup.  Eq (26)

From equations (25) and (26), the calibrated downlink and uplink channel responses with the correction matrices {tilde over (K)}aptx and {tilde over (K)}uttx may be expressed as:

{tilde over (H)}cdntx=kaptxRutHRap and {tilde over (H)}cuptx=kuttxRapHTRut.  Eq (27)

Since the scalars kaptx and kuttx do not disturb the reciprocal relationship of the downlink and uplink, {tilde over (H)}cdntx is equal to a scaled version of the transpose of:





  
    
      
        H
        _
      
      ~
    
    cuptx
  
  ,
  
    
  
  ⁢
  
    
      or
      ⁢
      
        
      
      ⁢
      
        
          
            H
            _
          
          ~
        
        cuptx
      
    
    =
    
      
        
          k
          uttx
        
        
          k
          aptx
        
      
      ·
      
        
          
            
              H
              _
            
            ~
          
          cdntx
          T
        
        .
      
    
  




In yet another embodiment, the responses of the transmit and receive chains at each wireless entity are accounted for by a single correction matrix applied on the receive path after the receive chain. The two equations in equation set (20) may also be combined to obtain the following:

HdnT=TapRap−1HupTut−1Rut=KaprxHupKutrx−1 or Hdn=(KaprxHupKutrx−1)T,  Eq (28)

where Kaprx=TapRap−1 and Kutrx=TutRut−1. Correction matrices {tilde over (K)}aprx and {tilde over (K)}utrx for the access point and user terminal, respectively, may be derived as:





  
    
      
        
          
            
              
                K
                ~
              
              _
            
            aprx
          
          =
          
            
              
                
                  
                    T
                    ~
                  
                  _
                
                ap
              
              ⁢
              
                
                  
                    R
                    _
                  
                  ~
                
                ap
                
                  -
                  1
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  (
                  
                    
                      r
                      
                        ap
                        ,
                        11
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ap
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      
                        t
                        
                          ap
                          ,
                          11
                        
                      
                      ·
                      
                        
                          R
                          _
                        
                        ap
                      
                    
                    )
                  
                  
                    -
                    1
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              =
              
                
                  
                    
                      
                        r
                        
                          ap
                          ,
                          11
                        
                      
                      
                        t
                        
                          ap
                          ,
                          11
                        
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ap
                    
                  
                  ⁢
                  
                    
                      R
                      _
                    
                    ap
                    
                      -
                      1
                    
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                
                =
                
                  
                    k
                    aprx
                  
                  ·
                  
                    
                      K
                      _
                    
                    aprx
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          29
          )
        
      
    
  
  
    
      
        
          
            
              
                K
                _
              
              ~
            
            utrx
          
          =
          
            
              
                
                  
                    T
                    ~
                  
                  _
                
                ut
              
              ⁢
              
                
                  
                    R
                    ~
                  
                  _
                
                ut
                
                  -
                  1
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  (
                  
                    
                      r
                      
                        ut
                        ,
                        11
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ut
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      
                        t
                        
                          ut
                          ,
                          11
                        
                      
                      ·
                      
                        
                          R
                          _
                        
                        ut
                      
                    
                    )
                  
                  
                    -
                    1
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              =
              
                
                  
                    
                      
                        r
                        
                          ut
                          ,
                          11
                        
                      
                      
                        t
                        
                          ut
                          ,
                          11
                        
                      
                    
                    ·
                    
                      
                        T
                        _
                      
                      ut
                    
                  
                  ⁢
                  
                    
                      R
                      _
                    
                    ut
                    
                      -
                      1
                    
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                
                =
                
                  
                    k
                    utrx
                  
                  ·
                  
                    
                      K
                      _
                    
                    utrx
                  
                
              
            
          
        
        ,
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          30
          )
        
      
    
  




where kaprx and kutrx two scalars defined as kaprx=rap,11/tap,11 and kutrx=rut,11/tut,11.

FIG. 7 shows the use of correction matrices on the receive paths to account for the responses of the transmit and receive chains at an access point 410c and a user terminal 450c. On the downlink, at access point 410c, the transmit vector xdn is processed by transmit chain 424 and transmitted from Nap antennas 428. At user terminal 450c, the Nap downlink signals are received by Nut antennas 452, processed by receive chain 454, and further multiplied with the correction matrix {tilde over (K)}utrx by a unit 756 to obtain the received vector ydn.
On the uplink, at user terminal 450c, the transmit vector xup is processed by transmit chain 464 and transmitted from Nut antennas 452. At access point 410c, the Nut uplink signals are received by Nap antennas 428, processed by receive chain 434, and further multiplied with the correction matrix {tilde over (K)}aprx by a unit 736 to obtain the received vector yup.
The received vectors for the downlink and uplink, with the correction matrices {tilde over (K)}aprx and {tilde over (K)}utrx applied on the receive path at the access point and user terminal, respectively, may be expressed as:

ydn={tilde over (K)}utrxRutHTapxdn=kutrxTutRut−1RutHTapxdn=kutrxTutHTapxdn, and  Eq (31)

yup={tilde over (K)}aprxRapHTTutxup=kaprxTapRap−1RapHTTutxup=kaprxTapHTTutxup.  Eq (32)

From equations (31) and (32), the calibrated downlink and uplink channel responses with the correction matrices {tilde over (K)}aprx and {tilde over (K)}utrx may be expressed as:

{tilde over (H)}cdnrx=kutrxTutHTap and {tilde over (H)}cuprx=kaprxTapHTTut.  Eq (33)

Again, the scalars kaprx and kutrx do not disturb the reciprocal relationship of the downlink and uplink, and {tilde over (H)}cdnrx is equal to a scaled version of the transpose of {tilde over (H)}cuprx, or:





  
    
      
        H
        _
      
      ~
    
    cuprx
  
  =
  
    
      
        k
        aprx
      
      
        k
        utrx
      
    
    ·
    
      
        
          
            H
            _
          
          ~
        
        cdnrx
        T
      
      .
    
  




As shown in FIGS. 5 through 7, the responses of the transmit and receive chains at the access point and user terminal may be accounted for with a correction matrix for each chain (as shown in FIG. 5), a correction matrix for the transmit path (as shown in FIG. 6), or a correction matrix for the receive path (as shown in FIG. 7). The embodiment shown in FIG. 5 may be used for both TDD and FDD MIMO systems. The embodiments shown in FIGS. 6 and 7 are typically used for a TDD MIMO system. For the TDD MIMO system, the use of the correction matrices allows the calibrated channel response for one link to be expressed by the calibrated channel response for the other link, which can simplify both channel estimation and spatial processing for data transmission over the MIMO channel.
The pre-calibration techniques may be used for a single-carrier MIMO system, as described above. These techniques may also be used for a multi-carrier MIMO system, which may utilize orthogonal frequency division multiplexing (OFDM) or some other multi-carrier modulation technique. OFDM effectively partitions the overall system bandwidth into multiple (NF) orthogonal subbands, which are also called tones, subcarriers, bins, and frequency channels. With OFDM, each subband is associated with a respective subcarrier that may be modulated with data. For a MIMO system that utilizes OFDM (a MIMO-OFDM system), the pre-calibration described above may be performed for each of multiple subbands (e.g., for each subband used for transmission).
The pre-calibration may also be performed for different operating points. The transmit and/or receive chains may have variable gains, and different responses may be obtained for the transmit/receive chains at different gain settings. The pre-calibration may be performed to obtain different correction matrices for different gain settings. The appropriate correction matrices would then be used based on the gain settings for these chains. In general, pre-calibration may be performed for one or multiple values of a given parameter (e.g., gain, temperature, and so on) to obtain correction matrices that can account for the responses of the transmit/receive chains at each parameter value.
Field Calibration
Field calibration may be performed to determine and account for the responses of the transmit/receive chains at the access point and user terminal. For field calibration, the access point transmits a MIMO pilot on the downlink, and the user terminal transmits a MIMO pilot on the uplink. A MIMO pilot is a pilot comprised of NT pilot transmissions sent from NT transmit antennas, where the pilot transmission from each transmit antenna is identifiable by the receiving entity. This may be achieved, for example, by using a different orthogonal sequence for the pilot transmission from each transmit antenna. The access point obtains the effective uplink channel response Hup based on the uplink MIMO pilot. The user terminal obtains the effective downlink channel response Hdn based on the downlink MIMO pilot. One entity (e.g., the access point) then sends to the other entity (e.g., the user terminal) its effective channel response. Correction matrices for both the access point and the user terminal may be computed from Hdn and Hup using, for example, matrix-ratio computation or minimum mean square error (MMSE) computation.
For the matrix-ratio computation to derive Kaptx and Kuttx for the transmit paths, an Nut×Nap matrix C is first computed as:




  
    
      
        
          
            C
            _
          
          =
          
            
              
                H
                _
              
              up
              T
            
            
              
                H
                _
              
              dn
            
          
        
        ,
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          34
          )
        
      
    
  




where the ratio is taken element-by-element.

The diagonal elements of Kaptx are set equal to the mean of the normalized rows of C. Each row of C is first normalized by scaling each of the Nap elements in that row with the first element in the row. The mean of the normalized rows (denoted by a vector  crow) is then computed as the sum of the Nut normalized rows divided by Nut. The Nap diagonal elements of Kaptx are set equal to the Nap elements of  crow. Because of the normalization, the lead element of Kaptx is equal to unity.
The diagonal elements of Kuttx are set equal to the mean of the inverses of the normalized columns of C. The j-th column of C, for j=1 . . . Nap, is first normalized by scaling each element in that column with the j-th diagonal element of Kaptx. The mean of the inverses of the normalized columns (denoted by a vector  ccol) is then computed by (1) taking the inverse of each normalized column, where the inversion is performed element-wise, (2) summing the Nap inverse normalized columns, and (3) dividing each element in the resultant column by Nap to obtain  ccol. The Nut diagonal elements of Kuttx are set equal to the Nut elements of  ccol.
For the MMSE computation, the correction matrices Kaptx and Kuttx for the transmit paths are derived from Hdn and Hup such that the mean square error (MSE) between the calibrated downlink and uplink channel responses is minimized. This condition may be expressed as:

min|(HdnKaptx)T−HupKuttx|2 or min|KaptxHdnT−HupKuttx|2,  Eq (35)

where KaptxT=Kaptx since Kaptx is a diagonal matrix. For equation (35), the lead element in the first row of the first column of Kaptx is set equal to unity so that the trivial solution, with all elements of Kaptx and Kuttx (set equal to zero, is not obtained.

To obtain Kaptx and Kuttx based on equation (35), the mean square error (or the square error since a divide by NapNut is omitted) may be computed as:




  
    
      
        
          MSE
          =
          
            
              ∑
              
                j
                =
                1
              
              
                N
                ut
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  N
                  ap
                
              
              ⁢
              
                
                  
                  
                    
                      
                        h
                        
                          dn
                          ,
                          ji
                        
                      
                      ·
                      
                        k
                        
                          ap
                          ,
                          i
                        
                      
                    
                    -
                    
                      
                        h
                        
                          up
                          ,
                          ij
                        
                      
                      ·
                      
                        k
                        
                          ut
                          ,
                          j
                        
                      
                    
                  
                  
                
                2
              
            
          
        
        ,
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          36
          )
        
      
    
  




where hdn,ji is the element in the j-th row and i-th column of Hdn; hup,ij is the element in the i-th row and j-th column of Hup; kap,i is the i-th diagonal element of Kaptx, where kap,1=1; and kut,j is the j-th diagonal element of Kuttx.

The minimum mean square error may be obtained by taking the partial derivatives of equation (36) with respect to kap,i and kut,j and setting the partial derivatives to zero. The results of these operations are the following equations:




  
    
      
        
          
            
              ∑
              
                j
                =
                1
              
              
                N
                ut
              
            
            ⁢
            
              
                (
                
                  
                    
                      h
                      
                        dn
                        ,
                        ji
                      
                    
                    ·
                    
                      k
                      
                        ap
                        ,
                        i
                      
                    
                  
                  -
                  
                    
                      h
                      
                        up
                        ,
                        ij
                      
                    
                    ·
                    
                      k
                      
                        ut
                        ,
                        j
                      
                    
                  
                
                )
              
              ·
              
                h
                
                  dn
                  ,
                  ji
                
                *
              
            
          
          =
          0
        
        ,
        
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            for
            ⁢
            
              
            
            ⁢
            i
          
          =
          
            2
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              
            
            ⁢
            
              N
              ap
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          
            37
            ⁢
            a
          
          )
        
      
    
  
  
    
      
        
          
            
              ∑
              
                i
                =
                1
              
              
                N
                ap
              
            
            ⁢
            
              
                (
                
                  
                    
                      h
                      
                        dn
                        ,
                        ji
                      
                    
                    ·
                    
                      k
                      
                        ap
                        ,
                        i
                      
                    
                  
                  -
                  
                    
                      h
                      
                        up
                        ,
                        ij
                      
                    
                    ·
                    
                      k
                      
                        ut
                        ,
                        j
                      
                    
                  
                
                )
              
              ·
              
                h
                
                  up
                  ,
                  ij
                
                *
              
            
          
          =
          0
        
        ,
        
          

        
        ⁢
        
          
            for
            ⁢
            
              
            
            ⁢
            j
          
          =
          
            1
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              
            
            ⁢
            
              
                N
                ut
              
              .
            
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          
            37
            ⁢
            b
          
          )
        
      
    
  




In equation (37a), kap,1=1 so there is no partial derivative for this case, and index i runs from 2 through Nap. The Nap+Nut−1 equations in equation sets (37a) and (37b) may be solved (e.g., using matrix operations) to obtain kap,i and kut,j, which are the elements of Kaptx and Kuttx that minimize the mean square error in the calibrated downlink and uplink channel responses.
Follow-on Calibration
The correction matrices obtained from the pre-calibration and/or field calibration may contain errors due to various sources such as (1) noise in the measurements for the pre-calibration, (2) imperfect channel estimates used for the field calibration, (3) changes in the transmit/receive chains at the access point and user terminal, and so on. Errors in the correction matrices cause errors in the transmissions sent and received using these matrices. Follow-on calibration may be performed to estimate and remove the errors in the correction matrices.
The channel response matrix H may be “diagonalized” to obtain NS eigenmodes of the MIMO channel, which may be viewed as orthogonal spatial channels. This diagonalization may be achieved by performing singular value decomposition of H. Table 1 shows (1) the effective and calibrated channel responses for the downlink and uplink for a reciprocal channel and (2) the singular value decomposition of the calibrated downlink and uplink channel response matrices.







TABLE 1







Singular Value Decomposition










Downlink
Uplink





Effective Channel Response
Hdn = RutHTap
Hup = RapHTTut


Calibrated Channel Response
Hcdn = HdnKap
Hcup = HupKut


Singular Value Decomposition
Hcdn = Vut*ΣTUapT
Hcup = UapΣVutH


Unnormalized Transmit Matrices
VutΣT = HcdnUap
UapΣ = HcupVut









In Table 1, Uap is an Nap×Nap unitary matrix of left eigenvectors of Hcup, Σ is an Nap×Nut diagonal matrix of singular values of Hcup, Vut is an Nut×Nut unitary matrix of right eigenvectors of Hcup, and “*” denotes the complex conjugate. A unitary matrix M is characterized by the property MHM=I, where I is the identity matrix. Because of the reciprocal channel, the matrices V*ut, and U*ap are also matrices of left and right eigenvectors, respectively, of Hcdn. The matrices Uap and Vut (which are also called transmit matrices) may be used by the access point and user terminal, respectively, for spatial processing and are denoted as such by their subscripts. Singular value decomposition is described in further detail by Gilbert Strang in a book entitled “Linear Algebra and Its Applications,” Second Edition, Academic Press, 1980.
For a reciprocal channel, the singular value decomposition may be performed by one wireless entity to obtain both matrices Uap and Vut. For example, the user terminal may obtain the calibrated downlink channel response Hcdn, perform decomposition of HcdnT, use Vut for spatial processing, and send Uap back to the access point via a steered reference. A steered reference (or steered pilot) is a pilot transmitted from all antennas and on the eigenmodes of the MIMO channel. For clarity, the following description assumes the correction matrices being applied on the transmit paths.
The user terminal may transmit an uplink steered reference, as follows:

xup,m=Kuttxvut,mpm,  Eq (38)

where pm is a pilot symbol transmitted on eigenmode m for the steered reference; xup,m is a transmit vector for the uplink steered reference for eigenmode m; and vut,m is the m-th eigenvector or column of Vut, where Vut=[vut,1 vut,2 . . . vut,Nut].

The received uplink steered reference at the access point may be expressed as:

yup,m=Hupxup,m=HupKuttxvut,mpm=Hcupvut,mpm=UapΣVutHvut,mpm=uap,mσmpm,  Eq (39)

where yup,m is a received vector for the uplink steered reference for eigenmode m; σm is the m-th diagonal element of Σ; and uap,m is the m-th eigenvector or column of Uap, where Uap=[uap,1 uap,2 . . . uap,Nap].

Equation (39) shows that the received uplink steered reference at the access point, in the absence of noise, is equal to uap,mσmpm. The access point may thus obtain the eigenvectors uap,m, for m=1 . . . NS, based on the uplink steered reference sent by the user terminal. Since the NS eigenvectors are obtained one at a time and because of noise, these NS eigenvectors may not be orthogonal to one another. The access point may perform QR factorization (e.g., using a Gram-Schmidt procedure) on the NS eigenvectors to obtain orthogonal eigenvectors. In any case, the access point obtains the transmit matrix Uap and uses it for spatial processing for downlink transmission.
Table 2 summarizes the spatial processing performed by the user terminal and access point for data transmission and reception on the eigenmodes of the MIMO channel.









TABLE 2






Uplink
Downlink



















User Terminal
Transmit:
Receive:




xup = KuttxVutsup
ŝdn = Σ−1VutTydn



Access Point
Receive:
Transmit:




ŝup = Σ−1UapHyup
xdn = KaptxUap*sdn









In Table 2, sdn is a vector of data symbols for the downlink, ŝdn is a vector of detected symbols for the downlink, sup is a vector of data symbols for the uplink, and ŝup is a vector of detected symbols for the uplink, where ŝdn and ŝup, are estimates of sdn and sup, respectively. A “detected symbol” is an estimate of a data symbol.
The access point and user terminal may use correction matrices {circumflex over (K)}aptx and {circumflex over (K)}uttx, respectively, which have errors from the ideal correction matrices Kaptx and Kuttx. The errors in {circumflex over (K)}aptx and {circumflex over (K)}uttx may be represented by diagonal calibration error matrices Q′ap and Q′ut, respectively. The matrices {circumflex over (K)}aptx and {circumflex over (K)}uttx may then be expressed as:

{circumflex over (K)}aptx=KaptxQ′ap and {circumflex over (K)}uttx=KuttxQ′ut.  Eq (40)

If the access point transmits a MIMO pilot with {circumflex over (K)}aptx, then the calibrated downlink response Ĥcdn by the user terminal may be expressed as:

Ĥcdn=Hdn{circumflex over (K)}aptx=HdnKaptxQ′ap=HcdnQ′ap, or  Eq (41)

Hcdn=ĤcdnQ′ap−1,

where Ĥcdn contains error due to the fact that {circumflex over (K)}aptx contains error. The singular value decomposition of ĤcdnT may be expressed as: ĤcdnT=Ûap{circumflex over (Σ)}{circumflex over (V)}utH, where Ûap and {circumflex over (V)}ut are estimates of Uap and Vut, respectively, due to errors in {circumflex over (K)}aptx.

For clarity, a specific embodiment of the follow-on calibration is described below. For this embodiment, the access point transmits a MIMO pilot on the downlink using {circumflex over (K)}aptx and also transmits a steered reference on the downlink using {circumflex over (K)}aptx and Ûap. The downlink steered reference may be expressed as: xdn,m={circumflex over (K)}aptxûap,mpm, where Ûap=[ûap,1 ûap,2 . . . ûap,Nap]. The user terminal can obtain {circumflex over (V)}ut{circumflex over (Σ)}T (which is called an unnormalized transmit matrix {tilde over (V)}a) based on the downlink steered reference.
The matrices Q′ap and Q′ut contain “true” errors in {circumflex over (K)}aptx and {circumflex over (K)}uttx, respectively. A guess of Q′ap and Q′ut may be denoted as Qap and Qut, respectively. A “hypothesized” downlink channel may be defined as:

Hhyp=ĤcdnQap−1.  Eq (42)

The hypothesized downlink channel is a guess of Hcdn and is derived under the assumption that the errors in {circumflex over (K)}aptx is Qap.
If the user terminal transmits an uplink steered reference using {circumflex over (V)}ut (which is derived from Ĥcdn obtained from the downlink MIMO pilot) and {circumflex over (K)}uttx, then the transmit matrix Ûap obtained by the access point may be expressed as:

Ûap{circumflex over (Σ)}=Ĥcup{circumflex over (V)}ut=Hup{circumflex over (K)}uttx{circumflex over (V)}ut=HupKuttxQ′ut{circumflex over (V)}ut=HcupQ′ut{circumflex over (V)}ut.  Eq (43)

However, the user terminal does not have Q′ap and Q′ut, but only their guesses Qap and Qut. The user terminal thus computes an unnormalized transmit matrix Ũrx that hypothetically would have been obtained by the access point if the errors were Qap and Qut, as follows:

Ûrx=HhypTQut{circumflex over (V)}ut=(ĤcdnQap−1)TQut{circumflex over (V)}ut.  Eq (44)

Equation (44) is equal to equation (43) if Qap is a perfect guess of Q′ap (in which case HhypT=Hcup=HcdnT) and Qut is a perfect guess of Q′ut.
The user terminal then performs processing on Ũrx in the same manner that the access point would have performed on a received uplink steered reference (e.g., QR factorization) and obtains a transmit matrix Ug, which is a normalized transmit matrix that resembles Ûap. The user terminal emulates the processing performed by both the access point and user terminal for normal operation, albeit under an assumption of calibration errors represented by Qap and Qut. The matrix Ug would have been used by the access point to transmit the downlink steered reference.
If the access point transmits a downlink steered reference using Ug and {circumflex over (K)}aptx, then the transmit vector Vg obtained by the user terminal may be expressed as:

VgΣgT=ĤcdnUg=Hdn{circumflex over (K)}aptxUg=HdnKaptxQ′apUg=HcdnQ′apUg.  Eq (45)

Again, the user terminal does not have Q′ap and Q′ut, but only their guesses Qap and Qut. The user terminal thus computes a hypothesized transmit matrix {tilde over (V)}hyp as follows:

{tilde over (V)}hyp=HhypQapUg.  Eq (46)

Equation (46) is equal to equation (45) if Qap is a perfect guess of Q′ap (in which case Hhyp=Hcdn). The unnormalized transmit matrix {tilde over (V)}hyp includes a user terminal transmit matrix Vg (which corresponds to Ug) as well as a diagonal matrix Σg (which resembles Σ). The matrix {tilde over (V)}hyp is hypothesized to have been received by the user terminal with (1) the user terminal transmitting an uplink steered reference using {circumflex over (V)}ut and {circumflex over (K)}uttx, (2) the access point performing its normal processing on the received uplink steered reference to derive its transmit matrix Ug and {circumflex over (K)}aptx, (3) the access point transmitting a downlink steered reference using Ug, and (4) the correction matrices {circumflex over (K)}aptx and {circumflex over (K)}uttx having the errors indicated by the matrices Qap and Qut, respectively.
Equations (44) and (46) are correct if Qap and Qut indicate the true errors in {circumflex over (K)}aptx and {circumflex over (K)}uttx, respectively. The difference between {tilde over (V)}a obtained from the downlink steered reference and {tilde over (V)}hyp obtained from the downlink MIMO pilot may be computed as:

E={tilde over (V)}a−{tilde over (V)}hyp,  Eq (47)

where E is an Nut×Nap matrix of errors between {tilde over (V)}a and {tilde over (V)}hyp. The error matrix E gives an indication of the accuracy of the guesses for Qap and Qut. An adaptive procedure (e.g., an MMSE adaptive procedure or a steepest descent adaptive procedure) may be used to adjust Qap and Qut to drive E toward zero.

For the MMSE adaptive procedure, approximate partial derivatives of the elements of E are computed with respect to the elements of Qap and Qut. To facilitate the computation, the real and imaginary components of the diagonal elements of Qap and Qut (except for the lead elements, which are set to 1.0) may be stored in a real vector q of length 2(Nap+Nut−2). Similarly, the real and imaginary components of E may be stored in a real vector e of length 2NapNut. Approximate partial derivatives of the elements of e with respect to the elements of q may be expressed as:




  
    
      
        
          
            A
            
              j
              ,
              i
            
          
          =
          
            
              
                ∂
                
                  e
                  j
                
              
              
                ∂
                
                  q
                  i
                
              
            
            ≅
            
              
                
                  
                    e
                    j
                  
                  ⁡
                  
                    (
                    
                      
                        q
                        _
                      
                      +
                      
                        
                          Δ
                          _
                        
                        i
                      
                    
                    )
                  
                
                -
                
                  
                    e
                    j
                  
                  ⁡
                  
                    (
                    
                      q
                      _
                    
                    )
                  
                
              
              δ
            
          
        
        ,
        
          
        
        ⁢
        
          

        
        ⁢
        
          
            for
            ⁢
            
              
            
            ⁢
            i
          
          =
          
            
              1
              ⁢
              
                
              
              ⁢
              ...
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                (
                
                  
                    N
                    ap
                  
                  +
                  
                    N
                    ut
                  
                  -
                  2
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              j
            
            =
            
              1
              ⁢
              
                
              
              ⁢
              ...
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                N
                ap
              
              ⁢
              
                N
                ut
              
            
          
        
      
    
    
      
        Eq
        ⁢
        
          
        
        ⁢
        
          (
          48
          )
        
      
    
  




where Δj is a vector of length 2(Nap+Nut−2) and containing a small real value of δ for the j-th element and zeros elsewhere; and

Aj,i is the approximate partial derivative of the j-th element of e with respect to the i-th element of q.
The approximate partial derivative Aj,i may be obtained as follows. A vector qi is first computed as qi=q+Δi. The function defined by equations (42), (44), and (46) is then evaluated for qi (which contains Qap,i and Qut,i) to obtain a new hypothesized transmit matrix {tilde over (V)}hyp,i. {tilde over (V)}hyp,i is then subtracted from {tilde over (V)}a to obtain a new error matrix Ei={tilde over (V)}a−{tilde over (V)}hyp,i, which is used to form a new error vector ei. The j-th element of e, which is denoted as ej(q) in equation (48), is then subtracted from the j-th element of ei, which is denoted as ej(q+Δi) in equation (48). The result of the subtraction is divided by δ to obtain Aj,i.
If the relationships in equations (42), (44), (46), and (47) are approximately linear, then an estimate of the difference between the guess of the calibration errors in q and the actual calibration errors may be expressed as:

b=A−1e,  Eq (49)

where A is a matrix of approximate partial derivatives Aj,i obtained from equation (48) and b is an update vector. The calibration error vector may then be updated as follows:

q(n+1)=b(n)+q(n),  Eq (50)

where q(n) and q(n+1) are the calibration error vectors for the n-th and (n+1)-th iterations, respectively, and y(n) is the update vector for the n-th iteration.

The computation described above may be repeated for a number of iterations. Each iteration uses the updated calibration error vector q(n+1) obtained from the prior iteration. The procedure can terminate when the update vector b(n) is sufficiently small, e.g., if ∥b(n)∥2<yth1, where ∥b(n)∥2 is the sum of the squares of the magnitude of the elements of b(n) and yth1 is a threshold value. After all iterations have been completed, the matrices for the final estimates of the calibration errors are denoted as Qap,final and Qut,final. The correction matrix for the user terminal may be updated to account for the calibration errors, as follows:

{circumflex over (K)}uttx,new={circumflex over (K)}uttxQut,final−1.  Eq (51)

The user terminal thereafter uses {circumflex over (K)}uttx,new for spatial processing for uplink transmission, as shown in FIG. 6. The user terminal may send Qap,final to the access point, which may then update its correction matrix as {circumflex over (K)}aptx,new={circumflex over (K)}aptxQap,final−1.
System
FIG. 8 shows a block diagram of an access point 810 and a user terminal 850 in a MIMO system 800. On the downlink, at access point 810, a TX data processor 814 receives traffic data from a data source 812 and signaling and other data from a controller 830. TX data processor 814 formats, encodes, interleaves, and modulates (or symbol maps) the different types of data and provides data symbols. A TX spatial processor 820 performs spatial processing on the data symbols from TX data processor 814, multiplexes in pilot symbols as appropriate (e.g., for channel estimation, calibration, and so on), performs scaling with a correction matrix (if applicable), and provides Nap streams of transmit symbols to Nap transmitter units 824a through 824ap. Each transmitter unit 824 conditions a respective transmit symbol stream to generate a corresponding downlink signal. Nap downlink signals from transmitter units 824a through 824ap are then transmitted from Nap antennas 826a through 826ap, respectively.
At user terminal 850, Nut antennas 852a through 852ut receive the downlink signals, and each antenna provides a received signal to a respective receiver unit 854. Each receiver unit 854 performs processing complementary to that performed at transmitter units 824 and provides received symbols. An RX spatial processor 860 may perform scaling with a correction matrix (if applicable) and further performs receiver spatial processing on the received symbols from all Nut receiver units 854 to obtain detected symbols, which are estimates of the data symbols sent by the access point. An RX data processor 870 demodulates (or symbol demaps), deinterleaves, and decodes the detected symbols and provides decoded data to a data sink 872 for storage and/or a controller 880 for further processing.
The processing for the uplink may be the same or different from the processing for the downlink. Data and signaling are encoded, interleaved, and modulated by a TX data processor 888, and further spatially processed, multiplexed with pilot symbols, and scaled with a correction matrix (if applicable) by TX spatial processor 890 to obtain transmit symbols. The transmit symbols are further processed by transmitter units 864a through 864ut to obtain Nut uplink signals, which are then transmitted via Nut antennas 852a through 852ut to the access point. At access point 810, the uplink signals are received by antennas 826, conditioned by receiver units 834, and processed by an RX spatial processor 840 and an RX data processor 842 in a manner complementary to that performed at the user terminal.
Controllers 830 and 880 control the operation of various processing units at the access point and user terminal, respectively. Controllers 830 and/or 880 may also perform processing for pre-calibration, field calibration, and/or follow-on calibration. Memory units 832 and 882 store data and program codes used by controllers 830 and 880, respectively. Channel estimators 828 and 878 estimate the channel response based on pilots received on the uplink and downlink, respectively.
For pre-calibration at user terminal 850, test signals may be sent by TX spatial processor 890 and measured by RX spatial processor 860 to determine overall gains for different combinations of transmitter and receiver units at the user terminal, as described above. Controller 880 may (1) obtain matrices {tilde over (T)}ut and {tilde over (R)}ut of gains for the transmitter and receiver units and (2) derive one or more correction matrices for the user terminal based on {tilde over (T)}ut and {tilde over (R)}ut. For field calibration, controller 880 may obtain the effective downlink and uplink channel responses Hup and Hdn and may derive correction matrices for both the user terminal and access point based on Hup and Hdn, as described above. For follow-on calibration, controller 880 may obtain the downlink steered reference and downlink MIMO pilot, determine the calibration error matrices Qap,final and Qut,final, update the correction matrix for the user terminal with Qut,final, and send Qap,final back to the access point. At access point 810, controller 830 may perform processing for pre-calibration, field calibration, and/or follow-on calibration.
The calibration techniques described herein may be implemented by various means. For example, these techniques may be implemented in hardware, software, or a combination thereof For a hardware implementation, the processing units used to perform pre-calibration, field calibration, and/or follow-on calibration may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof.
For a software implementation, the calibration techniques may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. The software codes may be stored in a memory unit (e.g., memory unit 832 or 882 in FIG. 8) and executed by a processor (e.g., controller 830 or 880). The memory unit may be implemented within the processor or external to the processor, in which case it can be communicatively coupled to the processor via various means as is known in the art.
Headings are included herein for reference and to aid in locating certain sections. These headings are not intended to limit the scope of the concepts described therein under, and these concepts may have applicability in other sections throughout the entire specification.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

## Claims

1. A computer-program apparatus for calibrating transmitter units and receiver units at a wireless entity in a multiple-input multiple-output (MIMO) communication system comprising a memory unit having instructions stored thereon, the instructions being executable by one or more processors and the instructions comprising:
instructions for obtaining a plurality of first overall gains for a first receiver unit and a plurality of transmitter units, one first overall gain for each transmitter unit, each first overall gain indicative of a combined response for the first receiver unit and the associated transmitter unit, each first overall gain proportional to a ratio of the signal level of a baseband signal sent by the associated transmitter unit to the first receiver unit, to the signal level of the baseband signal received by the first receiver unit from the associated transmitter unit, wherein the first receiver unit is one of a plurality of receiver units;
instructions for obtaining a plurality of second overall gains for a first transmitter unit and the plurality of receiver units, one second overall gain for each receiver unit, each second overall gain indicative of a combined response for the first transmitter unit and the associated receiver unit, each second overall gain proportional to a ratio of the signal level of a baseband signal received by the associated receiver unit from the first transmitter unit, to the signal level of the baseband signal sent by the first transmitter unit to the associated receiver unit, wherein the first transmitter unit is one of the plurality of transmitter units;
instructions for determining a gain of each of the plurality of transmitter units based on the plurality of first overall gains; and
instructions for determining a gain of each of the plurality of receiver units based on the plurality of second overall gains.

2. The computer-program apparatus of claim 1, wherein the instructions for obtaining the plurality of first overall gains for the first receiver unit and the plurality of transmitter units includes, for each one of the plurality of transmitter units:
instructions for the transmitter unit to send a test signal to the first receiver unit,
instructions for the first receiver unit to receive the test signal from the transmitter unit, and
instructions for determining the first overall gain for the first receiver unit and the transmitter unit based on a ratio of the received test signal to the sent test signal.

3. The computer-program apparatus of claim 1, wherein the gain of each transmitter unit is normalized by the gain of the first transmitter unit, and wherein the gain of each receiver unit is normalized by the gain of the first receiver unit.

4. The computer-program apparatus of claim 1, further comprising:
instructions for deriving at least one correction matrix based on the gains of the plurality of transmitter units and the gains of the plurality of receiver units, wherein the at least one correction matrix is used to account for responses of the plurality of transmitter units and responses of the plurality of receiver units.

5. The computer-program apparatus of claim 1, further comprising:
instructions for deriving a first correction matrix based on the gains of the plurality of transmitter units, wherein the first correction matrix is used to account for responses of the plurality of transmitter units; and
instructions for deriving a second correction matrix based on the gains of the plurality of receiver units, wherein the second correction matrix is used to account for responses of the plurality of receiver units.

6. The computer-program apparatus of claim 5, wherein the first correction matrix is an inverse of a first diagonal matrix with the gains of the plurality of transmitter units, and wherein the second correction matrix is an inverse of a second diagonal matrix with the gains of the plurality of receiver units.

7. The computer-program apparatus of claim 1, further comprising:
instructions for deriving a correction matrix based on the gains of the plurality of transmitter units and the gains of the plurality of receiver units, wherein the correction matrix is applied on a transmit path and is used to account for responses of the plurality of transmitter units and responses of the plurality of receiver units.

8. The computer-program apparatus of claim 7, wherein each element of the correction matrix is a ratio of a corresponding element of a first diagonal matrix and a corresponding element of a second diagonal matrix, and wherein the first diagonal matrix has the gains of the plurality of receiver units as its elements, and the second diagonal matrix has the gains of the plurality of transmitter units as its elements.

9. The computer-program apparatus of claim 1, further comprising:
instructions for deriving a correction matrix based on the gains of the plurality of transmitter units and the gains of the plurality of receiver units, wherein the correction matrix is applied on a receive path and is used to account for responses of the plurality of transmitter units and responses of the plurality of receiver units.

10. The computer-program apparatus of claim 9, wherein each element of the correction matrix is a ratio of a corresponding element of a first diagonal matrix and a corresponding element of a second diagonal matrix, and wherein the first diagonal matrix has the gains of the plurality of transmitter units as its elements, and the second diagonal matrix has the gains of the plurality of receiver units as its elements.

11. The computer-program apparatus of claim 1, wherein the MIMO communication system utilizes orthogonal frequency division multiplexing (OFDM), and wherein the instructions for obtaining a plurality of first overall gains, instructions for obtaining a plurality of second overall gains, instructions for determining a gain of each of the plurality of transmitter units, and instructions for determining a gain of each of the plurality of receiver units are performed for a plurality of sub bands.

12. The computer-program apparatus of claim 1, wherein the gains of the plurality of transmitter units and the gains of the plurality of receiver units are determined for a plurality of operating points.

13. The computer-program apparatus of claim 12, wherein each operating point corresponds to a different gain setting or a different temperature.

14. A computer-program apparatus for calibrating transmitter units and receiver units at a first wireless entity in a multiple-input multiple-output (MIMO) communication system comprising a memory unit having instructions stored thereon, the instructions being executable by one or more processors and the instructions comprising:
instructions for performing a first calibration to obtain a gain of each of a plurality of transmitter units at the wireless entity and to obtain a gain of each of a plurality of receiver units at the wireless entity, wherein the first calibration is performed based on a plurality of test signals sent via the plurality of transmitter units and received via the plurality of receiver units,
wherein the gain of each transmitter unit is proportional to a ratio of the signal level of one of the test signals sent by that transmitter unit and received by an associated receiver unit, and
wherein the gain of each receiver unit is proportional to a ratio of the signal level of one of the test signals received by that receiver unit and sent by an associated transmitter unit; and
instructions for deriving at least one correction matrix based on the gains of the plurality of transmitter units and the gains of the plurality of receiver units, wherein the at least one correction matrix is used to account for responses of the plurality of transmitter units and responses of the plurality of receiver units.

15. The computer-program apparatus of claim 14, further comprising:
instructions for performing a second calibration to determine at least one updated correction matrix for the wireless entity, wherein the second calibration is performed based on a channel response estimate for a downlink and a channel response estimate for an uplink in the MIMO system.

16. The computer-program apparatus of claim 15, further comprising:
instructions for performing a third calibration to determine errors in the at least one correction matrix, wherein the third calibration is performed using two different pilots transmitted by the first wireless entity to a second wireless entity, the second wireless entity configured to estimate the errors in the correction matrix based on the two different pilots; and
instructions for updating the at least one correction matrix based on the determined errors in the at least one correction matrix.