Patent Publication Number: US-8982987-B2

Title: Paired OFDM pilot symbols

Description:
RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Patent Applications No. 61/618,624, titled “Receiver-Side Estimation of and Compensation for Signal Impairments,” filed Mar. 30, 2012, and No. 61/719,326, titled “Receiver-Side Estimation of and Compensation for Signal Impairments,” filed Oct. 26, 2012, both of which are hereby incorporated by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The present embodiments relate generally to communication systems, and specifically to communication systems that use pilot symbols to compensate for signal impairments. 
     BACKGROUND OF RELATED ART 
     Transceivers are sensitive to various signal impairments that affect the quality of the transmitted and received signals. Signal impairments may result from non-idealities in the RF front-ends of the transceivers. For example, mismatched active and passive elements (e.g., quadrature mixers, filters, digital-to-analog converters, and/or analog-to-digital converters) in the I and Q (in-phase and quadrature) signal paths introduce I/Q mismatch impairments in transmitted and received signals. I/Q mismatch, which also may be referred to as I/Q offset, is present in both the transmitter and receiver. In another example, carrier frequency offset in the receiver impairs received signals. Channel effects may also impair signals. 
     I/Q mismatch introduces an image signal that degrades signal quality. The signal-to-image ratio is typically around 25-30 dB, making I/Q mismatch an issue for systems targeting high spectral efficiency. I/Q mismatch is also frequency dependent, making I/Q mismatch an issue for wideband communication systems. 
     Accordingly, there is a need for techniques to estimate and compensate for signal impairments. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present embodiments are illustrated by way of example and are not intended to be limited by the figures of the accompanying drawings. Like numbers reference like elements throughout the drawings and specification. 
         FIG. 1A  illustrates a communications system in accordance with some embodiments. 
         FIG. 1B  illustrates sources of signal impairment in the communications system of  FIG. 1A . 
         FIG. 2A  is a block diagram of a direct-conversion transceiver in accordance with some embodiments. 
         FIGS. 2B-2E  are block diagrams illustrating signal impairments in accordance with some embodiments. 
         FIG. 3A  is a flowchart illustrating a method of estimating and compensating for signal impairments in accordance with some embodiments. 
         FIG. 3B  is a flowchart illustrating a two-phased method of estimating and compensating for signal impairments in accordance with some embodiments. 
         FIGS. 4A-4D  illustrate a periodic signal used for receiver-side I/Q mismatch estimation and compensation in accordance with some embodiments. 
         FIG. 5  is a block diagram of a communication device in accordance with some embodiments. 
         FIG. 6  illustrates a channel matrix used for frequency-domain signal impairment estimation and compensation in accordance with some embodiments. 
         FIG. 7  illustrates successive pairs of OFDM symbols with pilot symbols on different sets of subcarriers in accordance with some embodiments. 
         FIG. 8  is a flowchart showing a method of communicating between an OFDM transmitter and an OFDM receiver in accordance with some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Techniques are disclosed for transmitting and receiving pilot symbols that may be used to compensate for signal impairments. 
     In some embodiments, an orthogonal frequency-division multiplexing (OFDM) transmitter transmits successive pairs of OFDM symbols. The successive pairs include a first pair of OFDM symbols. First and second OFDM symbols of the first pair both include pilot symbols on two subcarriers that are symmetric about a center carrier frequency. The two subcarriers are the same for the first and second OFDM symbols. The pilot symbols on the two subcarriers for the first and second OFDM symbols compose an orthogonal matrix. 
     In some embodiments, an OFDM receiver receives successive pairs of OFDM symbols. The successive pairs include a first pair of OFDM symbols. First and second OFDM symbols of the first pair both include pilot symbols on two subcarriers that are symmetric about a center carrier frequency. The two subcarriers are the same for the first and second OFDM symbols. The pilot symbols on the two subcarriers for the first and second OFDM symbols compose an orthogonal matrix. The OFDM receiver estimates frequency responses at frequencies including the frequencies of the two subcarriers and compensates for signal impairment based at least in part on the estimated frequency responses. 
     In the following description, numerous specific details are set forth such as examples of specific components, circuits, and processes to provide a thorough understanding of the present disclosure. Also, in the following description and for purposes of explanation, specific nomenclature is set forth to provide a thorough understanding of the present embodiments. However, it will be apparent to one skilled in the art that these specific details may not be required to practice the present embodiments. In other instances, well-known circuits and devices are shown in block diagram form to avoid obscuring the present disclosure. The term “coupled” as used herein means connected directly to or connected through one or more intervening components or circuits. Any of the signals provided over various buses described herein may be time-multiplexed with other signals and provided over one or more common buses. Additionally, the interconnection between circuit elements or software blocks may be shown as buses or as single signal lines. Each of the buses may alternatively be a single signal line, and each of the single signal lines may alternatively be buses, and a single line or bus might represent any one or more of a myriad of physical or logical mechanisms for communication between components. The present embodiments are not to be construed as limited to specific examples described herein but rather to include within their scopes all embodiments defined by the appended claims. 
       FIG. 1A  illustrates a communications system  100  in accordance with some embodiments. A transmitter  102  transmits a signal onto a channel  104 , and a direct-conversion receiver  106  receives the signal from the channel  104 . In some embodiments, the channel  104  is wireless. In other embodiments, the channel  104  is a wired link (e.g., a coaxial cable or other physical connection). 
       FIG. 1B  illustrates sources of signal impairment, and thus signal degradation, in the communications system  100  of  FIG. 1A . Transmitter-side (Tx) I/Q mismatch  122  in the transmitter  102  causes signal impairment, as does receiver-side (Rx) I/Q mismatch  128  in the receiver  106 . The channel  104  introduces channel distortion  124 , which may be linear distortion. Carrier frequency offset  126  in the receiver  106 , which results from the frequency of a local oscillator in the receiver  106  differing from the frequency of a corresponding local oscillator in the transmitter  102 , also causes signal impairment. In some embodiments, channel distortion  124  includes multi-path effects and Additive White Gaussian Noise (AWGN). 
       FIG. 2A  is a block diagram of a direct-conversion transceiver  200  using quadrature amplitude modulation (QAM) in accordance with some embodiments. The transceiver  200  may be included within a communication device (e.g., communication device  500 ,  FIG. 5 ), such as a wireless (e.g., WLAN) device or a device with a wired network connection. As illustrated, the transceiver  200  includes a transmitter unit  210  and a receiver unit  250 . The transmitter unit  210  of a first transceiver  200  corresponds to the transmitter  102  ( FIG. 1A ) and the receiver unit  250  of a second transceiver  200  corresponds to the receiver  106  ( FIG. 1A ), where the first transceiver  200  and second transceiver  200  are separated by the channel  104  ( FIG. 1A ). 
     In some embodiments, the transmitter unit  210  includes an antenna  202 , a transmitter analog front end (AFE)  220 , and a transmitter baseband processor  240 . The receiver unit  250  includes an antenna  201 , a receiver AFE  260 , and a receiver baseband processor  280 . In some embodiments, the receiver baseband processor  280  includes a signal impairment compensation unit  285  for estimating and compensating for signal impairments introduced both in the transmitter (e.g., transmitter  102 ,  FIG. 1A ) and receiver (e.g., receiver  106 ,  FIG. 1A ). 
     In the example of  FIG. 2A , the transmitter AFE  220  includes a digital-to-analog converter (DAC)  221 A for the I signal path, amplifier/filter circuitry  222 A for the I signal path, a local oscillator (LO) mixer  224 A for the I signal path, a DAC  221 B for the Q signal path, amplifier/filter circuitry  222 B for the Q signal path, an LO mixer  224 B for the Q signal path, a combiner  272 , a variable gain amplifier (VGA)  226 , and a power amplifier (PA)  228 . The mixers  224 A and  224 B up-convert the I and Q signals from baseband directly to the carrier frequency by mixing the I and Q signals with local oscillator signals LO(I) and LO(Q), where the frequency of the local oscillator signal is the carrier frequency. Mismatch between mixers  224 A and  224 B, between amplifiers/filters  222 A and  222 B, and/or between DACs  221 A and  221 B results in transmitter-side I/Q mismatch. The combiner  272  combines the up-converted I and Q signals. 
     The receiver AFE  260  includes a low-noise amplifier (LNA)  261 , a VGA  262 , an LO mixer  264 A for the I signal path, amplifier/filter circuitry  266 A for the I signal path, an analog-to-digital converter (ADC)  268 A for the I signal path, an LO mixer  264 B for the Q signal path, amplifier/filter circuitry  266 B for the Q signal path, and an ADC  268 B for the Q signal path. The mixers  264 A and  264 B directly down-convert the received signal into baseband I and Q signals by mixing the received signal with local oscillator signals LO(I) and LO(Q), where the frequency of the local oscillator signals (as generated by a local oscillator, not shown) is ideally the carrier frequency. Mismatch between mixers  264 A and  264 B, between amplifiers/filters  266 A and  266 B, and/or between ADCs  268 A and  268 B results in receiver-side I/Q mismatch. A difference between the frequency of the local oscillator signals in the receiver unit  250  of a receiver  106  ( FIG. 1A ) and the corresponding frequency of local oscillator signals in the transmitter unit  210  of a transmitter  102  ( FIG. 1A ) results in carrier frequency offset. 
     The components described with reference to  FIG. 2A  are exemplary only. In various embodiments, one or more of the components described may be omitted, combined, or modified, and additional components may be included. For instance, in some embodiments, the transmitter unit  210  and receiver unit  250  may share a common antenna, or may have various additional antennas and transmitter/receiver chains. In other embodiments, there may be no antenna; instead, the transmitter unit  210  and receiver unit  250  connect to a wired link. In some implementations, the transceiver  200  may include less or more filter and/or amplifier circuitry (e.g., blocks  222  and  266  of  FIG. 2A ). 
     Attention is now directed to mathematical modeling of I/Q mismatch.  FIG. 2B  illustrates I/Q mismatch in a transmitter front end (e.g., transmitter AFE  220 ,  FIG. 2A ) in accordance with some embodiments. A signal component x R [n] is provided to the I signal path, which includes components  270 A (e.g., DAC  221 A and amplifier/filter  222 A,  FIG. 2A ) and a mixer  224 A. A signal x I [n] is provided to the Q signal path, which includes components  270 B (e.g., DAC  221 B and amplifier/filter  222 B,  FIG. 2A ) and a mixer  224 B. (The R and I subscripts refer to real and imaginary components, and thus respectively to the I and Q components, of a signal x[n].) The signal path components  270 A and  270 B have corresponding functions f[n]+g[n] and f[n]−g[n], respectively, where f[n] is common between the components  270 A and  270 B and g[n] includes an amplitude mismatch (e.g., a frequency-dependent amplitude mismatch) between components  270 A and  270 B. A phase mismatch Δφ is introduced during up-conversion by mixers  224 A and  224 B. The I/Q mismatch of  FIG. 2B  thus is represented by g[n] and Δφ. The combiner  272  combines the up-converted I and Q signal to generate an RF transmitted signal. 
     The baseband equivalent of the RF transmitted signal, as affected by the I/Q mismatch of  FIG. 2B , is
 
 r[n ]=[cos Δφ( f[n]+g[n ])* x   R   [n ]−sin Δφ( f[n]−g[n ])* x   I   [n]]+j[−sin Δφ(   f[n]+g[n ])* x   R   [n ]+cos Δφ( f[n]−g[n ])* x   I   [n]] 
 
     With some change in notation, this expression may be written in matrix form as 
               [             r   R     ⁡     [   n   ]                   r   I     ⁡     [   n   ]             ]     =       [           cos   ⁢           ⁢   Δ   ⁢           ⁢     ϕ   ⁡     (       f   ⁡     [   n   ]       +     g   ⁡     [   n   ]         )                 -   sin     ⁢           ⁢   Δ   ⁢           ⁢     ϕ   ⁡     (       f   ⁡     [   n   ]       -     g   ⁡     [   n   ]         )                     -   sin     ⁢           ⁢   Δ   ⁢           ⁢     ϕ   ⁡     (       f   ⁡     [   n   ]       +     g   ⁡     [   n   ]         )               cos   ⁢           ⁢   Δ   ⁢           ⁢     ϕ   ⁡     (       f   ⁡     [   n   ]       -     g   ⁡     [   n   ]         )               ]     *     [             x   R     ⁡     [   n   ]                   x   I     ⁡     [   n   ]             ]             
Expanding the matrix results in
 
               [             r   R     ⁡     [   n   ]                   r   I     ⁡     [   n   ]             ]     =         [             f   ⁡     [   n   ]       ⁢   cos   ⁢           ⁢   Δ   ⁢           ⁢   ϕ             g   ⁡     [   n   ]       ⁢           ⁢   sin   ⁢           ⁢   Δ   ⁢           ⁢   ϕ                 -     g   ⁡     [   n   ]         ⁢   sin   ⁢           ⁢   Δ   ⁢           ⁢   ϕ             f   ⁡     [   n   ]       ⁢   cos   ⁢           ⁢   Δ   ⁢           ⁢   ϕ           ]     *     [             x   R     ⁡     [   n   ]                   x   I     ⁡     [   n   ]             ]       +             [             g   ⁡     [   n   ]       ⁢   cos   ⁢           ⁢   Δ   ⁢           ⁢   ϕ             f   ⁡     [   n   ]       ⁢   sin   ⁢           ⁢   Δ   ⁢           ⁢   ϕ                 -     f   ⁡     [   n   ]         ⁢   sin   ⁢           ⁢   Δ   ⁢           ⁢   ϕ             g   ⁡     [   n   ]       ⁢   cos   ⁢           ⁢   Δ   ⁢           ⁢   ϕ           ]     *     [             x   R     ⁡     [   n   ]                 -       x   I     ⁡     [   n   ]               ]                 
This equation corresponds to the following relation in complex notation:
 
 r[n ]=( f[n ] cos Δφ−jg[n] sin Δφ)* x[n ]+( g[n ] cos Δφ−jf[n] sin Δφ)* x*[n] 
 
If we define
 
 a[n]=f[n] cos Δφ−jg[n] sin Δφ 
 
 b[n]=g[n] cos Δφ−jf[n] sin Δφ 
 
then we can write compactly
 
 r[n]=a[n]*x[n]+b[n]*x*[n] 
 
     As this result indicates, I/Q mismatch causes interference between I and Q components in the time domain. Equivalently, I/Q mismatch causes interference between mirror frequencies in the frequency domain, as shown by transforming the equation for r[n] into the frequency domain:
 
 R ( f )= A ( f ) X ( f )+ B ( f ) X *(− f ).
 
     I/Q mismatch can be perfectly compensated in the frequency domain via the following linear combination of the signal R(f) and its conjugate: 
                     Y   ⁡     (   f   )       =       ⁢       A   *     (     -   f     )     ⁢     R   ⁡     (   f   )         -       B   ⁡     (   f   )       ⁢   R   *     (     -   f     )                     =       ⁢       [       A   *     (     -   f     )     ⁢     A   ⁡     (   f   )         -     B   *     (     -   f     )     ⁢     B   ⁡     (   f   )           ]     ⁢     X   ⁡     (   f   )                     
where {tilde over (H)}(f)=A*(−f)A(f)−B*(−f)B(f) is the equivalent transmitter shaping filter, which acts as a scaling factor to be corrected for. Dividing Y(f) by {tilde over (H)}(f) recovers the original signal X(f).
 
     This expression for the compensated signal Y(f) can be simplified by expressing complex signals in terms of their real and imaginary parts. Also, it is possible to define an alternative correction formula, 
                     Y   ⁡     (   f   )       =       ⁢       R   ⁡     (   f   )       -         B   ⁡     (   f   )           A   *     ⁡     (     -   f     )         ⁢   R   *     (     -   f     )                     =       ⁢       A   ⁡     (   f   )       ⁢     (     1   -         B   ⁡     (   f   )       ⁢       B   *     ⁡     (     -   f     )             A   ⁡     (   f   )       ⁢       A   *     ⁡     (     -   f     )             )     ⁢     X   ⁡     (   f   )                     
Note that the equivalent shaping filter is different in this case.
 
     If the I/Q mismatch is small, we can approximate f[n]≈1, sin Δφ≈Δφ, cos Δφ≈1, g[n] sin Δφ≈0, so that the complex envelope of the transmitted signal reads
 
 r[n]=x[n]+b[n]*x*[n],  
 
where b[n]=g[n]−jΔφ. Also, the interference due to I/Q mismatch can be subtracted directly as y[n]=r[n] b[n]*x*[n].
 
     The above mathematics model I/Q mismatch in the transmitter (e.g., transmitter  102 ,  FIG. 1A ). Receiver-side I/Q mismatch (e.g., in the receiver  106 ,  FIG. 1A ) may be modeled in the same fashion. 
     Assuming the only sources of distortion in the communication system are transmitter (Tx) I/Q mismatch, receiver (Rx) I/Q mismatch, and multi-path effects in the channel, the received signal in the frequency domain is
 
 Z ( f )= Ã ( f ) X ( f )+ {tilde over (B)} ( f ) X *(− f )+ A   Rx   W ( f )+ B   Rx ( f ) W *(− f )
 
where
 
 Ã ( f )= A   Rx ( f ) H ( f ) A   Tx ( f )+ B   Rx ( f ) H *(− f ) B   Tx *(− f )
 
 {tilde over (B)} ( f )= A   Rx ( f ) H ( f ) B   Tx ( f )+ B   Rx ( f ) H *(− f ) A   Tx *(− f ).
 
W(f) is the spectrum of the additive Gaussian noise and H(f) is the frequency response of the channel. (H(f) is thus unrelated to {tilde over (H)}(f)). The corrected signal {circumflex over (Z)}(f) is defined as
 
     
       
         
           
             
               
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     After straightforward calculations, it can be shown that the corrected signal reads 
                 Z   ^     ⁡     (   f   )       =           A   ~     ⁡     (   f   )       ⁢     (     1   -           B   ~     ⁡     (   f   )       ⁢         B   ~     *     ⁡     (     -   f     )               A   ~     ⁡     (   f   )       ⁢         A   ~     *     ⁡     (     -   f     )             )     ⁢     X   ⁡     (   f   )         -       W   ~     ⁡     (   f   )               
where {tilde over (W)}(f) is the equivalent noise after the correction filter has been applied.
 
     In some embodiments, transmitted signals are narrowband signals. For narrowband signals, the filters in the I and Q signal paths (e.g., filters  222 A-B and  266 A-B,  FIG. 2A ) function as scalar multipliers. For example, functions a[n] and b[n] reduce to scalars a and b. Also, signal impairments due to multi-path effects in the channel may be neglected in narrowband embodiments. 
       FIG. 2C  illustrates signal impairments in a system (e.g., system  100 ,  FIG. 1A ) in which the received signal is affected by AWGN, carrier frequency offset  126 , and receiver-side I/Q mismatch  128 . Noise w[n] is mixed into the signal x[n] in the channel. 
     The received sampled signal z[n] can be expressed as
 
 z[n]=a[e   jΔωn   x[n]]+b[e   −jΔωn   x*[n]]+aw[n]+bw*[n] 
 
where Δω is the carrier frequency offset (CFO)  126 . The Rx I/Q mismatch  128  creates two different components rotating in opposite direction.
 
     There are algorithms for CFO estimation based on the autocorrelation of the received signal, r zz [M]=z[n+M]z*[n]. The autocorrelation at lag M for the received signal affected by Rx I/Q mismatch  128  is
 
 r   zz   [M]=|a|   2   |x[n]|   2   e   jΔωM   +|b|   2   |x[n]|   2   e   −jΔωM   +ab*e   j2Δω(n+M) ( x[n ]) 2   +a*be   −j2Δω(n+M) ( x*[n ]) 2  
 
The second term significantly degrades the accuracy of CFO estimation based on autocorrelation of the received signal.
 
     However, the CFO  126  may be estimated using other estimation techniques. For example, the CFO  126  may be estimated using classical non-linear least squares (NLS) techniques in receivers that process multiple symbols at a time (e.g., when the system employs a repeating training sequence or a cyclic prefix of an OFDM symbol, which repeats twice and changes from OFDM symbol to OFDM symbol). 
     Once Δω is known (or estimated), the (scalar) I/Q mismatch parameters a, b may be estimated. Defining {tilde over (z)}[n]=e jΔωn x[n], we can write the following matrix equation: 
               [           z   ⁡     [   n   ]                 z   ⁡     [     n   +   1     ]             ]     =         [             x   ~     ⁡     [   n   ]               x   ~     *     [   n   ]                   x   ~     ⁡     [     n   +   1     ]               x   ~     *     [     n   +   1     ]             ]     ⁡     [         a           b         ]       +     [             w   ~     ⁡     [   n   ]                   w   ~     ⁡     [     n   +   1     ]             ]             
The I/Q imbalance parameters can then be estimated by inverting the 2×2 system matrix in this equation.
 
     The accuracy of the estimate of the I/Q mismatch parameters can be improved by stacking samples received over a whole period of M samples. Defining the input signal vector {tilde over (x)}[n]=[{tilde over (x)}[n] {tilde over (x)}[n+1] . . . {tilde over (x)}[n+M−1]] and the received signal vector z[n]=[z[n] z[n+1] . . . z[n+M−1]], the received signal vector may be rewritten as 
     
       
         
           
             
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     Solving for a, b now entails the inversion of an M×2 matrix. However, because of the linearity of the problem, classical low-complexity adaptive filtering techniques, such as least mean squares (LMS), may be used. 
     Once the I/Q mismatch parameters a, b have been found, the Rx I/Q mismatch  128  can then be corrected by a linear combination of z[n] and z*[n], 
                       z   ^     ⁡     [   n   ]       =       ⁢         a   *     ⁢     z   ⁡     [   n   ]         -       bz   *     ⁡     [   n   ]                     =       ⁢           ⅇ     j   ⁢           ⁢   Δ   ⁢           ⁢   ω   ⁢           ⁢   n       ⁡     (            a        2     -          b        2       )       ⁢     x   ⁡     [   n   ]         +     w   ⁡     [   n   ]                     
As this equation indicates, the CFO  126  can now be corrected using phase rotation.
 
     In some embodiments, iterative estimations of the CFO  126  and Rx I/Q mismatch  128  are performed.  FIG. 3A  illustrates a method  350  of iteratively estimating CFO and I/Q mismatch. The method  350  is performed in a receiver (e.g., receiver unit  250  of transceiver  200 ,  FIG. 2A ) in accordance with some embodiments. 
     In the method  350 , a repeating or periodic signal is received ( 351 ). In some embodiments, the signal is a repeating training sequence. In some embodiments, the signal is a narrowband signal. In some embodiments, the signal is a cyclic prefix of an OFDM symbol. 
     CFO (e.g., CFO  126 ,  FIG. 2C ) is estimated ( 352 ) based on the signal assuming no I/Q mismatch. For example, the CFO is estimated using an autocorrelation technique; the iterative nature of the method  350  accommodates the inaccuracy associated with using autocorrelation. Alternatively, the CFO is estimated using an NLS technique based on the signal. Using the estimated CFO, I/Q mismatch (e.g., receiver-side I/Q mismatch  128 ,  FIG. 2C ) is estimated ( 354 ). The estimated I/Q mismatch is compensated for ( 356 ), and the CFO is then re-estimated ( 358 ). The I/Q mismatch is re-estimated ( 360 ) using the re-estimated CFO and is compensated for ( 362 ), and the re-estimated CFO is compensated for ( 364 ). 
     While the method  350  includes a number of operations that appear to occur in a specific order, it should be apparent that the method  350  can include more or fewer operations. An order of two or more operations may be changed and two or more operations may be combined into a single operation. 
     In some embodiments, the method  350  is an example of a first phase of a two-phase process of estimating and compensating for signal impairments. In the first phase, estimation of and compensation for receiver-side I/Q mismatch and carrier frequency offset are performed. In a subsequent second phase, estimation of and compensation for transmitter-side I/Q mismatch and channel distortion (e.g., linear channel distortion) are performed. Impairments  126  and  128  ( FIG. 1B ) thus are estimated and compensated for in the first phase, and impairments  122  and  124  ( FIG. 1B ) are estimated and compensated for in the second phase. This two-phase approach provides for computational simplicity compared to approaches that jointly estimate and correct for transmitter-side I/Q mismatch, receiver-side I/Q mismatch, channel distortion, and carrier frequency offset. The two-phase approach thus is easier to implement than joint approaches. 
     In some embodiments, a two-phased approach is used in communications systems in which the physical layer employs a repetitive known signal (e.g., a training sequence, preamble, or prefix). For example, the two-phased approach may be implemented in systems compatible with one of the IEEE 802.11 family of protocols. In some embodiments, a two-phased approach is used in communications systems that perform multicarrier modulation based on orthogonal frequency-division multiplexing (OFDM). Examples include systems compatible with one of the IEEE 802.11 family of protocols and systems compatible with the 3GPP E-UTRAN (LTE) standard. 
       FIG. 3B  is a flowchart illustrating a two-phased method  300  of estimating and compensating for signal impairments in accordance with some embodiments. The method  300  is performed in a receiver (e.g., receiver  106 ,  FIG. 1A ). In some embodiments, the method  300  is performed by the receiver baseband processor  280  ( FIG. 2A ) (e.g., by the signal impairment compensation unit  285 ,  FIG. 2A ). In some embodiments, the method  300  uses a training signal or dedicated repeating preamble but in general the method  300  is not so limited. The method  300  thus is performed in the digital domain in baseband in accordance with some embodiments. During phase one, carrier frequency offset (e.g., CFO  126 ,  FIG. 2C ) and receiver-side I/Q mismatch (e.g., Rx I/Q mismatch  128 ,  FIG. 2C ) are repeatedly estimated for a predefined number of iterations. Phase one is terminated, however, if the estimated carrier frequency offset is determined to be less than a specified threshold, even if the predefined number of iterations has not been completed. In response to a determination that the estimated carrier frequency offset is less than the specified threshold determination, the method  300  proceeds to phase two, in which channel distortion is estimated and equalized and transmitter-side I/Q mismatch is estimated and compensated for. 
     At the start  302  of the method  300 , an iteration counter (N_iter) is set to zero. An estimate of carrier frequency offset is made ( 304 ) using any known technique. For example, non-linear least-squares (NLS) techniques that process multiple symbols at a time (e.g., from a repeating training sequence or the cyclic prefix of an OFDM symbol) may be used to estimate the carrier frequency offset. Alternately, autocorrelation techniques may be used. In some embodiments, the estimation of carrier frequency offset is made assuming no receiver-side I/Q mismatch. For example, the estimation of carrier frequency offset is agnostic toward receiver-side I/Q mismatch. 
     The estimated carrier frequency offset is compared ( 306 ) to a predefined threshold (CFO_thresh). The predefined threshold is system-dependent. In some embodiments implemented in OFDM systems, the predefined threshold is set to be comparable to the sub-carrier spacing. 
     If the estimated carrier frequency offset is less than the predefined threshold ( 306 —Yes), phase one is terminated and the method  300  proceeds to operations  320  and  322  of phase two (described below). 
     If the estimated carrier frequency offset is not less than the predefined threshold ( 306 —No), an estimate is made ( 308 ) of the receiver-side I/Q mismatch. This estimate is made, for example, as described below with regard to  FIGS. 4A-4D  and equations (1)-(5). 
     The iteration counter is incremented ( 310 ) and compared ( 312 ) to the predefined number of iterations (N_max). In some embodiments, N_max equals two. In some embodiments, N_max has a value in the range of 2-10. 
     If the incremented iteration counter is not greater than N_max ( 312 —No), compensation is performed ( 314 ) for the receiver-side I/Q mismatch estimated at  308 , and the method  300  returns to operation  304 . 
     If the incremented iteration counter is greater than N_max ( 312 —Yes), compensation is performed ( 316 ) for the receiver-side I/Q mismatch estimated at  308 , and compensation is performed ( 318 ) for carrier frequency offset estimated at  304 . The carrier frequency offset compensation is performed using any known technique (e.g., phase rotation). At this point, phase one is complete. The estimation operations  304  and  308  and compensation operations  314 / 316  thus are performed a number of times equal to N_max, assuming that the estimated carrier frequency offset is not determined to be less than the predefined threshold during one of the iterations. Note that if N_max=2 and the determination at  306  is “No” in both iterations, phase one is an example of the method  350  ( FIG. 3A ). If N_max (i.e., the predefined number of iterations) is greater than two, portions of phase one are an example of the method  350  ( FIG. 3A ): for example, operations  352 - 356  of the method  350  may correspond to a first iteration and operations  358 - 362  of the method  350  may correspond to a final iteration. 
     In phase two of method  300 , a joint estimate is made ( 320 ) of channel distortion (e.g., distortion  124 ,  FIG. 1B ) and transmitter-side I/Q mismatch (e.g., mismatch  122 ,  FIG. 1B ). Joint equalization of the estimated channel distortion and compensation for the estimated transmitter-side I/Q mismatch is performed ( 322 ), at which point the method  300  ends ( 324 ). Equalization of channel distortion and compensation for transmitter-side I/Q mismatch thus are performed after receiver-side I/Q mismatch and carrier frequency offset have been compensated for in accordance with some embodiments. (Equalization of channel distortion compensates for the channel distortion.) In some embodiments, equalization of channel distortion and compensation for transmitter-side I/Q mismatch are performed in the frequency domain, as described below. 
     While the method  300  includes a number of operations that appear to occur in a specific order, it should be apparent that the method  300  can include more or fewer operations. An order of two or more operations may be changed and two or more operations may be combined into a single operation. 
     Attention is now directed to estimating and compensating for receiver-side I/Q mismatch. A repeating or periodic narrowband signal (e.g., a training signal) z[n] is used, as shown in  FIG. 2D  and  FIG. 4A  in accordance with some embodiments, and the carrier frequency offset is estimated beforehand (e.g., the CFO is estimated in operation  304 ,  FIG. 3B , prior to estimating the receiver-side I/Q mismatch in operation  308 ). The variable n indexes samples of the signal. Each period  402  of the signal z[n] includes M samples, as shown in  FIG. 4A , where M is an integer greater than or equal to one. For example, a first period  402 - 1  includes M samples, as does a second period  402 - 2 , a third period  402 - 3 , and an Nth period  402 -N. 
     The signal z[n] may be expressed as:
 
 z[n ]=( a   Rx   a   Tx ) e   jΔωn   x[n ]+( a   Rx   b   Tx ) e   jΔωn   x*[n ]+( b   Rx   a   Tx *) e   −jΔωn   x*[n]+(   b   Rx   b   Tx *) e   −jΔωn   x[n]+a   Rx   w[n]+b   Rx   w*[n] 
 
     The received signal over two successive periods of length M is:
 
 z[n]=a   Rx   [e   jΔωn   y[n]]+b   Rx   [e   −jΔωn   y*[n]]+{tilde over (w)}[n] 
 
 z[n+M]=a   Rx   e   jΔωM   [e   jΔωn   y[n]]+b   Rx   e   −jΔωM   [e   −jΔωn   y*[n]]+{tilde over (w)}[n+M] 
 
The signal y[n] includes the (unknown) transmitter-side I/Q mismatch  122  ( FIG. 2D ).
 
     Receiver-side I/Q mismatch can be compensated by performing a linear transformation involving a scalar correction factor q to generate a compensated received signal {circumflex over (z)}[n]. Specifically: 
                       z   ^     ⁡     [   n   ]       =       [           z   ⁡     [   n   ]               z   *     ⁡     [   n   ]             ]     ⁡     [         1           q         ]               (   1   )               
where z*[n] is the complex conjugate of z[n].
 
     Assuming perfect compensation and thus perfect correction, 
             q   =       -     b   RX         a   Rx   *             
and the corrected received signal over two successive training sequences reads
 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   z 
                   ^ 
                 
                 ⁡ 
                 
                   [ 
                   n 
                   ] 
                 
               
               = 
               
                 
                   
                     
                       a 
                       Rx 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             
                                
                               
                                 b 
                                 Rx 
                               
                                
                             
                             2 
                           
                           
                             
                                
                               
                                 a 
                                 Rx 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅇ 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                   ⁢ 
                   
                     y 
                     ⁡ 
                     
                       [ 
                       n 
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
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                     ⁡ 
                     
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                                
                             
                             2 
                           
                         
                       
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                   ⁢ 
                   
                     w 
                     ⁡ 
                     
                       [ 
                       n 
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 z 
                 ^ 
               
               ⁡ 
               
                 [ 
                 
                   n 
                   + 
                   M 
                 
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             = 
             
               
                 
                   a 
                   Rx 
                 
                 ⁢ 
                 
                   
                     ⅇ 
                     
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                       ⁢ 
                       
                           
                       
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                       ⁢ 
                       
                           
                       
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                       ⁢ 
                       
                           
                       
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                 ⁢ 
                 
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                     ⁢ 
                     
                         
                     
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                     ⁢ 
                     
                         
                     
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                     ⁢ 
                     
                         
                     
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                   ⁡ 
                   
                     ( 
                     
                       1 
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     Neglecting the noise and assuming perfect compensation for receiver-side I/Q mismatch, it holds that:
 
 {circumflex over (z)}[n+M]=e   jΔωM   {circumflex over (z)}[n]   (2)
 
where Δω is the carrier frequency offset (e.g., as estimated in operation  304 ,  FIG. 3B ) in units of radians/sample. Relationship (2) thus holds that a compensated sample in a given period (e.g., period  402 - 2 ) equals the compensated sample in a previous period (e.g., period  402 - 1 ) multiplied by a phase factor determined by the product of the carrier frequency offset and the number of samples in a period.
 
     Relationship (2) holds for all samples within two successive periods.  FIG. 4B  illustrates samples in two successive periods, including a vector z 1  of samples in period  402 - 1  and a vector z 2  of samples in period  402 - 2 . Relationship (2) holds between vectors z 1  and z 2 , such that: 
                       ⌈           z   2           z   2   *           ⌉     ⁡     [         1           q         ]       =         ⅇ     jΔω   ⁢           ⁢   M       ⁡     [           z   1           z   1   *           ]       ⁡     [         1           q         ]               (   3   )               
Solving equation (3) for the correction factor q yields:
 
                   q   =         -       (       z   2   *     -       ⅇ     jΔω   ⁢           ⁢   M       ⁢     z   1   *         )     H       ⁢     (       z   2     -       ⅇ     jΔω   ⁢           ⁢   M       ⁢     z   1         )                  z   2   *     -       ⅇ     jΔω   ⁢           ⁢   M       ⁢     z   1   *              2               (   4   )               
In equation (4), H refers to the Hermitian transpose operation (i.e., taking the transpose conjugate of the vector by transposing the vector and taking the complex conjugate of each element). The correction factor q is determined using equation (4) and is then used to determine the compensated received signal {circumflex over (z)}[n] in accordance with equation (1).
 
     In some embodiments, accuracy of the receiver-side I/Q mismatch estimation is improved by averaging multiple estimates from adjacent periods. For example, a first estimate is made based on periods  402 - 1  and  402 - 2  and a second estimate is made based on periods  402 - 3  and  402 - 4 , as illustrated in  FIG. 4C  in accordance with some embodiments. Each of the two estimates is made, for example, using equation (4). The first and second estimates are then averaged, and the resulting value of q is used to compensate for the receiver-side I/Q mismatch using equation (1). 
     In some embodiments, conditioning of the problem of estimating the receiver-side I/Q mismatch is improved by using non-consecutive periods. Using non-consecutive periods helps to ensure a reasonably large value of the carrier frequency offset (Δω).  FIG. 4D  illustrates an example in which vectors z 1  and z k  for respective non-consecutive periods  402 - 1  and  402 - k  are used, where the index k indicates the separation between the two periods (i.e., there are k−1 periods between period  402 - 1  and period  402 - k ). In this example, equation (3) is modified to: 
                       ⌈           z   k           z   k   *           ⌉     ⁡     [         1           q         ]       =         ⅇ     jΔω   ⁢           ⁢   k   ⁢           ⁢   M       ⁡     [           z   1           z   1   *           ]       ⁡     [         1           q         ]               (   5   )               
Equation (5) is then solved for the correction factor q, and the resulting value of q is used to compensate for the receiver-side I/Q mismatch in accordance with equation (1).
 
     In some embodiments, instead of using a periodic training signal, the cyclic prefix of an OFDM symbol is used to estimate the receiver-side I/Q mismatch. While the cyclic prefix changes from OFDM symbol to OFDM symbol, it repeats twice and thus may be considered a repeating signal. 
     Equations (1)-(5) describe performing estimation of and compensation for receiver-side I/Q mismatch in the time domain. In some embodiments, after time-domain compensation of receiver-side I/Q mismatch (e.g., in phase one of method  300 ,  FIG. 3B ), transmitter-side I/Q mismatch and channel distortion are compensated for in the frequency domain (e.g., in phase two of method  300 ,  FIG. 3B ). The frequency-domain compensation may also compensate for residual receiver-side I/Q mismatch that was not compensated for in the time domain. Frequency-domain compensation is performed, for example, in the receiver of an OFDM system. 
     In the frequency domain, OFDM transmissions may be modeled by a complex transmit symbol vector x c , a complex diagonal channel matrix H c , a complex receive symbol vector y c , and a complex additive noise vector n c :
 
 y   c   =H   c   x   c   +n   c .
 
An equivalent real-valued notation is:
 
     
       
         
           
             y 
             = 
             
               Hx 
               + 
               n 
             
           
         
       
       
         
           where 
         
       
       
         
           
             x 
             = 
             
               
                 
                   x 
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                 ⊗ 
                 
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                         1 
                       
                     
                     
                       
                         0 
                       
                     
                   
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                 ⊗ 
                 
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             = 
             
               
                 
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           and 
         
       
       
         
           
             H 
             = 
             
               
                 
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                   cR 
                 
                 ⊗ 
                 
                   [ 
                   
                     
                       
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               + 
               
                 
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                 ⊗ 
                 
                   
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                     ] 
                   
                   . 
                 
               
             
           
         
       
     
     Here,   denotes the Kronecker product. H becomes a block diagonal matrix with blocks of size 2×2, as illustrated in  FIG. 6  in accordance with some embodiments. The real and imaginary parts of each entry are stacked on top of each other in x, y, and n. 
     Transmitter IQ mismatch and receiver IQ mismatch cause interference between a frequency f and its mirror frequency −f. The real-valued model allows us to redefine H such that it includes the modeling of transmitter IQ mismatch and receiver IQ mismatch.  FIG. 6  illustrates the structure of the effective channel matrix H including transmitter IQ mismatch and receiver IQ mismatch. The interference between f and −f adds blocks on the main skew diagonal of H. In the center, the 2×2 matrix  600  corresponds to the zero frequency. There is no interference between symbols that are affected by entries with different fill patterns in the matrix. For example, symbols affected by entries  602 - 1  through  602 - 4  do not interfere with symbols affected by entries  600 ,  604 - 1  through  604 - 4 , and  606 - 1  through  606 - 4 , and so on. Therefore, it is sufficient to consider a submatrix composed of the entries marked with a single fill pattern in  FIG. 6 . 
     The transmitter IQ offsets and receiver IQ offsets are respectively modeled by 
             [           cos   ⁡     (     φ   T     )               A   T     ⁢     sin   ⁡     (     φ   T     )                 -     A   T       ⁢     cos   ⁡     (     φ   T     )               sin   ⁡     (     φ   T     )                   -     A   T       ⁢     sin   ⁡     (     φ   T     )               cos   ⁡     (     φ   T     )             sin   ⁡     (     φ   T     )               A   T     ⁢     cos   ⁡     (     φ   T     )                     -     A   T       ⁢     cos   ⁡     (     φ   T     )               sin   ⁡     (     φ   T     )             cos   ⁡     (     φ   T     )               A   T     ⁢     sin   ⁡     (     φ   T     )                   sin   ⁡     (     φ   T     )               A   T     ⁢     cos   ⁡     (     φ   T     )                 -     A   T       ⁢     sin   ⁡     (     φ   T     )               cos   ⁡     (     φ   T     )             ]               and   ⁢     
     [           cos   ⁡     (     φ   R     )               -     A   R       ⁢     sin   ⁡     (     φ   R     )                 A   R     ⁢     cos   ⁡     (     φ   R     )               sin   ⁡     (     φ   R     )                   A   R     ⁢     sin   ⁡     (     φ   R     )               cos   ⁡     (     φ   R     )             sin   ⁡     (     φ   R     )               -     A   R       ⁢     cos   ⁡     (     φ   R     )                     -     A   R       ⁢     cos   ⁡     (     φ   R     )               sin   ⁡     (     φ   R     )             cos   ⁡     (     φ   R     )               A   R     ⁢     sin   ⁡     (     φ   R     )                   sin   ⁡     (     φ   R     )               A   R     ⁢     cos   ⁡     (     φ   R     )                 -     A   R       ⁢     sin   ⁡     (     φ   R     )               cos   ⁡     (     φ   R     )             ]         
where A T /2 and A R /2 are respectively the transmitter and receiver amplitude offsets and φ T /2 and φ R /2 are respectively the transmitter and receiver phase offsets in terms of percent. Assuming that the frequency response is represented by the complex numbers h f  and h −f  we define a symmetric component
 
 h   s =( h   f   +h   −f )/2 =|h   s |(cos(φ cs )+ j  sin(φ cs ))
 
and an asymmetric component
 
 h   a =( h   f   −h   −f )/2 =|h   a |(cos(φ ca )+ j  sin(φ ca ))
 
Combining the transmitter IQ offset, the channel matrix, and the receiver IQ offset, we obtain
 
                   Z   =       ⁢     [           Z   1           -     Z   2             Z   7           Z   8               Z   2           Z   1           Z   8           -     Z   7                 Z   3           Z   4           Z   5           -     Z   6                 Z   4           -     Z   3             Z   6           Z   5           ]                 =       ⁢       [           X   1           -     X   2             X   3           X   4               X   2           X   1           X   4           -     X   3                 X   3           X   4           X   1           -     X   2                 X   4           -     X   3             X   2           X   1           ]     +     [           Y   1           -     Y   2             -     Y   3             -     Y   4                 Y   2           Y   1           -     Y   4             Y   3               Y   3           Y   4           -     Y   1             Y   2               Y   4           -     Y   3             -     Y   2             -     Y   1             ]                   with   X   1   =|h   s |[cos(φ cs )(1 +A   R   A   T )cos(φ T −φ R )−sin(φ cs )( A   T   +A   R )sin(φ T −φ R )],
 
 Y   1   =|h   a |[cos(φ ca )(1 −A   R   A   T )cos(φ T +φ R )−sin(φ ca )( A   T   −A   R )sin(φ T +φ R )],
 
 X   2   =|h   s |[cos(φ cs )( A   T   −A   R )sin(φ T +φ R )+sin(φ cs )(1 −A   R   A   T )cos(φ T +φ R )],
 
 Y   2   =|h   a |[cos(φ ca )( A   T   +A   R )sin(φ T −φ R )+sin(φ ca )(1 +A   R   A   T )cos(φ T −φ R )],
 
 X   3   =|h   s |[cos(φ cs )( A   T   +A   R )cos(φ T −φ R )−sin(φ cs )(1 +A   R   A   T )sin(φ T +φ R )],
 
 Y   3   =|h   a |[cos(φ ca )( A   T   −A   R )cos(φ T +φ R )−sin(φ ca )(1 −A   R   A   T )sin(φ T +φ R )],
 
 X   4   =|h   s |[cos(φ cs )(1 −A   R   A   T )sin(φ T +φ R )+sin(φ cs )( A   T   −A   R )cos(φ T +φ R )],
 
and
 
 Y   4   =|h   a |[cos(φ ca )+(1 +A   R   A   T )sin(φ T −φ R )+sin(φ ca )( A   T   +A   R )cos(φ T −φ R )].
 
Note that X denotes the contribution due to h s  and Y denotes the contribution due to h a .
 
     An estimate of H may be obtained by estimating the frequency response at certain frequencies and subsequently interpolating between these estimates. The estimated frequency responses are spaced closely enough with respect to the coherence bandwidth. The frequency response is jointly estimated for f and its mirror frequency −f. This is done using pilot symbols that are symmetric with respect to frequency zero. Considering Z we observe that eight parameters are to be estimated jointly. At least two OFDM symbols are used to get a valid estimate. Examples of valid pilot matrices include 
     
       
         
           
             
               P 
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               P 
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
                   P   =     [           p   1           -     p   3                 p   2           -     p   4                 p   3           p   1               p   4           p   2           ]             (   6   )               
or more generally
 
where the first column specifies a symbol transmitted in time slot 1 and the second column specifies a symbol transmitted in time slot 2. Also, every entry in a respective column may be multiplied by −1 without changing the relevant properties. In some embodiments, the interpolation is done independently for the diagonal elements and the skew diagonal elements.
 
     Assuming that P was transmitted, R=ZP is observed at the receiver. Based on P and 
             R   =     [           r   1           r   5               r   2           r   6               r   3           r   7               r   4           r   8           ]           
we construct the following matrices
 
               R   eff     =     [           r   1           -     r   2             r   5           -     r   6                 r   2           r   1           r   6           r   5               r   3           r   4           r   7           r   8               r   4           -     r   3             r   8           -     r   7             ]           
and the orthogonal pilot matrix
 
               P   eff     =       [           p   1           -     p   2             -     p   3             p   4               p   2           p   1           -     p   4             -     p   3                 p   3           p   4           p   1           p   2               p   4           -     p   3             p   2           -     p   1             ]     .           
From this we can estimate Z:
 
 Z   est   =R   eff   P   eff   H ( P   eff   *P   eff   H ) −1 .  (7)
 
     Note that (P eff *P eff   H ) −1  is a diagonal matrix and thus can be represented as a scaling factor. To apply a zero-forcing approach, Z est  is inverted. The inversion of Z has the same complexity as the inversion of a 2×2 complex matrix even though there does not exist a complex-valued equivalent. Hence, no inversions of arbitrary 4×4 matrices are involved in compensating for Z. 
     This frequency-domain compensation technique includes the estimation and correction of a frequency-dependent IC) offset. In particular, frequency-selective IC) offset compensation is performed at the edge frequencies. 
     Also, this frequency-domain compensation technique assumes no carrier frequency offset (e.g., as for the phase two of method  300 ,  FIG. 3B ). If there were a carrier frequency offset, H would result in a fully occupied matrix, which reflects the inter-subcarrier interference caused by the carrier frequency offset and makes CFO compensation in the frequency domain computationally inefficient. Accordingly, in some embodiments CFO is corrected in the time domain prior to the conversion of the received signal into the frequency domain (e.g., in phase one of method  300 ,  FIG. 3B ). 
     In some embodiments, correction of the carrier frequency offset involves correction of the (estimated) receiver IQ offset (e.g., as illustrated in methods  350  and  300 ,  FIGS. 3A and 3B ). For example, method  350  or  300  ( FIGS. 3A and 3B ) is used to correct the carrier frequency offset and receiver IQ mismatch if a carrier frequency offset is present. Frequency-domain estimation and correction are then used to compensate for the influence of the channel, transmitter IQ offset, and any residual receiver IQ offset. If no carrier frequency offset is present, a joint estimate of the channel, transmitter IQ offset, and receiver IQ offset is obtained. 
       FIG. 7  illustrates successive pairs of OFDM symbols transmitted by a transmitter  102  ( FIG. 1A ) and received by a receiver  106  ( FIG. 1A ). The OFDM symbols include known pilot symbols  702  on different sets of subcarriers (the placing of the pilot symbols  702  is indicated by the boxes in  FIG. 7 ). These pilot symbols  702  are known modulation symbols placed on respective subcarriers. The OFDM symbols are indexed by a symbol index (“symbol idx”) and the subcarriers are indexed by a subcarrier index (“subcarrier idx”). Successive pairs of OFDM symbols compose respective subframes, which are indexed by a subframe index (“subframe idx”). For example, OFDM symbols  0  and  1  compose subframe  0 , OFDM symbols  2  and  3  compose subframe  1 , and so on. A specified number of subframes (e.g., 16 subframes) compose a frame. Frames are indexed by a frame index (“frame idx”).  FIG. 7  shows only a snapshot (i.e., a portion) of the available subcarriers. For example, there may be 4096, 8192, or 16,384 subcarriers. 
     Pilot symbols  702  are placed on the same subcarriers in both OFDM symbols of a subframe (e.g., in accordance with Equation (6) for pilot matrices) and are symmetric (e.g., mirrored) about a center carrier frequency, which is the DC subcarrier in baseband. The subcarriers used for pilot symbols  702  are evenly spaced on each side of the center carrier frequency. For example, 1 of every 64 subcarriers is used for a pilot symbol  702 . For OFDM symbols  0  and  1  (i.e., subframe  0 ), subcarriers 2, 66, 130, and so on, and also subcarriers −2, −66, −130, and so on, are used for pilot symbols  702 . For OFDM symbols  2  and  3  (i.e., subframe  1 ), subcarriers 34, 98, 162, and so on, and also subcarriers −34, −98, −162, and so on, are used for pilot symbols  702 . The pilot symbol overhead is thus 1/64 (or more generally, a predefined fraction), resulting in a corresponding reduction in spectral efficiency; the remaining 63 of each 64 subcarriers may be used for data transmission. 
     Within a given subframe, the pilot symbols  702  are generated in accordance with equation (6) or a variant of equation (6) in which one of the columns of the orthogonal pilot matrix of equation (6) is multiplied by −1 (thus maintaining the orthogonality). In the pilot matrix of equation (6), or variants thereof, the first column corresponds to the first OFDM symbol in the subframe and the second column corresponds to the second OFDM symbol in the subframe. The first two entries in each column correspond to the real and imaginary components of a pilot symbol  702  above the center carrier frequency and the second two entries in each column correspond to the real and imaginary components of a pilot symbol  702  below the center carrier frequency. For example, the first OFDM symbol includes a first pilot symbol  702  on a subcarrier above the center carrier frequency (i.e., a positive subcarrier) and a second pilot symbol  702  on a subcarrier below the center carrier frequency (i.e., a negative subcarrier), and the second OFDM symbol includes the first pilot symbol  702  on the subcarrier below the center carrier frequency and the negative of the second pilot symbol  702  on the subcarrier above the center carrier frequency. Alternatively, the second OFDM symbol includes the second pilot symbol  702  on the subcarrier above the center carrier frequency and the negative of the first pilot symbol  702  on the subcarrier below the center carrier frequency. In these examples, the subcarrier above the center carrier frequency and the subcarrier below the center carrier frequency (i.e., the negative and positive subcarriers) are symmetric about the center carrier frequency. 
     The pilot symbol subcarriers in the different subframes of a frame are staggered with respect to each other, such that a predefined fraction of subcarriers are used for pilot symbols  702  somewhere in the frame. The subcarriers to be used for pilot symbols  702  in a given subframe may be determined by averaging the indices of the subcarriers used in two previous subframes, resulting in a pattern that is staggered in time as shown in  FIG. 7 . In the example of  FIG. 7 , every fourth subcarrier is used for pilot symbols  702  in some subframe within the frame. Each frame in this example thus includes OFDM symbols with 16 different sets of pilot positions. 
     For each subframe, the receiver  106  ( FIG. 1A ) estimates frequency responses for the subcarriers of the pilot symbols  702  (e.g., using Equation (7)) and compensates accordingly for the estimated frequency response, which represents signal impairment. Thus, in the example of  FIG. 7 , frequency response estimation is performed for every fourth subcarrier during reception of each frame. Frequency responses for the remaining subcarriers (i.e., the subcarriers not used for pilot symbols  702  anywhere in the frame) may be interpolated and compensated for accordingly. 
     Including pilot symbols  702  in each OFDM symbol (e.g., staggered in a pattern such as the pattern of  FIG. 7 , or alternatively in the same subcarriers from OFDM symbol to OFDM symbol) enables continuous tracking of phase noise and carrier frequency offset, while using only a small amount of overhead (e.g., 1/64). Furthermore, the symmetry of the pilot symbol subcarriers about the center carrier frequency and the use of the same subcarriers for pilot symbols  702  in a pair of OFDM symbols (e.g., in accordance with Equation (6)) enables correction of transmitter and/or receiver IQ mismatch (e.g., in accordance with Equation (7)). 
     The spacing of subcarriers used for pilot symbols  702  within a frame may be determined based on the minimal coherence bandwidth, which corresponds to a maximal delay spread equal to the cyclic prefix length. For example, for a cyclic prefix length of 4 us, and thus a maximal delay spread of 4 us, the minimal coherence bandwidth is 250 kHz. The spacing of pilot symbol subcarriers may be a specified fraction of the coherence bandwidth. For example, if the spacing between subcarriers in  FIG. 7  is 12.5 kHz, the spacing between subcarriers used for pilot symbols  702  somewhere within a frame is 50 kHz, or one-fifth of the coherence bandwidth, since every fourth subcarrier is used for pilot symbols  702  at some point within a frame. 
     Attention is now directed to joint estimation of and correction for both transmitter-side I/Q mismatch and receiver-side I/Q mismatch. In some embodiments, carrier frequency offset is estimated, after which joint estimation of transmitter-side I/Q mismatch and receiver-side I/Q mismatch is performed. 
     Let us define p 1 =a Rx a Tx , p 2 =a Rx b Tx , p 3 =b Rx a Tx *, p 4 =b Rx b Tx *. These four unknown parameters can be estimated in a similar fashion as in the case of receiver-only IQ mismatch. The received signal vector z[n] can be written as 
               z   ⁡     [   n   ]       =         [               x   ~     1     ⁡     [   n   ]                 x   ~     2     ⁡     [   n   ]                 x   ~     3     ⁡     [   n   ]                 x   ~     4     ⁡     [   n   ]             ]     ⁡     [           p   1               p   2               p   3               p   4           ]       +       w   ~     ⁡     [   n   ]               
where {tilde over (x)} 1 [n]=e jΔωn x[n], {tilde over (x)} 2 [n]=e jΔωn x*[n], {tilde over (x)} 3 [n]=e −jΔωn x*[n], {tilde over (x)} 4 [n]=e −jΔωn x[n]. Inversion of the N×4 system matrix to solve for the four unknown parameters can be avoided by employing adaptive filtering techniques such as LMS.
 
     Once the signal parameters have been estimated, CFO, transmitter-side I/Q mismatch, and receiver-side I/Q mismatch can be compensated. The received signal can be re-written as
 
 z[n ]=( p   1   e   jΔωn   +p   4   e   −jΔωn ) x[n ]+( p   2   e   jΔωn   +p   3   e   −jΔωn ) x*[n]+a   Rx   w[n]+b   Rx   w*[n] 
 
     Transmitter-side I/Q mismatch and receiver-side I/Q mismatch can be compensated jointly via the following transformation: 
                 z   ^     ⁡     [   n   ]       =       1              u   ⁡     [   n   ]            2     -            s   ⁡     [   n   ]            2         ⁡     [           u   *     ⁡     [   n   ]       ⁢     z   ⁡     [   n   ]         -       s   ⁡     [   n   ]       ⁢       z   *     ⁡     [   n   ]           ]             
where u[n]=p 1 e jΔωn +p 4 e −jΔωn  and s[n]=p 2 e jΔωn +p 3 e −jΔωn . The signal after joint CFO and I/Q compensation reads
 
 {circumflex over (z)}[n]=x[n]+{tilde over (w)}[n] 
 
     For small I/Q mismatch, such that we can approximate f[n]≈1, sin Δφ≈Δφ, cos Δφ≈1, g[n] sin Δφ≈0, the signal model simplifies to
 
 z[n]=e   jΔωn   x[n]+b   Tx   e   jΔωn   x*[n]+b   Rx   e   −jΔωn   x*[n]+w[n]+b   Rx   w*[n] 
 
     This simplification allows the transmitter-side I/Q mismatch and receiver-side I/Q mismatch to be estimated separately. In some embodiments, the correction procedure would first compensate for receiver-side I/Q mismatch, then compensate for the CFO (e.g., via phase rotation), and then compensate for the transmitter-side I/Q mismatch. For example, the method  300  ( FIG. 3B ) would be used for the correction procedure. 
     In some embodiments the method  350  ( FIG. 3A ) is used for joint compensation, wherein operations  354  and  360  estimate the joint receiver-side and transmitter-side I/Q mismatch and operations  356  and  362  compensate for the joint I/Q mismatch. 
     Joint estimation of and correction for both transmitter-side I/Q mismatch and receiver-side I/Q mismatch may be performed for wideband signals (e.g., after the carrier frequency offset has been estimated). In some embodiments, if signals occupy a wide bandwidth, multipath channel effects are considered, as is the frequency dependency of I/Q mismatch at both the transmitter side and receiver side.  FIG. 2E  illustrates signal impairments for wideband signals, including a channel transfer function h[n] for a channel  130 . 
     The overall received signal at the receiver can be expressed as the sum of four signal components and an effective noise component
 
 z[n]=a   Rx   [n]*[e   jΔωn ( h[n]*a   Tx   [n]*x[n ])]+ a   Rx   [n]*[e   jΔωn ( h[n]*b   Tx   [n]*x*[n ])]+ b   Rx   [n]*[e   −jΔωn ( h*[n]*a   Tx   *[n]*x*[n ])]+ b   Rx   [n]*[e   −jΔωn ( h*[n]*b   Tx   *[n]*x[n ])]+ a   Rx   [n]*w[n]+b   Rx   [n]*w*[n] 
 
     The signal components other than the first constitute the interference due to transmitter I/Q mismatch and receiver I/Q mismatch. The last interference term is presumably quite weak as compared to the preceding two. 
     Before compensating for I/Q mismatch, the CFO is estimated by employing a narrowband signal (e.g., a narrowband training signal). In some embodiments, the I/Q mismatch is then estimated separately for the receiver side and the transceiver side, by analogy for the narrowband case, using convolution matrices to account for the frequency dependency of the I/Q mismatch. 
     Alternatively, joint estimation and correction of transmitter-side I/Q mismatch and receiver-side I/Q mismatch is performed. If the CFO is small compared to the coherence bandwidth of the channel and transmitter/receiver I/Q filters, the frequency shift due to CFO can be corrected before filtering operations. Therefore, the received signal in the frequency domain can be expressed as 
               Z   ⁡     (   f   )       =           P   1     ⁡     (   f   )       ⁢     X   ⁡     (     f   +       Δ   ⁢           ⁢   ω       2   ⁢   π         )         +         P   2     ⁡     (   f   )       ⁢       X   *     ⁡     (       -   f     +       Δ   ⁢           ⁢   ω       2   ⁢   π         )         +         P   3     ⁡     (   f   )       ⁢       X   *     ⁡     (       -   f     -       Δ   ⁢           ⁢   ω       2   ⁢   π         )         +         P   4     ⁡     (   f   )       ⁢     X   ⁡     (     f   -       Δ   ⁢           ⁢   ω       2   ⁢   π         )         +       W   ~     ⁡     (   f   )               
Estimation and compensation are performed using the method described above with respect to  FIG. 6  that results in equation (7), in accordance with some embodiments.
 
     Attention is now directed to a method of communication using pilot symbols, such as the pilot symbols  702  ( FIG. 7 ).  FIG. 8  is a flowchart showing a method  800  of communicating between an OFDM transmitter (e.g., the transmitter  102 ,  FIG. 1A ) and an OFDM receiver (e.g., the receiver  106 ,  FIG. 1A ) in accordance with some embodiments. 
     The OFDM transmitter transmits ( 802 ) successive pairs of OFDM symbols (e.g., successive subframes that include pilot symbols  702  on both OFDM symbols of each subframe, as shown in  FIG. 7 ). Both OFDM symbols of a respective pair include pilot symbols on one or more groups of two subcarriers. The two subcarriers of each group are the same for both OFDM symbols and are symmetric about a center carrier frequency. The pilot symbols on the two subcarriers compose an orthogonal matrix (e.g., in accordance with equation (6) or a variant of equation (6) in which one of the columns is multiplied by −1). 
     In some embodiments, the OFDM symbols of the successive pairs include ( 804 ) pilot symbols on respective subsets of a plurality of subcarriers (e.g., as shown in  FIG. 7 ). Both OFDM symbols of each of the successive pairs have pilot symbols on a respective subset of the plurality of subcarriers. The respective subsets of the plurality of subcarriers are distinct. 
     In some embodiments, the respective subsets of the plurality of subcarriers include ( 805 ) subcarriers that are evenly spaced on each side of a center carrier frequency and have mirror symmetry about the center carrier frequency. For example, the pilot symbols  702  have mirror symmetry about the DC subcarrier and may be placed on a predefined fraction of subcarriers in an evenly spaced manner (e.g., on every 64th subcarrier in an OFDM symbol). 
     In some embodiments, the respective subsets of the plurality of subcarriers are staggered ( 806 ) with respect to each other. For example, the index of a subcarrier on which pilot symbols are placed for a pair of OFDM symbols may be determined by averaging the indices of subcarriers on which pilot symbols were placed for two previous pairs of subcarriers. In  FIG. 7 , the pair of OFDM symbols in subframe  0  includes pilot symbols on subcarriers ±2, the pair of OFDM symbols in subframe  1  includes pilot symbols on subcarriers ±34, and the pair of OFDM symbols in subframe  2  includes pilot symbols on subcarriers ±18. The subcarrier indices for the pilot symbols in subframe  2  are determined by averaging the subcarrier indices for the pilot symbols in subframes  0  and  1 : 18 is the average of 2 and 34, and −18 is the average of −2 and −34. Similarly, the pair of OFDM symbols in subframe  4  includes pilot symbols on subcarriers ±10, as determined by averaging subcarrier indices for subframes  0  and  2 . 
     In some embodiments, pilot symbols for respective pairs (e.g., each pair) of OFDM symbols are generated in accordance with equation (6) or a variant of equation (6) in which one of the columns of the orthogonal pilot matrix of equation (6) is multiplied by −1 (thus maintaining the orthogonality). For example, the first OFDM symbol of a pair includes a first pilot symbol on a first subcarrier above a center carrier frequency and a second pilot symbol on a second subcarrier below the center carrier frequency and symmetric with the first subcarrier about the center carrier frequency. The second OFDM symbol of the pair includes the first pilot symbol on the second subcarrier and the negative of the second pilot symbol on the first subcarrier. Alternatively, the second OFDM symbol of the pair includes the negative of the first pilot symbol on the second subcarrier and the second pilot symbol on the first subcarrier. 
     The OFDM receiver receives ( 808 ) the successive pairs of OFDM symbols. Using the pilot symbols, the OFDM receiver estimates ( 810 ) frequency responses at frequencies corresponding to the subcarriers carrying pilot symbols (e.g., to the respective subsets of the plurality of subcarriers). This estimation is performed, for example, using equation (7). In some embodiments, the OFDM receiver interpolates ( 812 ) frequency responses for subcarriers not carrying pilot symbols (e.g., for subcarriers not included in the respective subsets of the plurality of subcarriers), based on the estimated frequency responses. The OFDM receiver compensates ( 814 ) for signal impairment based at least in part on the estimated frequency responses. In some embodiments, the OFDM receiver compensates ( 816 ) for the signal impairment based further on the interpolated frequency responses. 
     While the method  800  includes a number of operations that appear to occur in a specific order, it should be apparent that the method  800  can include more or fewer operations, which can be executed serially or in parallel. Performance of two or more operations may overlap and two or more operations may be combined into a single operation. For example, all of the operations of the method  800  may be performed in an ongoing basis. 
       FIG. 5  is an example of a block diagram of a communication device  500  that performs signal impairment estimation and compensation. In some embodiments, the device  500  is a wireless device (e.g., a WLAN device, such as a personal computer, laptop or tablet computer, mobile phone, personal digital assistant, GPS device, wireless access point, or other electronic device). In some embodiments, the device  500  has a wired network connection. 
     The device  500  includes a processor unit  501 , memory unit  507 , network interface  505 , and transceiver  200  ( FIG. 2A ) coupled by a bus  503 . The processor unit  501  includes one or more processors and/or processor cores. In some embodiments, the network interface  505  includes at least one wireless network interface (e.g., a WLAN interface, a Bluetooth® interface, a WiMAX interface, a ZigBee® interface, a Wireless USB interface, etc.). In some embodiments, the device  500  includes at least one wired network interface (e.g., to interface with a coaxial cable or other physical medium). 
     The memory unit  507  includes a non-transitory computer-readable storage medium (e.g., one or more nonvolatile memory elements, such as EPROM, EEPROM, Flash memory, a hard disk drive, and so on) that stores a signal impairment estimation and compensation software module  510 . In some embodiments, the software module  510  includes one or more programs with instructions that, when executed by processor unit  501  and/or by the receiver baseband processor  280  ( FIG. 2A ), cause the mobile device  500  to perform the methods  300  and/or  350  ( FIGS. 3A-3B ). In some embodiments, these instructions include instructions for performing time-domain compensation (e.g., as described with regard to  FIGS. 4A-4D  and equations 1-5) and/or frequency domain compensation. In some embodiments, these instructions include instructions for separately estimating and compensating for transmitter IQ mismatch and receiver IQ mismatch and/or for jointly estimating and compensating for transmitter IQ mismatch and receiver IQ mismatch, using any technique described herein. In some embodiments, these instructions include instructions for performing all or part of the transmitter-side and/or receiver-side portions of the method  800  ( FIG. 8 ). 
     In the foregoing specification, the present embodiments have been described with reference to specific exemplary embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the disclosure as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense.