Patent Publication Number: US-7711329-B2

Title: Adaptive filter for transmit leakage signal rejection

Description:
This application claims the benefit of provisional U.S. Application Ser. No. 60/519,561, entitled “Adaptive Filtering of TX Leakage in CDMA Receivers,” filed Nov. 12, 2003. 

   BACKGROUND 
   I. Field 
   The present invention relates generally to electronics, and more specifically to techniques for mitigating the deleterious effects of a transmit (TX) leakage signal in a wireless full-duplex communication system. 
   II. Background 
   A wireless device in a wireless full-duplex communication system can simultaneously transmit and receive data for two-way communication. One such full-duplex system is a Code Division Multiple Access (CDMA) system. On the transmit path, a transmitter within the wireless device (1) modulates data onto a radio frequency (RF) carrier signal to generate an RF modulated signal and (2) amplifies the RF modulated signal to obtain a transmit signal having the proper signal level. The transmit signal is routed via a duplexer and transmitted from an antenna to one or more base stations. On the receive path, a receiver within the wireless device (1) obtains a received signal via the antenna and duplexer and (2) amplifies, filters, and frequency downconverts the received signal to obtain baseband signals, which are further processed to recover data transmitted by the base station(s). 
   For a full-duplex wireless device, the RF circuitry in the receiver is often subjected to interference from the transmitter. For example, a portion of the transmit signal typically leaks from the duplexer to the receiver, and the leaked signal (which is commonly referred to as a “TX leakage” signal or a “TX feed-through” signal) may cause interference to a desired signal within the received signal. Since the transmit signal and the desired signal typically reside in two different frequency bands, the TX leakage signal can normally be filtered out and does not pose a problem in itself. However, the TX leakage signal may interact with a “jammer” (which is a large amplitude undesired signal close in frequency to the desired signal) to generate “cross modulation” distortion components on both sides of the jammer, as described below. Distortion components that fall within the signal band of the desired signal and which are not filtered out act as additional noise that may degrade performance. 
   A surface acoustic wave (SAW) filter is often used to filter out the TX leakage signal and mitigate its deleterious effects. The use of a SAW filter for TX leakage rejection is undesirable for several reasons. First, the SAW filter is normally a discrete component that is not fabricated on an RF integrated circuit (RFIC) and thus occupies space on a circuit board. Second, the SAW filter typically requires other discrete components for input and output impedance matching. Third, the SAW filter and its impedance matching circuitry increase the cost of the wireless device. 
   There is therefore a need in the art for techniques to mitigate the deleterious effects of a TX leakage signal without using a SAW filter. 
   SUMMARY 
   Adaptive filters that can attenuate a TX leakage signal in wireless full-duplex communication systems (e.g., a CDMA system) are described herein. An adaptive filter may be fabricated on an RFIC, along with other circuit blocks for a receiver such as a low noise amplifier (LNA) for amplification, a mixer for frequency downconversion, and so on. The adaptive filters can avoid the disadvantages described above for SAW filters. 
   In an embodiment, an adaptive filter suitable for use for TX leakage rejection includes a summer and an adaptive estimator. The summer receives an input signal having TX leakage signal and an estimator signal having an estimate of the TX leakage signal, subtracts the estimator signal from the input signal, and provides an output signal having the TX leakage signal attenuated. The adaptive estimator receives the output signal and a reference signal having a portion or version of the signal being transmitted, estimates the TX leakage signal in the input signal based on the output signal and the reference signal, and provides the estimator signal having the TX leakage signal estimate. 
   The adaptive estimator may utilize a least mean squared (LMS) algorithm to minimize a mean square error (MSE) between the TX leakage signal in the input signal and the TX leakage signal estimate in the estimator signal. In this case, the adaptive estimator may include (1) a first multiplier that multiplies the output signal with an in-phase reference signal and provides a first in-phase signal, (2) a first integrator that integrates the first in-phase signal and provides a second in-phase signal, (3) a second multiplier that multiplies the second in-phase signal with either the in-phase reference signal or a quadrature reference signal and provides a third in-phase signal, (4) a third multiplier that multiplies the output signal with the quadrature reference signal and provides a first quadrature signal, (5) a second integrator that integrates the first quadrature signal and provides a second quadrature signal, and (6) a fourth multiplier that multiplies the second quadrature signal with either the in-phase or quadrature reference signal and provides a third quadrature signal, and (7) a summer that sums the third in-phase signal and the third quadrature signal and provides the estimator signal. The multipliers may be implemented with mixers. The adaptive estimator may further include other circuit blocks/elements for improved performance, as described below. A quadrature splitter receives the reference signal and provides the in-phase and quadrature reference signals for the adaptive estimator. 
   Various aspects and embodiments of the invention are described in further detail below. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The features and nature of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference characters identify correspondingly throughout and wherein: 
       FIG. 1  shows an RF portion of a wireless device; 
       FIGS. 2A through 2C  show signals at various points in a receiver within the wireless device; 
       FIG. 3  shows an implementation of the receiver with an RF SAW filter; 
       FIG. 4  shows the RF portion of a wireless device with an adaptive filter for TX leakage rejection; 
       FIGS. 5 and 6  show two embodiments of the adaptive filter; 
       FIGS. 7 and 8  show two more detailed embodiments of the adaptive filter; 
       FIG. 9  shows a simplified model of the adaptive filter for stability analysis; 
       FIG. 10  shows the frequency responses of the adaptive filter for three different damping factors; and 
       FIGS. 11A through 11D  show the generation of cross modulation distortion due to double mixing within the adaptive filter. 
   

   DETAILED DESCRIPTION 
   The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment or design described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments or designs. 
   The adaptive filters described herein may be used for various wireless full-duplex communication systems. These adaptive filters may also be used for various frequency bands such as a cellular band from 824 to 894 MHz, a Personal Communication System (PCS) band from 1850 to 1990 MHz, a Digital Cellular System (DCS) band from 1710 to 1880 MHz, an International Mobile Telecommunications-2000 (IMT-2000) band from 1920 to 2170 MHz, and so on. For clarity, the following description is for the cellular band, which includes (1) an uplink frequency band from 824 to 849 MHz and (2) a downlink frequency band from 869 to 894 MHz. The uplink and downlink frequency bands are transmit (TX) and receive (RX) frequency bands, respectively, for a wireless device. 
     FIG. 1  shows a block diagram of an RF portion of a wireless device  100 . On the transmit path, a power amplifier (PA)  112  within a transmitter  110  receives and amplifies a TX modulated signal and provides a transmit signal. The transmit signal is routed through a duplexer  116  and transmitted via an antenna  118  to one or more base stations. A portion of the transmit signal also couples or leaks through duplexer  116  to the receive path. The amount of TX leakage is dependent on the isolation between the transmit and receive ports of the duplexer, which may be approximately 50 dB for a SAW duplexer at the cellular band. A lower TX-RX isolation results in higher level of TX leakage. 
   On the receive path, a received signal containing a desired signal and possibly a jammer is received via antenna  118 , routed through duplexer  116 , and provided to an LNA  122  within a receiver  120 . LNA  122  also receives a TX leakage signal from the transmit path, amplifies the receiver input signal at its input, and provides an amplified RF signal, x(t). A filter  130  receives and filters the amplified RF signal to remove out of band signal components (e.g., the TX leakage signal) and provides a filtered RF signal, y(t). A mixer  132  receives and frequency downconverts the filtered RF signal with a local oscillator (LO) signal and provides a downconverted signal. 
     FIG. 2A  shows the received signal, which includes a desired signal  212  and a jammer  214 . Jammer  214  is an undesired signal and may correspond to, for example, a signal transmitted by a nearby base station in an Advanced Mobile Phone Service (AMPS) system. The jammer may have an amplitude that is much higher than that of the desired signal and may be located close in frequency to the desired signal. 
     FIG. 2B  shows the signal at the input of LNA  122 . This signal contains desired signal  212  and jammer  214  in the received signal as well as a TX leakage signal  216  from the transmit path. The TX leakage signal may have a large amplitude relative to the desired signal because the transmit signal is often much larger in amplitude than the desired signal. 
     FIG. 2C  shows the signal at the output of mixer  132 . Non-linearity in LNA  122  and mixer  132  can cause the modulation on TX leakage signal  216  to be transferred to (narrowband) jammer  214 , which then results in a widened spectrum  218  around the jammer. This spectral widening is referred to as cross modulation and is described in detail below. As shown in  FIG. 2C , a portion  220  of widened spectrum  218  (which is shown with shading) may fall within the desired signal band. Portion  220  acts as additional noise that degrades the performance of the wireless device. This noise further degrades the receiver sensitivity so that the smallest desired signal that can be reliably detected by the receiver needs to have a larger amplitude. 
     FIG. 3  shows a schematic diagram of a conventional implementation of receiver  120  with an RF SAW filter  330 . SAW filter  330  has various desirable characteristics such as sharp transition band edges and large attenuation of out-of-band signal components. SAW filter  330  is often used to reject the TX leakage signal at the input of mixer  132 , which then reduces the amount of cross modulation distortion generated by the mixer. 
   LNA  122  couples to SAW filter  330  via an input impedance matching network  310  formed by a resistor  312 , inductors  314  and  316 , and a capacitor  318 . SAW filter  330  couples to mixer  132  via an output impedance matching network  340  formed by capacitors  342 ,  344 , and  346  and inductors  348  and  350 . A capacitor  320  provides filtering of the power supply, VCC, for LNA  122 . 
   The use of an RF SAW filter for TX leakage signal filtering has several disadvantages. First, if LNA  122  and mixer  132  are implemented within a single RFIC for reduced cost and improved reliability, then SAW filter  330  is implemented off-chip and requires three IC package pins for interface to the LNA and mixer. Second, SAW filter  330  and the discrete components for matching networks  310  and  340  require extra board space and further add cost to the wireless device. Third, the insertion losses of SAW filter  330  and matching networks  310  and  340  degrade the cascaded gain and noise figure of the receiver. 
   An adaptive filter may be used to reject the TX leakage signal and avoid the disadvantages described above for the SAW filter. The adaptive filter may be implemented on an RFIC (e.g., the same RFIC used for the LNA and mixer) so that no additional board space is needed for external components and cost is reduced. The adaptive filter may be designed to achieve the desired rejection of the TX leakage signal and consume low power. 
     FIG. 4  shows a block diagram of the RF portion of a wireless device  400  with an adaptive filter  430  for TX leakage rejection. On the transmit path, a TX modulated signal is amplified by a power amplifier  412  within a transmitter  410 , routed through a duplexer  416 , and transmitted via an antenna  418  to one or more base stations. A coupler  414  receives the transmit signal from power amplifier  412  and provides a portion of this transmit signal as a reference signal, r(t). 
   On the receive path, a received signal is received via antenna  418 , routed through duplexer  416 , and provided to an LNA  422  within a receiver  420 . LNA  422  also receives the TX leakage signal from the transmit path, amplifies the signal at its input, and provides an amplified RF signal, x(t). Adaptive filter  430  receives and filters the amplified RF signal to attenuate/reject the TX leakage signal and provides a filtered RF signal, y(t). A mixer  432  frequency downconverts the filtered RF signal with an LO signal and provides a downconverted signal. 
   In general, adaptive filter  430  may be located at any point on the received path prior to mixer  432 . For example, adaptive filter  430  may be placed either before or after LNA  422 . Improved noise performance can typically be achieved with adaptive filter  430  placed after LNA  422 . 
     FIG. 5  shows a block diagram of an adaptive filter  430   a , which is an embodiment of adaptive filter  430  within receiver  420 . Adaptive filter  430   a  generates an estimate of the TX leakage signal, e(t), based on the r(t) reference signal and further subtracts the TX leakage signal estimate from the x(t) signal to obtain the y(t) signal for mixer  432 . The x(t) signal is also referred to as the filter input signal, and the y(t) signal is also referred to as the filter output signal. 
   For the embodiment shown in  FIG. 5 , adaptive filter  430   a  utilizes an LMS algorithm to minimize the mean square error between the TX leakage signal in the filter input signal and the TX leakage signal estimate. Adaptive filter  430   a  includes a quadrature splitter  508 , an LMS adaptive estimator  510   a , and a summer  540 . Quadrature splitter  508  receives the reference signal, r(t), and provides an in-phase reference signal, i(t), and a quadrature reference signal, q(t). The i(t) and q(t) signals respectively contain the in-phase and quadrature components of the reference signal, with the i(t) signal leading the q(t) signal by 90°. 
   LMS estimator  510   a  includes an in-phase section  520   a , a quadrature section  520   b , and a summer  530 . Within in-phase section  520   a , a multiplier  522   a  receives and multiplies the i(t) signal with the y(t) signal and provides an m i (t) signal, which is m i (t)=y(t)·i(t). An integrator  524   a  receives and integrates the m i (t) signal and provides an in-phase integrated signal, w i (t). A multiplier  528   a  receives and multiplies the i(t) signal with the w i (t) signal and provides a z i (t) signal, which is z i (t)=w i (t)·i(t). Similarly, within quadrature section  520   b , a multiplier  522   b  receives and multiplies the q(t) signal with the y(t) signal and provides an m q (t) signal, which is m q (t)=y(t)·q(t). An integrator  524   b  receives and integrates the m q (t) signal and provides a quadrature integrated signal, w q (t). A multiplier  528   b  receives and multiplies the q(t) signal with the w q (t) signal and provides a z q (t) signal, which is z q (t)=w q (t)·q(t). Multipliers  522   a ,  522   b ,  528   a , and  528   b  are four-quadrant multipliers. Summer  530  receives and sums the z i (t) and z q (t) signals and provides an estimator signal, e(t), which contains the TX leakage signal estimate obtained based on the LMS algorithm. The w i (t) and w q (t) signals are effectively weights used for estimating the TX leakage signal. 
   Summer  540  receives the estimator signal, e(t), from LMS estimator  510   a  and the filter input signal, x(t), which contains the received signal as well as the TX leakage signal. Summer  540  subtracts the estimator signal from the filter input signal and provides the filter output signal, y(t). 
   For the LMS algorithm, the estimator signal from LMS estimator  510   a  may be expressed as: 
                     e   ⁡     (   t   )       =   μ     ⁣       ·     [         i   ⁡     (   t   )       ·       ∫     τ   =   0     t     ⁢     i   ⁡     (   τ   )           ⁣         ·     y   ⁡     (   τ   )         ⁢           ⁢     ⅆ   τ       +       q   ⁡     (   t   )       ·       ∫     τ   =   0     t     ⁢     q   ⁡     (   τ   )             ⁣       ·     y   ⁡     (   τ   )         ⁢           ⁢     ⅆ   τ         ]       ,             Eq   ⁢           ⁢     (   1   )                 
where μ is the unity-gain angular frequency of LMS estimator  510   a , which is the angular frequency at which the overall gain from the output of summer  540  to the inverting input of summer  540  is equal to one. The parameter μ includes the gains of all circuit blocks in the feedback loop from the output to the inverting input of summer  540  and is given in units of rad/sec/V 2 . Equation (1) assumes that the integrators are ideal with a single pole at DC.
 
   The filter output signal from adaptive  430   a  may be expressed as: 
                   y   ⁡     (   t   )       =         x   ⁡     (   t   )       -     e   ⁡     (   t   )         =       x   ⁡     (   t   )       -     μ   ·       [         i   ⁡     (   t   )       ·       ∫     τ   =   0     t     ⁢     i   ⁡     (   τ   )           ⁣         ·     y   ⁡     (   τ   )         ⁢           ⁢     ⅆ   τ       +       q   ⁡     (   t   )       ·       ∫     τ   =   0     t     ⁢     q   ⁡     (   τ   )             ⁣       ·     y   ⁡     (   τ   )         ⁢           ⁢     ⅆ   τ         ]     .                   Eq   ⁢           ⁢     (   2   )                 
The filter output signal, y(t), is often referred to as an error signal. For simplicity, the following analysis assumes that the x(t) signal contains only the TX leakage signal. The TX leakage signal and the in-phase and quadrature reference signals may also be assumed to be sinusoids with the following form:
   x ( t )= A ·sin(ω t +φ),  i ( t )= B ·sin(ω t ), and  q ( t )= B ·cos(ω t ),   Eq (3) 
where A is the amplitude of the TX leakage signal;
         φ is a random angle of the TX leakage signal;   B is the amplitude of the r(t) reference signal; and   ω is the angular frequency of the transmit signal and the reference signal.
 
A frequency f and its angular frequency ω are related by a factor of 2π, or ω=2π·f. Equation (2) may be converted to a linear second-order ordinary differential equation with the signals shown in equation (3), as follows:
       
   
     
       
         
           
             
               
                 
                   
                     
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   Equation (4) may be solved using Laplace transform, as follows:
 
 s   2   ·Y ( s )− s·y (0)− y ′(0)+μ· B   2   [s·Y ( s )− y (0)]+ω 2   ·Y ( s )=0,   Eq (5)
 
where y(0) and y′(0) are initial conditions for y(t) and d y(t)/dt, respectively. If no reference signal is applied for t≦0 (i.e., i(t)=0 and q(t)=0 for t≦0), then y(t)=x(t) for t≦0, and the initial conditions may be expressed as:
 
 y (0)= x (0)= A ·sin(φ), and
 
 y ′(0)= x ′(0)= A ·ω·cos(φ).   Eq (6)
 
With the initial conditions as shown in equation (6), the Laplace transform of the adaptive filter output, y(t), may be expressed as:
 
                     Y   ⁡     (   s   )       =     A   ·         s   ·     sin   ⁡     (   ϕ   )         +     ω   ·     cos   ⁡     (   ϕ   )         +     2   ⁢     ζω   ·     sin   ⁡     (   ϕ   )                 s   2     +     2   ⁢     ζω   ·   s       +     ω   2             ,           Eq   ⁢           ⁢     (   7   )                 
where ζ is a damping factor, which is ζ=μ·B 2 /(2ω). The Laplace transform of the adaptive filter input, x(t), may be expressed as:
 
   
     
       
         
           
             
               
                 
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   The transfer function of adaptive  430   a  may then be expressed as: 
                     H   ⁡     (   s   )       =         Y   ⁡     (   s   )         X   ⁡     (   s   )         =           s   ·     sin   ⁡     (   ϕ   )         +     ω   ·     cos   ⁡     (   ϕ   )         +     2   ⁢     ζω   ·     sin   ⁡     (   ϕ   )                 s   ·     sin   ⁡     (   ϕ   )         +     ω   ·     cos   ⁡     (   ϕ   )             ·         s   2     +     ω   2           s   2     +     2   ⁢     ζω   ·   s       +     ω   2               ,             Eq   ⁢           ⁢     (   9   )       ⁢                       
where s=jω x  and ω x  is a variable for angular frequency.
 
     FIG. 10  shows the frequency responses of adaptive  430   a  for three different damping factors. The frequency responses are given for a TX leakage signal composed of a single tone at frequency of ω/2π=835 MHz. Plot  1012  in  FIG. 10  shows the frequency response for a damping factor of ζ=0.001 (under-damping), which has the narrowest notch and the least amount of attenuation in the RX frequency band of 869 to 894 MHz for the cellular band. Plots  1014  and  1016  in  FIG. 10  show the frequency responses for damping factors of ζ=0.01 and ζ=0.1, respectively. As the damping factor increases, the notch widens and the amount of attenuation in the RX frequency band increases. An ideal adaptive filter can achieve infinite attenuation of the TX leakage signal. The amount of TX leakage attenuation achieved by a practical adaptive filter is dependent on imperfections in the adaptive filter, as described below. 
   The inverse Laplace transform of Y(s) in equation (7) may be expressed as: 
                   y   ⁡     (   t   )       =     A   ·     ⅇ       -   ζω     ⁢           ⁢   t       ·       [             cos   ⁡     (   ϕ   )       +     ζ   ·     sin   ⁡     (   ϕ   )               1   -     ζ   2           ·     sin   ⁡     (     ω   ⁢           ⁢   t   ⁢       1   -     ζ   2           )         +       sin   ⁡     (   ϕ   )       ·     cos   ⁡     (     ω   ⁢           ⁢   t   ⁢       1   -     ζ   2           )           ]     .               Eq   ⁢           ⁢     (   10   )                 
The exponential term e −ζωt  controls the settling time and thus the convergence speed of the LMS algorithm. Since the damping factor ζ needs to be much smaller than one (i.e., ζ&lt;&lt;1) to reduce filter distortion and attenuation, as shown in  FIG. 10 , equation (10) may be simplified as follows:
   y ( t )≅ x ( t )· e   −ζωt .   Eq (11) 
Equation (11) indicates that the filter output signal is simply an exponentially decaying version of the filter input signal. For 30 dBc of TX leakage rejection, e −ζωt =10 −30/20 , and the settling time may be expressed as:
 
   
     
       
         
           
             
               
                 
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   Adaptive  430   a  generates cross modulation distortion even if all of the circuit blocks of the adaptive filter are perfectly linear. The cross modulation distortion is generated by the frequency mixing function of multipliers  522  and  528 , as illustrated in  FIGS. 11A through 11D . 
     FIG. 11A  shows a case in which the filter input signal, x(t), contains TX leakage signal  1112  centered at frequency f TX  and a single-tone jammer  1114  located at frequency f J . For this example, the jammer frequency is close to the signal band of the desired signal and f J −f TX ≈45 MHz, which is the separation between the TX and RX frequency bands for the cellular band. 
     FIG. 11B  shows the signal components at the output of each of multipliers  522   a  and  522   b . Signal component  1122  at DC and signal component  1126  at 2f TX  are generated by the mixing between the TX leakage signal and the i(t) and q(t) reference signals. Signal component  1124  at f J −f TX  and signal component  1128  at f J +f TX  are generated by the mixing between the jammer and the reference signals. 
     FIG. 11C  shows the signal components at the output of each of integrators  524   a  and  524   b . For this analysis, each integrator  524  has an ideal transfer function, which is a single pole at DC. Signal component  1124  at f J −f TX  is attenuated by a particular amount, and signal components  1126  and  1128  at the higher frequencies are attenuated by larger amounts and are negligible. Signal component  1124  represents an undesired component that contains the convolved spectra of the jammer and the transmit signal. 
     FIG. 11D  shows the signal components at the output of adaptive filter  430   a . Signal component  1144  centered at f J  is generated by the mixing of signal component  1124  centered at f J −f TX  and the reference signals centered at f TX . The double mixing actions of multipliers  522   a / 522   b  and multipliers  528   a / 528   b  result in the transmit signal component centered at f TX  being transferred to the jammer frequency f J . Signal component  1144  represents the cross modulation distortion that is added to the filter input signal by summer  540 . The filter output signal contains an attenuated/rejected TX leakage signal  1112 , an unattenuated jammer  1114 , and signal component  1144 . 
   The cross modulation distortion generated by adaptive  430   a  may be analyzed by a triple beat distortion. For the analysis, the transmit signal (and thus the reference signal) contains two closely spaced tones at frequencies of f TX ±Δf/2. The filter input signal contains (1) the TX leakage signal with the two transmit tones and (2) an inband single-tone jammer at a frequency of f J . If the adaptive filter completely rejects the TX leakage signal such that the filter output signal, y(t), contains only the jammer, then its triple beat distortion, d(t), may be derived as: 
                     d   ⁡     (   t   )       =         μ       ω   J     -     ω   TX         ⁡     [           i   2     ⁡     (   t   )       ·     y   ⁡     (   t   )         +         q   2     ⁡     (   t   )       ·     y   ⁡     (   t   )           ]         ω   =       ω   j     ±   Δω           ,           Eq   ⁢           ⁢     (   13   )                 where  i ( t )= B ·[sin((ω−Δω/2)· t )+sin((ω+Δω/2)· t )];   q ( t )= B ·[cos((ω−Δω/2)· t )+cos((ω+Δω/2)· t )]; and   y ( t )= C ·cos(ω J   t ), where C is the amplitude of the jammer. 
Equation (13) indicates that two triple beat distortion terms are generated at frequencies of f J ±Δf, as described above for  FIGS. 11A through 11D .
 
   A triple beat rejection ratio (TBRR) is defined as the ratio of the jammer amplitude to the amplitude of the cross modulation distortion. The TBRR may be obtained by performing simple trigonometric manipulations on equation (13) and taking the ratio of the jammer amplitude to the triple beat distortion amplitude. The TBRR may be expressed as: 
                   TBRR   =       10   ⁢     log   ⁡     (         ω   J     -     ω   TX         μ   ·     B   2         )         =     20   ⁢           ⁢     log   ⁡     [       1     2   ⁢   ζ       ⁢     (         ω   J       ω   TX       -   1     )       ]             ,           Eq   ⁢           ⁢     (   14   )                 
where ζ=μ·B 2 /(2ω TX ) is the damping factor. Equation (14) indicates that a TBRR of 68 dBc may be obtained with a damping factor of ζ≦8.1×10 −6  for f TX =849 MHz and f J =894 MHz. The settling time is 81 μsec with this damping factor.
 
     FIG. 6  shows a block diagram of an adaptive filter  430   b , which is another embodiment of adaptive filter  430  within receiver  420 . Adaptive filter  430   b  includes an additional pole used to (1) reduce the amplitude of the cross modulation distortion generated by the adaptive filter (2) achieve faster LMS algorithm convergence, and (3) shorten the settling time. 
   Adaptive filter  430   b  includes an LMS adaptive estimator  510   b  and summer  540 . LMS estimator  510   b  includes all of the circuit blocks for LMS estimator  510   a  in  FIG. 5 . LMS estimator  510 b further includes (1) a single-pole or first-order lowpass filter (LPF)  526   a  placed between the output of integrator  524   a  and the input of multiplier  528   a  and (2) a single-pole lowpass filter  526   b  placed between the output of integrator  524   b  and the input of multiplier  528   b . Each lowpass filter  526  may be implemented, for example, with an RC lowpass network composed of a series resistor and a shunt capacitor to circuit ground. The frequency of the single pole is selected such that the adaptive filter is unconditionally stable. As an example, with the additional pole placed at 318 KHz, the damping factor may be increased to ζ=1×10 −4 , the settling time may be reduced to 6 μsec, and the TBRR may be improved to better than 80 dBc. All of these improvements are achieved with an unconditionally stable adaptive filter. Lowpass filters of higher order and/or with poles placed at different frequencies may also be used for lowpass filters  526   a  and  526   b.    
   An ideal adaptive filter provides infinite rejection of the TX leakage signal so that the filter output signal contains no TX leakage signal. However, various imperfections in a practical/realizable adaptive filter limit the amount of TX leakage rejection that may be achieved. Such imperfections may include, for example, finite gain for the integrators and non-zero DC offsets for the circuit blocks of the LMS estimator. 
   A TX leakage rejection ratio (TXRR) is the ratio of the TX leakage signal power at the adaptive filter output to the TX leakage signal power at the adaptive filter input. The TXRR requirement for adaptive filter  430  is dependent on various factors such as, for example, (1) the maximum TX leakage signal power expected at the output of LNA  422  and (2) the maximum acceptable TX leakage signal power at the input of mixer  432 . It can be shown that an adaptive filter with a TXRR of approximately 30 dB can provide performance comparable to that achieved by a receiver with an RF SAW filter (e.g., the receiver shown in  FIG. 3 ). In general, the TXRR requirement for the adaptive filter is dependent on various factors, such as those noted above and possibly other factors. 
   Adaptive filters  430   a  and  430   b  have similar TXRR performance. The actual TXRR achieved by adaptive  430   a  is dependent on various factors such as, for example, (1) the overall gain of the integrators and multipliers and (2) the DC offsets of the multipliers and integrators. An inadequate overall gain limits the TXRR that can be achieved by the adaptive filter. The overall gain is thus selected such that the required TXRR can be achieved and is appropriately distributed among the integrator and multipliers. 
   DC offsets can also adversely affect the TXRR performance of adaptive filter  430   a . Multipliers  522   a ,  522   b ,  528   a , and  528   b  typically have DC responses due to imbalances on the two inputs. Integrator  524   a  and  524   b  have systematic as well as random input DC offsets. The DC offsets introduce an error that reduces the amount of TX leakage rejection by the filter. Also, due to their large DC gains, the integrators may be saturated initially by the combined DC offsets. Once saturated, the integrators have very low gains that result in a long settling time for the adaptive filter. To prevent saturation due to DC offsets, the output of each integrator may be reset (e.g., by shorting together the differential output of each integrator) prior to enabling the adaptive filter and then released thereafter. 
   Various techniques may be used to achieve low combined DC offsets for the in-phase and quadrature paths. The combined DC offsets may be reduced by:
         Increasing the gain of multipliers  522   a  and  522   b  and reducing their DC offsets;   Increasing the reference signal power (i.e., increase B); and/or   Using dynamic offset cancellation techniques such as chopper stabilization and/or auto-zeroing techniques.       

   The multiplier gain may be increased by converting multipliers  522   a  and  522   b  into mixers and using the in-phase and quadrature reference signals as strong LO signals. The high gain of the mixers (e.g., approximately 50 dB for an exemplary mixer design) can significantly reduce the DC offset contribution of the integrators. The output DC offset of the mixers is low due to the inherent chopping action of the mixers. 
   Chopper stabilization techniques may be able to achieve low input DC offset voltages (e.g., below 10 μV). Auto-zeroing techniques, such as correlated double sampling techniques, typically increase the noise floor, which may then contaminate the RX frequency band. The auto-zeroing techniques should thus be used with care. 
   Adaptive filter  430  inherently introduces additional noise that degrades the noise figure of the receiver. Adaptive filter  430  may be designed to minimize noise contribution by using various circuit design techniques known in the art. This way, system requirements can be met even with the additional noise contribution from adaptive filter  430 . 
   Adaptive filter  430  is a feedback system and is unstable if the total phase delay along the feedback loop is 180° and the loop gain is greater than one. For an ideal adaptive filter, the only delay along the feedback loop is 90° introduced by the integrators. For a practical adaptive filter, a delay is introduced by each circuit block within the adaptive filter. 
     FIG. 7  shows a block diagram of an adaptive filter  430   c , which is a more detailed embodiment of adaptive filter  430 . A quadrature splitter  708  receives the reference signal, r(t), and provides a differential in-phase reference signal, i′(t), and a differential quadrature reference signal, q′(t). 
   A pre-amplifier  718  receives and amplifies the filter input signal, x(t), and provides a differential output signal, y′(t), to an in-phase section  720   a  and a quadrature section  720   b . Within in-phase section  720   a , a multiplier  722   a  receives and multiplies the y′(t) signal with the i′(t) signal and provides a differential m i ′(t) signal. An integrator  724   a  receives and integrates the m i ′(t) signal and provides a differential w i ′(t) signal. Integrator  724   a  is implemented with an amplifier and two capacitors coupled between the differential output and the differential input of the amplifier, as shown in  FIG. 7 . A multiplier  728   a  receives and multiplies the w i ′(t) signal with the i′(t) signal and provides a z i ′(t) signal. Within quadrature section  720   b , a multiplier  722   b , an integrator  724   b , and a multiplier  728   b  similarly process the y′(t) signal with the q′(t) signal and provide a z q ′(t) signal. The z i ′(t) and z q ′(t) signals are current outputs and may be combined by tying these outputs together to obtain the estimator signal, e(t). The e(t) signal is subtracted from the x(t) signal by tying these signals together at the input of pre-amplifier  718 . For the circuit embodiment shown in  FIG. 7 , the summer is just a node labeled by a black dot at the input of pre-amplifier  718 , and the input of the adaptive filter is also its output (i.e., y(t)=x(t)). 
   Pre-amplifier  718  has a delay of Δφ 1  at the frequency of the TX leakage signal. Multipliers  722   a  and  722   b  each has a delay of Δφ 2  due to unequal delays of the RF and LO inputs. Multipliers  728   a  and  728   b  each has a delay of Δφ 3  from the reference signal to the multiplier output at the frequency of the TX leakage signal. The total delay Δφ for adaptive filter  430   c  may be computed as: Δφ=Δφ 1 +Δφ 2 +Δφ 3 . 
     FIG. 9  shows a simplified model  900  for the adaptive filter which is suitable for stability analysis. A summer  912  receives and subtracts the output of an integrator  916  from the filter input signal, V in , and provides the filter output signal, V out . A delay element  914  delays the V out , signal by a delay of Δφ. Integrator  916  integrates the delayed signal with a transfer function of G o /(s/p+1). The transfer function between the filter output signal to the filter input signal may be expressed as: 
                     v   out       v   in       =           s   /   p     +   1         s   /   p     +   1   +       G   o     ·     ⅇ   jΔφ           .             Eq   ⁢           ⁢     (   15   )                 
A step response for equation (15) may be expressed as:
   V   out ( t )≈ e   −(1+G     o     ·e     jΔφ     )·pt   =e   −(1+G     o     ·cos(Δφ))·pt .  Eq (16) 
Equation (16) indicates that the filter output signal is an oscillatory signal having an exponential decay of e −pt  due to the integrator pole at p. The presence of the delay Δφ introduces oscillations in the filter output signal. The amplitude of the oscillations can either decay or grow depending on the delay Δφ. It can be shown that the adaptive filter is (1) stable if Δφ is in the range of −90° to +90° and (2) unstable if |Δφ| exceeds 90°. For example, if Δφ 1 =40°, Δφ 2 =0°, and Δφ 3 =60°, then Δφ=100° and the adaptive filter oscillates.
 
     FIG. 8  shows a block diagram of an adaptive filter  430   d , which is another more detailed embodiment of adaptive filter  430 . Adaptive filter  430   d  utilizes an architecture that can compensate for phase delays (e.g., Δφ 1  and Δφ 3 ) at RF frequencies. Adaptive filter  430   d  includes all of the circuit blocks of adaptive filter  430   c  in  FIG. 7 . However, adaptive filter  430   d  uses different reference signals for the two multipliers in each of sections  720   a  and  720   b . For in-phase section  720   a , multiplier  722   a  is driven by the i′(t) signal and multiplier  728   a  is driven by the q′(t) signal (instead of the i′(t) signal). For quadrature section  720   b , multiplier  722   b  is driven by the q′(t) signal and multiplier  728   b  is driven by the i′(t) signal (instead of the q′(t) signal). The LO signals for multipliers  728   a  and  728   b  thus lead the LO signals for multipliers  722   a  and  722   b , respectively, by 90°. The total delay Δφ is also correspondingly reduced by 90°. For the example described above, the total delay is Δφ=10° instead of 100°, and the adaptive filter is stable. 
   The adaptive filters shown in  FIGS. 5 and 6  may be implemented in various manners. Two exemplary implementations are shown in  FIGS. 7 and 8 . The circuit blocks for the adaptive filters may also be implemented in various manners. For example, the multipliers may be implemented with mixers, summers may be implemented by tying current outputs together, and so on. The adaptive filters may also be implemented with differential or single-ended circuit designs.  FIGS. 7 and 8  show exemplary differential designs for adaptive  430   a  in  FIG. 5 . A differential design may provide certain advantages over a single-ended design such as better immunity to noise. 
   The adaptive filters described herein utilize an LMS adaptive estimator to estimate the TX leakage signal. Other types of estimators may also be used to estimate the TX leakage signal, and this is within the scope of the invention. For example, the transmit signal may be stepped across the TX frequency band and weight values w i  and w q  may be determined for the in-phase and quadrature sections, respectively, and used in place of w i (t) and w q ( t ) in  FIG. 5  to estimate the TX leakage signal. 
   The adaptive filters may also be trained in various manners. For example, an adaptive filter may be enabled at the beginning of a training burst (which contains a known training signal) and weight values may be derived based on this burst. The weight values may thereafter be fixed and used for estimating the TX leakage signal during the signaling interval. The weight values may be updated whenever the training bursts are available. To speed up convergence, the conditions for the integrators may be determined and stored prior to tuning away from an RF channel, and the integrators may be initialized with the stored conditions the next time this RF channel is selected. 
   The adaptive filters described herein may also be used for various systems and applications. For example, the adaptive filters may be used in wireless full-duplex communication systems such as cellular systems, OFDM systems, orthogonal frequency division multiple access (OFDMA) systems, multiple-input multiple-output (MIMO) systems, wireless local area networks (LANs), and so on. The full-duplex cellular systems include CDMA system and some versions of Global System for Mobile Communications (GSM) systems, and the CDMA systems include IS-95, IS-2000, IS-856, and Wideband-CDMA (W-CDMA) systems. The adaptive filters may be used for a wireless device as well as a base station in a wireless full-duplex communication system. 
   The adaptive filters described herein may be implemented within an integrated circuit (IC), an RF integrated circuit, an application specific integrated circuit (ASIC), or other electronic units designed to perform the functions described herein. The adaptive filters may also be fabricated with various IC process technologies such as complementary metal oxide semiconductor (CMOS), bipolar junction transistor (BJT), bipolar-CMOS (BiCMOS), silicon germanium (SiGe), gallium arsenide (GaAs), and so on. 
   The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.