Abstract:
Apparatus, systems, and methods implementing techniques for estimating a relative rotation between a first complex signal and a second complex signal. The first complex signal is quantized to produce a quantized signal, and the quantized signal and an additional signal are combined, where the additional signal corresponds to the second complex signal. An estimate of a relative rotation between the first complex signal and the second complex signal is generated in accordance with the combined signal.

Description:
BACKGROUND 
     The following disclosure relates to electrical circuits and signal processing. 
     Aligning the phases of two or more signals in a communications system can be useful. The signals can be information signals at multiple points in a signal path. For example, in a communications system that uses feedback, the phase of a feedback signal may need to be aligned with the phase of a forward path signal for the system to operate correctly or efficiently. 
     A complex signal (represented, for example, by an in-phase component and a quadrature component) can be used to transmit information using symbols that are specific combinations of the components of the complex signal. For example, an in-phase (I) and quadrature (Q) representation of a complex signal can have four symbols, corresponding to the following combinations of the amplitudes of (I,Q): (1,1); (1,−1); (−1,1); and (−1,−1).  FIG. 1A  shows a plot  100  of the in-phase and quadrature components of two complex signals. Two constellations of symbols, each constellation corresponding to a complex signal, are shown. Each of the constellations includes four symbols that represent the symbols that can be transmitted by the corresponding complex signal. One constellation is represented by four circles (e.g., circle  110 ), and the other constellation is represented by four crosses (e.g., cross  120 ). The constellation represented by the crosses is rotated relative to the constellation represented by the circles by an angle of φ, indicating that the corresponding complex signals are rotated relative to each other. 
     A relative rotation between two complex signal constellations can have several causes. In a typical scenario, a first (I,Q) signal pair is used to modulate a radio-frequency (RF) carrier. The modulated carrier, an RF signal, undergoes analog processing steps, after which the modulated carrier is demodulated. A second (I,Q) signal pair results from the demodulation. Any difference in phase between the modulated carrier before the processing steps and the modulated carrier after the processing steps is manifested as a relative rotation between the baseband constellations corresponding to the first and second (I,Q) signal pairs. 
       FIG. 1B  shows a plot  105  of the in-phase and quadrature components of the two complex signals from  FIG. 1A  where the phase of the complex signal with the constellation represented by the crosses (e.g., cross  120 ) has been rotated by −φ to align the two constellations and therefore align the two complex signals. Aligning two complex signals is equivalent to making the relative rotation between the signals substantially zero. 
     An example of a component in a communications system that uses feedback is a Cartesian feedback transmitter. In a Cartesian feedback transmitter, a complex feedback signal is subtracted from a complex input signal to produce a complex error signal. The complex error signal is amplified and filtered to produce an intermediate signal, which is then modulated for transmission. The modulated signal is also demodulated in the transmitter to produce the complex feedback signal. Using Cartesian feedback in a transmitter improves the linearity of the transmitter, but properly aligning the phases of the complex intermediate signal and the complex feedback signal is important. 
     The complex feedback signal typically has a different phase than the complex intermediate signal because of, for example, delays in the RF signal path or a phase difference between the oscillator signal used during modulation and the oscillator signal used during demodulation. A change in output power level or a change in carrier frequency can also cause a relative rotation between the complex intermediate signal and the complex feedback signal. The phase of the complex intermediate signal can be adjusted (e.g., by using a rotator circuit) to align the complex intermediate signal and the feedback signal. The adjustment of the phase of the complex intermediate signal can be controlled based on, for example, an estimate of the relative rotation between the complex intermediate signal and the complex feedback signal. 
     One technique that can be used to estimate the phase difference between the complex intermediate signal and the complex feedback signal is to multiply the in-phase component of the intermediate signal (I IN ) by the quadrature component of the feedback signal (Q FB ) and to multiply the quadrature component of the intermediate signal (Q IN ) by the in-phase component of the feedback signal (I FB ), all multiplication being done in the analog domain. The second product (Q IN  I FB ) is then subtracted from the first product (I IN  Q FB ), and the result is integrated. A rotator circuit uses the integrated result to rotate the phase of the intermediate signal with respect to the feedback signal. 
     In practice, use of the I IN  Q FB −Q IN  I FB  phase alignment technique described above may require analog multipliers capable of handling signals that have a very large dynamic range. The analog multipliers can be difficult to design, and may require an unacceptable amount of power and/or complexity to implement. 
     SUMMARY 
     In one aspect, a method for estimating a relative rotation between a first complex signal and a second complex signal is presented, in which the first complex signal is quantized to produce a quantized signal. The quantized signal and an additional signal are combined, where the additional signal corresponds to the second complex signal. An estimate of a relative rotation between the first complex signal and the second complex signal is generated in accordance with the combined signal. 
     In another aspect, a rotation-estimation circuit is presented, which includes a quantizer that receives a first complex signal and produces a quantized signal. A combiner receives the quantized signal and an additional signal and produces a combined signal, where the additional signal corresponds to a second complex signal. An estimate generator receives the combined signal and produces an estimate of a relative rotation between the first complex signal and the second complex signal. 
     In yet another aspect, a wireless transmitter is presented that transmits a complex input signal. The wireless transmitter includes a rotator that receives a first complex signal and adjusts a phase of the first complex signal relative to a phase of a second complex signal to produce an adjusted signal. The rotator adjusts the phase of the first complex signal responsive to an estimate of a relative rotation between the first complex signal and the second complex signal. The wireless transmitter also includes a rotation-estimation circuit, which includes a quantizer operable to receive the first complex signal and produce a quantized signal. A combiner receives the quantized signal and an additional signal and produces a combined signal, where the additional signal corresponds to the second complex signal. An estimate generator receives the combined signal and produces the estimate of the relative rotation between the first complex signal and the second complex signal. 
     Particular implementations may include one or more of the following features. The second complex signal can be quantized to produce the additional signal. The additional signal can be the second complex signal. Generating an estimate can include filtering the combined signal, which can include digital filtering of the combined signal and/or analog filtering of the combined signal. Digital filtering can include integrating, filtering synchronously, and/or filtering asynchronously. Filtering the combined signal can include low-pass filtering the combined signal, which can include integrating. Filtering can also include setting a filter to a predetermined state. 
     The first complex signal and the second complex signal can be substantially identical except for the relative rotation. The first complex signal can include a first in-phase component and a first quadrature component, and the second complex signal can include a second in-phase component and a second quadrature component. Quantizing the first complex signal can include quantizing the first in-phase component and the first quadrature component, and combining can include combining multiplicative products of any of the quantized first in-phase component, the quantized first quadrature component, the second in-phase component, and the second quadrature component. The second in-phase component and the second quadrature component can be quantized, and combining can include combining multiplicative products of any of the quantized first in-phase component, the quantized first quadrature component, the quantized second in-phase component, and the quantized second quadrature component. Combining can include summing and/or differencing. 
     Quantizing can include multi-bit quantizing and/or one-bit quantizing. The first complex signal can include a first in-phase component and a first quadrature component, and one-bit quantizing can include quantizing the first in-phase component and/or the first quadrature component using one bit per component. The first complex signal and/or the second complex signal can be a continuous-time signal. The first complex signal and/or the second complex signal can be a discrete-time signal. Either of the first complex signal and the second complex signal can be sampled in time. 
     At least one of the first complex signal and the second complex signal can be rotated in accordance with the estimate of the relative rotation between the first complex signal and the second complex signal. Rotating can include shifting the phase of the first complex signal and/or the second complex signal. The first complex signal can include a first in-phase component and a first quadrature component, the second complex signal can include a second in-phase component and a second quadrature component, and rotating can include producing a weighted sum of any of the first in-phase component, the first quadrature component, the second in-phase component, and the second quadrature component. The estimate of the relative rotation between the first complex signal and the second complex signal can be quantized. Combining can include a digital combination and/or an analog combination. Either of the first complex signal and the second complex signal can be filtered. Filtering can include high-pass, low-pass, and/or band-pass filtering. 
     The invention can be implemented to realize one or more of the following advantages. An estimate of relative rotation between complex signals can be provided in a communications system whose design, manufacture, and operation are simplified. The rotation estimate can be provided for signals having a wide range of amplitudes without special signal processing steps. The rotation estimate can be produced using less power and without limiting the dynamic range of a transmitter. 
     These general and specific aspects may be implemented using an apparatus, a method, a system, or any combination of apparatus, methods, and systems. 
     The details of one or more implementations are set forth in the accompanying drawings and the description below. Other features and advantages will become apparent from the description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a plot of symbols of misaligned complex signals. 
         FIG. 1B  is a plot of symbols of aligned complex signals. 
         FIG. 2  is a diagram of a Cartesian feedback transmitter. 
         FIG. 3  is a schematic of a quantizer and a rotation estimator. 
         FIG. 4  is a flowchart of a process for estimating and adjusting a relative rotation between complex signals. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
       FIG. 2  shows a Cartesian feedback transmitter  200 , hereafter referred to as transmitter  200 . Transmitter  200  receives a complex input signal  201  from, for example, a baseband circuit and subtracts a complex feedback signal  202  from input signal  201 . A summer  210  subtracts an in-phase component  204  of feedback signal  202  from an in-phase component  206  of input signal  201  to produce an in-phase component  203  of a complex error signal  207 . A summer  215  subtracts a quadrature component  212  of feedback signal  202  from a quadrature component  214  of input signal  201  to produce a quadrature component  205  of error signal  207 . Filter  218  filters in-phase component  203  of error signal  207  to produce an in-phase component  208  of a complex intermediate signal  209  and filter  219  filters quadrature component  205  of error signal  207  to produce a quadrature component  216  of intermediate signal  209 . Filters  218  and  219  provide a gain to error signal  207 . Any of input signal  201 , feedback signal  202 , error signal  207 , and intermediate signal  209  can be a continuous-time signal or a discrete-time signal. 
     Rotator  230  receives in-phase component  208  and quadrature component  216  of intermediate signal  209  and rotates intermediate signal  209  responsive to a rotation signal  222  representing the relative rotation between feedback signal  202  and intermediate signal  209  to produce a complex rotated signal  211 . Rotator  230  can rotate the phase of intermediate signal  209  by computing an in-phase component  232  of rotated signal  211  and a quadrature component  234  of rotated signal  211  as weighted sums of in-phase component  208  and quadrature component  216  of intermediate signal  209 . In one implementation, rotator  230  rotates the phase of intermediate signal  209  by shifting the phase of first local-oscillator signals  236  and  238  relative to the phase of second local-oscillator signals  262  and  266 . In another implementation, rotator  230  rotates the phase of feedback signal  202  by shifting the phase of second local-oscillator signals  262  and  266  relative to the phase of first local-oscillator signals  236  and  238 . In one implementation, rotator  230  is placed in the feedback path of transmitter  200  (e.g., between summers  210  and  215  and mixers  260  and  266 ). In this implementation, rotator  230  rotates feedback signal  202  instead of intermediate signal  209 . Alternatively, in one implementation, rotator  230  can be placed anywhere in the baseband signal path to the right of or below summers  210  and  215 . 
     A mixer  240  mixes in-phase component  232  of rotated signal  211  with a first in-phase local-oscillator signal  236  and a mixer  245  mixes quadrature component  234  of rotated signal  211  with a first quadrature local-oscillator signal  238  to produce a modulated signal  247 . 
     Modulated signal  247  is amplified by a power amplifier  250  and is transmitted via antenna  255 . A mixer  260  receives a modulated signal  258  that corresponds to the signal transmitted via antenna  255 . Mixer  260  mixes modulated signal  258  with a second in-phase local-oscillator signal  262  to produce in-phase component  204  of the feedback signal  202 . A mixer  265  also receives modulated signal  258  and mixes modulated signal  258  with a second quadrature local-oscillator signal  266  to produce quadrature component  212  of feedback signal  202 . Feedback signal  202  typically has a different phase than intermediate signal  209  because of, for example, delays in the signal path (e.g., the signal path between the outputs of mixers  240  and  245  and the inputs of mixers  260  and  265 ) or a phase difference between the local-oscillator signals (i.e., first local-oscillator signals  236  and  238 ) provided to mixers  240  and  245  and the local-oscillator signals (i.e., second local-oscillator signals  262  and  266 ) provided to mixers  260  and  265 . All sources of relative rotation between feedback signal  202  and intermediate signal  209  can be modeled (e.g., by the phase φ in second local-oscillator signals  262  and  266 ) as being caused by a phase difference between first local-oscillator signals  236  and  238  and second local-oscillator signals  262  and  266 . Feedback signal  202  is provided to summers  210  and  215 . 
     Intermediate signal  209  is provided to a quantizer  270  after in-phase component  208  of intermediate signal  209  has been filtered by a filter  220  and quadrature component  216  of intermediate signal  209  has been filtered by a filter  221 . Feedback signal  202  is also provided to quantizer  270  after in-phase component  204  of feedback signal  202  has been filtered by a filter  226  and quadrature component  212  of feedback signal  202  has been filtered by a filter  227 . Filters  220 ,  221 ,  226 , and  227  can be any type of filter. For example, filters  220 ,  221 ,  226 , and  227  can be high-pass, low-pass, or band-pass filters and can filter in the analog domain or in the digital domain. Filters  220 ,  221 ,  226 , and  227  are not required to be identical filters. In one implementation, one or more of filters  220 ,  221 ,  226 , and  227  may be removed (i.e., no filtering is performed on intermediate signal  209  and/or feedback signal  202 ). Quantizer  270  quantizes intermediate signal  209  and feedback signal  202  and provides quantized signal components  272   a - d  to a combiner  282 . In one implementation, quantizer  270  coarsely quantizes intermediate signal  209  and feedback signal  202 . In another implementation, quantizer  270  quantizes in-phase component  208  of intermediate signal  209 , quadrature component  216  of intermediate signal  209 , in-phase component  204  of feedback signal  202 , and quadrature component  212  of feedback signal  202  using one bit for each component. In one implementation, quantizer  270  only quantizes the components of intermediate signal  209  or the components of feedback signal  202  and does not quantize the components of the other signal. Quantizer  270  can sample any or all of the components of intermediate signal  209  and of feedback signal  202  in time. In this description, quantized signal components will be indicated by using a “′” symbol. For example, I′ IN  is the quantized version of I IN . 
     Combiner  282  receives quantized signal components  272   a - d  and combines quantized signal components  272   a - d  to produce a rotation estimate  284 . Rotation estimate  284  is provided to a filter  286  that filters rotation estimate  284  to produce rotation signal  222 . Combiner  282  and filter  286  are included in a rotation estimator  280 . When rotation estimator  280  reaches the steady state, rotation signal  222  represents the value of the relative rotation between feedback signal  202  and intermediate signal  209 . Quantized signal components  272   a - d  can be combined using linear or nonlinear operations, and can be combined using digital circuitry or analog circuitry. In one implementation, a quantized in-phase component of the intermediate signal  272   a  (I′ IN ) is multiplied by a quantized quadrature component of the feedback signal  272   d  (Q′ FB ) to produce a first product (I′ IN Q′ FB ). A quantized quadrature component of the intermediate signal  272   b  (Q′ IN ) is multiplied by a quantized in-phase component of the feedback signal  272   c  (I′ FB ) to produce a second product (Q′ IN I′ FB ), and the second product is subtracted from the first product (I′ IN Q′ FB −Q′ IN I′ FB ) to produce rotation estimate  284 . In this implementation, when the average value of the sum-of-products (I′ IN Q′ FB −Q′ IN I′ FB ) is 0, intermediate signal  209  and feedback signal  202  are aligned or are 180 degrees out of phase. 
     In another implementation, the quantized in-phase component of the intermediate signal  272   a  and the quantized in-phase component of the feedback signal  272   c  are multiplied to form a first product (I′ IN I′ FB ), the quantized quadrature component of the intermediate signal  272   b  and the quantized quadrature component of the feedback signal  272   d  are multiplied to form a second product (Q′ IN Q′ FB ). The two products are added (I′ IN I′ FB +Q′ IN Q′ FB ) to produce rotation estimate  284 . In this implementation, when the average value of the sum-of-products (I′ IN I′ FB +Q′ IN Q′ FB ) is 0, intermediate signal  209  and feedback signal  202  have a relative rotation of 90 degrees, so when the average value of rotation estimate  284  is 0, one of the signals can be rotated 90 degrees relative to the other signal to align the two signals. 
     The combinations I′ IN Q′ FB −Q′ IN I′ FB  and I′ IN I′ FB +Q′ IN Q′ FB  can be used together in one implementation to produce rotation signal  222 , as will be discussed below in the context of  FIG. 3 . In one implementation, when the combinations I′ IN Q′ FB −Q′ IN I′ FB  and I′ IN I′ FB +Q′ IN Q′ FB  are used together and one of the combinations indicates that intermediate signal  209  and feedback signal  202  either are aligned or are 180 degrees out of phase, the other combination can be used to detect whether the signals are aligned or are 180 degrees out of phase. For example, the sign of the other combination can indicate whether the signals are aligned or are 180 degrees out of phase. 
     In another implementation, filter  286  filters rotation estimate  284  synchronously or asynchronously with an analog or digital filter. For example, filter  286  can low-pass filter rotation estimate  284 . Filter  286  can also integrate rotation estimate  284  (e.g., using one-step or two-step integration). 
     As was discussed above, quantizer  270  can quantize in-phase component  208  of intermediate signal  209 , quadrature component  216  of intermediate signal  209 , in-phase component  204  of feedback signal  202 , and quadrature component  212  of feedback signal  202  using one bit for each component. Using one-bit quantization simplifies the design and manufacture of transmitter  200  by simplifying the circuitry used to combine intermediate signal  209  with feedback signal  202  in combiner  282 . One-bit quantization of each of the four signal components yields an instantaneous value of rotation estimate  284  that is accurate to within substantially ±45 degrees of the actual relative rotation. When one-bit quantization is combined with filtering (e.g., using filter  286 ), however, greater accuracy can be obtained. For example, when all symbols are transmitted with uniform probability and the symbol constellation is symmetric about the real and imaginary axes, individual rotation estimates (e.g., rotation estimates from several clock cycles of a synchronous system) can be filtered (e.g., averaged) to provide a very accurate rotation signal  222 . Using one bit to quantize each of the in-phase and quadrature components  204 ,  208 ,  212 , and  216  of each of the intermediate and feedback signals  209  and  202  in quantizer  270  simplifies the implementation of combiner  282 . In one implementation, the components of only one of intermediate signal  209  and feedback signal  202  are quantized, while the other signal is processed in the analog domain. In this implementation, combiner  282  can be implemented more simply than when neither intermediate signal  209  nor feedback signal  202  are quantized (i.e., multiplication of a component from one signal with a component of the other signal will involve multiplying an analog value by ±1 instead of by another analog value). 
     In one implementation, rotation estimator  280  can quantize rotation signal  222 . In another implementation, rotator  230  is not used in transmitter  200 , and rotation signal  222  instead controls the phase of second local-oscillator signals  262  and  266 , the phase of first local-oscillator signals  236  and  238 , or both. By adjusting the phase of second local-oscillator signals  262  and  266  relative to the phase of first local-oscillator signals  236  and  238 , feedback signal  202  can be rotated relative to intermediate signal  209 . 
       FIG. 3  shows an implementation of quantizer  270  and rotation estimator  280  from  FIG. 2 . The in-phase component  208  of the intermediate signal  209  is provided to a comparator  310 , and the quadrature component  216  of the intermediate signal  209  is provided to a comparator  314 . The in-phase component  204  of the feedback signal  202  is provided to a comparator  316 , and the quadrature component  212  of the feedback signal  202  is provided to a comparator  312 . In one implementation, comparators  310 ,  312 ,  314 , and  316  compare their respective input signals to ground and output synchronous one-bit quantized representations of the input signals. The output of comparators  310 ,  312 ,  314 , and  316  represents the sign of the respective input signal. Hereafter, when the output from comparators  310 ,  312 ,  314 , or  316  is discussed, a positive output will be referred to as a 1, and a negative output will be referred to as a −1. Quantizer  270  provides quantized signals  272   a - d  to rotation estimator  280 . 
     The quantized in-phase component of the intermediate signal  272   a  and the quantized quadrature component of the feedback signal  272   d  are provided to an exclusive-OR (XOR) gate  320 . The quantized quadrature component of the intermediate signal  272   b  and the quantized in-phase component of the feedback signal  272   c  are provided to an XOR gate  325 . The quantized in-phase component of the intermediate signal  272   a  and the quantized in-phase component of the feedback signal  272   c  are also provided to an XOR gate  350 , while the quantized quadrature component of the intermediate signal  272   b  and the quantized quadrature component of the feedback signal  272   d  are provided to an XOR gate  355 . XOR gates  320 ,  325 ,  350 , and  355  perform an exclusive-OR logic operation on their respective input signals. The input signals can have a positive value (1) or a negative value (−1). The output of XOR gates  320 ,  325 ,  350 , and  355  is the sign-inverted, scaled, and shifted product of the two input signals. For example, when both input signals to an XOR gate are 1 or both are −1, the output of the XOR gate is low (0). When one input signal is −1 and one input signal is 1, the output of the XOR gate is high (1). 
     The output  322  of XOR gate  320  is received by a counter  330 . Counter  330  includes some of the functionality of combiner  282  and filter  286  ( FIG. 2 ). In one implementation, counter  330  is a modulo-N counter. Increasing the value of N increases the amount of time that rotation estimator  280  takes to update rotation signal  222  and causes the phase alignment system to respond more slowly. When the output  322  of XOR gate  320  is high (e.g., I′ IN Q′ FB =−1) and the output  323  of XOR gate  325  is low (e.g., Q′ IN I′ FB =1), counter  330  increments. When the output  322  of XOR gate  320  is low (e.g., I′ IN Q′ FB =1) and the output  323  of XOR gate  325  is high (e.g., Q′ IN I′ FB =−1), counter  330  decrements. When the outputs  322  and  323  of XOR gates  320  and  325  are both high or both low, counter  330  neither increments nor decrements. When counter  330  increments to N, an overflow flag  332  is set and counter  330  is reset to a value between 0 and N (e.g., N/2). When counter  330  decrements beyond 0, an underflow flag  334  is set and counter  330  is reset to the value between 0 and N. The combination of XOR gates  320  and  325  and counter  330  combines the quantized components of the intermediate signal  272   a  and  272   b  and the quantized components of the feedback signal  272   c  and  272   d , and counter  330  filters the combined signal (e.g., by integrating the combined signal to produce overflow flag  332  and underflow flag  334 ). The combination of quantizer  270  and rotation estimator  280  provides a nonlinear combination of intermediate signal  209  and feedback signal  202 . 
     A counter  340  increments when counter  330  sets overflow flag  332  and decrements when counter  330  sets underflow flag  334 . The number stored in counter  340  corresponds to rotation signal  222 , and counter  340  provides rotation signal  222  as an output. Rotation signal  222  is increased or decreased when counter  340  is incremented or decremented. Counter  340  can be preset to a value by asserting a “preset” signal  342  and providing the value on a preset bus  346  either serially or in parallel. Counter  340  can also be frozen at a value by asserting a “freeze” signal  344 . While freeze signal  344  is asserted, counter  340  remains at the same value regardless of the states of the inputs other than freeze signal  344 . 
     The outputs  352  and  357  of XOR gates  350  and  355  are provided to an AND gate  360  and an AND gate  365 . The inputs of AND gate  360  are both inverting inputs. When the output  352  of XOR gate  350  is high (e.g., I′ IN I′ FB =−1) and the output  357  of XOR gate  355  is also high (e.g., Q′ IN Q′ FB =−1), an output  366  of AND gate  365  is high, and a counter  370  decrements. When the output  352  of XOR gate  350  is low (e.g., I′ IN I′ FB =1) and the output  357  of XOR gate  355  is also low (e.g., Q′ IN Q′ FB =1), an output  362  of AND gate  360  is high, and counter  370  increments. When the output  352  or  357  of one of XOR gates  350  and  355  is high and the output  352  or  357  of the other XOR gate is low, the outputs  362  and  366  of AND gates  360  and  365  are both low and counter  370  neither increments nor decrements. In one implementation, counter  370  is a modulo-M counter. When the counter reaches M, an overflow flag  372  is set and counter  370  is reset to a value between 0 and M (e.g., M/2). Setting overflow flag  372  of counter  370  indicates that transmitter  200  ( FIG. 2 ) and/or rotation estimator  280  is in an undesirable state (e.g., the relative rotation is 180 degrees), and rotation estimator  280  can, for example, be put into a predetermined desirable state. As M is increased, the amount of time that rotation estimator  280  must be in an undesirable state before overflow flag  372  is set increases. 
     In another implementation, counters  330  and/or  370  are reset at fixed, predetermined intervals instead of resetting and triggering following circuit blocks (e.g., counter  340 ) based on overflow or underflow events. The value of counter  330  or  370  immediately before resetting determines the action of the following block (e.g., whether counter  340  increments, decrements, or stays the same). 
       FIG. 4  shows a process  400  for estimating and correcting a relative rotation between a first complex signal and a second complex signal. One or both of the first complex signal and the second complex signal is quantized (step  410 ). The quantization in step  410  can be coarse. For example, the quantization in step  410  can quantize the in-phase and quadrature components of the first complex signal and/or the second complex signal with one bit per component. When both complex signals are quantized, the quantized signal components can be combined to form a rotation estimate (step  420 ). When only one of the complex signals is quantized, the quantized signal components can be combined with the non-quantized signal components to form the rotation estimate. 
     Optionally, the rotation estimate is filtered (step  430 ). The filtering in step  430  can be performed using, for example, a low-pass filter or an integrator. In one implementation, the rotation estimate is not filtered. In another implementation, the rotation estimate is quantized before or after filtering. The phase of the first complex signal and/or the phase of the second complex signal are adjusted based on the rotation estimate (step  440 ). The phase of a complex signal can be adjusted, for example, by using a rotator circuit in the signal path of the complex signal or by adjusting the phase of a signal used to modulate the complex signal. 
     The invention and all of the functional operations described in this specification can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. 
     Method steps of the invention can be performed by one or more programmable processors executing a computer program to perform functions of the invention by operating on input data and generating output. Method steps can also be performed by, and apparatus of the invention can be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit). 
     The invention has been described in terms of particular embodiments. Other embodiments are within the scope of the following claims. The described apparatus and method can be used in many different types of digital or analog systems. For example, the apparatus or method can be used in any electronic communication system whose complex signal path includes at least two points between which phase alignment is useful for operation.