Patent Publication Number: US-8126086-B2

Title: System and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation

Description:
BACKGROUND 
     Many communications systems employ communication protocols in which data to be communicated is converted to a system that includes in-phase (I) and quadrature-phase (Q) components. The in-phase and the quadrature-phase components are shifted in phase by an angular amount, such as, for example, 90 degrees. The data signal is impressed on the in-phase and the quadrature-phase signal components by a transmitter and recovered by a receiver. One example of a communication system that employs in-phase and quadrature-phase components is what is referred to as a multiple input multiple output (MIMO) communication system that uses orthogonal frequency division multiplexing (OFDM) in a multiple-antenna arrangement and that complies with communication standard IEEE 802.11n. 
     In any communication system that uses in-phase and quadrature-phase signal components, the quality of the signal transmission is dependent upon the relationship between the in-phase and the quadrature-phase components. If the in-phase and the quadrature-phase components are not properly matched, also referred to as “I/Q mismatch,” degradation of the communication signal occurs. There are typically three types of I/Q mismatch. These are gain imbalance, quadrature error and I/Q offset. Gain imbalance is caused by a difference in gain between the in-phase component and the quadrature-phase component. Quadrature error is caused by an error in the phase rotation between the between the in-phase component and the quadrature-phase component. I/Q offset is caused by the leakage of a direct current (DC) signal at the receiver input. 
     I/Q mismatch occurs because of non-ideal modulator and demodulator elements in which the upconverters and attenuators of the I and Q channels are mismatched. I/Q mismatch results in serious performance degradation because it causes inter-carrier interference. A MIMO system is more likely to be influenced by I/Q mismatch than a single input single output (SISO) communication system because there are multiple modulators and demodulators, each of which can introduce I/Q mismatch. Further, the I/Q mismatch may result in inter-channel interference. 
     The measurement of I/Q mismatch for OFDM and MIMO OFDM systems is challenging in that the I/Q mismatch will result in large channel estimation error if the I/Q mismatch impairment is not correctly compensated. Further, the accurate measurement of I/Q mismatch requires decoupling the influence of I/Q mismatch on the channel estimation. This is especially true for a MIMO communication system under a fading channel situation. 
     Prior attempts of I/Q mismatch compensation fail to address each of the I/Q mismatch parameters, and further, fail to address I/Q mismatch compensation under a MIMO fading channel situation. Prior attempts at I/Q mismatch measurement and compensation only considered the measurement of the ratio of gain imbalance to quadrature error. Other prior attempts at I/Q mismatch measurement and compensation provided measurement methods for gain imbalance and quadrature error for a SISO communication system without considering the influence of channel fading. Furthermore, these prior attempts all require a specially designed test signal. 
     Therefore, it would be desirable to have a way to measure and compensate I/Q mismatch, including measuring gain imbalance, quadrature error and I/Q offset under a MIMO fading channel situation without using specially designed signals. 
     SUMMARY 
     An embodiment of a system for determining in-phase and quadrature-phase mismatch in a multiple-input, multiple-output (MIMO) communication architecture includes at least one transmitter coupled to at least one receiver and an in-phase (I) signal, quadrature-phase (Q) signal mismatch element configured to receive I and Q signal components over at least one communication channel, the I/Q signal mismatch element also configured to provide a signal representing gain imbalance, a signal representing quadrature error and a signal representing I/Q offset. 
     Other embodiments and methods of the invention will be discussed with reference to the figures and to the detailed description. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       The invention will be described by way of example, in the description of exemplary embodiments, with particular reference to the accompanying figures. 
         FIG. 1  is a schematic diagram illustrating a basic communication system. 
         FIG. 2  is a schematic diagram illustrating the basic components of the transmitter of  FIG. 1 . 
         FIG. 3  is a schematic diagram illustrating the basic components of the receiver of  FIG. 1 . 
         FIG. 4  is a schematic diagram illustrating an exemplary modulator of  FIG. 2 . 
         FIG. 5  is a schematic diagram illustrating an exemplary demodulator of  FIG. 3 . 
         FIG. 6  is a schematic diagram illustrating an embodiment of the receive baseband module of  FIG. 3 . 
         FIG. 7  is a schematic diagram illustrating an example of the measurement of gain imbalance and quadrature error. 
         FIG. 8  is a schematic diagram illustrating an example of a first embodiment of the measurement of I/Q offset. 
         FIG. 9  is a schematic diagram illustrating an example of an alternative embodiment of the measurement of I/Q offset. 
         FIG. 10  is a flowchart showing the operation of an embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation. 
         FIG. 11  is a schematic diagram illustrating the measurement of gain imbalance and quadrature error with general channel fading, where the iteration follows the flow in  FIG. 10 . 
         FIG. 12  is a schematic diagram illustrating an example of one of the iteration blocks of  FIG. 1 . 
         FIG. 13  is a block of an alternative embodiment of the Rx baseband module of  FIG. 6 . 
         FIG. 14  is a block diagram illustrating the equalizer of  FIG. 13 . 
         FIG. 15  is a graphical illustration showing the results of the operation of the first embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation. 
         FIG. 16  is a graphical illustration showing the results of the operation of the second embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation. 
         FIG. 17  is a graphical illustration showing the results of the operation of the third embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation. 
     
    
    
     DETAILED DESCRIPTION 
     The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation can be implemented on a measurement instrument or can be implemented in a receiver of a communication device for compensating for I/Q mismatch, including the joint measurement of gain imbalance, quadrature error and I/Q offset. 
     As will be described below, the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation decouples the influence of the fading channel on the measurement of the I/Q mismatch parameters. The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation can be implemented on a measurement instrument for measuring MIMO I/Q mismatch or can be implemented in a MIMO-OFDM receiver to measure and compensate for the influence of I/Q mismatch. 
     Further, the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation can use either a standard signal or a specially designed signal to perform the measurement. 
     The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation can be used in the scenario of a test and measurement case in which the receiver of the user device is directly connected to the test instrument through a cable; and in the scenario of a MIMO OFDM communication environment having multiple transmitters and receivers. 
     For the test and measurement case in which the receiver of the user device is directly connected to the test instrument through a cable, a fast efficient I/Q mismatch measurement method is derived with the assumption of no coupling between different channels. With the assumption that the channel is quasi-stationary (stable for two successive OFDM symbols), the influence of the channel fading can be eliminated by determining a ratio of two consecutive symbols. After measuring the I/Q mismatch parameters, further accurate measurement results of channel response can be obtained without the influence of I/Q mismatch. 
     For a MIMO OFDM communication environment including multiple transmitters and receivers and experiencing channel fading, an iterative method derived from the SAGE algorithm performs the joint estimation of channel response and I/Q mismatch parameters. The SAGE algorithm (Jeffrey A. Fessler, and Alfred O. Hero, “Space-Alternating Generalized Expectation-Maximization Algorithm”, Signal Processing IEEE Trans. on, pp. 2664-2677, Vol. 42, No. 10, October 1994.) is well known in the field of array processing. The SAGE algorithm is a general method for joint estimation of multiple parameters. The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation is derived from the basic idea of the SAGE algorithm and is applied to the joint estimation of I/Q mismatch and channel response for MIMO systems under fading channel conditions. The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation obtains accurate measurement of I/Q mismatch and channel parameters, while taking into account coupling between multiple transmit-receive channels. 
     Further, the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation works with both standard signals and specially designed test signals. 
       FIG. 1  is a schematic diagram illustrating a basic communication system. The communication system includes a transmitter  110 , a channel  120  and a receiver  130 . The transmitter  110  can be any transmitter and can be located in, for example, a portable cellular communication device, a personal computer (PC), a personal digital assistant (PDA), a portable game player, a wireless local area network (LAN) device, or any other communication device. In an embodiment, the transmitter has a plurality of transmit channels, each channel having a corresponding antenna, which are referred to as Tx- 1  through Tx-P. 
     The receiver  130  can be any receiver adapted to receive the transmissions from the transmitter  110  and can be located in, for example, a portable cellular communication device, a personal computer (PC), a personal digital assistant (PDA), a portable game player, a wireless local area network (LAN) device, or any other communication device. In an embodiment, the receiver  130  has a plurality of receive channels, each channel having a corresponding antenna, which are referred to as Rx- 1  through Rx-L, where the number of transmitters and the number of receivers are greater than 1. Further, the transmitter  110  and the receiver  130  can be incorporated into a transceiver and the transceiver can communicate with another transceiver according to the principles described below. In some nomenclature, and in some of the equations below, there are said to be “P” transmitter antennas and “L” receiver antennas, associated with the transmitter  110  and receiver  130 . 
     The channel  120  represents the transmission environment between the transmitter and the receiver, and can be any wired or wireless communication channel. Due to the various application scenarios, there are many channel types. Most channel types will have a direct influence on I/Q mismatch. Each of a number of individual transmitters within the transmitter  110  can be wirelessly coupled to a corresponding one of a number of receivers in the receiver  130 . In such a MIMO configuration, each transmitter-receiver pair communicates over a separate wireless communication channel  120 - 1  through  120 -P. 
       FIG. 2  is a schematic diagram illustrating the basic components of the transmitter  110  of  FIG. 1 . The transmitter  110  includes a transmit baseband module  210 , a plurality of modulators  220 - 1  through  220 -P, and a plurality of transmit radio frequency (RF) modules  230 - 1  through  230 -P, corresponding to the number of channels. The transmit baseband module  210  performs baseband signal processing and provides in-phase and quadrature-phase components of a transmit signal to each modulator  220 . In an embodiment, the transmit signal provided by the transmit baseband module  210  includes an in-phase component and a quadrature-phase component. Each modulator  220  modulates the baseband signal and optionally upconverts the transmit signal to an intermediate frequency (IF) or to an RF signal level. The modulator provides the modulated signal to a corresponding transmit RF module  230 . Each transmit RF module  230  generally includes one or more amplification stages to amplify the modulated signal for transmission over the channel  120  ( FIG. 1 ). 
       FIG. 3  is a schematic diagram illustrating the basic components of the receiver  130  of  FIG. 1 . The receiver  130  includes a plurality of receive RF modules  310 - 1  through  310 -L, and a plurality of demodulators  320 - 1  through  320 -L, corresponding to the corresponding to the number of channels. The receiver  130  also includes a receive baseband module  330 . Each receive RF module  310  receives and filters the transmit signal of a respective channel received over the channel  120  ( FIG. 1 ). Each receive RF module  310  provides the received signal to a corresponding demodulator  320 . Each demodulator  320  demodulates the received signal to recover the information signal. The information signal includes an in-phase component, I rx (t) and a quadrature-phase component Q rx (t) for each channel. The in-phase component, I rx (t) and the quadrature-phase component, Q rx (t), are provided to the receive baseband module  330  where the information contained in the in-phase component and the quadrature-phase component are recovered to baseband level. 
       FIG. 4  is a schematic diagram illustrating an exemplary modulator  220  of  FIG. 2 . The modulator  220  includes a mixer  412  and a mixer  414 . The mixer  412  receives the in-phase signal on connection  402  and the mixer  414  receives the quadrature-phase signal on connection  404 . The mixer  412  also receives a local oscillator (LO) signal from a local oscillator  406 . The local oscillator  406  can be any frequency generations device, such as a crystal oscillator, a frequency synthesizer, etc., that generates the desired reference frequency. The signal provided by the local oscillator  406  is a reference signal that is used by the mixer  412  to upconvert the in-phase signal on connection  402  to a transmit frequency. The mixer  412  can perform direct upconversion, in which the transmit signal is upconverted directly to a radio frequency (RF) transmit signal, or may perform upconversion in which the transmit signal is first converted to an intermediate frequency (IF) prior to upconversion to the RF transmit frequency. 
     The local oscillator also supplies the LO signal to a phase shifter (also referred to as a phase rotator)  408 . The phase shifter  408  alters the phase of the LO signal received from the local oscillator  406  by a predetermined amount. In an embodiment, the phase shifter  408  alters the LO signal by 90 degrees to correspond to the 90 degree phase difference between the in-phase and quadrature signals. In an alternative embodiment, the phase shifter  408  may alter the phase of the LO signal by an amount other than 90 degrees, such as, for example, 45 degrees. The amount of phase offset generated by the phase shifter is determined by specific application. 
     The phase-shifted LO signal is supplied to the mixer  414 . The mixer  414  uses the phase-shifted LO signal as a reference signal to upconvert the quadrature-phase signal on connection  402  to a transmit frequency. The mixer  414  can perform direct upconversion, in which the transmit signal is upconverted directly to a radio frequency (RF) transmit signal, or may perform upconversion in which the transmit signal is first converted to an intermediate frequency (IF) prior to upconversion to the RF transmit frequency. 
     The output of the mixer  412  is provided to a filter  416  and the output of the mixer  414  is provided to a filter  418 . The filter  416  filters the in-phase signal to remove unwanted components and supplies the filtered signal to an attenuator  422 . The attenuator  422  modifies the level of the in-phase signal and supplies the in-phase signal to an adder  426 . 
     The filter  418  filters the quadrature-phase signal to remove unwanted components and supplies the filtered signal to an attenuator  424 . The attenuator  424  modifies the level of the quadrature-phase signal and supplies the quadrature—phase signal to the adder  426 . 
     The adder  426  combines the in-phase signal and the quadrature-phase signal and generates a transmit signal, TxP. 
       FIG. 5  is a schematic diagram illustrating an exemplary demodulator  320  of  FIG. 3 . The demodulator  320  receives a receive signal, Rx- 1  on connection  502 . The receive signal is provided to a mixer  512  and a mixer  514 . The mixer  512  also receives a local oscillator (LO) signal from a local oscillator  506 . The local oscillator  506  can be any frequency generation device, such as a crystal oscillator, a frequency synthesizer, etc., that generates the desired reference frequency and can be similar to the local oscillator  406  ( FIG. 4 ). The signal provided by the local oscillator  506  is a reference signal that is used by the mixer  512  to downconvert the receive signal on connection  502  to obtain the in-phase signal. The mixer  512  can perform direct downconversion, in which the received signal is downconverted directly to baseband, or near baseband, or may perform downconversion in which the receive signal is first converted to an intermediate frequency (IF) prior to downconversion to baseband. 
     The local oscillator also supplies the LO signal to a phase shifter (also referred to as a phase rotator)  508 . The phase shifter  508  alters the phase of the LO signal received from the local oscillator  506  by a predetermined amount. In an embodiment, the phase shifter  508  alters the LO signal by 90 degrees to correspond to the 90 degree phase difference between the in-phase and quadrature signals. In an alternative embodiment, the phase shifter  508  may alter the phase of the LO signal by an amount other than 90 degrees, such as, for example, 45 degrees. The amount of phase offset generated by the phase shifter is determined by specific application. 
     The phase-shifted LO signal is supplied to the mixer  514 . The mixer  514  uses the phase-shifted LO signal as a reference signal to downconvert the receive signal on connection  502  to obtain the quadrature-phase signal. The mixer  514  can perform direct downconversion, in which the received signal is downconverted directly to baseband, or near baseband, or may perform downconversion in which the receive signal is first converted to an intermediate frequency (IF) prior to downconversion to baseband. 
     The output of the mixer  512  is provided to a filter  516  and the output of the mixer  514  is provided to a filter  518 . The filter  516  filters the in-phase signal to remove unwanted components and supplies the filtered signal to an attenuator  522 . The attenuator  522  modifies the level of the in-phase signal and supplies an in-phase information signal, I. 
     The filter  518  filters the quadrature-phase signal to remove unwanted components and supplies the filtered signal to an attenuator  524 . The attenuator  524  modifies the level of the quadrature-phase signal and supplies the quadrature—phase information signal, Q. 
     The system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation addresses the three main types of I/Q mismatch. These are gain imbalance, quadrature error and I/Q offset. Gain imbalance is caused by different gain of the attenuators,  422  and  424 , in the modulator  220  and the attenuators,  522  and  524 , in the demodulator  320 . Quadrature error is cause by phase rotation error with respect to 90 degrees and can be imparted by the phase shifter  408  in the modulator  220  and by the phase shifter  508  in the demodulator  320 . I/Q offset has several sources such as, for example, DC (Direct Current) leakage of the filters  416  and  418  in the modulator  220  and the filters  516  and  518  in the demodulator  320 . 
       FIG. 6  is a schematic diagram illustrating an embodiment of the receive baseband module  330  of  FIG. 3 . 
     For each channel, the in-phase (I) and quadrature-phase (Q) signals received from the demodulator  320  ( FIG. 3 ) are combined into complex form (I+jQ) and provided to a synchronizer  606 . In the example shown in  FIG. 6 , the signal on connection  602 - 1  is provided to a synchronizer  606 - 1  and the signal on connection  602 -L is provided to a synchronizer  606 -L. The number of synchronizers corresponds to the number of channels (i.e., the number of transmitters P ( FIG. 1 ) and receivers L ( FIG. 1 )). 
     After synchronization, each receive signal is provided to a corresponding cyclic prefix removal element  608 - 1  through  608 -L. The cyclic prefix removal elements  608 - 1  through  608 -L remove the cyclic prefix from the OFDM symbol so that the original signal can be recovered in the time domain. 
     After the cyclic prefix is removed from each receive signal, the receive signals are supplied to corresponding fast Fourier transform (FFT) elements  612 . 
     The FFT elements  612 - 1  through  612 -L convert the receive signals to the frequency domain. The output of the FFT element  612 - 1  is supplied over connection  614 - 1  and is referred to as Y 1,j . The output of the FFT element  612 -L is supplied over connection  614 -L and is referred to as Y L,j . As used in  FIG. 6 , the first term in the subscript (Y 1,j ) refers to the first channel in  FIG. 1  and the first term in the subscript (Y L,j ) refers to the last channel, L, in  FIG. 1 . 
     The output of the FFT elements  612 - 1  through  612 -L is used for I/Q mismatch measurement and channel estimation. 
     The signals on connections  614 - 1  and  614 -L are supplied to an equalizer  632 , to a channel estimation element  620  and to an I/Q mismatch element  630 . 
     The channel estimation element  620  measures the response of the channel and provides an indicator, H n,i , that is a signal having a term that defines the channel response. The measurement results can be sent to both the channel estimation element  620  and equalizer  632  for compensation of I/Q mismatch. The I/Q mismatch element  630  receives the signals Y 1,j  and Y L,j , on connections  614 - 1  and  614 -L, and provides the terms {circumflex over (β)} {circumflex over (φ)} and ĉ that are signals representing estimates for I/Q imbalance, quadrature error and I/Q offset, respectively, over connection  622  to the channel estimation element  620  and to the equalizer  632  over connection  628 . The terms β, φ, c indicate real value while the terms {circumflex over (β)} {circumflex over (φ)} and ĉ indicate estimation results which may contain error. 
     The output of the channel estimation element  620  on connection  626 , H n,i , and the output of the I/Q mismatch element  630  on connection  628 , {circumflex over (β)} ĉ {circumflex over (φ)}, is supplied to the equalizer  632 . The equalizer  632  recovers the transmitted symbols “X” from received symbols “Y.” For example, after the estimation results of β, φ, c and H are obtained, the equalizer  632  obtains the symbols “X” by solving equation Eq. (1). 
     The compensation can be performed using either the real time measurement results or using results stored some time before. The compensation can be performed in different ways. For example, after obtaining the estimation results {circumflex over (β)} ĉ {circumflex over (φ)} and substituting them into Eq. 1, X becomes the only unknown variable in Eq. 1 and then X could be solved using Eq. 1. The process of solving for X from Eq. 1 is one type of compensation for I/Q mismatch. Using real time measurement results means that the estimation results of I/Q mismatch are updated once for every OFDM symbol. Using stored results means estimation results of I/Q mismatch are updated once for every several OFDM symbols. 
     Embodiment 1 
     Transmitter and Receiver Directly Connected 
     If the transmitter and receiver are directly connected using, for example, a cable or other direct connection, such that the output port of each transmitter is connected to an input port of the receiver, the output of the FFT  612  at the n th  Rx stream can be expressed as 
     
       
         
           
             
               
                 
                   
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     where X k,n,i  is the transmitted symbol carried by the k th  subcarrier of the i th  OFDM symbol at the n th  transmitter antenna. The terms, β, φ, c represent I/Q imbalance, quadrature error and I/Q offset, respectively. The term H k,n,i  is the frequency response at the k th  subcarrier of the i th  OFDM symbol transmitted from the n th  transmitter antenna to the n th  input port of test instrument. The term W k,n,i  is the noise. N is the size of FFT. Given that φ is small enough, Eq. (1) can be rewritten by approximation as 
     
       
         
           
             
               
                 
                   
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                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 W 
                 
                   n 
                   , 
                   i 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       W 
                       
                         1 
                         , 
                         n 
                         , 
                         i 
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       W 
                       
                         M 
                         , 
                         n 
                         , 
                         i 
                       
                     
                   
                   ] 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               J 
               = 
               
                 
                   [ 
                   
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         0 
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         ⋰ 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         0 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                   
                   ] 
                 
                 
                   M 
                   × 
                   M 
                 
               
             
           
         
       
     
     M is the number of the subcarrier used. J is an M×M exchange matrix with nonzero elements only on its orthogonal diagonal. 
     The term H n,i  varies slowly among consecutive OFDM symbols, i.e. H n,i ≅H n,i+1 . As a result, the unknown channel response can be eliminated by the following operation: 
     
       
         
           
             
               
                 
                   
                     Z 
                     
                       n 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           Y 
                           
                             n 
                             , 
                             i 
                           
                         
                         . 
                       
                       / 
                       
                         Y 
                         
                           n 
                           , 
                           
                             i 
                             - 
                             1 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 H 
                                 
                                   n 
                                   , 
                                   i 
                                 
                               
                               . 
                             
                             / 
                             
                               H 
                               
                                 n 
                                 , 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         . 
                       
                       * 
                       
                           
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     X 
                                     
                                       n 
                                       , 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       2 
                                       + 
                                       β 
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         φ 
                                       
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     JX 
                                     
                                       n 
                                       , 
                                       i 
                                     
                                     * 
                                   
                                   ⁢ 
                                   
                                     
                                       β 
                                       - 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         φ 
                                       
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       W 
                                       
                                         n 
                                         , 
                                         i 
                                       
                                     
                                     . 
                                   
                                   / 
                                   
                                     
                                       H 
                                       
                                         n 
                                         , 
                                         i 
                                       
                                     
                                     . 
                                   
                                 
                               
                               ] 
                             
                             . 
                           
                           / 
                           
                               
                             
                               
                                 [ 
                                 
                                   
                                     
                                       X 
                                       
                                         n 
                                         , 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         2 
                                         + 
                                         β 
                                         + 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           φ 
                                         
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       JX 
                                       
                                         n 
                                         , 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       * 
                                     
                                     ⁢ 
                                     
                                       
                                         β 
                                         - 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           φ 
                                         
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         W 
                                         
                                           n 
                                           , 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                       
                                       . 
                                     
                                     / 
                                     
                                       H 
                                       
                                         n 
                                         , 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ≈ 
                               
                                   
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             X 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               2 
                                               + 
                                               β 
                                               + 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 φ 
                                               
                                             
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             JX 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                             * 
                                           
                                           ⁢ 
                                           
                                             
                                               β 
                                               - 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 φ 
                                               
                                             
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               W 
                                               
                                                 n 
                                                 , 
                                                 i 
                                               
                                             
                                             . 
                                           
                                           / 
                                           
                                             H 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     . 
                                   
                                   / 
                                   
                                       
                                     
                                       [ 
                                       
                                         
                                           
                                             X 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               2 
                                               + 
                                               β 
                                               + 
                                               jφ 
                                             
                                             2 
                                           
                                         
                                         + 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             JX 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                             * 
                                           
                                           ⁢ 
                                           
                                             
                                               β 
                                               - 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 φ 
                                               
                                             
                                             2 
                                           
                                         
                                         + 
                                         
                                             
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               
                                                 W 
                                                 
                                                   n 
                                                   , 
                                                   
                                                     i 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               . 
                                             
                                             / 
                                             
                                               H 
                                               
                                                 n 
                                                 , 
                                                 
                                                   i 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     where ‘./’ indicates the operation of elementary quotient. Eq. (4) can be rewritten in the standard form with respect to β and φ as 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         
                           n 
                           , 
                           i 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               β 
                             
                           
                           
                             
                               φ 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         b 
                         
                           n 
                           , 
                           i 
                         
                       
                       + 
                       
                         U 
                         
                           n 
                           , 
                           i 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                      
                   
                   ⁢ 
                   
                     
                       A 
                       
                         n 
                         , 
                         i 
                       
                     
                     = 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 
                                   Re 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           X 
                                           
                                             n 
                                             , 
                                             
                                               i 
                                               + 
                                               1 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         JX 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                         * 
                                       
                                       - 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           ( 
                                           
                                             JX 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                             * 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     Im 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           X 
                                           
                                             n 
                                             , 
                                             i 
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               Z 
                                               
                                                 n 
                                                 , 
                                                 i 
                                               
                                             
                                             . 
                                           
                                           * 
                                           
                                             X 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                         
                                         - 
                                         
                                           JX 
                                           
                                             n 
                                             , 
                                             i 
                                           
                                           * 
                                         
                                         + 
                                         
                                           
                                             
                                               Z 
                                               
                                                 n 
                                                 , 
                                                 i 
                                               
                                             
                                             . 
                                           
                                           * 
                                           
                                             ( 
                                             
                                               JX 
                                               
                                                 n 
                                                 , 
                                                 
                                                   i 
                                                   + 
                                                   1 
                                                 
                                               
                                               * 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   Im 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           X 
                                           
                                             n 
                                             , 
                                             
                                               i 
                                               + 
                                               1 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         JX 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                         * 
                                       
                                       - 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           ( 
                                           
                                             JX 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                             * 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   Re 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           X 
                                           
                                             n 
                                             , 
                                             
                                               i 
                                               + 
                                               1 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         JX 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                         * 
                                       
                                       + 
                                       
                                         
                                           
                                             Z 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           . 
                                         
                                         * 
                                         
                                           ( 
                                           
                                             JX 
                                             
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                             * 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
 
                         
                         ⁢ 
                         
                           
                             b 
                             
                               n 
                               , 
                               i 
                             
                           
                           = 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       Re 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 Z 
                                                 
                                                   n 
                                                   , 
                                                   i 
                                                 
                                               
                                               . 
                                             
                                             * 
                                             
                                               X 
                                               
                                                 n 
                                                 , 
                                                 
                                                   i 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                           
                                           - 
                                           
                                             X 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Im 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 Z 
                                                 
                                                   n 
                                                   , 
                                                   i 
                                                 
                                               
                                               . 
                                             
                                             * 
                                             
                                               X 
                                               
                                                 n 
                                                 , 
                                                 
                                                   i 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                           
                                           - 
                                           
                                             X 
                                             
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             · 
                             2 
                           
                         
                         , 
                         
                           and 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             U 
                             
                               n 
                               , 
                               i 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           is 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           the 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           noise 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             term 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     The least squares (LS) solution for Eq. (5) is 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             β 
                             ^ 
                           
                         
                       
                       
                         
                           
                             φ 
                             ^ 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             A 
                             
                               n 
                               , 
                               i 
                             
                             H 
                           
                           ⁢ 
                           
                             A 
                             
                               n 
                               , 
                               i 
                             
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       A 
                       
                         n 
                         , 
                         i 
                       
                       H 
                     
                     ⁢ 
                     
                       b 
                       
                         n 
                         , 
                         i 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     After obtaining {circumflex over (β)} and {circumflex over (φ)}, the I/Q offset c can be derived from Eq. (2) by the following operation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Z 
                           
                             0 
                             , 
                             n 
                             , 
                             i 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             Y 
                             
                               0 
                               , 
                               n 
                               , 
                               i 
                             
                           
                           
                             Y 
                             
                               0 
                               , 
                               n 
                               , 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               H 
                               
                                 0 
                                 , 
                                 n 
                                 , 
                                 i 
                               
                             
                             
                               H 
                               
                                 0 
                                 , 
                                 n 
                                 , 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                             
                           
                           · 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           + 
                                           c 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           2 
                                           + 
                                           β 
                                           + 
                                           jφ 
                                         
                                         2 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                             * 
                                           
                                           + 
                                           
                                             c 
                                             * 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           β 
                                           - 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             φ 
                                           
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         W 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           i 
                                         
                                       
                                       / 
                                       
                                         H 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           i 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                           + 
                                           c 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           2 
                                           + 
                                           β 
                                           + 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             φ 
                                           
                                         
                                         2 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               
                                                 i 
                                                 + 
                                                 1 
                                               
                                             
                                             * 
                                           
                                           + 
                                           
                                             c 
                                             * 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           β 
                                           - 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             φ 
                                           
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         W 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                       
                                       / 
                                       
                                         H 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     X 
                                     
                                       0 
                                       , 
                                       n 
                                       , 
                                       i 
                                     
                                   
                                   + 
                                   c 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   2 
                                   + 
                                   β 
                                   + 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     φ 
                                   
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     X 
                                     
                                       0 
                                       , 
                                       n 
                                       , 
                                       i 
                                     
                                     * 
                                   
                                   + 
                                   
                                     c 
                                     * 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   β 
                                   - 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     φ 
                                   
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 W 
                                 
                                   0 
                                   , 
                                   n 
                                   , 
                                   i 
                                 
                               
                               / 
                               
                                 H 
                                 
                                   
                                     0 
                                     , 
                                     n 
                                     , 
                                     i 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           X 
                                           
                                             0 
                                             , 
                                             n 
                                             , 
                                             
                                               i 
                                               + 
                                               1 
                                             
                                           
                                         
                                         + 
                                         c 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         2 
                                         + 
                                         β 
                                         + 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           φ 
                                         
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           X 
                                           
                                             0 
                                             , 
                                             n 
                                             , 
                                             
                                               i 
                                               + 
                                               1 
                                             
                                           
                                           * 
                                         
                                         + 
                                         
                                           c 
                                           * 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         β 
                                         - 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           φ 
                                         
                                       
                                       2 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     W 
                                     
                                       0 
                                       , 
                                       n 
                                       , 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                   / 
                                   
                                     H 
                                     
                                       0 
                                       , 
                                       n 
                                       , 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     Eq. (7) can be rewritten in the standard form of equation with respect to c as 
     
       
         
           
             
               
                 
                   
                     
                       
                         D 
                         
                           n 
                           , 
                           i 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 Re 
                                 ⁡ 
                                 
                                   ( 
                                   c 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 Im 
                                 ⁡ 
                                 
                                   ( 
                                   c 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         e 
                         
                           n 
                           , 
                           i 
                         
                       
                       + 
                       
                         
                           V 
                           
                             n 
                             , 
                             i 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         where 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       D 
                       
                         n 
                         , 
                         i 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               Re 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         Z 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       β 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 Im 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           Z 
                                           
                                             0 
                                             , 
                                             n 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           φ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               Im 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         Z 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       β 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               Re 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         Z 
                                         
                                           0 
                                           , 
                                           n 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         φ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       e 
                       
                         n 
                         , 
                         i 
                       
                     
                     = 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 
                                   Re 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                             + 
                                             β 
                                             + 
                                             
                                               j 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               φ 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   Z 
                                                   
                                                     0 
                                                     , 
                                                     n 
                                                     , 
                                                     i 
                                                   
                                                 
                                                 . 
                                               
                                               * 
                                               
                                                 X 
                                                 
                                                   0 
                                                   , 
                                                   n 
                                                   , 
                                                   
                                                     i 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                             
                                             - 
                                             
                                               X 
                                               
                                                 0 
                                                 , 
                                                 n 
                                                 , 
                                                 i 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             β 
                                             - 
                                             jφ 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   Z 
                                                   
                                                     0 
                                                     , 
                                                     n 
                                                     , 
                                                     i 
                                                   
                                                 
                                                 . 
                                               
                                               * 
                                               
                                                 X 
                                                 
                                                   0 
                                                   , 
                                                   n 
                                                   , 
                                                   
                                                     i 
                                                     + 
                                                     1 
                                                   
                                                 
                                                 * 
                                               
                                             
                                             - 
                                             
                                               X 
                                               
                                                 0 
                                                 , 
                                                 n 
                                                 , 
                                                 i 
                                               
                                               * 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Im 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                             + 
                                             β 
                                             + 
                                             
                                               j 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               φ 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   Z 
                                                   
                                                     0 
                                                     , 
                                                     n 
                                                     , 
                                                     i 
                                                   
                                                 
                                                 . 
                                               
                                               * 
                                               
                                                 X 
                                                 
                                                   0 
                                                   , 
                                                   n 
                                                   , 
                                                   
                                                     i 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                             
                                             - 
                                             
                                               X 
                                               
                                                 0 
                                                 , 
                                                 n 
                                                 , 
                                                 i 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             β 
                                             - 
                                             
                                               j 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               φ 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   Z 
                                                   
                                                     0 
                                                     , 
                                                     n 
                                                     , 
                                                     i 
                                                   
                                                 
                                                 . 
                                               
                                               * 
                                               
                                                 X 
                                                 
                                                   0 
                                                   , 
                                                   n 
                                                   , 
                                                   
                                                     i 
                                                     + 
                                                     1 
                                                   
                                                 
                                                 * 
                                               
                                             
                                             - 
                                             
                                               X 
                                               
                                                 0 
                                                 , 
                                                 n 
                                                 , 
                                                 i 
                                               
                                               * 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         · 
                         
                           ( 
                           0.5 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
     and V n,i  is the noise term. The LS solution for Eq. (8) is 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             Re 
                             ⁡ 
                             
                               ( 
                               
                                 c 
                                 ^ 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             Im 
                             ⁡ 
                             
                               ( 
                               
                                 c 
                                 ^ 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 D 
                                 
                                   n 
                                   , 
                                   i 
                                 
                                 H 
                               
                               ⁢ 
                               
                                 D 
                                 
                                   n 
                                   , 
                                   i 
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           D 
                           
                             n 
                             , 
                             i 
                           
                           H 
                         
                         ⁢ 
                         
                           e 
                           
                             n 
                             , 
                             i 
                           
                         
                       
                       ] 
                     
                     
                       
                         β 
                         = 
                         
                           β 
                           ^ 
                         
                       
                       , 
                       
                         φ 
                         = 
                         
                           φ 
                           ^ 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
     Eq. (9) can only be used if X 0,n,i+1 ≠X 0,n,i , otherwise e n,i →0 and only zero solution can be obtained from Eq. (9). If X 0,n,i+1 =X 0,n,i , estimation of c can be done after compensation of I/Q imbalance and quadrature error and channel estimation. After that, c can be directly obtained from Eq. (2). 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               Re 
                               ⁡ 
                               
                                 ( 
                                 
                                   c 
                                   ^ 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               Im 
                               ⁡ 
                               
                                 ( 
                                 
                                   c 
                                   ^ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   G 
                                   
                                     n 
                                     , 
                                     i 
                                   
                                   H 
                                 
                                 ⁢ 
                                 
                                   G 
                                   
                                     n 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             G 
                             
                               n 
                               , 
                               i 
                             
                             H 
                           
                           ⁢ 
                           
                             f 
                             
                               n 
                               , 
                               i 
                             
                           
                         
                         ] 
                       
                       
                         
                           β 
                           = 
                           
                             β 
                             ^ 
                           
                         
                         , 
                         
                           φ 
                           = 
                           
                             φ 
                             ^ 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         G 
                         
                           n 
                           , 
                           i 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 + 
                                 β 
                               
                             
                             
                               
                                 - 
                                 φ 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         
                           n 
                           , 
                           i 
                         
                       
                       = 
                       
                           
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     Re 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             Y 
                                             
                                               k 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           
                                             H 
                                             
                                               k 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               1 
                                               + 
                                               β 
                                               + 
                                               
                                                 ⅇ 
                                                 jφ 
                                               
                                             
                                             2 
                                           
                                           ⁢ 
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               1 
                                               + 
                                               β 
                                               - 
                                               
                                                 ⅇ 
                                                 jφ 
                                               
                                             
                                             2 
                                           
                                           ⁢ 
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                             * 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     Im 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             Y 
                                             
                                               k 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                           
                                             H 
                                             
                                               k 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               1 
                                               + 
                                               β 
                                               + 
                                               
                                                 ⅇ 
                                                 jφ 
                                               
                                             
                                             2 
                                           
                                           ⁢ 
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               1 
                                               + 
                                               β 
                                               - 
                                               
                                                 ⅇ 
                                                 jφ 
                                               
                                             
                                             2 
                                           
                                           ⁢ 
                                           
                                             X 
                                             
                                               0 
                                               , 
                                               n 
                                               , 
                                               i 
                                             
                                             * 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             ( 
                             0.5 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
       FIG. 7  is a schematic diagram illustrating an example of the measurement of gain imbalance and quadrature error using Eq. (6). The signal Y n,i  is supplied to a delay element  706  via connection  702 . The delay element  706  registers a symbol received in a previous time slot so that the symbols received in the previous time slot (Y n,i−1 ) and current time slot (Y n,i ) can be combined and sent to a divider  712  simultaneously. The output of the delay element  706  is the function Y n,i−1  on connection  708  and is provided to the divider  712 . The signal Y n,i  is also supplied to the divider  712 . The output of the divider  712  is the signal Z n,i−1  on connection  714  and is supplied to a gain imbalance and quadrature error computation element  716 . The signal X n,i  is supplied to the gain imbalance and quadrature error computation element  716  via connection  704 . The gain imbalance and quadrature error computation element  716  performs the computation provided in Equation (6) and provides signals representing the terms {circumflex over (β)} and {circumflex over (φ)} on connection  718 . 
       FIG. 8  is a schematic diagram illustrating an example of a first embodiment of the measurement of I/Q offset using Eq. (9). The signal Y 0,n,i  is supplied to a delay element  806  via connection  802 . The delay element  806  is similar to the delay element  706 . The output of the delay element  806  is the function Y 0,n,i−1  on connection  808  and is provided to a divider  812 . The signal Y 0,n,i  is also supplied to the divider  812 . The output of the divider  812  is the signal Z 0,n,i−1  on connection  814  and is supplied to an I/Q offset element  816 . The signal X 0,n,I  is supplied to the I/Q offset element  816  via connection  804 . The signals representing the terms {circumflex over (β)} and {circumflex over (φ)} are supplied to the I/Q offset element  816  via connection  718 . The I/Q offset element  816  performs the computation provided in Equation (9) and provides a signal having the term c on connection  818 . In the embodiment shown in  FIGS. 7 and 8 , the measurement of I/Q mismatch is performed prior to channel estimation. 
       FIG. 9  is a schematic diagram illustrating an example of an alternative embodiment of the measurement of I/Q offset using Eq. (10). The signal Y n,i  is supplied to an I/Q offset element  916  via connection  902 , the signal H n,i  is supplied to the I/Q offset element  916  via connection  904  and the signals {circumflex over (β)} and {circumflex over (φ)} are supplied to the I/Q offset element  916  via connection  906 . The I/Q offset element  916  performs the computation provided in Equation (10) and provides the signal c on connection  918 . In the embodiment shown in  FIGS. 7 and 9 , the measurement of I/Q mismatch is performed after the channel estimation. 
     Embodiment 2 
     Transmitter and Receiver Wirelessly Connected (General Application) 
     If the channel is a general fading channel rather than a cable connection, the received signal at the n th  Rx stream involves contributions from all the transmitter antennas. 
     
       
         
           
             
               
                 
                   
                     Y 
                     
                       k 
                       , 
                       n 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         1 
                       
                       P 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             H 
                             
                               k 
                               , 
                               n 
                               , 
                               m 
                               , 
                               i 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   1 
                                   + 
                                   
                                     β 
                                     m 
                                   
                                   + 
                                   
                                     ⅇ 
                                     
                                       jφ 
                                       m 
                                     
                                   
                                 
                                 2 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   k 
                                   , 
                                   m 
                                   , 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   1 
                                   + 
                                   
                                     β 
                                     m 
                                   
                                   - 
                                   
                                     ⅇ 
                                     
                                       jφ 
                                       m 
                                     
                                   
                                 
                                 2 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   
                                     N 
                                     - 
                                     k 
                                   
                                   , 
                                   m 
                                   , 
                                   i 
                                 
                                 * 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         1 
                                         + 
                                         
                                           β 
                                           m 
                                         
                                         + 
                                         
                                           ⅇ 
                                           
                                             jφ 
                                             m 
                                           
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       c 
                                       m 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         1 
                                         + 
                                         
                                           β 
                                           m 
                                         
                                         - 
                                         
                                           ⅇ 
                                           
                                             jφ 
                                             m 
                                           
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       c 
                                       m 
                                       * 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 δ 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           W 
                           
                             k 
                             , 
                             m 
                             , 
                             i 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
     where H k,n,m,i  is the frequency response at the k th  subcarrier of the i th  OFDM symbol transmitted from the m th  transmitter antenna to the n th  Rx stream. P is the number of transmitter antennas. 
     Because on the right side of Eq. (11) there are several different channel responses H k,n,m,i  Eq. (4) cannot be employed to eliminate the influences of channel responses. Direct joint estimation of channel response and I/Q mismatch parameters is a possible solution for the problem, but is very complex. 
     In accordance with an embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation, the received signal is defined as an incomplete observation set for estimation of parameters and the first step is to construct a complete observation set based on the incomplete set to simplify the estimation of parameters. Eq. (11) is rewritten in vector mode as 
     
       
         
           
             
               
                 
                   
                     
                       Y 
                       
                         n 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             1 
                           
                           P 
                         
                         ⁢ 
                         
                           Y 
                           
                             n 
                             , 
                             m 
                             , 
                             i 
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             1 
                           
                           P 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               T 
                               
                                 n 
                                 , 
                                 m 
                                 , 
                                 i 
                               
                             
                             ⁢ 
                             
                               W 
                               H 
                             
                             ⁢ 
                             
                               h 
                               
                                 n 
                                 , 
                                 m 
                                 , 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     2 
                     ⁢ 
                     
                       T 
                       
                         n 
                         , 
                         m 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             β 
                             m 
                           
                           + 
                           
                             ⅇ 
                             
                               jφ 
                               m 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         X 
                         
                           n 
                           , 
                           m 
                           , 
                           i 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             β 
                             m 
                           
                           - 
                           
                             ⅇ 
                             
                               jφ 
                               m 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         JX 
                         
                           n 
                           , 
                           m 
                           , 
                           i 
                         
                         * 
                       
                       ⁢ 
                       J 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
     The signal representing the term Y n,i  defines the incomplete set of received vectors at the n th  Rx stream of the i th  OFDM symbol, the term h n,m,i  is the time domain channel response vector between the m th  transmitter antenna and the n th  Rx stream, the terms W and J are IDFT (Inverse Discrete Fourier Transform) matrix and exchange matrix, respectively. In accordance with an embodiment, the complete set Ŷ l,n,m,i   (s)  is defined as 
     
       
         
           
             
               
                 
                   
                     
                       Y 
                       ^ 
                     
                     
                       l 
                       , 
                       n 
                       , 
                       m 
                       , 
                       i 
                     
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               T 
                               ^ 
                             
                             
                               n 
                               , 
                               m 
                               , 
                               i 
                             
                             
                               ( 
                               
                                 s 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               W 
                               ⁡ 
                               
                                 ( 
                                 
                                   l 
                                   , 
                                   
                                     : 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         H 
                       
                       ⁢ 
                       
                         
                           h 
                           ^ 
                         
                         
                           l 
                           , 
                           m 
                           , 
                           n 
                           , 
                           i 
                         
                         
                           ( 
                           
                             s 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         α 
                         
                           l 
                           , 
                           n 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             Y 
                             
                               n 
                               , 
                               i 
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 1 
                               
                               P 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 L 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       
                                         T 
                                         ^ 
                                       
                                       
                                         n 
                                         , 
                                         m 
                                         , 
                                         i 
                                       
                                       
                                         ( 
                                         
                                           s 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         W 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             l 
                                             , 
                                             
                                               : 
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   H 
                                 
                                 ⁢ 
                                 
                                   
                                     h 
                                     ^ 
                                   
                                   
                                     l 
                                     , 
                                     m 
                                     , 
                                     n 
                                     , 
                                     i 
                                   
                                   
                                     ( 
                                     
                                       s 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
     where {circumflex over (T)} n,m,i   (s) =[T n,m,i ] β={circumflex over (β)}     m       (s)     ,φ={circumflex over (φ)}     m       (s)    and s is the iteration index. Ŷ l,n,m,i   (s)  is the received vector of the i th  OFDM symbol from the u th  path between the m th  transmitter antenna and n th  Rx stream at the s th  iteration, α l,n  are constant factors with respect to 
                 ∑     l   =   1     L     ⁢     α     l   ,   n         =   1.         
The terms P and U are the number of transmitter antennas and paths of channel responses, respectively.
 
     The objective is to obtain {circumflex over (β)} m   (s) , {circumflex over (φ)} m   (s)  and ĥ l,m,n,i   (s)  based on Y l,n,m,i =Ŷ l,m,n,i   (s)  using an ML (Maximum Likelihood) approach, update the current values of parameters from estimation results, and then estimate another subset of parameters until values of all the parameters are obtained. 
     The steps of proposed method are described as below: 
     Expectation Step: 
     Evaluate (14) with {circumflex over (β)} m   (s−1) , {circumflex over (φ)} m   (s−1)  and ĥ l,m,n,i   (s−1)    
     Maximization Step: 
     Resolve 
               {         β   ^     m     (   s   )       ,       φ   ^     m     (   s   )       ,       h   ^       l   ,   m   ,   n   ,   i       (   s   )         }     =       min       β   m     ,     φ   m     ,     h     l   ,   m   ,   n   ,   i           ⁢     {              Y     l   ,   n   ,   m   ,   i       -       Y   ^       l   ,   n   ,   m   ,   i       (   s   )              2     }             
with ML approach.
 
     Return to expectation step with estimation results from maximization step. 
     Proceed until results converge. 
     Using wireless connection measurement, the estimation of c m  (I/Q offset) is performed after compensation of gain imbalance and quadrature error with estimation results in last iteration and channel estimation as described in Eq. (10). 
       FIG. 10  is a flowchart showing the operation of an embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation. 
     In block  1002 , initial values of gain imbalance, quadrature error and channel response are obtained. Typically, the initial values for gain imbalance and quadrature error are set to zero. The initial value for channel response is obtained from known parts of the received signal, such as from a preamble or pilot portion of the signal. 
     In block  1004 , the estimation results of gain imbalance, quadrature error and channel response from the previous iteration are used to evaluate T n,m,i  using Eq. 13 and then T n,m,i  used to evaluate Eq. 14. In block  1006 , estimates for gain imbalance, quadrature error and channel response using the equation 
               {         β   ^     m     (   s   )       ,       φ   ^     m     (   s   )       ,       h   ^       l   ,   m   ,   n   ,   i       (   s   )         }     =       min       β   m     ,     φ   m     ,     h     l   ,   m   ,   n   ,   i           ⁢     {              Y     l   ,   n   ,   m   ,   i       -       Y   ^       l   ,   n   ,   m   ,   i       (   s   )              2     }             
are obtained.
 
     The process of blocks  1004  and  1006  are graphically represented in  FIGS. 11 and 12 . 
       FIG. 11  is a schematic diagram illustrating the measurement of gain imbalance and quadrature error with general channel fading, where the iteration follows the flow in  FIG. 10 . The blocks  1110 ,  1120  and  1130  depict iterations of blocks  1004 ,  1006 ,  1008  and  1012  of  FIG. 10 . 
       FIG. 12  is a schematic diagram illustrating an example of one of the blocks  1110 ,  1120  and  1130  of  FIG. 11 . Block  1210  performs the functionality of block  1004  of  FIG. 10  using Eq. 14. In block  1220 , estimates for gain imbalance, quadrature error and channel response using the equation 
               {         β   ^     m     (   s   )       ,       φ   ^     m     (   s   )       ,       h   ^       l   ,   m   ,   n   ,   i       (   s   )         }     =       min       β   m     ,     φ   m     ,     h     l   ,   m   ,   n   ,   i           ⁢     {              Y     l   ,   n   ,   m   ,   i       -       Y   ^       l   ,   n   ,   m   ,   i       (   s   )              2     }             
are obtained.
 
     In block  1008 , it is determined whether the entire observation set has been evaluated. If the entire evaluation set has not been evaluated, then, in block  1012 , the indices l, m and n are incremented and the process returns to block  1004 . 
     If, in block  1008  it is determined that the entire observation set has been evaluated, then, in block  1014 , it is determined whether the estimation results converge with increasing iterations of the index, s. If the estimation results do not converge with increasing iterations of the index, s, then, in block  1016 , the index, s, is incremented and the process returns to block  1004 . 
     If it is determined in block  1014  that the estimation results converge with increasing iterations of the index, s, then, in block  1018 , the estimation results for gain imbalance and quadrature error for the last iteration are accepted. In block  1022  the gain imbalance and the quadrature error are compensated using the estimation results of block  1018 . In block  1024 , the term X 0,n,i  is detected and the I/Q offset is estimated using Eq. 10. 
     Embodiment 3 
     Transmitter and Receiver Wirelessly Connected (No Spatial Extension) 
     If spatial extension is not applied in a diversity MIMO system, a simpler approach to determine and compensate I/Q mismatch measurement can be employed. For example, in a MIMO OFDM system based on the IEEE 802.11n standard, with P transmitters and L receivers, the k th  received subcarrier of training symbols at the l th  receiver can expressed as
 
 R   k   =P   T ( Q   k   T   α[H   k ( q ,:)] T   T   k   +Q   N−k   H   b[H   k ( q ,:)] T   T   N−k *)+ V   k   Eq. (15)
 
     where R k =[R k,1  . . . R k,p ] is observed in P symbol durations at the 1 th  receiver. 
     The term a=diag([a 1  . . . a P ]), b=diag([b 1  . . . b P ]) is the I/Q mismatch vector. T k  refers to the training symbols at k th  subcarriers. 
     The estimated channel response using training sequences can represented as
 
[ Ĥ   k ( q ,:)] T =( P*P   T ) −1   P*P   k   T   k   1   ≅Q   N k   H   b[H   k ( q ,:)] T   T   N k   *T   k   1   Eq. (16)
 
     The channel matrix for symbol detection is obtained as 
     
       
         
           
             
               
                 
                   
                     
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               where 
               ⁢ 
               
                   
               
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     At the receiver, received symbols can be represented as
 
 Y   k   =H   k ( aQ   k   X   k   +bQ   N−k   *X   N−k )+ V   k   Eq. (17)
 
     where X k =[X k,1  . . . X k,S ] T  is the transmitted symbols at S spatial streams, Y k =[Y k,1  . . . Y k,Q ] T  are the received symbols at the L receivers. 
     If the estimated channel is used for symbol detection, recovered symbols can be written as
 
 Z   k =( Ĥ   k   H   Ĥ   k ) −1   Ĥ   k   H   Y   k ≅( G   k   H   H   k   H   H   k   G   k ) −1   G   k   H   H   k   H   H   k ( aQ   k   X   k   +bQ   N−k   *X   N−k )  Eq. (18)
 
     If G k  is a square matrix, i.e. Q k  is a square matrix, then
 
 Z   k =( Ĥ   k   H   Ĥ   k ) −1   Ĥ   k   H   Y   k   ≅G   k   −1 ( aQ   k   X   k   +bQ   N−k   *X   N−k *)  Eq. (19)
 
     It can be obtained from previous analysis that a=I+g and b=g*, where 
     
       
         
           
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     From Eq. (19), it can derived that 
     
       
         
           
             
               
                 
                   
                     
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                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
           
         
       
     
     The solution to Eq. (20) is 
     
       
         
           
             
               
                 
                   
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                                 g 
                                 ^ 
                               
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                             ⁡ 
                             
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                   = 
                   
                     
                       
                         ( 
                         
                           
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                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
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                       H 
                     
                     ⁢ 
                     D 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     21 
                     ) 
                   
                 
               
             
           
         
       
     
     Then gain imbalance and quadrature error can obtained by
 
{circumflex over (β)}(:)=2 g   r (:)  Eq. (22)
 
{circumflex over (φ)}(:)=2 g   t (:)  Eq. (23)
 
     The structure is of the measurement of gain imbalance and quadrature error using this approach is given in  FIG. 13 .  FIG. 13  is a block diagram of an alternative embodiment of the Rx baseband module of  FIG. 6 . In  FIG. 13 , the I/Q mismatch measurement is implemented in an equalizer. 
     For each channel, the in-phase (I) and quadrature-phase (Q) signals received from the demodulator  320  ( FIG. 3 ) are combined into complex form (I+jQ) and provided to a synchronizer  1306 . In the example shown in  FIG. 13 , the signal on connection  1302 - 1  is provided to a synchronizer  1306 - 1  and the signal on connection  1302 -L is provided to a synchronizer  1306 -L. The number of synchronizers corresponds to the number of channels (i.e., the number of transmitters, P ( FIG. 1 ) and receivers, L ( FIG. 1 )). 
     After synchronization, each receive signal is provided to a corresponding cyclic prefix removal element  1308 - 1  through  1308 -L. The cyclic prefix removal element  1308 - 1  is similar to the cyclic prefix removal element  608 - 1  described above. 
     After the cyclic prefix is removed from each receive signal, the receive signals are supplied to corresponding fast Fourier transform (FFT) elements  1312 . 
     The FFT elements  1312 - 1  through  1312 -L convert the receive signals to the frequency domain. The output of the FFT element  1312 - 1  is supplied over connection  1314 - 1  and is referred to as Y 1,j . The output of the FFT element  1312 -L is supplied over connection  1314 -L and is referred to as Y L,j . As used in  FIG. 13 , the first term in the subscript (Y 1,j ) refers to the first channel in  FIG. 1  and the first term in the subscript (Y L,j ) refers to the last channel, L, in  FIG. 1 . 
     The output of the FFT elements  1312 - 1  through  1312 -L is used for I/Q mismatch measurement and channel estimation. However, the I/Q mismatch is performed in the equalizer  1332 , as will be described below. 
     The signals on connections  1314 - 1  and  1314 -L are supplied to an equalizer  1332  and to a channel estimation element  1320 . The channel estimation element  1320  measures the response of the channel and provides an indicator, H n,i , that is a term that defines the channel response. The measurement results can be sent to the equalizer  1332  for compensation of I/Q mismatch. The compensation can be performed using either the real time measurement results or using results stored some time before, as described above with the exception that Eq. 17 is used in this example. 
       FIG. 14  is a block diagram illustrating an embodiment of the equalizer of  FIG. 13 . The term Y n,i  is supplied via connection  1402  to a symbol detection element  1408 . In the term Y n,i,  n is a variable that can have any value from 1 to L, so it covers Y 1,i  to Y L,i . The symbol detection element  1408  operates similar to an equalizer, but it only removes the effects of the channel and ignores the effect of I/Q mismatch. The symbol detection element  1408  provides the term Z n,i  over connection  1412  to the I/Q mismatch compensation element  1416 . 
     The term X n,i  is a known part of the transmitted signal and can be saved in registers in the equalizer  1332 . The term X n,i  is shown schematically supplied via connection  1404  to the I/Q mismatch measurement element  1414 , and the term H n,i  is supplied via connection  1406  to the I/Q mismatch measurement element  1414 . The term H n,i  is also supplied via connection  1406  to symbol detection element  1408 ; and the terms X n,i  and Z n,i  are supplied over respective connections  1404  and  1412  to the I/Q mismatch measurement element  1414 . The I/Q mismatch measurement element  1414  provides the terms {circumflex over (β)} ĉ and {circumflex over (φ)} to the I/Q compensation element  1416  via connection  1418 . The I/Q mismatch measurement element  1414  obtains the terms {circumflex over (β)} and {circumflex over (φ)} using equations  21 ,  22  and  23 . The terms {circumflex over (β)} and {circumflex over (φ)} are then provided to equation 10 to obtain the term ĉ. 
     The compensation performed by the I/Q mismatch compensation element  1416  can be performed as described above, with the exception that Eq. 17 is used in this example. 
     The output of the I/Q compensation element  1416  is provided over connection  1422  to a symbol-to-bit converter  1424 . The symbol-to-bit converter converts the symbols on connection  1422  to bits. 
       FIG. 15  is a graphical illustration showing the results of the operation of the first embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation.  FIG. 15  shows an example of real values for I/Q impairments set in an I/Q waveform that is downloaded to an Agilent E4438c ESG (Electronic Signal Generator). 
       FIG. 16  is a graphical illustration showing the results of the operation of the second embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation.  FIG. 16  shows an example of real values for I/Q impairments set in an I/Q waveform that is downloaded to an Agilent E4438c ESG. 
       FIG. 17  is a graphical illustration showing the results of the operation of the third embodiment of the system and method for in-phase/quadrature-phase (I/Q) mismatch measurement and compensation.  FIG. 17  shows an example of real values for I/Q impairments set in an I/Q waveform that is downloaded to an Agilent E4438c ESG. 
     The foregoing detailed description has been given for understanding exemplary implementations of the invention and no unnecessary limitations should be understood therefrom as modifications will be obvious to those skilled in the art without departing from the scope of the appended claims and their equivalents.