Patent ID: 12224893

DETAILED DESCRIPTION

Overview

This disclosure encompasses numerous inventive principles relating to in-phase (I) and quadrature (Q) mismatch (IQMM) in quadrature transceivers. Some of the principles involve techniques for jointly estimating IQMM, which may include frequency-dependent IQMM (FD-IQMM), in both the transmit (TX) and receive (RX) paths of a quadrature transceiver. In some embodiments, two or more pilot signals may be applied at baseband to the TX path which may be coupled to the RX path through a loopback path. Different phase shifts may be applied to the pilot signals, for example, through the use of a phase shifter in the loopback path. The received pilot signals may be observed at baseband of the RX path and analyzed using various disclosed algorithms to estimate the IQMM in both the TX and RX paths.

The observed signals may be formulated as one or more functions of IQMM parameters and/or ratios or other combinations thereof. In some embodiments, systems of equations may be formulated using IQMM parameters to provide enough equations to estimate or solve for the number of unknown variables provided by the parameters. Different equations may be formulated, for example, based on different phase shift values and/or frequencies of the pilot signals. The equations may be estimated and/or solved using techniques such as block coordinate descent (BCD), gradient descent, Newton's method, and/or the like. In some embodiments, IQMM parameters may be estimated and/or solved using division and matrix inversion, for example, in implementations that may ignore one or more image of image signals. In some embodiments, one or more estimates obtained using one technique may be used as initial estimates for another estimation technique.

In various embodiments, one or more of the IQMM parameters may be related to physical aspects of a transceiver such as the frequency response of a filter, a scaling factor that may account for a gain and/or delay associated with transmission and/or receiver links, the gain and/or phase mismatch of mixers, and/or the like. In some embodiments, one or more unknown variables may account for a phase shift applied to a pilot signal. This may enable IQ mismatch to be estimated without knowing one or more values of the phase shift, for example, by estimating or solving for the phase shift.

Some of the inventive principles relate to techniques for enhancing an estimate of a cross multiplication factor for a real-valued compensator (RVC) for the RX path of a quadrature transceiver. Some embodiments may begin with an initial estimate of the cross multiplication factor, and then adjust the cross multiplication factor by correcting for inaccuracies caused by one or more residual RX phase mismatches due to an incorrect initial estimate of the cross multiplication factor. In some embodiments, this may be accomplished by re-estimating one or more values of RX phase mismatch using, for example, one or more RVC compensated pilot signals and an estimated TX phase mismatch.

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.

Joint TX and RX Calibration

FIG.1illustrates an embodiment of a quadrature transceiver that may be used to implement joint TX and RX IQMM calibration according to this disclosure. The embodiment illustrated inFIG.1may include a TX path100, an RX path102, a loopback path104, and a signal processing unit106. The TX path100may include a pre-compensator108, a digital-to-analog converter (DAC)110, an up-converter114, and a radio frequency (RF) transmission block116. The RX path102may include an RF reception block118, a down-converter120, an analog-to-digital converter (ADC)124, and a compensator126. The signal processing unit106may include a signal generator128, a signal capture unit130, and a signal processor132.

The loopback path104may be coupled between the TX path100and the RX path102. The loopback path104may include a phase shifter, but in various embodiments, the phase shifter may alternatively be located in the TX path100, in the RX path102, or may have functionality distributed between multiple paths and/or components. In some embodiments, the phase shifter and/or phase shifting functionality may be located downstream the up-converter114and upstream of the down-converter120. In some embodiments, some or all of the loopback path104may be integral with the TX path100and/or the RX path102.

The TX path100and the RX path102may each include an I path and a Q path. Imbalances or mismatches between the I and Q paths (IQMM) may degrade system performance, for example, by creating interference between the mirror frequencies after down-conversion to baseband in the RX path102and/or after up-conversion to radio frequency (RF) or intermediate frequency (IF) in the TX path100.

IQMM in the TX path100may be corrected by the IQMM pre-compensator108, while IQMM in the RX path102may be corrected by the IQMM compensator126.

Providing effective IQMM compensation, however, may involve obtaining accurate values of coefficients for the pre-compensator108and the compensator126. Obtaining values of these coefficients may involve first determining values of IQMM parameters for the TX path100(TX-IQMM) and the RX path102(RX-IQMM). The TX-IQMM parameters may then be used to obtain coefficients for the pre-compensator108, and the RX-IQMM parameters may be used to obtain coefficients for the compensator126.

To determine values of IQMM parameters, one or more pilot signals (e.g., single-tone or multi-tone signals) may be sent through a pilot signal path which may include the TX path100, the loopback path104, and the RX path102. The pilot signal received at the RX path102may have TX-IQMM and RX-IQMM parameters entangled with each other.

In some embodiments according to this disclosure, RX-IQMM may be estimated from the received pilot signals by ignoring TX-IQMM in the pilot signals. Likewise, TX-IQMM may be estimated by ignoring RX-IQMM in the pilot signals. These estimates may provide sufficient accuracy in some applications. However, in more demanding applications, performance may be degraded to an unacceptable level by ignoring the mismatch in one of the paths. For example, in some applications in which a common local oscillator (LO) may be used to provide both the LOTXsignal to the up-converter114and the LORXsignal to the down-converter120, the transmit signal may suffer from TX-IQMM, which may create an error in the estimation of coefficients for the RX compensator126, which in turn, may degrade the receive path performance. Similarly, the received pilot signal may suffer from RX-IQMM which may create an error in the estimation of the coefficients for the TX pre-compensator108which may also degrade performance.

A further difficulty in determining TX and RX mismatch may arise from the frequency-dependency of some IQMM parameters. For example, in some applications, one or more filters/components in the up-converter114and or down-converter120may be frequency dependent.

In some embodiments according to this disclosure, two or more pilot signals (e.g., single-tone signals) may be sent through the pilot signal path which may include the TX path100, the loopback path104, and the RX path102. Different phase shifts may be applied to different pilot signals, for example, by a phase shifter in the loopback path104. This may enable the system to capture multiple independent observations at the receive path, for example, at both the principle frequency of the pilot signal and its mirror frequency, which may then be used to solve a system of equations (e.g., nonlinear equations) to jointly obtain estimates of both TX-IQMM parameters and RX-IQMM parameters from the same observations.

In some embodiments, the gain and/or phase shift applied by the phase shifter may be estimated as one or more unknowns in the system of equations. This may enable TX and RX IQMM parameters to be estimated without a priori knowledge of the phase shift.

FIG.2illustrates an embodiment of a method compensating for IQMM in a transceiver according to this disclosure. The method illustrated inFIG.2may be used, for example, with the embodiment illustrated inFIG.1. The method may begin at operation200. At operation202, the method may send first and second signals from a transmit path through a loopback path, using a phase shifter to introduce a phase shift in at least one of the first and second signals, to obtain first and second signals received by a receive path. At operation204, the method may use the first and second signals received by the receive path to obtain joint estimates of transmit and receive IQMM, at least in part, by estimating the phase shift. At operation206, the method may compensate for IQMM using the estimates of IQMM. The method may terminate at operation208. In some embodiments, the method may additionally include sending a third signal from the transmit path through the loopback path, using the phase shifter to introduce a phase shill in at least two of the first, second, and third signals, to obtain a third signal received by the receive path, and using the first, second, and third signals received by the receive path to obtain joint estimates of the transmit and receive IQMM, at least in part, by estimating the phase shift.

The operations and/or components described with respect to the embodiments illustrated inFIGS.1and2, 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.

Referring again toFIG.1, the RF transmission block116may include various components to transmit an RF signal such as an additional up-converter for a system having an architecture that may use intermediate frequency (IF) structures, as well as a power amplifier, a band-pass filter, an antenna, and/or the like. The RF reception block118may 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. In an IF system, the RF reception block118may further include an additional down-converter to convert an RF signal to IF.

Although various components illustrated inFIG.1are 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 generator128and/or signal capture unit130may 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 inFIG.1.

The components of the signal processing unit106may 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, 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 unit106.

In some embodiments, the processor132may manage and/or control the overall operation of the system illustrated inFIG.1including controlling the application of one or more pilot signals to the TX path100, capturing observations one or more received pilot signals from the RX path102, controlling the application of one or more phase shifts to a pilot signal, for example, through a phase control signal, performing calculations and/or offloading calculations to other resources, providing estimated coefficients to the TX pre-compensator108and the RX compensator126, controlling the TX pre-compensator108and the RX compensator126during transmission and/or sending of pilot signals, for example, by disabling the pre-compensator108and/or the compensator126, placing them in a transparent or pass-through state, and/or the like.

Some example embodiments of systems, processes, methods, and/or the like illustrating some possible implementation details according to this disclosure are described below. These examples are provided for purposes of illustrating the principles of this disclosure, but the principles are not limited to these embodiments, implementation details, and/or the like. For example, although the embodiment illustrated inFIG.4may be described in the context of a system that implements direct-conversion to RF (zero-IF), the principles may be applied to any other type of quadrature transceiver, for example, a system that may implement IF processing. As a further example, some embodiments may be described in the context of systems having a specific mix of digital and analog signals. However, the principles may also be applied to systems and/or methods having a different mix of digital and analog signals, as well as all analog implementations, and/or the like.

FIG.3illustrates an example embodiment of a phase shifter that may be used to apply a phase shift to one or more pilot signals according to this disclosure. The embodiment illustrated inFIG.3may include an inductor L and a capacitor C arranged in a parallel configuration with a load RL. The phase shifter may be switched into and out of the pilot signal path to provide one or more phase shifts. The amount of phase shift may be controlled by switching one or more phase shifter circuits, switching and/or varying the values of one or more components, and/or the like. The embodiment illustrated inFIG.3may be located in the loopback path104, in the TX path100and/or the RX path102. The phase shifter may be located, for example, after the up-converter114and before the down-converter120. In some implementations, the phase shifter may always be located after the up-converter114and before the down-converter120.

FIG.4illustrates an example embodiment of a quadrature transceiver that may be used to implement joint TX and RX calibration according to this disclosure. The embodiment illustrated inFIG.4may include a TX path400, an RX path402, and a loopback path403. The TX path400may include an I signal path including a DAC404, a low-pass filter408having an impulse response h1TX(t), and a mixer412. The TX path400may also include a Q signal path including a DAC406, a low-pass filter410having an impulse response h2TX(t), and a mixer414. The mixers412,414and filters408,410, along with summing circuit416, may collectively form an up-converter. The TX path400may further include an IQMM pre-compensator418.

The RX path402may include an I signal path including a mixer426, a low-pass filter430having an impulse response h1RX(t), and an ADC434. The RX path402may also include a Q signal path including a mixer428, a low-pass filter432having an impulse response h2RX(t), and an ADC436. The mixers426,428and filters430,432may collectively form a down-converter. The RX path402may further include an IQMM compensator444.

In the embodiment illustrated inFIG.4, that same local oscillator (LO) may drive the mixers in both the TX path400and the RX path402. In the TX path400, gTX≠1 and ϕTX≠0 may denote the TX gain and phase mismatches, respectively, that may create frequency-independent IQ mismatch (FI-IQMM) at the up-converter. The mismatch between the overall impulse responses h1TX(t) and h2TX(t) may create frequency-dependent IQ mismatch (FD-IQMM) in the TX path, that is, h1TX(t)≠h2TX(t).

In the RX path402, the gain and phase mismatches at the RX mixers426and428may be donated by gRX≠1 and ϕRX≠0, respectively, and may create the FI-IQMM at the down-converter. The FD-IQMM on RX path may be caused by mismatch between the impulse responses of h1RX(t) and h2RX(t).

IQMM Parameter Estimation (First Algorithm)

The baseband equivalent of the upconverted signal in the TX path400(at the output of mixers) in frequency-domain may be given by
ZTX(f)=G1TX(f)U(f)+G2TX(f)U*(−f),  (1)

where U(f) may be the frequency response of the clean IQMM-free signal at the input of the analog baseband (ABB) filters408,410in the TX path, and G1TX(f) and G2TX(f) may be defined as

G1⁢TX(f)=H1⁢TX(f)+gTX⁢ej⁢ϕTX⁢H2⁢TX(f)2,G2⁢TX(f)=H1⁢TX(f)-gTX⁢ej⁢ϕTX⁢H2⁢TX(f)2.(2)

In Equations (2), H1TX(f) and H2TX(f) may denote the frequency responses of filter408(h1TX(t)) and filter410(h2TX(t)), respectively. In Equation (1), G1TX(f)U(f) may represent a desired TX signal, and G2TX(f)U*(−f) may represent a TX image signal. Without any IQMM, (gTX=1, ϕTX=0, and h1TX(t)=h2TX(t)), G2TX(f), and consequently, the second term in Equation (1) may become zero.

The effect of IQMM at the output of low-pass filters in the RX path402may be expressed as the addition of a filtered mirror image of the principal frequency and the ideal signal as follows
R(f)=G1RX(f)ZRX(f)+G2RX(f)Z*RX(−f),  (3)

where R(f) may be the received signal at the output of ABB filters430and432, ZRX(f) may be the baseband equivalent of the received signal at the input of the mixers426and428, and G1RX(f) and G2RX(f) may be defined as

G1⁢RX(f)=H1⁢RX(f)+gRX⁢e-j⁢ϕRX⁢H2⁢RX(f)2,G2⁢RX(f)=H1⁢RX(f)-gRX⁢e+j⁢ϕRX⁢H2⁢RX(f)2.(4)

In Equations (4), H1RX(f) and H2RX(f) may denote the frequency responses of filter430(h1RX(t)) and filter432(h2RX(t)), respectively. In Equation (3), G1RX(f)ZRX(f) may represent a desired RX signal, and G2BX(f)Z*RX(−f) may represent interfering image signal due to RX IQMM.

Some embodiments according to this disclosure may obtain coefficients for the IQMM pre-compensator418that may fully or partially cancel the second term in Equation (1). Similarly, some embodiments according to this disclosure may obtain coefficients for the IQMM compensator444that may fully or partially cancel the second term in Equation (3).

To obtain coefficients that may compensate for the effects of IQMM in the TX path400and/or the RX path402, some embodiments according to this disclosure may first estimate the IQMM parameters in TX path400and RX path402and then use the estimated parameters to obtain estimates of IQMM compensator coefficients for the TX and/or RX paths. An example embodiment of such a method may be described as follows.

A pilot signal may be generated and applied at baseband to the TX path400, sent through the loopback path403, and received at baseband in the RX path402. In some embodiments, during the calibration phase, the captured observed signal received at the receive path may be assumed to only be coming from the loopback path, and thus, there may be no signal coming from upstream of the loopback path403shown inFIG.4.

By using the same LO for the mixers in both the TX path400and the RX path402, the received signal may be denoted by RBB(f) and may be given by
RBB(f)=ARXG1RX(f){G1TX(f)U(f)+G2TX(f)U*(−f)}+A*RXG2RX(f){G*1TX(−f)U*(−f)+G*2TX(−f)U(f)},  (5)

where ARXmay denote the gain and delay from the RX ABB filter430,432to the output of ADC434,436, and U(f) may denote the frequency response of the signal at the input of TX ABB filters408,410. This may be used to estimate IQMM parameters, which may then be used to obtain IQMM compensator coefficients. To this end, parameters G1RX(±f), G2BX(±f), G1TX(±f), and G2TX(±f) in Equation (5) may be estimated at pre-selected frequencies f=f1, . . . , fK. To estimate IQMM parameters at the preselected frequencies ±fk, a single-tone pilot signal at frequency fkmay be generated and applied to the TX path400at baseband, upconverted and fed through the loopback path403and received at baseband of the RX path402. A phase shift applied between the mixers in the TX and RX paths400and402may cause an arbitrary gain and/or phase shift, which may be frequency dependent, in the signal. In some embodiments, this phase shift may be applied anywhere after the mixers in the TX path400and before the mixers in the RX path402.

The transmitted signal at frequency fkmay be denoted by U(f)=ATXδ(f−fk), where ATXmay be an unknown scaling factor that may account for gain and/or delay of the path between the TX baseband signal generation and input of the TX ABB filter408,410(h1TX(t) and h2TX(t)). One or more first observations may be made with a phase shift value θ0. Without loss of generality, and to simplify the resulting equations, the phase shift value may be set to θ0=0. However, any other unknown value may be used. The observations at the principal and mirror frequencies fkand −fkat the RX baseband may be denoted as R1,kand R2,k. Another single-tone signal may be sent at frequency −fk, for which observations at the principal and mirror frequencies −fkand fkmay be denoted by R3,kand R4,krespectively. Next, single-tone signals at frequencies fkand −fkmay be sent with applied phase shift values θ1and θ2separately to obtain eight additional observations denoted by R5,k, . . . , R12,k, which may be given by
R1,k=ARXATXG1RX(fk)G1TX(fk)+A*RXATXG2RX(fk)G*2TX(−fk)
R2,k=ARXA*TXG1RX(−fk)G2TX(−fk)+A*RXA*TXG2RX(−fk)G*1TX(fk)
R3,k=ARXA*TXG1RX(−fk)G1TX(−fk)A*RXA*TXG2RX(−fk)G2TX(fk)
R4,k=ARXATXG1RX(fk)G2TX(fk)+A*RXATXG2RX(fk)G*1TX(−fk)
R5,k=β1,kARXATXG1RX(fk)G1TX(fk)+β*2,kA*RXATXG2RX(fk)G*2TX(−fk)
R6,k=β2,kARXA*TXG1RX(−fk)G2TX(−fk)+β*1,kA*RXA*TXG2RX(−fk)G*1TX(fk)
R7,k=β2,kARXA*TXG1RX(−fk)G1TX(−fk)+β*1,kA*RXA*TXG2RX(−fk)G*1TX(fk)
R8,k=β1,kARXATXG1RX(fk)G2TX(fk)+β*2,kA*RXATXG2RX(fk)G*1TX(−fk)
R9,k=β3,kARXATXG1RX(fk)G1TX(fk)+β*4,kA*RXATXG2RX(fk)G*2TX(−fk)
R10,k=β4,kARXA*TXG1RX(−fk)G2TX(−fk)+β*3,kA*RXA*TXG2RX(−fk)G*1TX(fk)
R11,k=β4,kARXA*TXG1RX(−fk)G1TX(−fk)+β*3,kA*RXA*TXG2RX(−fk)G*2TX(fk)
R12,k=β3,kARXATXG1RX(fk)G2TX(fk)+β*4,kA*RXATXG2RX(fk)G*1TX(−fk)  (6)

where β1,k, . . . , β4,kaccount for phase and/or gain change due to a phase shifter. Different factors may be used for positive and negative frequencies since the phase shifter may not have a symmetric frequency response around the center frequency. However, the phase shifter effect in the first four equations of Equations (6) may be omitted since any gain and phase variation may be absorbed by G1TX(±fk) and G2TX(±fk) in Equations (6). In some embodiments, Equations (6) may include one or more nonlinear equations.

FIG.5provides example embodiments of spectral plots of transmitted and captured (observed) signals corresponding to the Equations (6).

In some embodiments, ARXand/or ATXmay account, for example, for gain and delay effects related to the DAC404,406, and/or ADC434,436, such as up sampling, down sampling, anti-aliasing filtering, and/or the like. In some embodiments, β1,k, . . . , β4,kmay be related to the phase and/or gain change due to a phase shifter such that β=rejθwhere r may represent a gain of a phase shifter and θ may represent a phase change due to the phase shifter.

The following parameters may be defined

γ1,k=△ATX⁢G1⁢TX(fk),γ2,k=△ATX⁢G2⁢TX*(-fk),γ3,k=△ATX*⁢G1⁢TX(-fk),γ4,k=△ATX*⁢G2⁢TX*(fk),γ5,k=△ARX⁢G1⁢RX(fk),γ6,k=△ARX*⁢G2⁢RX(fk),γ7,k=△ARX⁢G1⁢RX(-fk),γ8,k=△ARX*⁢G2⁢RX(-fk).(7)

Observations at principal and mirror frequencies may then be rewritten as follows
R1,k=γ1,kγ5,k+γ2,kγ6,k
R2,k=γ*2,kγ7,k+γ*1,kγ8,k
R3,k=γ3,kγ7,k+γ4,kγ8,k
R4,k=γ*4,kγ5,k+γ*3,kγ6,k
R5,k=β1,kγ1,kγ5,k+β*2,kγ2,kγ6,k
R6,k=β2,kγ*2,kγ7,k+β*1,kγ*1,kγ8,k
R7,k=β2,kγ3,kγ7,k+β*1,kγ4,kγ8,k
R8,k=β1,kγ*4,kγ5,k+β*2,kγ*3,kγ6,k
R9,k=β3,kγ1,kγ5,k+β*4,kγ2,kγ6,k
R10,k=β4,kγ*2,kγ7,k+β*3,kγ*1,kγ8,k
R11,k=β4,kγ3,kγ7,k+β*3,kγ4,kγ8,k
R12,k=β3,kγ*4,kγ5,k+β*4,kγ*3,kγ6,k.  (8)

Thus, Equations (8) may provide 12 complex equations with 12 complex unknowns. In some embodiments, the phases θ0, θ1, and θ2may be distinct (e.g., may have sufficient spacing) such that β1,k, . . . , β4,kmay be sufficiently different from 1 and from each other so that the Equations (8) represent independent observations.

For a given set of observations R1,k, . . . , R12,k, the problem of solving γ1,k, . . . , γ8,k, β1,k, . . . , β4,kmay be formulated as follows

xopt=argminx∈ℂ12f⁡(x)2,f⁡(x)=[γ1,k⁢γ5,k+γ2,k⁢γ6,k-R1,kγ2,k*⁢γ7,k+γ1,k*⁢γ8,k-R2,kγ3,k⁢γ7,k+γ4,k⁢γ8,k-R3,kγ4,k*⁢γ5,k+γ3,k*⁢γ6,k-R4,kβ1,k⁢γ1,k⁢γ5,k+β2,k*⁢γ2,k⁢γ6,k-R5,kβ2,k⁢γ2,k*⁢γ7,k+β1,k*⁢γ1,k*⁢γ8,k-R6,kβ2,k⁢γ3,k⁢γ7,k+β1,k*⁢γ4,k⁢γ8,k-R7,kβ1,k⁢γ4,k*⁢γ5,k+β2,k*⁢γ3,k*⁢γ6,k-R8,kβ3,k⁢γ1,k⁢γ5,k+β4,k*⁢γ2,k⁢γ6,k-R9,kβ4,k⁢γ2,k*⁢γ7,k+β3,k*⁢γ1,k*⁢γ8,k-R10,kβ4,k⁢γ3,k⁢γ7,k+β3,k*⁢γ4,k⁢γ8,k-R11,kβ3,k⁢γ4,k*⁢γ5,k+β4,k*⁢γ3,k*⁢γ6,k-R12,k](9)

where the vector of unknowns may be given by
x=[γ1,k, . . . ,γ8,k,β1,k, . . . ,β4,k]T.  (10)

The problem represented by Equations (9) may be solved, for example, using a block coordinate descent (BCD) algorithm, which may minimize the cost function ∥ƒ(x)∥2iteratively along a coordinate block while the other coordinates remain fixed.

In some embodiments using this method, descent in the cost function may be ensured at every iteration. In the BCD algorithm, the unknown variables may be divided into the following three blocks
x1=[γ5,k,γ6,k,γ7,k,γ8,k]T,
x2=[γ1,k,γ2,k,γ3,k,γ4,k]T,
x3=[β1,k,β2,k,β3,k,β4,k]T.  (11)

Each iteration of the BCD solver may consist of three steps (i=1,2,3). At the ith step of the (+1)th iteration, ∥f(, . . . ,, xi,, . . . ,)∥2may be minimized with respect to xi. With the specific choice of coordinate blocks in Equations (11), the cost function at the ith step may become a least squares (LS) problem in terms of xi, which, may be readily solved.

The following parameters may be defined

A1(ℓ)=[γ1,k(ℓ)γ2,k(ℓ)γ4,k(ℓ)*γ3,k(ℓ)*β1,k(ℓ)⁢γ1,k(ℓ)β2,k(ℓ)*⁢γ2,k(ℓ)β1,k(ℓ)⁢γ4,k(ℓ)*β2,k(ℓ)*⁢γ3,k(ℓ)*β3,k(ℓ)⁢γ1,k(ℓ)β4,k(ℓ)*⁢γ2,k(ℓ)β3,k(ℓ)⁢γ4,k(ℓ)*β4,k(ℓ)*⁢γ3,k(ℓ)*],A2(ℓ)=[γ2,k(ℓ)*γ1,k(ℓ)*γ3,k(ℓ)γ4,k(ℓ)β2,k(ℓ)⁢γ2,k(ℓ)*β1,k(ℓ)*⁢γ1,k(ℓ)*β2,k(ℓ)⁢γ3,k(ℓ)β1,k(ℓ)*⁢γ4,k(ℓ)β4,k(ℓ)⁢γ2,k(ℓ)*β3,k(ℓ)*⁢γ1,k(ℓ)*β4,k(ℓ)⁢γ3,k(ℓ)β3,k(ℓ)*⁢γ4,k(ℓ)],(12)B1(ℓ)=[γ5,k(ℓ+1)γ6,k(ℓ+1)γ8,k(ℓ+1)*γ7,k(ℓ+1)*β1,k(ℓ)⁢γ5,k(ℓ+1)β2,k(ℓ)*⁢γ6,k(ℓ+1)β1,k(ℓ)⁢γ8,k(ℓ+1)*β2,k(ℓ)*⁢γ7,k(ℓ+1)*β3,k(ℓ)⁢γ5,k(ℓ+1)β4,k(ℓ)*⁢γ6,k(ℓ+1)β3,k(ℓ)⁢γ8,k(ℓ+1)*β4,k(ℓ)*⁢γ7,k(ℓ+1)*],B2(ℓ)=[γ7,k(ℓ+1)γ8,k(ℓ+1)γk,6(ℓ+1)*γ5,k(ℓ+1)*β2,k(ℓ)⁢γ7,k(ℓ+1)β1,k(ℓ)*⁢γ8,k(ℓ+1)β2,k(ℓ)⁢γ6,k(ℓ+1)*β1,k(ℓ)*⁢γ5,k(ℓ+1)*β4,k(ℓ)⁢γ7,k(ℓ+1)β3,k(ℓ)*⁢γ8,k(ℓ+1)β4,k(ℓ)⁢γ6,k(ℓ+1)*β3,k(ℓ)*⁢γ5,k(ℓ+1)*],(13)C(ℓ)=[γ1,k(ℓ+1)⁢γ5,k(ℓ+1)j⁢γ1,k(ℓ+1)⁢γ5,k(ℓ+1)γ2,k(ℓ+1)⁢γ6,k(ℓ+1)-j⁢γ2,k(ℓ+1)⁢γ6,k(ℓ+1)γ1,k(ℓ+1)*⁢γ8,k(ℓ+1)-j⁢γ1,k(ℓ+1)*⁢γ8,k(ℓ+1)γ2,k(ℓ+1)*⁢γ7,k(ℓ+1)j⁢γ2,k(ℓ+1)*⁢γ7,k(ℓ+1)γ4,k(ℓ+1)⁢γ8,k(ℓ+1)-j⁢γ4,k(ℓ+1)⁢γ8,k(ℓ+1)γ3,k(ℓ+1)⁢γ7,k(ℓ+1)j⁢γ3,k(ℓ+1)⁢γ7,k(ℓ+1)γ4,k(ℓ+1)*⁢γ5,k(ℓ+1)j⁢γ4,k(ℓ+1)*⁢γ5,k(ℓ+1)γ3,k(ℓ+1)*⁢γ6,k(ℓ+1)-j⁢γ3,k(ℓ+1)*⁢γ6,k(ℓ+1)],(14)RA1=[R1,k,R4,k,R5,k,R8,k,R9,k,R12,k]T,RA2=[R2,k,R3,k,R6,k,R7,k,R10,k,R11,k]T,RB1=[R1,k,R2,k*,R5,k,R6,k*,R9,k,R10,k*]T,RB2=[R3,k,R4,k*,R7,k,R8,k*,R11,k,R12,k*]T,RC1=[R5,k,R6,k,R7,k,R8,k]T,RC2=[R9,k,R10,k,R11,k,R12,k]T.(15)

These parameters may then be used to solve Equations (9) using a BCD solver procedure as summarized in Table 1.

TABLE 11. Choose initial guess x(0)∈122. For= 0 to itermax− 1 reapeat,a. Obtain,,, andas follows[γ5,k(ℓ+1)γ6,k(ℓ+1)]=pin⁢v⁡(A1(ℓ))⁢RA1,[γ7,k(ℓ+1)γ8,k(ℓ+1)]=pinv⁡(A2(ℓ))⁢RA2b. Obtain,,, andas follows[γ1,k(ℓ+1)γ2,k(ℓ+1)]=pinv⁡(B1(ℓ))⁢RB1,[γ3,k(ℓ+1)γ4,k(ℓ+1)]=pinv⁡(B2(ℓ))⁢RB2c. Obtain, . . . ,as follows= [1, 1j, 0, 0] × (pinv()RC1),= [0, 0, 1, 1j] × (pinv()RC1),= [1, 1j, 0, 0] × (pinv()RC2),= [0, 0, 1, 1j] × (pinv()RC2)

To speed up the convergence of BCD solver in Table 1, the final solution x(itermax)of a frequency tone may be used as an initial point of the next adjacent tone (x(0)) since the IQMM parameters of the adjacent tones (if the tones are relatively close to each other in frequency) may be similar.

In some embodiments, Equations (6) may be formulated in different manners, and/or different methods may be used to solve Equations (9), for example, gradient descent or Newton's method.

TX Pre-Compensation

In some embodiments, estimates of γ1,k, . . . , γ4,kfor k=1, . . . , K, obtained using, for example, the techniques described above, may be used to obtain estimates of coefficients for a TX pre-compensator such as the pre-compensator418illustrated inFIG.4. An example of a pre-compensator for which coefficients may be estimated according to this disclosure is illustrated inFIG.6.

FIG.6illustrates an example embodiment of a complex-valued pre-compensator (CVPC) according to this disclosure. The embodiment illustrated inFIG.6may include an integer delay element600having a delay TD, a complex conjugate block602, a complex-valued filter604having an impulse response wTX[n], and a summing circuit606.

Values for coefficients that may fully or partially remove TX FD-IQMM from the TX path400illustrated inFIG.4for the pre-compensator illustrated inFIG.6may then be given by

WTXopt(f)=-G2⁢TX(f)T1⁢TX(f)⁢e-j⁢2⁢π⁢fTD.(16)

where WTX(f) denotes the frequency responses of filter wTX[n].

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 my provide optimal values.

For a given delay element TD, WTXoptmay be estimated at frequencies f=±f1, . . . , ±fKas

W^TXopt(fk)=-γ4,k*γ1,k⁢e-j⁢2⁢π⁢fkFs⁢TD,W^TXopt(-fk)=-γ2,k*γ3,k⁢ej⁢2⁢π⁢fkFs⁢TD,(17)

where Fsmay be the sampling rate over which the pre-compensation block is operating.

In some embodiments, a finite impulse response (FIR) filter604(wTX[n]) of length L may be used in the embodiment illustrated inFIG.6to obtain an L-tap filter wTX=[wTX[0], . . . , wTX[L−1]]T∈L×1that may reduce or minimize the least squared (LS) error between WTX(f) and ŴTXopt(f) at frequencies f=±f1, . . . , ±fKas

minwTX,TDW^opt-FwTX2,(18)

where Ŵopt=[ŴTXopt(−fK), . . . , ŴTXopt(−f1), ŴTXopt(f1), . . . , ŴTXopt(fK)]Tand F=[F0, . . . , FL−1] may be a Discrete Fourier Transform (DFT) matrix of size 2K×L, wherein TDmay take values in (0, . . . , L−1). For a fixed TD, wTXmay be found as ŵTX,TD=pinv(F)Ŵoptwith a least squared error of LSETD=∥Ŵopt−FŵTX,TD∥2. Then TDand filter coefficients ŵTXoptmay be given by

TDopt=arg⁢minTD⁢LSETD,w^TXopt=w^TX,TDopt.(19)

In some embodiments, and depending on the implementation details, this process may provide an optimal L-tap filter and/or optimal values for TDand filter coefficients ŵTXopt.

In some embodiments, other pre-compensator structures may be used, and the calibration algorithm may be applied to other IQ mismatch pre-compensation structures as well. Furthermore, techniques other than LS may be used to obtain filter coefficients for the pre-compensation structures.

RX Compensation

In some embodiments, estimates of γ5,k, . . . , γ8,kfor k=1, . . . , K, obtained using, for example, the techniques described above, may be used to obtain estimates of coefficients for an RX compensator such as the compensator444illustrated inFIG.4. Some examples of compensators for which estimates of coefficients may be obtained according to this disclosure include complex valued compensators (CVC) such as those illustrated inFIGS.7and8and real-valued compensators such as that illustrated inFIG.9Alternatively, the estimates of γ5,k, . . . , γ8,kfor k=1, . . . , K may be used to compensate IQMM in frequency-domain rather than time-domain, for example, compensation of IQMM in orthogonal frequency-division multiplexing (OFDM) systems.

FIG.7illustrates an example embodiment of a CVC according to this disclosure. The embodiment illustrated inFIG.7may include an integer delay element700having a delay TD, a complex conjugate block702, a complex-valued filter704having an impulse response w1,RX[n], and a summing circuit706.

FIG.8illustrates another example embodiment of a complex-valued compensator (CVC) according to this disclosure. The embodiment illustrated inFIG.8may include an integer delay element800having a delay TD, a real value block802, a complex-valued filter804having an impulse response w2,RX[n], and a summing circuit806.

FIG.9illustrates an example embodiment of a real-valued compensator (RVC) according to this disclosure. The embodiment illustrated inFIG.9may include a real-valued filter900having an impulse response dRX[n], an integer delay element902having a delay TD, a real-valued cross multiplication factor α, a multiplier904and a summing circuit906.

Compensator coefficients that may reduce or remove RX FD-IQMM for the complex-valued compensators illustrated inFIGS.7and8may be given by

W1,RXopt(f)=-G2⁢RX(f)G1⁢RX*(-f)⁢e-j⁢2⁢π⁢fTD,W2,RXopt(f)=-2⁢G2⁢RX(f)G1⁢RX*(-f)+G2⁢RX(f)⁢e-j⁢2⁢π⁢fTD,(20)

where W1,RX(f) and W2,RX(f) may denote the frequency responses of filters704(w1,RX[n]) and804(w2,RX[n]) respectively.

For the RVC illustrated inFIG.9, a method may begin by defining

R⁡(f)=△G1⁢RX(f)+G2⁢RX*(-f)G1⁢RX(f)-G2⁢RX*(-f).

Compensator coefficients that reduce or remove RX FD-IQMM may then be given by

αRXopt=tan⁢ϕRX=tan⁡(12×∡⁡(R⁡(f)R*(-f))),(21)DRXopt(f)=1gRX⁢cos⁢ϕRX⁢H1⁢RX(f)H2⁢RX(f)⁢e-j⁢2⁢π⁢fTD=(1-j⁢αopt)⁢R⁡(f)⁢e-j⁢2⁢π⁢fTD,

wheremay denote the angle operator, and DRX(f) may denote the frequency response of filter900(dRX).

In some embodiments, and depending on the implementation details, the expressions in Equations (20) and/or (21), may provide optimal values.

In some embodiments, after obtaining estimates of γ5,k, γ6,k, γ7,k, and γ8,kfor k=1, . . . , K, Equations (20) and (21) may be used to obtain the coefficients for the compensators illustrated inFIGS.7-9(for a given TD) at selected frequencies as follows

W^1,RXopt(fk)=-γ6,kγ7,k*⁢e-j⁢2⁢π⁢fkFs⁢TD,W^1,RXopt(-fk)=-γ8,kγ5,k*⁢ej⁢2⁢π⁢fkFs⁢TD,W^2,RXopt(fk)=-2⁢γ6,kγ7,k*+γ6,k⁢e-j⁢2⁢π⁢fkFs⁢TD,W^2,RXopt(-fk)=-2⁢γ8,kγ5,k*+γ8,k⁢ej⁢2⁢π⁢fkFs⁢TD,α^RXopt=tan⁡(12×∑k=1K∡⁡(R^(fk)R^*(-fk))),D^RXopt(fk)=(1-j⁢α^RXopt)⁢R^(fk)⁢e-j⁢2⁢π⁢fkFs⁢TD,D^RXopt(-fk)=(1-j⁢α^RXopt)⁢R^(-fk)⁢ej⁢2⁢π⁢fkFs⁢TD,(22)

where

R^(fk)=γ5,k+γ8,k*γ5,k-γ8,k*,R^(-fk)=γ7,k+γ6,k*γ7,k-γ6,k*.(23)

To obtain values for filters704(w1,RX[n]),804(w2,RX[n]), and900(dRX[n]), a least squares approach as described above may be used to obtain an FIR approximation of the filter responses w1,RXopt(f), w2,RXopt(f), and/or DRXopt(f).

FIG.10illustrates an embodiment of method for joint TX and RX IQMM calibration according to this disclosure. The method illustrated inFIG.10may begin at operation1000. At operation1001, a counter k may be initialized to 1. At operation1002, 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 operation1004where a counter p is initialized to zero. At operation1006, the method may check the value of the counter p. If the counter p is less than or equal to the maximum value P−1, the method may proceed to operation1008where the phase shift value may be set to θp.

At operation1010, a single-tone pilot signal may be generated at frequency fkand applied at baseband to the TX path400. At operation1012, the received pilot signal may be captured at frequencies fkand f−kat baseband of the RX path402and denoted by R4p+1,kand R4p+2,k, respectively. At operation1014, a single-tone pilot signal may be generated at frequency −fkand applied at baseband to the TX path400. At operation1016the received pilot signal may be captured at frequencies −fkand fkat baseband of the RX path402and denoted by R4p+3,kand R4p+4,k, respectively.

At operation1018, the counter p may be incremented, and the method may return to operation1006where the method may check the value of the counter p. If the counter p is greater than the maximum value P−1, the method may proceed to operation1020where the observations R1,k, . . . R4p,kmay be used to solve for γ1,k, . . . γ8,k, for example, using a BCD algorithm.

At operation1022, the method may increment the value of the counter k and return to operation1002, where the method may check the value of the counter k. If k greater than the maximum value K, the method may proceed to operations1024and1026. At operation1024, using γ1,k, . . . , γ4,kfor ∀k, the method may estimate coefficients for TX IQMM pre-compensator418.

At operation1026, using γ5,k, . . . γ8,kfor ∀k, the method may estimate coefficients for RX IQMM compensator444. The method may then terminate at operation1028.

In some embodiments, P=3 phase shifters may be enough to obtain 12 independent observations as used in some of the examples herein. However, any number of phase shifts may be used to obtain and solve fewer or more equations. In some embodiments, R1,k, . . . , R4p,kmay be obtained by capturing the time-domain signal at baseband (BB) of the RX path402and converting it to a frequency-domain signal, for example, using a Fast Fourier transform (FFT).

Alternative IQMM Parameter Estimations

In some alternative embodiments, the following parameters, which may depend on gain and filter mismatches, may be defined

VTX(f)=ΔH1⁢TX(f)gTX⁢H2⁢TX(f),VRX(f)=ΔH1⁢RX(f)gRX⁢H2⁢RX(f).(24)

TX IQMM parameters ϕTX, VTX(f) and RX IQMM parameters ϕRX, VRX(f) may be jointly estimated for a set of selected continuous-time frequencies ±f1, . . . , ±fKover the desired frequency band. These estimates may then be used to obtain coefficients for a one or more compensators and/or pre-compensator using, for example, one or more of the algorithms described below.

Simplified BCD Estimation (Second Algorithm)

In some embodiments, the following parameters may be defined

z1,k=ΔARX⁢ATX⁢G1⁢RX(fk)⁢G1⁢TX(fk),(25)z2,k=ΔARX⁢ATX*⁢G1⁢RX(-fk)⁢G2⁢TX(-fk),z3,k=ΔARX*⁢ATX*⁢G2⁢RX(-fk)⁢G1⁢TX*(fk),z4,k=ΔARX⁢ATX*⁢G1⁢RX(-fk)⁢G1⁢TX(-fk),z5,k=ΔARX⁢ATX⁢G1⁢RX(fk)⁢G2⁢TX(fk),z6,k=ΔARX*⁢ATX⁢G2⁢RX(fk)⁢G1⁢TX*(-fk),z7,k=ΔARX*⁢ATX⁢G2⁢RX(fk)⁢G2⁢TX*(-fk),z8,k=ΔARX*⁢ATX*⁢G2⁢RX(-fk)⁢G2⁢TX*(fk),

which may used to reformulate equations (6) as follows:
R1,k=ARXATXG1RX(fk)G1TX(fk)+A*RXATXG2RX(fk)G*2TX(−fk)=z1,k+z7,k
R2,k=ARXA*TXG1RX(−fk)G2TX(−fk)+A*RXA*TXG2RX(−fk)G*1TX(fk)=z2,k+z3,k
R3,k=ARXA*TXG1RX(−fk)G1TX(−fk)+A*RXA*TXG2RX(−fk)G*2TX(fk)=z4,k+z8,k
R4,k=ARXATXG1RX(fk)G2TX(fk)+A*RXATXG2RX(fk)G*1TX(−fk)=z5,k+z6,k
R5,k=β1,kARXATXG1RX(fk)G1TX(fk)+β*2,kA*RXATXG2RX(fk)G*2TX(−fk)=β1,kz1,k+β*2,kz7,k
R6,k=β2,kARXA*TXG1RX(−fk)G2TX(−fk)+β*1,kA*RXA*TXG2RX(−fk)G*1TX(fk)=β2,kz2,k+β*1,kz3,k
R7,k=β2,kARXA*TXG1RX(−fk)G1TX(−fk)+β*1,kA*RXA*TXG2RX(−fk)G*2TX(fk)=β2,kz4,k+β*1,kz8,k
R8,k=β1,kARXATXG1RX(fk)G2TX(fk)+β*2,kA*RXATXG2RX(fk)G*1TX(−fk)=β1,kz5,k+β*2,kz6,k
R9,k=β3,kARXATXG1RX(fk)G1TX(fk)+β*4,kA*RXATXG2RX(fk)G*2TX(−fk)=β3,kz1,k+β*4,kz7,k
R10,k=β4,kARXA*TXG1RX(−fk)G2TX(−fk)+β*3,kA*RXA*TXG2RX(−fk)G*1TX(fk)=β4,kz2,k+β*3,kz3,k
R11,k=β4,kARXA*TXG1RX(−fk)G1TX(−fk)+β*3,kA*RXA*TXG2RX(−fk)G*2TX(fk)=β4,kz4,k+β*3,kz8,k
R12,k=β3,kARXATXG1RX(fk)G2TX(fk)+β*4,kA*RXATXG2RX(fk)G*1TX(−fk)=β3,kz5,k+β*4,kz6,k(26)

The Equations (26) may provide 12 complex equations R1,k, . . . , R12,kwith 12 complex unknowns z1,k, . . . , z8,k, β1,k, . . . , β4,k. The problem of solving the equations may then be formulated as follows:

xopt=argminx∈ℂ12⁢f⁡(x)2,(27)f⁡(x)=[z1,k+z7,k-R1,kz2,k+z3,k-R2,kz4,k+z8,k-R3,kz5,k+z6,k-R4,kβ1,k⁢z1,k+β2,k*⁢z7,k-R5,kβ2,k⁢z2,k+β1,k*⁢z3,k-R6,kβ2,k⁢z4,k+β1,k*⁢z8,k-R7,kβ1,k⁢z5,k+β2,k*⁢z6,k-R8,kβ3,k⁢z1,k+β4,k*⁢z7,k-R9,kβ4,k⁢z2,k+β3,k*⁢z3,k-R10,kβ4,k⁢z4,k+β3,k*⁢z8,k-R11,kβ3,k⁢z5,k+β4,k*⁢z6,k-R12,k],

where the vector of unknowns may be given by
x=[z1,k, . . . ,z8,k,β1,k, . . . ,β4,k]T.  (28)

In some embodiments, all 12 equations in Equations (26) may be time-aligned for correct estimation of IQMM parameters. A problem represented by Equations (27) may be solved, for example, using a BCD algorithm which may minimize the cost function ∥ƒ(x)∥2iteratively along a coordinate block while the other coordinates remain fixed. In some embodiments using this method, descent in the cost function may be ensured at every iteration.

In the BCD algorithm, the unknown variables may be divided into the following two blocks
x1=[z1,k, . . . ,z8,k]T,
x2=[β1,k,β2,k,β3,k,β4,k]T.  (29)

Each iteration of the BCD solver may include two steps (i=1, 2). At the ith step of the (+1)th iteration, ∥f(, . . . ,, xi,, . . . ,)∥2may be minimized with respect to xi. With the specific choice of coordinate blocks in Equations (29), the cost function at the ith step may become a least-squares problem in terms of xi, which may be readily solved.

The following parameters may be defined

A1(ℓ)=[11β1,k(ℓ)β2,k(ℓ)β3,k(ℓ)β4,k(ℓ)],A2(ℓ)=[11β2,k(ℓ)β1,k(ℓ)β4,k(ℓ)β3,k(ℓ)],(30)B(ℓ)=[z1,k(ℓ+1)jz1,k(ℓ+1)z7,k(ℓ+1)-jz7,k(ℓ+1)z3,k(ℓ+1)-jz3,k(ℓ+1)z2,k(ℓ+1)jz2,k(ℓ+1)z8,k(ℓ+1)-jz8,k(ℓ+1)z4,k(ℓ+1)jz4,k(ℓ+1)z5,k(ℓ+1)jz5,k(ℓ+1)z6,k(ℓ+1)-jz6,k(ℓ+1)]RA1=[R1,k,R5,k,R9,k]T,RA2=[R2,k,R6,k,R10,k]T,RA3=[R3,k,R7,k,R11,k]T,RA4=[R4,k,R8,k,R12,k]T,RB1=[R5,k,R6,k,R7,k,R8,k]T,RB2=[R9,k,R10,k,R11,k,R12,k]T.

These parameters may then be used for a BCD solver procedure as summarized in Table 2.

TABLE 23. Choose initial guess x(0)∈124. For= 0 to itermax− 1 reapeat,a. Obtain, . . . ,as follows[z1,k(ℓ+1)z7,k(ℓ+1)]=pinv⁡(A1(ℓ))⁢RA1,[z2,k(ℓ+1)z3,k(ℓ+1)]=pinv⁡(A2(ℓ))⁢RA2[z4,k(ℓ+1)z8,k(ℓ+1)]=pinv⁡(A2(ℓ))⁢RA3,[z5,k(ℓ+1)z6,k(ℓ+1)]=pinv⁡(A1(ℓ))⁢RA4b. Obtain, . . . ,as follows= [1, 1j, 0, 0] × (pinv()RB1),= [0, 0, 1, 1j] × (pinv()RB1),= [1, 1j, 0, 0] × (pinv()RB2),= [0, 0, 1, 1j] × (pinv()RB2)

In some embodiments, and depending on the implementation details, the method described above with respect to Equations (25) through (30) and Table 2 may provide a relatively simple solution and/or may involve lower computational complexity compared to the first algorithm.

In some embodiments, to speed up the convergence of the BCD solver in Table 2, the final solution x(itermax)of a frequency tone may be used as an initial point of the next adjacent tone (x(0)) since the IQMM parameters of adjacent tones (if the tones are relatively close to each other in frequency) may be similar.

After sweeping frequency for k=1, . . . , K and obtaining estimates of z1,k, . . . , z8,k, estimates for ϕTX/RXand VTX/RX(f) may be obtained as

ϕ^TX=12⁢K⁢∑k=1Kangle(TTX*(-fk)TTX(fk)),(31)ϕ^RX=12⁢K⁢∑k=1Kangle(TRX(fk)TRX*(-fk)),V^TX(f)=TTX(f)⁢e+j⁢ϕ^TX,f=±f1,…,±fK,V^RX(f)=TRX(f)⁢e-j⁢ϕ^RX,f=±f1,…,±fK,whereTTX(fk)=z1,k+z5,kz1,k-z5,k,TTX(-fk)=z4,k+z2,kz4,k-z2,k,k=1,…⁢K,(32)TRX(fk)=z1,k+z3,k*z1,k-z3,k*,TRX(-fk)=z4,k+z6,k*z4,k-z6,k*,k=1,…⁢K.

Estimation with Reduced Equation Set (Third Algorithm)

In some embodiments, one or more image of image signals maybe ignored, which my be equivalent to assuming that G2RX(f)G2TX(f)=0 for every f. Thus, the last four equations in Equations (6) may be omitted, and only the first eight equations used.

A single tone pilot signal at frequency fkmay be used to obtain the observed outputs at the principal and mirror frequencies (fkand −fk) which may be denoted by R1,kand R2,k, respectively. Since the number of unknowns is larger than the number of equations, another single-tone pilot signal may be sent at frequency −fk, for which observations at the principal and mirror frequencies (−fkand fk) may be denoted by R3,kand R4,k, respectively. Next, single-tone pilot signals at frequencies fkand −fkmay be sent separately and another phase shift value θ1applied to obtain four additional observations R5,k, . . . , R8,k. Based on the assumption that G2RX(±f)G2TX(±f)=0, the following equations may be obtained
R1,k=ARXATXG1RX(fk)G1TX(fk)=z1,k
R2,k=ARXA*TXG1RX(−fk)G2TX(−fk)+A*RXA*TXG2RX(−fk)G*1TX(fk)=z2,k+z3,k
R3,k=ARXA*TXG1RX(−fk)G1TX(−fk)=z4,k
R4,k=ARXATXG1RX(fk)G2TX(fk)+A*RXATXG2RX(fk)G*1TX(−fk)=z5,k+z6,k
R5,k=β1,kARXATXG1RX(fk)G1TX(fk)=β1,kz1,k
R6,k=β2,kARXA*TXG1RX(−fk)G2TX(−fk)+β*1,kA*RXA*TXG2RX(−fk)G*1TX(fk)=β2,kz2,k+β*1,kz3,k
R7,k=β2,kARXA*TXG1RX(−fk)G1TX(−fk)=β2,kz4,k
R8,k=β1,kARXATXG1RX(fk)G2TX(fk)+β*2,kA*RXATXG2RX(fk)G*1TX(−fk)=β1,kz5,k+β*2,kz6,k(33)

where β1,kand β2,kmay represent the gain and phase changes that may be caused by using the phase shifter with different phase values and may not be the same for positive and negative baseband frequencies as the response of phase shifter might not be symmetric around the center frequency. With Equations (33), there may be 8 equations with 8 unknowns z1,k, . . . , z6,k, β1,k, β2,kwhich may be solved as follows

β1,k=R5,kR1,k,β2,k=R7,kR3,k,(34)z1,k=R1,k,[z2,kz3,k]=[11β2,kβ1,k*]-1[R2,kR6,k],z4,k=R3,k,[z5,kz6,k]=[11β1,kβ2,k*]-1[R4,kR8,k].
After sweeping the frequencies for k=1, . . . , K and obtaining estimates of z1,k, . . . , z6,k, ϕTX/RXand VTX/RX(f) may be estimated as follows

ϕ^TX=12⁢K⁢∑k=1Kangle(TTX*(-fk)TTX(fk)),(35)ϕ^RX=12⁢K⁢∑k=1Kangle(TRX(fk)TRX*(-fk)),V^TX(f)=TTX(f)⁢e+j⁢ϕ^TX,f=±f1,…,±fK,V^RX(f)=TRX(f)⁢e-j⁢ϕ^RX,f=±f1,…,±fK,whereTTX(fk)=z1,k+z5,kz1,k-z5,k,TTX(-fk)=z4,k+z2,kz4,k-z2,k,k=1,…⁢K,(36)TRX(fk)=z1,k+z3,k*z1,k-z3,k*,TRX(-fk)=z4,k+z6,k*z4,k-z6,k*,k=1,…⁢K.

Initial Estimate for BCD Algorithm

In some embodiments, a result obtained using one algorithm may be used as an initial estimate for another algorithm. For example, as described below, estimates obtained by ignoring one or more image of image signals using the third algorithm may be used as initial estimates for a BCD algorithm using the second algorithm.

Specifically, assuming the image of image signal is zero, the first 8 equations of Equations (26) may provide equations for use with Equations (33). The parameters z1,k, . . . , z6,k, β1,k, β2,kmay be estimated using Equations (34). Then using the last four equations in Equations (26), the parameters z7,k, z8,k, β3,k, β4,kmay be estimated to be used as the initial point x(0)in the BCD solver in the second algorithm. A procedure for obtaining x(0)for frequency fkis summarized in Table 3.

TABLE 31. Perform measurements and obtain R1,k, . . . , R12,k2. Solve for z1,k(0), . . . , z4,k(0), β1,k(0), β2,k(0)asβ1,k(0)=R5,kR1,k,β2,k(0)=R7,kR3,k,z1,k(0)=R1,k,[z2,k(0)z3,k(0)]=[11β2,k(0)β1,k(0)*]-1[R2,kR6,k],z4,k(0)=R3,k,[z5,k(0)z6,k(0)]=[11β1,k(0)β2,k(0)*]-1[R4,kR8,k]3. Solve for z7,k(0), z8,k(0), β3,k(0), β4,k(0)asβ3(0)= [1, 1j, 0, 0] × c, β4(0)= [0, 0, 1, 1j] × c,z7,k(0)=R9,k-β3,k(0)⁢z1,k(0)β4,k(0)*,z8,k(0)=R11,k-β4,k(0)⁢z4,k(0)β3,k(0)*,wherec=[Re⁢{z3,k(0)}Im⁢{z3,k(0)}Re⁢{z2,k(0)}-Im⁢{z2,k(0)}Im⁢{z3,k(0)}-Re⁢{z3,k(0)}Im⁢{z2,k(0)}Re⁢{z2,k(0)}Re⁢{z5,k(0)}-Im⁢{z5,k(0)}Re⁢{z6,k(0)}Im⁢{z6,k(0)}Im⁢{z5,k(0)}Re⁢{z5,k(0)}Im⁢{z6,k(0)}-Re⁢{z6,k(0)}]-1[Re⁢{R10,k}Im⁢{R10,k}Re⁢{R12,k}Im⁢{R12,k}]4. The intial guess is equal to x(0)= [z1,k(0), . . . z8,k(0), β1,k(0), . . . β4,k(0)]T

In some embodiments, and depending on the implementation details, using an initial estimate obtained through the method of Table 3 as an initial estimate for a BCD solver in Table 2 may increase the speed of convergence and/or reduce the total number of required iterations.

TX Pre-Compensation

After obtaining estimates of ϕTX/RXand VTX/RX(f) for f=±f1, . . . , ±fK, for example, as described above using the second and third algorithms, these parameters may be used to compensate for FD-IQMM in the TX path.

An examples of pre-compensator for which coefficients may be estimated according to this disclosure include the one illustrated inFIG.6.

Some pre-compensator coefficients that may partially or fully remove TX FD-IQMM for the pre-compensator shown inFIG.6may be given by

WTXopt(f)=1-VTX(f)⁢e-j⁢ϕTX1+VTX(f)⁢e-j⁢ϕTX⁢ej⁢2⁢π⁢f⁢TD,(37)
where WTX(f) may denote the frequency response of filter wTX[n]. In some embodiments, and depending on the implementation details, this expression may provide optimal values for the coefficients.

After obtaining estimates of ϕTXand VTX(f) for f=±f1, . . . , ±fK, by any calibration algorithm, they may be used to obtain coefficients for the compensator illustrated inFIG.6(for a given TD) at the selected frequencies using an LS approach as described above in Equations (18) and (19) to obtain an FIR approximation of the optimal filter WTXopt(f). In some embodiments, other pre-compensator structures may be used, and the calibration algorithm may be applied to other IQ mismatch pre-compensation structures as well. Furthermore, techniques other than LS may be used to obtain filter coefficients for the pre-compensation structures.

RX Compensation

In some embodiments, after obtaining estimates of ϕRXand VRX(f) for f=±f1, . . . , ±fK, these estimates may be used to compensate FD-IQMM in the RX path. Some examples of compensators for which coefficients may be estimated using estimates of ϕRXand VRX(f) according to this disclosure include those illustrated inFIGS.7,8, and9. Alternatively, estimates of ϕRXand VRX(f) for f=±f1, . . . , ±fKmay be used to compensate IQMM in the frequency-domain rather than the time-domain, for example, for compensation of IQMM in OFDM systems.

In some embodiments, compensator coefficients that may partially or fully remove RX FD-IQMM for the compensators shown inFIGS.7and8may be given by

W1,RXopt(f)=(VRX(f)-ej⁢ϕRXVRX(f)+ej⁢ϕRX)⁢e-j⁢2⁢π⁢f⁢TD,(38)W2,RXopt(f)=(-1+ej⁢ϕRXVRX(f))⁢e-j⁢2⁢π⁢f⁢TD,

where W1,RX(f) and W2,RX(f) may denote the frequency responses of filters w1,RX[n] and w2,RX[n], respectively. Compensator coefficients that partially or fully remove RX FD-IQMM for the RVC shown inFIG.9may be given by

αRXopt=tan⁢ϕRX,(39)DRXopt(f)=VRX(f)cos⁢ϕRX⁢e-j⁢2⁢π⁢f⁢TD,

where DRX(f) may denote the frequency response of filter dRX[n]. In some embodiments, and depending on the implementation details, these expressions may provide optimal values for the coefficients.

Thus, after obtaining estimates of ϕRXand VRX(±fk) for k=1, . . . , K, Equations (38) and (39) may be used to obtain coefficients for the compensators illustrated inFIGS.7,8, and9(for a given TD) at the selected frequencies using an LS approach as described above in Equations (18) and (19) to obtain an FIR approximation of the optimal filters W1,RXopt(f), W2,RXopt(f), or DRXopt(f). In some embodiments, other approaches may be used to find coefficients for filters w1,RX[n], w2,RX[n], dRX[n] and factor α.

FIG.11illustrates an alternative embodiment of method for joint TX and RX IQMM calibration according to this disclosure. The method illustrated inFIG.11may begin at operation1100. At operation1101, a counter k may be initialized to 1. At operation1102, 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 operation1104where a counter p is initialized to zero. At operation1106, the method may check the value of the counter p. If the counter p is less than or equal to the maximum value P−1, the method may proceed to operation1108where the phase shift value may be set to θp.

At operation1110, a single-tone pilot signal may be generated at frequency fkand applied at baseband to the TX path400. At operation1112, the received pilot signal may be captured at frequencies fkand −fkat baseband of the RX path402and denoted by R4p+1,kand R4p+2,k, respectively. At operation1114, a single-tone pilot signal may be generated at frequency −fkand applied at baseband to the TX path400. At operation1116the received pilot signal may be captured at frequencies −fkand fkat baseband of the RX path402and denoted by R4p+3,kand R4p+4,k, respectively.

At operation1118, the counter p may be incremented, and the method may return to operation1106where the method may check the value of the counter p. If the counter p is greater than the maximum value P−1, the method may proceed to operation1120where the observations R1,k, . . . R4p,kmay be used to solve for z1,k, . . . , z8,kusing, for example, the BCD algorithm of Table 2, or to solve for z1,k, . . . , z6,kusing, for example, Equations (34). At operation1122, the method may increment the value of the counter k and return to operation1102, where the method may check the value of the counter k. If k greater than the maximum value K, the method may proceed to operation1124where, using z1,k, . . . , z6,k, the method may estimate ϕTX/RXand VTX/RX(±fk) for ∀k. At operation1126, using the estimates of ϕTXand VTX(±fk) for ∀k, the method may estimate coefficients for TX IQMM pre-compensator418. At operation1128, using the estimates of ϕRXand VRX(±fk) for ∀k, the method may estimate coefficients for RX IQMM compensator444. The method may then terminate at operation1130.

In some embodiments, P=3 phase shifters may be sufficient for the second algorithm, and P=2 may be sufficient for the third algorithm. In some embodiments, R1,k, . . . , R4p,kmay be obtained by capturing the baseband time-domain signal at the RX path and converting it to a frequency-domain signal using, for example, a Fast Fourier transform (FFT).

The operations and/or components described with respect to the embodiments illustrated inFIGS.10and11, 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.

Enhanced Estimation of RX Cross Multiplication Factor

In some embodiments, the accuracy of a cross multiplication factor αRXfor an RX RVC may depend largely or only on the accuracy of an RX phase mismatch ϕRX. This may be apparent, for example, with reference to Equation (21). Thus, any inaccuracy in the initial or estimated value of ϕRXmay cause a corresponding inaccuracy in the estimation of αRX.

A method for re-estimating a cross multiplication factor according to this disclosure may involve adjusting an initial estimate of a cross multiplication factor by correcting for inaccuracies caused by one or more residual RX phase mismatches. Some embodiments may be described in the context of systems and/or methods that may estimate IQMM parameters using techniques described in this disclosure. However, the principles relating to re-estimating a cross multiplication factor according to this disclosure have independent utility and may be used with any other systems and/or methods for estimating IQMM parameters.

In some embodiments, a re-estimating technique for a cross multiplication factor according to this disclosure may begin, for example, using a pilot-based calibration method as described above to obtain RVC coefficients for a compensator in the RX path. The RVC coefficients may be used to compensate one or more pilot (e.g., single-tone) signals, which may have been measured already, sent from the TX path through the loopback path. The RX phase mismatch ϕRXmay then be re-estimated using the IQ compensated single-tone signals and the estimated TX phase mismatch. Some embodiments may be implemented as follows.

The received BB time-domain signal at RX for single-tone signals sent at frequencies fkand −fkwith phase shift value θpmay be denoted by rp(k)[n] and rp(−k)[n] respectively. The obtained RVC coefficients {circumflex over (α)}RXoptand {circumflex over (d)}RXoptmay be applied to the RX signals rp(k)[n] and rp(−k)[n] for p=0 only. The I and Q components of the compensated signal may be denoted by yI(k)[n], yQ(k)[n] and yI(−k)[n], yQ(−k)[n] respectively. If an estimation of αRX, i.e., {circumflex over (α)}RXopt, is not accurate, and the RX frequency-dependent and gain mismatches have been removed, then samples y(k)=yI(k)+jyQ(k)and y(−k)=yI(−k)+jyQ(−k)may still contain FI-IQMM due to the RX phase mismatch. Thus, the relationship between samples of y(k)and signal zRX(k)[n]=zRX,I(k)[n]+jzRX,Q(k)[n], which may denote the baseband signal without RX IQMM, may be written as
yI(k)[n]=zRX,I(k)[n],
yQ(k)[n]=cos ϕRX,krzRX,Q(k)[n]−sin ϕRX,krzRX,I(k)[n],(40)
where ϕRX,krmay denote the residual phase mismatch due to incorrect estimation of αRX. Equation (40) may be re-arranged to establish that

𝔼⁡(zRX,I(k)⁢zRX,Q(k))𝔼⁡((zRX,Q(k))2)=cos⁢ϕRX,kr⁢𝔼⁡(yI(k)⁢yQ(k))+cos⁢ϕRX,kr⁢sin⁢ϕRX,kr⁢𝔼⁡((yI(k))2)𝔼⁡((yQ(k))2)+sin2⁢ϕRX,kr⁢𝔼⁡((yI(k))2)+2⁢sin⁢ϕRX,kr⁢𝔼⁡(yI(k)⁢yQ(k)),(41)

where the expectation(.) may be taken over time n. Assuming TX IQMM is frequency independent (FI), the relation between clean baseband I/Q samples uI(k)/uQ(k)and the baseband equivalent impaired samples zTX,I(k)/zTX,Q(k)after mixers in TX path may be given by
zTX,I(k)[n]=uI(k)[n]−gTXsin ϕTXuQ(k)[n],
zTX,Q(k)[n]=gTXcos ϕTXuQ(k)[n].(42)

Assuming(uIuQ)=0 and(uI2)=(uQ2) may provide

𝔼⁡(zTX,I(k)⁢zTX,Q(k))𝔼⁡((zTX,Q(k))2)=-tan⁢ϕTX.(43)

Combining equations (41) and (43) and assuming that zRX(k)[n]=zTX(k)[n], which may be the case for p=0 (no phase shift), may provide

cos⁢ϕRX,kr⁢𝔼⁡(yI(k)⁢yQ(k))+cos⁢ϕRX,kr⁢sin⁢ϕRX,kr⁢𝔼⁡((yI(k))2)𝔼⁡((yQ(k))2)+sin2⁢ϕRX,kr⁢𝔼⁡((yI(k))2)+2⁢sin⁢ϕRX,kr⁢𝔼⁡(yI(k)⁢yQ(k))=-tan⁢ϕ^TX,(44)

which may be solved to obtain ϕRX,krand update the RX cross multiplication factor as

αRXre-estimate=tan(a⁢tan⁢αRXopt+12⁢K⁢∑k=-KKk≠0ϕ^RX,kr).(45)

In some implementations, there may be no phase ambiguity due to the a tan operator in Equation (45) because a tan αRXoptmay be an estimate of the phase mismatch4RX and hence may have a small value. In some embodiments, the nonlinear Equation (44) be solved using different methods such as Newton's method and/or the like, and(.) may be computed by averaging over time.

In some embodiments, and depending on the implementation details, a method as described, which may be characterized as a pilot-based calibration method to re-estimate αRXmore accurately, may only involve offline processing of measurements that may have already been obtained and, thus, may have the advantage of not involving additional measurements.

In some embodiments, an accurate estimation of the cross multiplication factor αRXfor RVC may be beneficial, for example, for obtaining a symmetric image rejection ratio (IRR) across frequency in RX, where IRR may be defined as the ratio of power of the primary signal to the power of the image signal.

FIG.12illustrates an embodiment of method for enhanced estimation of an RX cross multiplication factor according to this disclosure. The method illustrated inFIG.12may begin at operation1200. At operation1202, the method may use any calibration method as described above to obtain RVC parameters {circumflex over (α)}RXoptand {circumflex over (d)}RXoptand TX phase mismatch ϕTX. At operation1204, the method may set counter k=1. At operation1206, the method may check if the value of k is less than or equal to the maximum value K, the method may proceed to operation1208where, for r(k)[n] and r(−k)[n] denoting a captured baseband time-domain signal at RX without using a phase shifter for single-tone signals sent at frequencies fkand −fk, respectively, it may apply RVC with coefficients {circumflex over (α)}RXoptand {circumflex over (d)}RXopton r(k)[n] and r(−k)[n] and denote the compensated signals as y(k)[n] and y(−k)[n], respectively. K may be any integer greater than zero. In some embodiments, using a larger number of frequency tones (i.e., a larger value of K), may enable cross multiplication factor αRXto be re-estimated more accurately.

At operation1210, the method may solve for ϕRX,krusing Equation (44). At operation1212, the method may solve for ϕRX,−krin a similar manner using Equation (44). At operation1214, the counter k may be incremented, and method may return to operation1206where the value of the counter k may be checked. If k is greater than the maximum value K, the method may proceed to operation1216where the RX cross multiplication factor may be updated using Equation (45). The method may then end at operation1218.

This disclosure encompasses numerous inventive principles. 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.