Patent Publication Number: US-11646931-B2

Title: Methods and apparatus for transmit IQ mismatch calibration

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 17/092,214 filed Nov. 6, 2020 which is incorporated by reference and which claims priority to, and the benefit of, U.S. Provisional Patent Application Ser. No. 63/025,980 titled “Transmitter Frequency-Dependent In-Phase and Quadrature Mismatch Calibration” filed May 15, 2020 which is incorporated by reference. 
    
    
     TECHNICAL FIELD 
     This disclosure relates generally to quadrature transmitters, and more specifically to transmit IQ mismatch calibration. 
     BACKGROUND 
     A quadrature transmitter may include an in-phase (I) path and a quadrature (Q) path. Imbalances between the I and Q paths, which may be referred to as IQ mismatch (IQMM), may degrade the performance of the transmitter. 
     The above information disclosed in this Background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not constitute prior art. 
     SUMMARY 
     A method of pre-compensating for transmitter in-phase (I) and quadrature (Q) mismatch (IQMM) may include sending a signal through an up-converter of a transmit path to provide an up-converted signal, determining the up-converted signal through a down-converter of a receive feedback path, determining one or more IQMM parameters for the transmit path based on the determined up-converted signal, and determining one or more pre-compensation parameters for the transmit path based on the one or more IQMM parameters for the transmit path. Determining one or more IQMM parameters for the transmit path may include solving a system of equations, a rust one of the equations may include a first component of the up-converted signal and a first parameter representing, at least in part, a desired frequency response of the transmit path, and a second one of the equations may include a second component of the up-converted signal and a second parameter representing, at least in part, a frequency response of the transmit path due to transmit IQMM. The first one of the equations may further include a third parameter representing, at least in part, a gain and delay for the transmit path. The method may further include determining an IQMM for the receive feedback path by using a first local oscillator for the transmit path and a second local oscillator for the receive path, and determining one or more IQMM parameters for the transmit path based on the determined up-converted signal may include processing the up-converted signal to compensate for the IQMM in the receive path. A local oscillator for the transmit path may have a frequency shift from a local oscillator for the receive feedback path. The signal may include a first signal at a first frequency, the up-converted signal may include a first up-converted signal, and the method may further include sending a second signal at a second frequency through the up-converter of the transmit path to provide a second up-converted signal, determining the second up-converted signal through the down-converter of the receive feedback path, and determining one or more IQMM parameters for the transmit path based on the determined second up-converted signal. 
     A method of pre-compensating for transmitter in-phase (I) and quadrature (Q) mismatch (IQMM) may include sending a signal through an up-converter of a transmit path to provide an up-converted signal, determining the up-converted signal through an envelope detector, determining one or more IQMM parameters for the transmit path based on the determined up-converted signal, and determining one or more pre-compensation parameters for the transmit path based on the one or more IQMM parameters for the transmit path. Determining one or more IQMM parameters for the transmit path may include applying a first pre-compensation parameter to the transmit path, determining a first power of a component of the up-converted signal caused by transmit IQMM through the envelope detector based on the first pre-compensation parameter, applying a second pre-compensation parameter to the transmit path, and determining a second power of a component of the up-converted signal caused by transmit IQMM through the envelope detector based on the second pre-compensation parameter. Determining one or more IQMM parameters for the transmit path may further include selecting one of the first pre-compensation parameter or the second pre-compensation parameter based on a lower of the first power and the second power. The method may further include applying one or more additional pre-compensation parameters to the transmit path, and determining one or more additional powers of one or more components of the up-converted signal caused by transmit IQMM through the envelope detector based on the one or more additional pre-compensation parameters, and determining one or more IQMM parameters for the transmit path may include selecting one of the first pre-compensation parameter, the second pre-compensation parameter or the one or more additional pre-compensation parameters based on a lower of the first power, the second power, or the one or more additional powers. The signal may include a first signal at a first frequency, the up-converted signal may include a first up-converted signal, and the method may further include sending a second signal at a second frequency through the up-converter of the transmit path to provide a second up-converted signal, determining the second up-converted signal through the envelope detector, and determining one or more IQMM parameters for the transmit path based on the determined second up-converted signal. The method may further include applying first and second pre-compensation parameters to the transmit path for each of the first and second signals, and the first and second up-converted signals may be determined separately based on the first and second pre-compensation parameters. Determining one or more IQMM parameters for the transmit path may include solving a system of equations based on the determined first and second up-converted signals. A first one of the system of equations may include a function, at least in part, of the first and second pre-compensation parameters. The second frequency may be a negative of the first frequency at baseband. The method may further include sweeping the first and second frequencies for each of the first and second pre-compensation parameters, determining additional first and second up-converted signals based on sweeping the first and second frequencies, and determining one or more IQMM parameters for the transmit path over frequency based on the determined additional up-converted signals. The signal may include a first two-tone signal, the up-converted signal may include a first up-converted two-tone signal, and the method may further include sending a second two-tone signal through the up-converter of the transmit path to provide a second up-converted two-tone signal, determining the second up-converted two-tone signal through the envelope detector, and determining one or more IQMM parameters for the transmit path based on the determined second up-converted two-tone signal. Determining one or more IQMM parameters for the transmit path may include solving a system of equations based on the determined first and second up-converted two-tone signals, and at least one of the equations may include a first parameter of a first frequency of the first two-tone signal and a second parameter of a second frequency of the first two-tone signal. The method may further include sweeping first and second frequencies of at least one of the two-tone signals, determining additional first and second up-converted two-tone signals based on sweeping the first and second frequencies, and determining one or more IQMM parameters for the transmit path over frequency based on the determined additional up-converted two-tone signals. 
     A system may include an IQ transmit path comprising an up-converter, an envelope detector arranged to provide an envelope of an up-converted signal from the IQ transmit path, a signal generator arranged to apply a pilot signal to the IQ transmit path, a signal observer arranged to capture the envelope of the up-converted signal based on the pilot signal, and a processor configured to: estimate one or more IQ mismatch (IQMM) parameters for the IQ transmit path based on the captured envelope of the up-converted signal, and estimate one or more compensation coefficients for the IQ transmit path based on the estimated IQMM parameters. The signal observer may be arranged to capture the envelope of the up-converted signal through a branch of an IQ receiver. 
     A system may include an IQ transmit path comprising an up-converter, an IQ receive path comprising a down-converter, a feedback connection arranged to couple an up-converted signal from the IQ transmit path to the IQ receive path, a signal generator arranged to apply a pilot signal to the IQ transmit path, a signal observer arranged to capture the up-converted signal through the IQ receive path based on the pilot signal, and a processor configured to: estimate one or more IQ mismatch (IQMM) parameters for the IQ transmit path based on the captured up-converted signal, and estimate one or more compensation coefficients for the IQ transmit path based on the estimated IQMM parameters. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The figures are not necessarily drawn to scale and elements of similar structures or functions are generally represented by like reference numerals for illustrative purposes throughout the figures. The figures are only intended to facilitate the description of the various embodiments disclosed herein. The figures do not describe every aspect of the teachings disclosed herein and do not limit the scope of the claims. The accompanying drawings, together with the specification, illustrate example embodiments of the present disclosure, and, together with the description, serve to explain the principles of the present disclosure. 
         FIG.  1    illustrates an example embodiment of an IQ transmitter that may be used to implement any of the TX IQMM estimation and/or compensation techniques according to this disclosure. 
         FIG.  2    illustrates an example embodiment of complex-valued pre-compensator (CVC) structure for which coefficients may be estimated according to this disclosure. 
         FIG.  3    illustrates an embodiment of a system that may be used to implement TX FD-IQMM calibration using an RX feedback path according to this disclosure. 
         FIG.  4    illustrates an example embodiment of a system that may be used to implement TX FD-IQMM calibration using an RX feedback path according to this disclosure. 
         FIG.  5    illustrates example embodiments of spectral plots of transmitted and captured (observed) signals corresponding to some equations according to this disclosure. 
         FIG.  6    illustrates an embodiment of a system that may be used to implement TX FD-IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  7    illustrates an example embodiment of a system that may be used to implement TX FD-IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  8    illustrates some spectral plots of transmitted and captured (observed) signals using a first embodiment of a method for TX IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  9    illustrates some spectral plots of transmitted and captured (observed) signals using a third embodiment of a method for TX IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  10    illustrates an embodiment of method for TX IQMM calibration using an RX feedback path according to this disclosure. 
         FIG.  11    illustrates an embodiment of a first method for TX IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  12    illustrates an embodiment of a second method for TX IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  13    illustrates an embodiment of a third method for TX IQMM calibration using an envelope detector according to this disclosure. 
         FIG.  14    illustrates an embodiment of a method of pre-compensating for transmitter IQMM according to this disclosure. 
         FIG.  15    illustrates another embodiment of a method of pre-compensating for transmitter IQMM according to this disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     This disclosure encompasses numerous inventive principles relating to pre-compensation for in-phase (I) and quadrature (Q) mismatch (IQMM) in quadrature up-conversion transmitters. Pilot signals may be applied at baseband to a transmit (TX) path, and the IQMM impaired up-converted signals may be captured and processed using various disclosed techniques and algorithms to estimate the TX IQMM, which may include both frequency-independent IQMM (FI-IQMM) and frequency-dependent IQMM (FD-IQMM). The estimated IQMM may then be used to determine coefficients for a pre-compensator in the TX path. 
     In some embodiments, the IQMM impaired up-converted signals may be captured through a receive (RX) feedback path having a quadrature down-converter. Single-tone pilot signals may be applied at different frequencies, and primary and mirror components of the captured down-converted signals may be used in a system of equations to estimate IQMM parameters for the TX path. Effects of RX IQMM in the RX feedback path may be reduced or eliminated through various disclosed techniques, for example, by using separate local oscillators for the TX and RX paths and/or a frequency shift between the local oscillators for the TX and RX paths. 
     In some embodiments, the IQMM impaired up-converted signals may be captured through an envelope detector and processed using various disclosed techniques. In a first method using an envelope detector, a single-tone pilot signal may be applied while varying one or more pre-compensation parameters. A single-tone pilot signal applied at baseband may produce a signal at the output of the envelope detector having a component at twice the frequency of the pilot signal if there is IQMM in the TX path. Thus, the first method may sweep one or more pre-compensation parameters while applying a first single-tone pilot signal and selecting one or more of the parameters that provide the lowest output power from the envelope detector at twice the frequency of the pilot signal. A search may be performed by repeating this process at other frequencies to select one or more parameters for each frequency. The selected parameters may then be used to estimate IQMM parameters for the TX path. 
     In a second method using an envelope detector, one or more TX IQMM parameters for a given frequency may be estimated directly by separately sending the negative and positive frequencies of a single-tone pilot signal at baseband using two different sets of pre-compensator settings. The components at twice the given frequency at the output of the envelope detector may be combined in a set of equations and solved for the frequency-dependent gain and phase mismatch at the given frequency. This process may be repeated to determine the frequency-dependent gain and phase mismatch at other frequencies, which may then be used to estimate the IQMM parameters for the TX path. 
     In a third method using an envelope detector, various combinations of the negative and positive frequencies of two-tone pilot signals may be applied separately to the TX path at baseband. The outputs of the envelope detector at various frequencies may be combined and solved using a set of equations to obtain estimates of the TX IQMM parameters directly. 
     Once TX IQMM parameters are determined by any of these disclosed techniques, they may be used to determine coefficients for a pre-compensator in the TX path. 
     The principles disclosed herein may have independent utility and may be embodied individually, and not every embodiment may utilize every principle. Moreover, the principles may also be embodied in various combinations, some of which may amplify benefits of the individual principles in a synergistic manner. 
     TX Pre-Compensation 
     In quadrature up-conversion transmitters. IQMM between the I and Q branches may create interference between the mirror frequencies after up-conversion to radio frequency (RF) or intermediate frequency (IF). Thus, the IQMM may degrade system performance by reducing the effective signal-to-interference-plus-noise ratio (SINR). Frequency-independent IQMM (FI-IQMM) may originate from imbalances at mixers, while frequency-dependent IQMM (FD-IQMM) may be caused by mismatch between overall frequency responses on the I and Q paths. In some embodiments, only frequency-independent IQMM (FI-IQMM) may be compensated. However, in some applications such as wideband systems (e.g., mmWave systems). FI-IQMM compensation alone may not provide adequate performance. Thus, some of the inventive principles of this application relate to techniques for providing FD-IQMM compensation for quadrature up-converter transmitters. Moreover, TX IQMM may be different than RX IQMM. Therefore, in some embodiments, calibration methods for a TX path according to this disclosure may be different than that those for an RX path. 
       FIG.  1    illustrates an example embodiment of an IQ transmitter that may be used to implement any of the TX IQMM estimation and/or compensation techniques according to this disclosure. The transmitter  100  illustrated in  FIG.  1    may include an I signal path including a digital-to-analog converter (DAC)  104 , a low-pass filter  108  having an impulse response h ITX (t), and a mixer  112 . The transmitter  100 , which may also be referred to as a TX path, may also include a Q signal path including a DAC  106 , a low-pass filter  110  having an impulse response h QTX (t), and a mixer  114 . The mixers  112 ,  114  and filters  108 ,  110 , along with summing circuit  116 , may collectively form an up-converter. The transmitter  100  may further include an IQMM pre-compensator  118 . 
     In the transmitter, g TX ≠1 and ϕ TX ≠0 may denote the TX gain and phase mismatches, respectively, that may create frequency-independent IQ mismatch (FI-IQMM) at the transmitter. The mismatch between the overall frequency responses h ITX (t) and h QTX (t) in the I and Q paths of the TX path may create FD-IQMM in the TX path, that is, h ITX (t)≠h QTX (t). 
     The baseband equivalent of the upconverted signal in the TX path  100  (at the output of the mixers) in the frequency-domain may be given by
 
 Z   TX ( f )= G   1TX ( f ) U ( f )+ G   2TX ( f ) U *(− f ),  (1)
 
where U(f) may be the frequency response of the desired baseband (BB) signal at the input of the analog baseband (ABB) filters  108  and  110  in the TX path, and G 1TX (f) and G 2TX (f) may be defined as
 
     
       
         
           
             
               
                 
                   
                     
                       
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     In Equations (2). H ITX (f) and H QTX (f) may denote the frequency responses of filter  108  (h ITX (t)) and filter  110  (h QTX (t)), respectively. In Equation (1), G 1TX (f)U(f) may represent a desired TX signal, and G 2TX (f)U*(−f) may represent a TX image signal. Without any IQMM, (g TX =1, ϕ TX =0, and h 1TX (t)=h QTX (t)), G 2TX (f), and consequently, the second term in Equation (1) may become zero. Thus, In some embodiments, G 1TX (f) may represent a desired frequency response of the transmit path, and G 2TX (f) may represent a frequency response of the transmit path due to IQMM. 
     In some embodiments according to this disclosure, the effects of IQMM in the transmitter  100  may be compensated by estimating one or more IQMM parameters in the transmitter, and then using the estimated IQMM parameters to determine pre-compensation parameters. 
     The one or more IQMM parameters may include any parameters that may be affected by IQMM in the TX path such as gain mismatch g TX , phase mismatch ϕ TX , filters h ITX (t) and h QTX (t) (and/or their frequency responses H ITX (f) and H QTX (f)), G 1TX (f), G 2TX (f), V TX (f) (as described below), and/or the like. In some example embodiments described below, the parameters ϕ TX  and V TX (f) may be used as the IQMM parameters because, for example, they may reduce the complexity and/or effort involved with mathematical derivations. However, other IQMM parameters may be used according to this disclosure. For example, in some example embodiments, G 1TX (f) and G 2TX (f) may be used as IQMM parameters which may be estimated and then used to determine pre-compensation parameters. 
     The pre-compensation parameters may be any parameters that may determine how the IQMM pre-compensator  118  may affect the IQMM in the TX path  100 . An example of pre-compensation parameters may be coefficients for the IQMM pre-compensator  118  (IQMC coefficients) which may shape the BB signal s[n]=s I [n]+js Q [n] so as to reduce or eliminate an image component in the upconverted signal z TX (t). Examples of IQMC coefficients that may be obtained based on estimated IQMM parameters are described below. 
     In some embodiments, the IQMM parameter V TX (f) mentioned above, which may depend on the TX gain and filter mismatches, may be defined as follows 
     
       
         
           
             
               
                 
                   
                     
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     Various calibration algorithms described herein may be used to estimate phase mismatch ϕ TX  and V TX (f) for continuous-time frequencies f=±f 1 , . . . , ±f K  across a desired frequency band. The estimates of ϕ TX  and V TX (f) may then be used to obtain IQ mismatch compensator (IQMC) coefficients for the pre-compensator  118  to reduce TX FD-IQMM. 
       FIG.  2    illustrates an example embodiment of a complex-valued pre-compensator (CVC) structure for which coefficients may be estimated according to this disclosure. The embodiment illustrated in  FIG.  2    may include an integer delay element  200  having a delay T D , a complex conjugate block  202 , a complex-valued filter  204  having an impulse response w TX [n], and a summing circuit  206 . 
     Values for coefficients for the pre-compensator illustrated in  FIG.  2   , which may fully or partially remove TX FD-IQMM from the transmitter  100  illustrated in  FIG.  1   , may then be given by 
                       W   TX   opt     ⁡     (   f   )       =         -         G     2   ⁢   TX       ⁡     (   f   )           G     1   ⁢   TX       ⁡     (   f   )           ⁢     e       -     j   ⁢   2π       ⁢           ⁢     fT   D           =         1   -         V   TX     ⁡     (   f   )       ⁢     e     -       j   ⁢   ϕ     TX               1   +         V   TX     ⁡     (   f   )       ⁢     e     -       j   ⁢   ϕ     TX               ⁢     e       -     j   ⁢   2π       ⁢           ⁢     fT   D                     (   4   )               
where W TX   opt (f) may denote the frequency response of filter w TX [n]. From Equation (4), it may be apparent that optimal responses of IQMC coefficients may involve knowledge of ϕ TX  and/or V TX (f), which may be estimated, for example, using any of the techniques disclosed herein.
 
     In some embodiments, and depending on the implementation details, the methods, expressions, and/or the like disclosed herein may provide optimal values, and thus, the designator “opt” may be used. However, the inventive principles are not limited to embodiments in which optimal values may be obtained, and the use of “opt” or “optimal” is not limited to methods, expressions, and/or the like that may provide optimal values. 
     Some example embodiments of the CVC structure illustrated in  FIG.  2    may include any of the following implementation details. The complex-valued filter  204  having the impulse response w TX [n] may be implemented, for example, as a finite impulse response (FIR) filter. The complex conjugate block  202  may be configured to output the complex conjugate of the signal s[n] as, for example, s[n]*=s I [n]−js Q [n]. The integer delay element  200  having the delay T D  may be configured to create a delay in the input signal as, for example, s[n−T D ]. 
     The CVC structure illustrated in  FIG.  2    is provided as an example for purposes of illustrating the inventive principles of this disclosure, but other IQMM pre-compensation structures and/or combinations thereof, may be used. For example, in some embodiments, a real-valued compensator (RVC) architecture may be use. 
     RX Feedback Path 
       FIG.  3    illustrates an embodiment of a system that may be used to implement TX FD-IQMM calibration using an RX feedback path according to this disclosure. The embodiment illustrated in  FIG.  3    may include a TX path  300 , an RX path  302 , a feedback connection  304 , and a signal processing unit  306 . The TX path  300  may include a pre-compensator  308 , a digital-to-analog converter (DAC)  310 , an up-converter  314 , and a radio frequency (RF) transmission block  316 . The RX path  302  may include an RF reception block  318 , a down-converter  320 , and an analog-to-digital converter (ADC)  324 . In some embodiments, the RX path  302  may further include a compensator (not shown). The signal processing unit  306  may include a signal generator  328 , a signal capture unit  330 , and a signal processor  332 . 
     The feedback connection  304  may be implemented with any suitable apparatus such as switches, couplers, conductors, transmission lines, filters, and/or the like. The feedback connection  304  may be coupled to the TX path  300  at any location after the up-converter  314 . The feedback connection  304  may be coupled to the RX path  302  at any location before the down-converter  320 . In some embodiments, some or all of the feedback connection  304  may be integral with the TX path  300  and/or the RX path  302 . 
     The TX path  300  and the RX path  302  may each include an I signal path or branch and a Q signal path or branch. The RF transmission block  316  may include various components to transmit an RF signal such as a power amplifier, a band-pass filter, an antenna, and/or the like. The RF reception block  318  may include various components to receive an RF signal such as an antenna, a band-pass filter, a low noise amplifier (LNA) and/or the like. Depending on whether the system is in an operational mode or a calibration mode. IQMM in the TX path  300  may be corrected by the IQMM pre-compensator  308 . 
     In some embodiments, the processor  332  may manage and/or control the overall operation of the system illustrated in  FIG.  3   . This may include controlling the application of one or more pilot signals to the TX path  300 , capturing observations of the up-converted pilot signals through the RX path  302 , performing calculations and/or offloading calculations to other resources, providing estimated coefficients to the TX pre-compensator  308 , controlling the TX pre-compensator  308  during transmission and/or sending of pilot signals, for example, by disabling the pre-compensator  308 , placing it in a transparent or pass-through state, and/or the like. 
     Although various components illustrated in  FIG.  3    may be shown as individual components, in some embodiments, multiple components and/or their functionality may be combined into a smaller number of components. Likewise, a single component and/or its functionality may be distributed among, and/or integrated with, other components. For example, the signal generator  328  and/or signal capture unit  330  may be integrated with, and/or their functions may be performed by, one or more similar components in a modem that may be coupled to the transceiver shown in  FIG.  3   . 
     The components of the signal processing unit  306  may be implemented with hardware, software, and/or any combination thereof. For example, full or partial hardware implementations may include combinational logic, sequential logic, timers, counters, registers, gate arrays, amplifiers, synthesizers, multiplexers, modulators, demodulators, filters, vector processors, complex programmable logic devices (CPLDs), field programmable gate arrays (FPGAs), state machines, data converters such as ADCs and DACs, and/or the like. Full or partial software implementations may include one or more processor cores, memories, program and/or data storage, and/or the like, which may be located locally and/or remotely, and which may be programmed to execute instructions to perform one or more functions of the components of the signal processing unit  306 . 
       FIG.  4    illustrates an example embodiment of a system that may be used to implement TX FD-IQMM calibration using an RX feedback path according to this disclosure. The embodiment illustrated in  FIG.  4    may include a TX path  400 , an RX path  402 , and an RX feedback connection  403 . The TX path  400 , which may be similar to the transmitter  100  illustrated in  FIG.  1   , may include an I signal path including a DAC  404 , a low-pass filter  408  having an impulse response h ITX (t), and a mixer  412 . The TX path  400  may also include a Q signal path including a DAC  406 , a low-pass filter  410  having an impulse response h QTX (t), and a mixer  414 . The mixers  412  and  414  and filters  408  and  410 , along with summing circuit  416 , may collectively form an up-converter. The TX path  400  may further include an IQMM pre-compensator  418 . 
     The RX path  402  may include an I signal path including a mixer  426 , a low-pass filter  430  having an impulse response h IRX (t), and an ADC  434 . The RX path  402  may also include a Q signal path including a mixer  428 , a low-pass filter  432  having an impulse response h QRX (t), and an ADC  436 . The mixers  426  and  428  and filters  430  and  432  may collectively form a down-converter. In some embodiments, the RX path  402  may further include an IQMM compensator (not shown) which may be disabled or placed in a pass-through state during a calibration operation. 
     In some embodiments, during a calibration operation. IQMM pre-compensator  418  may be disabled or placed in a pass-through mode such that IQMC may be unity, and therefore U(f)=S(f). 
     To estimate the IQMM parameters ϕ TX  and V TX (f), a single-tone signal may be applied at baseband of the TX path  400  at frequency f k , that is, U(f)=A TX δ(f−f k ) where A TX  may be an unknown scaling factor that may account for gain and/or delay of the path between the TX baseband signal generation and the input of the ABB filters  408  and  410 . The IQMM impaired up-converted signal may be observed by capturing the frequency response of the down-converted signal through the RX feedback path at the principal and image frequencies (f k  and −f k ), which may be denoted by R 1,k   R(f k ) and R 2,k   R(−f k ). Next, a single-tone signal at frequency −f k , that is. U′(f)=A* TX δ(f+f k ), may be sent through the TX path  400 , and the down-converted signal at frequencies −f k  and f k  may be denoted by R 3,k =R′(−f k ) and R 4,k =R′(f k ), respectively. Collecting all of the observations may provide the following set of equations
 
 R   1,k   =A   TX   A   RX   G   1TX ( f   k )
 
 R   2,k   =A*   TX   A   RX   G   2TX (− f   k )
 
 R   3,k   =A*   TX   A   RX   G   1TX (− f   k )
 
 R   4,k   =A   TX   A   RX   G   2TX ( f   k )  (5)
 
where A RX  may denote the gain and/or delay from the RX ABB filters  430  and  432  to the RX BB. In some embodiments, the four Equations (5) may be time-aligned for correct estimation of IQMM parameters.
 
       FIG.  5    illustrates example embodiments of spectral plots of transmitted and captured (observed) signals corresponding to the Equations (5). 
     The single-tone signal (e.g., f k ) may be swept across the channel band for all selected frequencies to obtain estimates of ϕ TX  and V TX (f) using Equations (5) as follows 
                         ϕ   ^     TX     =       1     2   ⁢   K       ⁢       ∑     k   =   1     K     ⁢     angle   ⁡     (         T   *     ⁡     (     -     f   k       )         T   ⁡     (     f   k     )         )             ,     
     ⁢           V   ^     TX     ⁡     (   f   )       =       T   ⁡     (   f   )       ⁢     e       +   j     ⁢       ϕ   ^     TX             ,     
     ⁢     f   =     ±     f   1         ,   ⋯   ,     ±     f   K               (   6   )               
where
 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           f 
                           k 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           R 
                           
                             1 
                             , 
                             k 
                           
                         
                         + 
                         
                           R 
                           
                             4 
                             , 
                             k 
                           
                         
                       
                       
                         
                           R 
                           
                             1 
                             , 
                             k 
                           
                         
                         - 
                         
                           R 
                           
                             4 
                             , 
                             k 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             f 
                             k 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           R 
                           
                             3 
                             , 
                             k 
                           
                         
                         + 
                         
                           R 
                           
                             2 
                             , 
                             k 
                           
                         
                       
                       
                         
                           R 
                           
                             3 
                             , 
                             k 
                           
                         
                         - 
                         
                           R 
                           
                             2 
                             , 
                             k 
                           
                         
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   
                     ⋯ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       K 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In some implementations of the calibration algorithm described above, the IQMM at the RX feedback path may be assumed to be zero. In some other implementations, the RX feedback path may introduce RX IQMM into the observations as well, which may degrade the estimation accuracy of the TX IQMM parameters. 
     In some embodiments, either or both of the two techniques described below may reduce or eliminate the effects of IQMM in the RX feedback path on observations of up-converted pilot signals according to this disclosure. 
     In a first technique according to this disclosure, RX FD-IQMC may be calibrated using separate local oscillators (LOs) for the TX and RX paths in loopback mode (e.g., sweeping the TX LO and using a DC tone at BB of the TX path while keeping RX LO fixed). Next, BB TX tones may be swept across frequency keeping both the TX LO and RX LO fixed at the same frequency. The TX FD-IQMC coefficients may then be determined. In some embodiments, an additional step may be added to post-process the received signal R(f) to remove the effect of RX-IQMM before estimation of ϕ TX  and V TX (f). 
     In a second technique according to this disclosure, a frequency shift may be created between the LOs of the TX and RX paths such that the RX-IQMM may not interfere with the principal and mirror signals of the TX path. In some embodiments, the frequency shift between the LOs may be kept relatively small, for example, to preserve the approximate symmetry of the ABB filter response that the TX principal and image signals may observe. 
     Envelope Detector 
       FIG.  6    illustrates an embodiment of a system that may be used to implement TX FD-IQMM calibration using an envelope detector according to this disclosure. The system illustrated in  FIG.  6    may include a TX path  600  and a signal processing unit  606  that may be constructed and/or operate in a manner similar to those illustrated in  FIG.  3   . Specifically, the TX path  600  may include a pre-compensator  608 , a digital-to-analog converter (DAC)  610 , an up-converter  614 , and a radio frequency (RF) transmission block  616 . The signal processing unit  606  may include a signal generator  628 , a signal capture unit  630 , and a signal processor  632 . 
     The system illustrated in  FIG.  6    may further include an envelope detector  640  and a signal return path  642 . The envelope detector  640 , which may be implemented using any suitable apparatus including diodes, filters, and/or the like, may be coupled to the TX path  600  at any location after the up-converter  614 . The return signal path  642  may include any suitable apparatus such as switches, couplers, conductors, transmission lines, filters, data converters, and/or the like. 
     In some embodiments, the envelope detector  640  may provide an output having a form, for example, of y(t)=|z(t)| 2 . In some embodiments, some or all of the envelope detector  640  may be integral with the TX path  600 . 
     In some embodiments, the  640  envelope detector may output the envelope of the IQMM impaired up-converted signal and feed it back to the signal processing unit  606  without going through a mixer. Thus, the captured signal may only contain TX IQMM without any RX impairments. Although the return signal path  642  is not limited to any specific implementation details, in some embodiments, either the I or Q signal path downstream of a multiplier in a quadrature receiver may be used as the return signal path. This may be convenient, for example, in a transceiver system in which the RX path already exists. 
       FIG.  7    illustrates an example embodiment of a system that may be used to implement TX FD-IQMM calibration using an envelope detector according to this disclosure. The system illustrated in  FIG.  7    may include a TX path  700 , an RX path  702 , and an envelope detector  740 . 
     The TX path  700 , which may be similar to the TX path  400  illustrated in  FIG.  4   , may include an I signal path including a DAC  704 , a low-pass filter  708  having an impulse response h ITX (t), and a mixer  712 . The TX path  700  may also include a Q signal path including a DAC  706 , a low-pass filter  710  having an impulse response h QTX (t), and a mixer  714 . The mixers  712  and  714  and filters  708  and  710 , along with summing circuit  716 , may collectively form an up-converter. The TX path  700  may further include an IQMM pre-compensator  718 . 
     The RX path  702 , which may be similar to the RX path  402  illustrated in  FIG.  4   , may include an I signal path including a mixer  726 , a low-pass filter  730  having an impulse response h IRX (t), and an ADC  734 . The RX path  702  may also include a Q signal path including a mixer  728 , a low-pass filter  732  having an impulse response h QRX (t), and an ADC  736 . The mixers  726  and  728  and filters  730  and  732  may collectively form a down-converter. In some embodiments, the RX path  702  may further include an IQMM compensator which may be disabled or placed in a pass-through state during a calibration operation. 
     The envelope detector  740  may be connected to the TX path  700  at any location after the up-conversion unit. It may also be connected to the RX path  702  at any location after the mixers  726  and  728 . In the embodiment illustrated in  FIG.  7   , the envelope detector  740  is connected to the I path of the RX path, but it may be connected to the Q side as well. 
     Embodiments of three different methods of estimating TX IQMM using an envelope detector are described below in the context of the example embodiment illustrated in  FIG.  7   . These methods, however, are not limited to these or any other system implementation details. 
     Method 1 
     In some embodiments, this method may seek to obtain single-tap pre-compensator filter coefficients that may cancel IQMM at frequencies±f 1 , . . . , ±f K . These coefficients may then be used to estimate IQMM parameters ϕ TX  and V TX (f). 
     Referring to  FIG.  8   , in some embodiments, a single-tone signal at frequency −f k  sent at baseband may produce an output at the envelope detector having a component at frequency 2f k  if there is any IQMM in the TX chain. Therefore, single-tap pre-compensator coefficients (in some implementations, the best or optimal coefficients) for frequency f k  may be found by sending a single-tone signal from baseband of TX at frequency −f k , sweeping pre-compensator coefficients (in some implementations one tap may be adequate) and choosing the coefficients that may provide the lowest power at the output of the envelope detector at twice the frequency of the BB signal. i.e., 2f k . For a single-tone signal sent at frequency −f k , the output of the envelope detector path may be denoted by r(t), and its response (ignoring higher frequency components) may be given by: 
     
       
         
           
             
               
                 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                          
                         
                           Re 
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   z 
                                   TX 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     
                                       LO 
                                       TX 
                                     
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                             } 
                           
                         
                          
                       
                       2 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               A 
                               TX 
                             
                             ⁢ 
                             
                               
                                 G 
                                 
                                   1 
                                   ⁢ 
                                   TX 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     f 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       + 
                       
                         
                            
                           
                             
                               A 
                               TX 
                               * 
                             
                             ⁢ 
                             
                               
                                 G 
                                 
                                   2 
                                   ⁢ 
                                   TX 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   f 
                                   k 
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       + 
                       
                         2 
                         ⁢ 
                         Re 
                         ⁢ 
                         
                           
                             { 
                             
                               
                                 A 
                                 TX 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   G 
                                   
                                     1 
                                     ⁢ 
                                     TX 
                                   
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     
                                       f 
                                       k 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   G 
                                   
                                     2 
                                     ⁢ 
                                     TX 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     f 
                                     k 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     f 
                                     k 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                             } 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     The frequency response of the envelope detector output at frequency 2f k  may be given by
 
 R ( f )| f=2   fk=A   TX   2   G*   1TX (− f   k ) G   2TX ( f   k ).  (9)
 
     In the absence of IQMM, G 2TX (f k ) may be zero and thus R(2f k ) in Equation (9) may become zero. By performing a search of pre-compensator coefficients, one-tap pre-compensator settings. i.e., w TX [n]=w TX,0 ×δ[n], may be obtained such that R(2f k ) may become zero and cancel IQMM at frequency f k . After sweeping f k  and obtaining the IQMC coefficients (e.g., optimal coefficients) over all frequency tones denoted by w TX,0   opt (f) for T D =0, then ϕ TX  and V TX (f) may be estimated as follows for a CVC structure 
     
       
         
           
             
               
                 
                   
                     
                       
                         ϕ 
                         ^ 
                       
                       TX 
                     
                     = 
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           K 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           K 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           angle 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   1 
                                   + 
                                   
                                     
                                       w 
                                       
                                         TX 
                                         , 
                                         0 
                                       
                                       opt 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         f 
                                         k 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   1 
                                   + 
                                   
                                     
                                       w 
                                       
                                         TX 
                                         , 
                                         0 
                                       
                                       
                                         opt 
                                         * 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         - 
                                         
                                           f 
                                           k 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               × 
                               
                                 
                                   1 
                                   - 
                                   
                                     
                                       w 
                                       
                                         TX 
                                         , 
                                         0 
                                       
                                       
                                         opt 
                                         * 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         - 
                                         
                                           f 
                                           k 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   1 
                                   - 
                                   
                                     
                                       w 
                                       
                                         TX 
                                         , 
                                         0 
                                       
                                       opt 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         f 
                                         k 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           V 
                           ^ 
                         
                         TX 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           
                             
                               w 
                               
                                 TX 
                                 , 
                                 0 
                               
                               opt 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                         
                           1 
                           + 
                           
                             
                               w 
                               
                                 TX 
                                 , 
                                 0 
                               
                               opt 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                         e 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ϕ 
                               ^ 
                             
                             TX 
                           
                         
                       
                     
                   
                   , 
                   
                     f 
                     = 
                     
                       ± 
                       
                         f 
                         1 
                       
                     
                   
                   , 
                   ⋯ 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     ± 
                     
                       
                         f 
                         K 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In some embodiments, a search of pre-compensator coefficients may be implemented as an extensive or exhaustive search. For example, a search may be conducted over a wide range of pre-compensator settings and/or frequency tones at fixed intervals. In some embodiments, a search may be performed in stages. For example, an initial search may be conducted on a relatively coarse grid of pre-compensator settings and/or frequency tones over a wide range at wider intervals. One or more additional searches may then be performed on a finer grid at smaller intervals over one or more smaller ranges based on the results of the coarse search. 
     Method 2 
     In some embodiments, this method may estimate the IQMM parameters for a given frequency f k  directly, for example, by sending single-tone signals at f k  and −f k  separately using two different pre-compensator coefficients and/or settings. The envelope detector outputs at frequency 2f k  for these measurements may then be combined and solved using, for example, a quadratic equation in closed form to obtain the frequency-dependent gain and phase mismatches at f k . Then the IQMM parameters ϕ TX  and V TX (f) may be found, for example, as simple functions of the frequency-dependent gain and phase mismatches for each frequency f k . 
     Some example implementation details may be as follows. A single-tone signal at frequencies f k  and −f k  may be applied separately at BB to a TX path without any IQMC. e.g., W TX [n]=0, for the CVC architecture illustrated in  FIG.  2   , and the frequency response of the envelope detector output at frequency 2f k  may be denoted by Y 1,k  and Y 2,k  respectively. Another set of pre-compensation parameters w[n] with a delay element of T D =0 may be chosen and applied, and a single-tone signal at frequency f k  may be sent. The frequency response of the envelope detector output at frequency 2f k  may be denoted by Y 3,k . This may result in the following equations 
                           ⁢         Y     1   ,   k       =       A   TX   2     ⁢       G     1   ⁢   TX       ⁡     (     f   k     )       ⁢       G     2   ⁢   TX     *     ⁡     (     -     f   k       )           ,     
     ⁢           ⁢       Y     2   ,   k       =       A   TX   2     ⁢       G     1   ⁢   TX     *     ⁡     (     -     f   k       )       ⁢       G     2   ⁢   TX       ⁡     (     f   k     )           ,     
     ⁢       Y     3   ,   k       =         A   TX   2     4     [           G     1   ⁢   TX       ⁡     (     f   k     )       ⁢       G     2   ⁢   TX     *     ⁡     (     -     f   k       )       ⁢     J   1   2       +         G     2   ⁢   TX       ⁡     (     f   k     )       ⁢       G     1   ⁢   TX     *     ⁡     (     -     f   k       )       ⁢     J   2   2       +     (           G     1   ⁢   TX       ⁡     (     f   k     )       ⁢       G     1   ⁢   TX     *     ⁡     (     -     f   k       )         +         G     2   ⁢   TX       ⁡     (     f   k     )       ⁢       G     2   ⁢   TX     *     ⁡     (     -     f   k       )       ⁢     J   1     ⁢     J   2         ]                     (   11   )               
where J 1  and J 2  may be known values that may be defined as follows
 
 J   1 =1,
 
 J   2   =W*   TX (− f   k ).  (12)
 
     Equations (11) may be reformulated using the relationship V TX (f k )=V TX (−f k ) and Equations (2) and (3) as 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Y 
                         
                           1 
                           , 
                           k 
                         
                       
                       = 
                       
                         
                           γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   V 
                                   TX 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     f 
                                     k 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     TX 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 V 
                                 TX 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   f 
                                   k 
                                 
                                 ) 
                               
                             
                             - 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   TX 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         Y 
                         
                           2 
                           , 
                           k 
                         
                       
                       = 
                       
                         
                           γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   V 
                                   TX 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     f 
                                     k 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     TX 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 V 
                                 TX 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   f 
                                   k 
                                 
                                 ) 
                               
                             
                             - 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   TX 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         Y 
                         
                           3 
                           , 
                           k 
                         
                       
                       = 
                       
                         
                           γ 
                           4 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 J 
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 Y 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                             
                             + 
                             
                               
                                 J 
                                 2 
                                 2 
                               
                               ⁢ 
                               
                                 Y 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     J 
                                     1 
                                   
                                   ⁢ 
                                   
                                     J 
                                     2 
                                   
                                 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             V 
                                             TX 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               f 
                                               k 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               TX 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             V 
                                             TX 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               f 
                                               k 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           e 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               TX 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             V 
                                             TX 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               f 
                                               k 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               TX 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             V 
                                             TX 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               f 
                                               k 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           e 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               TX 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         where 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         γ 
                       
                       = 
                       
                         
                           A 
                           TX 
                           2 
                         
                         ⁢ 
                         
                           / 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               4 
                               ⁢ 
                               
                                 
                                   gH 
                                   QTX 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   f 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Equations (13) may provide six real equations with five real unknowns, i.e., Re{γ}, Im{γ}, Re{V TX (f k )}, Im{V TX (f k )}, ϕ TX , which may be solved to obtain estimates of V TX (f k ) and ϕ TX . IQMM parameter V TX (−f k ) may be estimated as {circumflex over (V)} TX (−f k )={circumflex over (V)}* TX (f k ), which may follow from h ITX (t) and h QTX (t) being real-valued filters that may be conjugate symmetric in the frequency domain. i.e., H ITX (f)=H* ITX (−f) and H QTX (f)=H* QTX (−f). 
     Method 3 
     In some embodiments, this method may involve sending two-tone pilot signals at frequencies f k     1   , f k     2    and −f k     1   , −f k     2    and f k     1   , −f k     2    separately. The envelope detector outputs for these measurements at frequencies 2f k     1   , 2f k     2   , f k     1   ±f k     2    may then be combined and solved in closed form, for example, using two quadratic equations to obtain estimates of ϕ TX  and V TX (f) at f=±f k     1   , ±f k     2    directly. 
     Referring to  FIG.  9   , in some embodiments of this method, a two-tone signal at frequencies f k     1   , f k     2    may be generated and sent at TX baseband, i.e., S(f)=A TX     1   δ(f−f k     1   )+A TX     2   δ(f−f k     2   ), and the time-domain signal may be captured at the output of the envelope detector. The frequency response of the time-domain signal may be denoted as 
                 Y     1   ,   k       =       R   ⁡     (   f   )       ⁢     |     f   =     2   ⁢     f     k   1                 ,       Y     2   ,   k       =       R   ⁡     (   f   )       ⁢     |     f   =     2   ⁢     f     k   2                 ,       Y     3   ,   k       =       R   ⁡     (   f   )       ⁢     |     f   =       f     k   1       +     f     k   2                 ,       Y     4   ,   k       =       R   ⁡     (   f   )       ⁢     |     f   =       f     k   1       -     f     k   2             .             
Next, a multi-tone signal may be sent at frequencies f k     1    and −f k     2   , i.e., S(f)=A TX     1   δ(f−f k     1   )+A* TX     2   δ(f+f k     2   ), and the frequency responses of the envelope detector output at frequencies f k     1   ±f k     2    may be denoted as
 
               Y     5   ,   k       =         R   ⁡     (   f   )       ⁢     ❘     f   =       f     k   1       +     f     k   2             ⁢           ⁢     and   ⁢           ⁢     Y     6   ,   k           =       R   ⁡     (   f   )       ⁢     ❘     f   =       f     k   1       -     f     k   2             .             
Then, a multi-tone signal may be sent at frequencies −f k     1    and −f k     2   . i.e., S(f)=A* TX     1   δ(f+f k     1   )+A* TX     2   δ(f+f k     2   ), and the captured frequency response of the envelope signal at frequencies f k     1   ±f k     2    may be denoted as
 
                   Y     7   ,   k       -     R   ⁡     (   f   )         ⁢     ❘     f   =       f     k   1       +     f     k   2             ⁢           ⁢     and   ⁢           ⁢     Y     8   ,   k           =       R   ⁡     (   f   )       ⁢     ❘     f   =       f     k   1       -     f     k   2             .           
The following parameters may be defined:
 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       1 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             1 
                             ⁢ 
                             TX 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             
                               k 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       x 
                       2 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             1 
                             ⁢ 
                             TX 
                           
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               
                                 f 
                                 k 
                               
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       y 
                       1 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             2 
                             ⁢ 
                             TX 
                           
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               f 
                               
                                 k 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       y 
                       2 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             2 
                             ⁢ 
                             TX 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             
                               k 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       z 
                       1 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             1 
                             ⁢ 
                             TX 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             
                               k 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       z 
                       2 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             1 
                             ⁢ 
                             TX 
                           
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               f 
                               
                                 k 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       1 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           G 
                           
                             2 
                             ⁢ 
                             TX 
                           
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               f 
                               
                                 k 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       2 
                     
                     = 
                     
                       
                         A 
                         
                           TX 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             G 
                             
                               2 
                               ⁢ 
                               TX 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               f 
                               
                                 k 
                                 2 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Combining all of the observations may provide the following set of non-linear equations:
 
 Y   1,k   =x   1   y   1 ,
 
 Y   2,k   =z   1   w   1 ,
 
 Y   3,k   =x   1   w   1   +y   1   z   1 ,
 
 Y   4,k   =x   1   z*   1   +y   1   w*   1 ,
 
 Y   5,k   =x   1   z   2   +y   1   w   2 ,
 
 Y   6,k   =x   1   w*   2   +y   1   z*   2 ,
 
 Y   7,k   =x   2   w   2   +y   2   z   2 ,
 
 Y   8,k   =x   2   z*   2   +y   2   w*   2 .  (15)
 
     This set of 8 equations with 8 unknowns in Equations (15) may be solved, for example, using the following steps: 
     1. 
     
         
         
           
             a. The following parameters may be calculated for l=1, 2 and i=1, 2 
           
         
       
    
     
       
         
           
             
               
                 β 
                 
                   1 
                   , 
                   l 
                 
               
               = 
               
                 
                   
                     Y 
                     
                       3 
                       , 
                       k 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       l 
                     
                     ⁢ 
                     
                       
                         
                           Y 
                           
                             3 
                             , 
                             k 
                           
                           2 
                         
                         - 
                         
                           4 
                           ⁢ 
                           
                             Y 
                             
                               1 
                               , 
                               k 
                             
                           
                           ⁢ 
                           
                             Y 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                       
                     
                   
                 
                 2 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 β 
                 
                   2 
                   , 
                   i 
                 
               
               = 
               
                 
                   
                     Y 
                     
                       4 
                       , 
                       k 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       i 
                     
                     ⁢ 
                     
                       
                         
                           Y 
                           
                             4 
                             , 
                             k 
                           
                           2 
                         
                         - 
                         
                           4 
                           ⁢ 
                           
                             Y 
                             
                               1 
                               , 
                               k 
                             
                           
                           ⁢ 
                           
                             Y 
                             
                               2 
                               , 
                               k 
                             
                             * 
                           
                         
                       
                     
                   
                 
                 
                   2 
                   ⁢ 
                   
                     Y 
                     
                       2 
                       , 
                       k 
                     
                     2 
                   
                 
               
             
           
         
       
         
         
           
             b. {circumflex over (x)} 1 , ŷ 1 , {circumflex over (z)} 1 , and ŵ 1  may be calculated as 
           
         
       
    
     
       
         
           
             
               
                 
                   x 
                   ^ 
                 
                 1 
               
               = 
               
                 
                    
                   
                     
                       β 
                       
                         1 
                         , 
                         
                           l 
                           k 
                         
                       
                     
                     ⁢ 
                     
                       β 
                       
                         2 
                         , 
                         
                           i 
                           k 
                         
                       
                       * 
                     
                   
                    
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   w 
                   ^ 
                 
                 1 
               
               = 
               
                 
                   β 
                   
                     1 
                     , 
                     
                       l 
                       k 
                     
                   
                 
                 
                   
                     x 
                     ^ 
                   
                   1 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   y 
                   ^ 
                 
                 1 
               
               = 
               
                 
                   Y 
                   
                     1 
                     , 
                     k 
                   
                 
                 
                   
                     x 
                     ^ 
                   
                   1 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   z 
                   ^ 
                 
                 1 
               
               = 
               
                 
                   Y 
                   
                     2 
                     , 
                     k 
                   
                 
                 
                   
                     w 
                     ^ 
                   
                   1 
                 
               
             
           
         
       
         
         
           
             where 
           
         
       
    
     
       
         
           
             
               i 
               k 
             
             = 
             
               
                 
                   
                     argmax 
                     i 
                   
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         β 
                         
                           2 
                           , 
                           i 
                         
                       
                        
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   l 
                   k 
                 
               
               = 
               
                 
                   argmin 
                   l 
                 
                 ⁡ 
                 
                   ( 
                   
                     lm 
                     ⁢ 
                     
                       { 
                       
                          
                         
                           
                             β 
                             
                               1 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             β 
                             
                               2 
                               , 
                               
                                 i 
                                 k 
                               
                             
                             * 
                           
                         
                          
                       
                       } 
                     
                   
                   ) 
                 
               
             
           
         
       
         
         
           
             c. {circumflex over (x)} 2 , ŷ 2 , {circumflex over (z)} 2 , and ŵ 2  may be calculated as 
           
         
       
    
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         
                           z 
                           ^ 
                         
                         2 
                       
                     
                   
                   
                     
                       
                         
                           w 
                           ^ 
                         
                         2 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               ^ 
                             
                             1 
                           
                         
                         
                           
                             
                               y 
                               ^ 
                             
                             1 
                           
                         
                       
                       
                         
                           
                             
                               y 
                               ^ 
                             
                             1 
                             * 
                           
                         
                         
                           
                             
                               x 
                               ^ 
                             
                             1 
                             * 
                           
                         
                       
                     
                     ] 
                   
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           Y 
                           
                             5 
                             , 
                             k 
                           
                         
                       
                     
                     
                       
                         
                           Y 
                           
                             6 
                             , 
                             k 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 [ 
                 
                   
                     
                       
                         
                           x 
                           ^ 
                         
                         2 
                       
                     
                   
                   
                     
                       
                         
                           y 
                           ^ 
                         
                         2 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               w 
                               ^ 
                             
                             2 
                           
                         
                         
                           
                             
                               z 
                               ^ 
                             
                             2 
                           
                         
                       
                       
                         
                           
                             
                               z 
                               ^ 
                             
                             2 
                             * 
                           
                         
                         
                           
                             
                               w 
                               ^ 
                             
                             2 
                             * 
                           
                         
                       
                     
                     ] 
                   
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           Y 
                           
                             7 
                             , 
                             k 
                           
                         
                       
                     
                     
                       
                         
                           Y 
                           
                             8 
                             , 
                             k 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
         
         
           
             d. T k     r    and T −k     r    may be calculated as 
           
         
       
    
                 T     k   1       ⁢     =   Δ     ⁢           x   ^     1     +       y   ^     2             x   ^     1     -       y   ^     2           ,     
     ⁢       T     -     k   1         ⁢     =   Δ     ⁢           x   ^     2   *     +       y   ^     1   *             x   ^     2   *     -       y   ^     1   *           ,     
     ⁢       T     k   2       ⁢     =   Δ     ⁢           z   ^     1     +       w   ^     1             z   ^     1     -       w   ^     1           ,     
     ⁢       T     -     k   2         ⁢     =   Δ     ⁢           z   ^     2   *     +       w   ^     1   *             z   ^     2   *     -       w   ^     1   *                 
2. After obtaining all T k     1   , T k     2   , T −k     1   , and T −k     2    where
 
     
       
         
           
             
               k 
               ∈ 
               
                 { 
                 
                   1 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     K 
                     2 
                   
                 
                 } 
               
             
             , 
           
         
       
         
         
           
             a. ϕ TX  may be estimated as 
           
         
       
    
     
       
         
           
             
               
                 ϕ 
                 ^ 
               
               TX 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   K 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   
                     K 
                     2 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       angle 
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             * 
                           
                           
                             T 
                             
                               k 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       angle 
                       ⁡ 
                       
                         ( 
                         
                           
                             T 
                             
                               - 
                               
                                 k 
                                 2 
                               
                             
                             * 
                           
                           
                             T 
                             
                               k 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
         
         
           
             b. Estimates of V TX (f) may be obtained as follows
 
 {circumflex over (V)}   TX ( f   k     r   )= T   k     r     e   j{circumflex over (ϕ)}     TX     ,{circumflex over (V)}   TX (− f   k     r   )= T   −k     r     e   j{circumflex over (ϕ)}     TX   , for  r= 1,2,
 
           
         
       
    
     In some embodiments, f k1 &gt;0 and f k     2   &gt;0 may be chosen such that the frequencies 2f k     1   , 2f k     2   , f k     1   +f k     2   , and |f k     1   −f k     2   | may be distinct. 
     The selection of two-tone pilot signals (and positive and negative frequencies thereof), as well as the resulting envelope detector output signals selected for analysis are for purposes of illustration only, and other combinations of pilot signals and/or output signals may be used. For example, in the second set of signals in  FIG.  9   , −f k     1    and f k     2    may be used instead of f k     1    and −f k     2   . Some unused signals are shown with dotted lines in  FIG.  9   , but in other embodiments, these signals may be used while others may be unused. Although some embodiments may be described in the context of two-tone pilot signals, pilot signals with any number of tones may be used. e.g., three-tone, four-tone, etc. 
     As described above, in some embodiments, one or more of the equations that may be obtained using method  3  may include one or more IQMM parameters of the two frequencies of a two-tone signal. In contrast, in some embodiments using method  2 , each equation may only contain the IQMM of a single-frequency. Thus, in some embodiments, and depending on the implementation details, different sets of equations may be obtained using different methods. 
     Obtaining IQMC Coefficients 
     In some embodiments, after obtaining estimates of ϕ TX  and V TX (f) for f=±f 1 , . . . , ±f K , these estimates may be used to compensate for FD-IQMM in the TX path. In some example embodiments, a least squares (LS) method may be implemented as follows: for a given delay element T D , the parameter W TX   opt (f) given in Equation (4) may be estimated at frequencies f=±f 1 , . . . , ±f K . For example, in an embodiment having a finite impulse response (FIR) filter w TX [n]=Σ i=0   L-1 w TX,i δ[n−i] of length L, the method may obtain the optimal L-tap filter w TX =[w TX,0 , . . . , w TX,L-1 ] T ∈   L×1  that may minimize the least squared (LS) error between W TX (f) and Ŵ TX   opt (f) at frequencies f=±f 1 , . . . , ±f K  as 
                       min       w   TX     ,     T   D         ⁢                W   ^     opt     -     Fw   TX            2       ,           (   16   )               
where Ŵ opt =[Ŵ TX   opt (−f K ), . . . , Ŵ TX   opt (−f 1 ), Ŵ TX   opt (f 1 ), . . . , Ŵ TX   opt (f K )] T  and F=[F 0 , . . . , F L-1 ] is the discrete Fourier transform (DFT) matrix of size 2K×L. In some embodiments, T D  may take values in {0, . . . , L−1}. For a fixed T D , w TX  may be found as ŵ TX,T     D   =pinv(F)Ŵ opt  with a least squared error of LSE T     D   =∥Ŵ opt −Fŵ TX,TD ∥ 2 . Then the optimal T D  and filter coefficients TX may be given by
 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       D 
                       opt 
                     
                     = 
                     
                       
                         argmin 
                         
                           T 
                           D 
                         
                       
                       ⁢ 
                       
                         LSE 
                         
                           T 
                           D 
                         
                       
                     
                   
                   , 
                   
                     
                       
                         w 
                         ^ 
                       
                       TX 
                       opt 
                     
                     = 
                     
                       
                         
                           w 
                           ^ 
                         
                         
                           TX 
                           , 
                           
                             T 
                             D 
                             opt 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Although some techniques have been described in the context of pre-compensator structures such as the one illustrated in  FIG.  2   , the inventive principles are not limited to these examples, and calibration algorithms according to this disclosure may be applied to other IQMC structures as well. Furthermore, techniques other than LS may be used to obtain filter coefficients for IQMC structures based on estimated IQMM parameters, and the methodologies described here are only examples for illustrating the inventive principles. 
     In any of the embodiments disclosed herein, frequency domain signals (e.g., signals R 1,k , . . . , R 4,k  in  FIG.  10   ) may be obtained by capturing the baseband time-domain signal and converting it to a frequency-domain signal using a fast Fourier transform (FFT) as an example. 
       FIG.  10    illustrates an embodiment of method for TX IQMM calibration using an RX feedback path according to this disclosure. The method illustrated in  FIG.  10    may be used, for example, with the system illustrated in  FIG.  4   . The method illustrated in  FIG.  10    may begin at operation  1000 . At operation  1002 , a counter k may be initialized to 1. At operation  1004 , the method may check the value of the counter k. If k is less than or equal to the maximum value K, the method may proceed to operation  1006  where a single-tone pilot signal may be generated at frequency f k  and applied at baseband to the TX path  400 . At operation  1008 , the received pilot signal may be captured at frequencies f k  and −f k  at baseband of the RX path  402  and denoted by R 1,k  and R 2,k , respectively. At operation  1010 , a single-tone pilot signal may be generated at frequency −f k  and applied at baseband to the TX path  400 . At operation  1012 , the received pilot signal may be captured at frequencies −f k  and f k  at baseband of the RX path  402  and denoted by R 3,k  and R 4,k , respectively. 
     At operation  1014 , the method may increment the value of the counter k and return to operation  1004 , where the method may check the value of the counter k. If k is greater than the maximum value K, the method may proceed to operation  1016  where, using the observations for R 1,k , . . . , R 4,k , ∀k, the method may estimate the IQMM parameters ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K . At operation  1018 , the method may use ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K  to estimate coefficients for the TX IQMM pre-compensator  418 . The method may then terminate at operation  1020 . 
     As mentioned above, in some embodiments. R 1,k , . . . , R 4,k  may be obtained by capturing the time-domain signal at BB of the RX path  402  and converting it to a frequency-domain signal, for example, using an FFT. 
       FIG.  11    illustrates an embodiment of a first method for TX IQMM calibration using an envelope detector according to this disclosure. The method illustrated in  FIG.  11    may be used, for example, with the system illustrated in  FIG.  7   . The method illustrated in  FIG.  11    may begin at operation  1100 . At operation  1102 , a counter k may be initialized to 1. At operation  1104 , the method may check the value of the counter k. If k is less than or equal to the maximum value K, the method may proceed to operation  1106  where a new pre-compensator setting may be selected from possible pre-compensator values. At operation  1108 , a single-tone pilot signal may be generated at frequency −f k  and applied at baseband to the TX path  700 . At operation  1110 , the signal at the output of the ABB filter in the envelope detector path at frequency 2f k  may be captured. At operation  1112 , the method may check the power of the captured signal. If the power is a non-negligible value, the method may return to operation  1106 . If the power is zero or a negligible value, the method may proceed to operation  1114  where the optimal value for the pre-compensator settings for frequency f k  may be set to the current settings. At operation  1116 , the procedure may be repeated for the single-tone signal generated at f k . 
     At operation  1118 , the method may increment the value of the counter k and return to operation  1104 , where the method may check the value of the counter k. If k is greater than the maximum value K, the method may proceed to operation  1120  where, using the pre-compensation settings for ±f 1 , . . . , ±f k , the method may estimate the IQMM parameters ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K . At operation  1122 , the method may use ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K  to estimate coefficients for the TX IQMM pre-compensator  718 . The method may then terminate at operation  1124 . 
       FIG.  12    illustrates an embodiment of a second method for TX IQMM calibration using an envelope detector according to this disclosure. The method illustrated in  FIG.  12    may be used, for example, with the system illustrated in  FIG.  7   . The method illustrated in  FIG.  12    may begin at operation  1200 . At operation  1202 , a counter k may be initialized to 1. At operation  1204 , the method may check the value of the counter k. If k is less than or equal to the maximum value K, the method may proceed to operation  1206  where a first pre-compensator setting, for example with no IQMC, may be selected. At operation  1208 , the method may generate and send a single-tone signal at frequency f k  at the BB of transmit path  700 . The signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequency 2f k  and denoted as Y 1,K . In some embodiments, the signal may be captured after the ADCs  734  and  736 . At operation  1210 , the method may generate and send a single-tone signal at frequency −f k  at the BB of transmit path  700 . The signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequency 2f k  and denoted as Y 2,k . At operation  1212 , the method may select a second pre-compensator setting to apply to the TX path  700 . At operation  1214 , the method may generate and send a single-tone signal at frequency f k  at the BB of transmit path  700 . The signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequency 2f k  and denoted as Y 3,k . 
     At operation  1216 , the method may increment the value of the counter k and return to operation  1204 , where the method may check the value of the counter k. If k is greater than the maximum value K, the method may proceed to operation  1218  where, using Y 1,k , Y 2,k , and Y 3,k , for every k, the method may estimate the IQMM parameters ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K . At operation  1220 , the method may use ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K  to estimate coefficients for the TX IQMM pre-compensator  718 . The method may then terminate at operation  1222 . 
       FIG.  13    illustrates an embodiment of a third method for TX IQMM calibration using an envelope detector according to this disclosure. The method illustrated in  FIG.  13    may be used, for example, with the system illustrated in  FIG.  7   . The method illustrated in  FIG.  13    may begin at operation  1300 . At operation  1302 , a counter k may be initialized to 1. At operation  1304 , the method may check the value of the counter k. If k is less than or equal to the maximum value K, the method may proceed to operation  1306  where a two-tone signal may be generated and sent at frequencies f k , f k     2    at baseband of the TX path  700 . At operation  1308 , the signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequencies 2f k     1   , 2f k     2   , f k     1   +f k     2   , f k     1   −f k     2   , and denoted as Y 1,k , Y 2,k , Y 3,k , Y 4,k , respectively. At operation  1310 , a two-tone signal may be generated and sent at frequencies f k     1   , −f k     2    at baseband of the TX path  700 . At operation  1312 , the signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequencies f k     1   +f k     2   , f k     1   −f k     2   , and denoted as Y 5,k , Y 6,k , respectively. At operation  1314 , a two-tone signal may be generated and sent at frequencies −f k     1   , −f k     2   , at baseband of the TX path  700 . At operation  1316 , the signal at the output of ABB filter  730  and  732  in the envelope detector path may be captured at frequencies f k     1   +f k     2   , f k     1   −f k     2   , and denoted as Y 7,k , Y 8,k , respectively. 
     At operation  1318 , the method may increment the value of the counter k and return to operation  1304 , where the method may check the value of the counter k. If k is greater than the maximum value K, the method may proceed to operation  1320  where, using Y 1,k , . . . , Y 8,k , for every k, the method may estimate the IQMM parameters ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K . At operation  1322 , the method may use ϕ TX  and V TX (f), f=±f 1 , . . . , ±f K  to estimate coefficients for the TX IQMM pre-compensator  718 . The method may then terminate at operation  1324 . 
       FIG.  14    illustrates an embodiment of a method of pre-compensating for transmitter IQMM according to this disclosure. The method may begin at operation  1400 . At operation  1402 , the method may send a signal through an up-converter of a transmit path to provide an up-converted signal. At operation  1404 , the method may determine the up-converted signal through a down-converter of a receive feedback path. At operation  1406 , the method may determine one or more IQMM parameters for the transmit path based on the determined up-converted signal, and at operation  1408 , the method may determine one or more pre-compensation parameters for the transmit path based on the one or more IQMM parameters for the transmit path. The method may end at operation  1410 . 
       FIG.  15    illustrates another embodiment of a method of pre-compensating for transmitter IQMM according to this disclosure. The method may begin at operation  1500 . At operation  1502 , the method may send a signal through an up-converter of a transmit path to provide an up-converted signal. At operation  1504 , the method may determine the up-converted signal through an envelope detector. At operation  1506 , the method may determine one or more IQMM parameters for the transmit path based on the determined up-converted signal, and at operation  1508 , the method may determine one or more pre-compensation parameters for the transmit path based on the one or more IQMM parameters for the transmit path. The method may end at operation  1510 . 
     The operations and/or components described with respect to the embodiments illustrated in  FIGS.  14  and  15   , as well as any other embodiments described herein, are example operations and/or components. In some embodiments, some operations and/or components may be omitted and/or other operations and/or components may be included. Moreover, in some embodiments, the temporal and/or spatial order of the operations and/or components may be varied. 
     This disclosure encompasses numerous inventive principles relating to association and authentication for multi access point coordination. These principles may have independent utility and may be embodied individually, and not every embodiment may utilize every principle. Moreover, the principles may also be embodied in various combinations, some of which may amplify the benefits of the individual principles in a synergistic manner. 
     The embodiments disclosed above have been described in the context of various implementation details, but the principles of this disclosure are not limited to these or any other specific details. For example, some functionality has been described as being implemented by certain components, but in other embodiments, the functionality may be distributed between different systems and components in different locations and having various user interfaces. Certain embodiments have been described as having specific processes, steps, etc., but these terms also encompass embodiments in which a specific process, step, etc. may be implemented with multiple processes, steps, etc., or in which multiple process, steps, etc. may be integrated into a single process, step, etc. A reference to a component or element may refer to only a portion of the component or element. 
     The use of terms such as “first” and “second” in this disclosure and the claims may only be for purposes of distinguishing the things they modify and may not indicate any spatial or temporal order unless apparent otherwise from context. A reference to a first thing may not imply the existence of a second thing. Various organizational aids such as section headings and the like may be provided as a convenience, but the subject matter arranged according to these aids and the principles of this disclosure are not limited by these organizational aids. 
     The various details and embodiments described above may be combined to produce additional embodiments according to the inventive principles of this patent disclosure. Since the inventive principles of this patent disclosure may be modified in arrangement and detail without departing from the inventive concepts, such changes and modifications are considered to fall within the scope of the following claims.