Patent Publication Number: US-7596184-B2

Title: Apparatus, system, and method for amplitude-phase synchronization in polar transmitter

Description:
BACKGROUND 
     Polar modulation is a technique whereby a signal, or carrier, having constant radian frequency ω, is time-varied in both magnitude and phase. Polar modulation transmitters transmit information that both the magnitude (R) and the phase (θ) of a signal simultaneously carry. There are many benefits to using polar modulation to transmit information. Polar transmitters receive baseband signals represented in Cartesian form as an in-phase (I) component and a quadrature (Q) component. The I/Q baseband signals are naturally symmetric at the source. The I/Q baseband signals are converted to polar form in terms of its magnitude R and phase θ signals. The magnitude R is referred to as the amplitude signal and the phase θ is referred to as the phase signal. A coordinate rotation digital computer (CORDIC) algorithm may be employed to convert the I/Q baseband signals to polar form amplitude R and phase θ signals. The amplitude R and phase θ signals are processed in separate amplitude and phase paths and may be recombined at the output of the power amplifier. The I/Q components may be reconstructed by additional processing downstream of the power amplifier output. 
     Circuits for processing the amplitude R and phase θ signals in the respective separate amplitude and phase paths are substantially different and may lead to timing misalignments between the amplitude and phase signals. Unlike the natural symmetry of the I/Q baseband signals at the source, the amplitude R and phase θ signals are asymmetric, and thus, there are timing misalignments between them. The timing misalignment between these signals is detrimental to the reconstructed I/Q component. Accordingly, in polar transmitters, there is a need to synchronize the amplitude R and phase θ signals to correct for the timing misalignments due to different delays encountered by these signals in the separate amplitude R and phase θ processing paths. Therefore, there is a need for techniques to determine and correct for delays in the amplitude R and phase θ processing paths. There is a need to estimate and correct for these timing misalignments in an accurate manner. 
     SUMMARY 
     In one embodiment, a modulation path synchronization apparatus in a polar transmitter includes a modulation path to receive a training waveform, a detector coupled to the modulation path, the detector to detect a modulated training waveform, and a processor coupled to the detector, the processor to determine a delay between the training waveform and the modulated training waveform. 
     In one embodiment, a method to synchronize modulation paths in a polar transmitter includes passing a baseband amplitude training waveform in an amplitude modulation path; detecting a carrier envelope of a first amplitude modulated training waveform associated with the amplitude training waveform; and determining an amplitude modulation path delay. 
     In one embodiment, a system to synchronize modulation paths in a polar transmitter includes an amplifier and a modulation path coupled to the amplifier to receive a training waveform, a detector coupled to the modulation path, the detector to detect a modulated training waveform, and a processor coupled to the detector, the processor to determine a delay between the training waveform and the modulated training waveform. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates one embodiment of an amplitude modulation path synchronization system. 
         FIG. 2  illustrates one embodiment of a phase modulation path delay synchronization system. 
         FIGS. 3A , B, C, D graphically illustrates various embodiments of waveforms in a phase modulation path synchronization system. 
         FIG. 4  illustrates one embodiment of a combined amplitude and phase modulation path synchronization system. 
         FIG. 5  illustrates one embodiment of a detector adapted for an amplitude and phase modulation path synchronization system. 
         FIG. 6  graphically illustrates one embodiment of an AM waveform. 
         FIG. 7  illustrates one embodiment of a combination amplitude and phase modulation path synchronization system. 
         FIG. 8  illustrates one embodiment of a combination amplitude and phase modulation path synchronization system. 
         FIG. 9  illustrates one embodiment of an amplitude and phase modulation path synchronization system. 
         FIG. 10  is a flow diagram of one embodiment of a synchronization process to determine a delay in an amplitude modulation path of an RF transmitter in a polar modulation topology. 
         FIG. 11  is a flow diagram of one embodiment of a synchronization process to determine a delay in a phase modulation path of an RF transmitter in a polar modulation topology. 
         FIG. 12  is a flow diagram of one embodiment of a process to estimate delay in an amplitude path and a phase path simultaneously. 
         FIG. 13  is a flow diagram of one embodiment of a process to estimate delay in an amplitude path and a phase path simultaneously. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of various implementations of a path delay estimation techniques to estimate and correct timing misalignments between amplitude and phase processing paths in a polar modulation topology transmitter are disclosed and claimed. Polar domain amplitude R and phase θ signal representations of I/Q baseband signals are employed to accurately estimate and correct amplitude and phase path timing misalignments (delay mismatches). These techniques may be integrated with circuits and modules in polar transmitter integrated circuit (IC) devices. 
     In one embodiment, a path delay estimation system calculates estimated amplitude modulation path delay and phase modulation path delay in separate steps (two-step method). In a first step, the system may estimate amplitude modulation path training mode and, in a second step, the system may estimate the phase modulation path delay. It will be appreciated that the order of the first and second steps may be reversed without limitation. In one embodiment, a path delay estimation system calculates estimated amplitude and phase modulation path delays in single-step or two-step method. In one embodiment, a path delay estimation system calculates estimated amplitude and phase modulation path delays simultaneously. In various embodiments, training waveforms may be passed through each amplitude and phase modulation paths simultaneously during the synchronization training period. Accordingly, amplitude and phase training waveforms are passed through respective amplitude and phase paths simultaneously. In the two-step method, input training waveforms are passed through the amplitude and phase paths of the polar transmitter separately. The estimated delay in each path is also obtained separately. The input training waveforms comprise a periodic waveform at a predetermined frequency, amplitude, and shape or form. The input training waveforms are amplitude-modulated (AM) or frequency-modulated (FM) at the power amplifier (PA) with a carrier waveform based on whether the path delay estimation system is operating in amplitude training mode or phase training mode, respectively. The carrier waveform comprises a periodic waveform at a much higher frequency than the input training waveform. At the output of the PA, the AM or FM training waveforms are detected and correlated with the respective input training waveform to estimate a delay for the respective amplitude or phase path. Detection and correlation techniques may differ based on whether the path delay estimation system is in amplitude path training mode or phase modulation path training mode. In one embodiment, the training waveform may be a sine wave at a frequency referred to as a tone. Although a “sine wave tone” may be referred to herein, it is to be understood that a sine wave tone is used merely to simplify the description. The embodiments are in no way intended to be limited to any particular waveform shape, magnitude, or frequency. Any waveform and/or frequency may be used to generate the training waveforms, carrier waveforms, modulation waveforms, and the like. 
     In the two-step method, each of the amplitude and phase modulation path delays are determined separately. In amplitude modulation path training mode, the input training waveform is an amplitude training waveform. The amplitude training waveform is driven through the amplitude path of the polar transmitter while no signal or waveform is driven through the phase path. The amplitude training waveform is amplified by the PA and amplitude modulated with a carrier waveform. The envelope of the amplified amplitude modulated training waveform is detected at the output of the PA. The detected envelope is correlated with the input amplitude training waveform to estimate the amplitude modulation path delay. 
     In phase modulation path training mode, the input training waveform is a frequency training waveform. The frequency training waveform is driven through the phase path of the polar transmitter while no signal or waveform is driven through the amplitude path. The frequency training waveform is frequency modulated at the PA with the carrier waveform. Accordingly, the output of the PA is a FM training waveform. In the phase modulation path training mode, prior to envelope detection, the FM training waveform may first be converted from an FM waveform to an AM waveform by an FM-to-AM converter. In one embodiment, the FM-to-AM converter may be implemented as a low pass filter (LPF). The FM-to-AM conversion module converts the output FM waveform (the first derivative of the phase signal) comprising frequency variations to an AM waveform comprising amplitude variations. An AM detector (e.g., envelope detector or a synchronous detector) is used to detect the envelope or amplitude variations in the AM waveform. The envelope of the AM waveform is correlated with the input frequency training waveform to estimate the phase modulation path delay. 
     In one embodiment, the path delay estimation system may calculate estimated amplitude and phase modulation path delays simultaneously (single-step method). Accordingly, amplitude and phase training waveforms are passed through respective amplitude and phase paths simultaneously. The output of the PA is down-converted with a local oscillator (LO) signal. A CORDIC process extracts amplitude and phase components, which are correlated with the respective input training waveforms to estimate the delays. 
     The baseband amplitude and frequency training waveforms used for the respective amplitude and phase modulation path delay estimations can be any suitable waveforms with satisfactory sensitivity to synchronization errors. For example, the baseband amplitude and frequency training waveforms may be Walsh codes or the likes, such as those on code division multiple access (CDMA) synchronization channels (e.g., in wideband code division multiple access or WCDMA) can be used. In one embodiment, the input training waveforms may be digitized waveforms generated by a digital waveform generator. In one embodiment, the input training waveforms may be in the form of a sine wave characterized by a frequency and amplitude. The embodiments, however, are not limited in this context as any periodic waveform may be used as a training waveform for delay estimation purposes. 
     In one embodiment, where an FM-to-AM converter is employed at the output of the PA to convert the FM waveform to an AM waveform, the FM-to-AM converter may comprise a negligible group delay or a delay that may be accurately ascertained with high certainty. For example, in embodiments where the FM-to-AM converter is implemented as a low-order LPF with bandwidth (BW) equal to half the carrier frequency, the group delay generally will be very small around the carrier frequency zone (e.g., on the order of picoseconds), and thus may be negligible relative to the amplitude or phase modulation path delays. 
     In one embodiment, the dynamic range of the baseband frequency training waveform may be selected to be as small as possible to maintain linearity of the FM-to-AM converter transfer function (i.e., to have a relatively small frequency-modulation depth). A low-frequency amplifier can be used to amplify the detected envelope of the AM waveform from the FM waveform if needed. Otherwise, an analog-to-digital converter (ADC) with sufficient bit-width may be employed at the output of the envelope detection module to obtain adequate numerical resolution. Then, for example, the mean of the detected envelope can be subtracted from the detected envelope and the difference can be numerically magnified (digital gain). 
       FIG. 1  illustrates one embodiment of an amplitude modulation path synchronization system  100 . System  100  may be implemented in a RF polar modulation transmitter topology. During an amplitude modulation path  106  synchronization training period, system  100  estimates delay τ α  in amplitude modulation path  106 . Delay τ α  is the time it takes an amplitude training waveform to propagate through amplitude modulation path  106 . System  100  uses estimated delay τ α  to synchronize amplitude modulation path  106  delays with phase modulation path  126  delays. Synchronization system  100  comprises baseband amplitude waveform generator  102  to generate baseband amplitude training waveform  104 . During the synchronization training period, amplitude training waveform  104  is passed through amplitude modulation path  106  of the polar transmitter. Amplitude training waveform  104  is delayed by τ α  seconds as it propagates through amplitude modulation path  106 . Amplitude training waveform  104  is received at an amplitude modulation node of PA  124 . Amplitude training waveform  104  is amplitude modulated with a RF carrier waveform at PA  124  to produce AM training waveform  110  at an output node of PA  124 . AM training waveform  110  comprises an envelope formed by the RF carrier waveform with the amplitude variations proportional to amplitude training waveform  104 . Envelope detection module  112  detects the envelope of the RF carrier waveform of AM training waveform  110 . Processor  116  receives detected envelope waveform  114 . Processor  116  correlates detected envelope waveform  114  with amplitude training waveform  104  and estimates the amplitude modulation path  106  delay τ α  between these two waveforms. In one embodiment, processor  116  estimates the amplitude modulation path  106  delay τ α  by normalizing the detected envelope waveform  114  and performing a slide-and-correlate process. Amplitude modulation path  106  delay τ α  may be recorded and/or stored in memory. To synchronize amplitude modulation path  106  with phase modulation path  126 , the delay mismatch (τ α −τ φ ), where τ φ  is the phase path time delay, is determined by way of calculation. (Determining the phase path delay τ φ  is described below with reference to  FIG. 2 .) As previously discussed, in this embodiment, the time delay estimates in each of amplitude modulation path  106  and phase modulation path  126  are done separately. Therefore, during the amplitude modulation path  106  synchronization training period, baseband frequency waveform generator  118  is turned off and does not generate baseband frequency training waveform  202 . Therefore, there are no baseband frequency training waveforms  202  in phase modulation path  126  during the amplitude modulation path  106  synchronization training period. 
       FIG. 2  illustrates one embodiment of a phase modulation path delay synchronization system  200 . System  200  may be implemented in a RF polar modulation transmitter topology. During a phase modulation path  126  synchronization training period, system  200  estimates delay τ φ  in phase modulation path  126 . Delay τ φ  is the time it takes a frequency training waveform to propagate through amplitude modulation path  126 . System  200  uses estimated delay τ φ  to synchronize phase modulation path  126  delays with amplitude modulation path  106  delays. Synchronization system  200  comprises baseband frequency waveform generator  118  to generate baseband frequency training waveform  202 . During the synchronization training period, frequency training waveform  202  is passed through phase modulation path  126  of the polar transmitter. Frequency training waveform  202  may be integrated by integrator  204 , which produces an integrated frequency training waveform  206 . Integrated frequency training waveform  202  is passed through polar transmitter phase modulation path  126 . Frequency training waveform  206  is delayed by τ φ  seconds as it propagates through phase modulation path  126 . Frequency training waveform  206  is received at an input node of PA  124 . Frequency training waveform  206  frequency modulated with a RF carrier waveform of PA  124  to produce FM training waveform  208  at an output node of PA  124 . FM training waveform  208  comprises an envelope formed by the RF carrier waveform with frequency variations proportional to frequency training waveform  202 . FM-to-AM converter  210  converts FM training waveform  208  to AM training waveform  212 . In various embodiments, FM-to-AM converter  210  may be implemented as a LPF or an FM slope-detector demodulator, for example. Envelope detection module  112  detects the envelope of the RF carrier waveform of AM training waveform  212 . Processor  218  correlates detected envelope training waveform  214  with frequency training waveform  202  and estimates the phase modulation path  126  delay τ φ  between these two waveforms. In one embodiment, processor  218  estimates the phase modulation path  126  delay τ φ  by normalizing the detected envelope waveform  214  and performing a slide-and-correlate process. Phase modulation path  126  delay τ φ  may be recorded and/or stored n memory. To synchronize phase modulation path  126  with amplitude modulation path  106 , the delay mismatch (τ α −τ φ ), where τ α  is the amplitude path time delay, is determined by way of calculation. (Determining the amplitude path delay τ α  is described with reference to  FIG. 1 .) As previously discussed, in this embodiment, the time delay estimates in each of amplitude modulation path  106  and phase modulation path  126  are done separately. Therefore, during the phase modulation path  126  synchronization training period, baseband amplitude waveform generator  102  is turned off and does not generate baseband amplitude training waveform  104 . Therefore, there are no baseband amplitude training waveforms  104  in amplitude modulation path  106  during the phase modulation path  126  synchronization training period. 
       FIGS. 3A , B, C, D graphically illustrates various embodiments of waveforms in phase modulation path  126  synchronization system  200 . Time (T) is displayed along the horizontal axis and voltage (V) or amplitude is displayed along the vertical axis.  FIG. 3A  graphically illustrates one embodiment of a baseband frequency training waveform  202 .  FIG. 3B  graphically illustrates one embodiment of an RF carrier waveform  310  to modulate baseband frequency training waveform  202  at PA  124 .  FIG. 3C  graphically illustrates one embodiment of AM training waveform  212  at the output of FM-to-AM converter  210 .  FIG. 3D  illustrates one embodiment of detected waveform  214  at the output of envelope detection module  112 . 
     This phase modulation path  126  delay measurement comprises a 3 rd -order LPF with 450 MHz 3 dB band-width (BW) as the FM-to-AM converter  210 . A 2 nd -order filter with the same BW would result in substantially the same performance. Baseband frequency training waveform  202  and detected waveform  214  can be subjected to a slide-and-correlate operation to find the timing mis-alignment and, hence, to find delay τ φ  of phase modulation path  126 . 
     The amplitude-phase synchronization techniques described herein estimate a time delay mismatch between an input training waveform and a detected waveform after passing through respective amplitude and phase modulation paths of a polar transmitter. To synchronize the amplitude and phase modulation paths, the process further comprises correcting the time delay mismatch based on the estimated delay. Estimates of the time delay mismatch should be as accurate as possible to comply with limits mandated by any predetermined system specifications. For example, in WCDMA applications the time delay mismatch should be about τ±2 ns, for example. Correcting the time delay mismatch also should be done accurately (i.e., apply the estimated delay) within the limits mandated by system specifications. 
     Several techniques are described to synchronize the amplitude and phase modulation paths based on calculated estimates of time delay mismatch between the two paths. Several estimation techniques may be employed to detect the phase difference between training waveforms and output detected waveforms by correlating the detected waveforms  114 ,  214  with the respective input training waveforms  104 ,  202  during a synchronization training period. In various embodiments, training waveforms may be passed through each amplitude and phase modulation paths simultaneously during the synchronization training period. In this mode of operation, phase detection may be implemented with a detector comprising a series of downmixers to mix-down the modulated waveforms at the output of PA  124  with an un-modulated LO signal. In other embodiments, training waveforms may be passed through each amplitude and phase modulation paths individually, separately, or simultaneously during first and second synchronization training periods as described above with reference to  FIGS. 1 and 2 , for example. 
     As described with reference to  FIG. 1 , first, the envelope of AM training waveform  110  is detected and correlated with amplitude training waveform  104  to determine an estimated amplitude path delay τ α . As described with reference to  FIG. 2 , second, FM training waveform  208  is converted to AM training waveform  212  and the envelope is detected and correlated with frequency training waveform  202  to determine an estimated phase path delay τ φ . Time delay mismatch between the amplitude and phase paths (τ α −τ φ ) can be determined by calculation. Alternatively, AM training waveform  110  can be correlated directly with FM training waveform  208  and time delay mismatch between the amplitude and phase paths (τ α −τ φ ) can be obtained directly. 
     In one embodiment, an ADC with a sufficient sampling rate may be coupled to the output node of envelope detection module  112 . The ADC may be adapted to detect the envelope of AM training waveform  110 ,  112  in a single high resolution conversion to obtain a high resolution estimate of the time delay mismatch (τ α −τ φ ). Alternatively, a multi-stage signal processor may be employed to determine the time delay mismatch (τ α −τ φ ) between the amplitude and phase modulation paths. In one embodiment, an ADC with a moderate sampling rate may be coupled to the output node of envelope detection module  112 . The ADC may be adapted to detect the envelope of AM training waveform  110 ,  112  in multiple conversions of varying resolutions. For example, the ADC may conduct a first “coarse” timing estimation computation followed by a second “fine” timing estimation computation. The second “fine” computation can be done via, for example, linear digital filter interpolation of the collected samples. These and various embodiments of other techniques are described more fully herein below. 
       FIG. 4  illustrates one embodiment of a combined amplitude and phase modulation path synchronization system  400 . In one embodiment, system  400  may be implemented in a RF polar modulation transmitter topology. System  400  may be configured to independently operate either in amplitude modulation path  106  training mode (determine amplitude path delay τ α ) or phase modulation path  126  training mode (determine phase path delay τ φ ). The training mode of operation may be selected via the state of several switches  404 ,  406 ,  408 . When the training mode is selected, the corresponding input training waveform  412  is generated. Accordingly, system  400  may be configured to switch between amplitude modulation path  106  training mode (e.g., system  100 ) or phase modulation path  126  training mode (e.g., system  200 ). Training waveform generator  402  generates the appropriate training waveform  412 , e.g., amplitude training waveform  104  or frequency training waveform  202 . In one embodiment, training waveform  412  may be a digital waveform, such as a digital sine wave waveform, for example. Switches  404 ,  406 ,  408 ,  410  comprise nodes “a” and “b” on one side and node c on the other side such that nodes a and c are coupled or node b and c are coupled based on the select state of the switch. Nodes a, b, and c may be employed as either input nodes or output nodes. Switches  404 ,  406 ,  408 , and  410  adapt system  400  in amplitude modulation path  106  training mode or phase modulation path  126  training mode. 
     Training waveform  412  is coupled to input node c of switch  404 . Training waveform  412  may be coupled to either amplitude modulation path  106  or phase modulation path  126  based on the training mode of system  400 . If system  400  is in amplitude modulation path  106  training mode, training waveform  412  is amplitude training waveform  104  and is coupled to amplitude modulation path  106  via node a of switch  404 . In one embodiment, training waveform  104  may be coupled to the amplitude modulation node of PA  124 . Training waveform  104  is amplitude modulated with a carrier waveform at PA  124 . AM training waveform  420  is produced at the output node of PA  124 . In amplitude modulation path  106  training mode, AM training waveform  420  is equivalent to AM training waveform  110  previously described. AM training waveform  420  is coupled to node a of switch  408 . In one embodiment, PA  124  may be bypassed by coupling amplitude training waveform  104  to node a of switch  406 . Signal  422  at output node c of switch  406  is coupled to node b of switch  408 . Either signal  422  (amplitude training waveform  104 ) or AM training waveform  420  (e.g., AM training waveform  110 ) may be coupled to node c of switch  408 . Switch  408  may be employed to select whether AM training waveform  420  (AM training waveform  110 ) or signal  422  (amplitude training waveform  104 ) is passed through switch  408 . In amplitude modulation path  106  training mode, AM training waveform  420  (e.g., signal  424 ) bypasses FM-to-AM converter  210  and is coupled to node c of switch  410 . Signal  428  at the output node of FM-to-AM converter  210  (e.g., AM training waveform  212 ) is coupled to input node a of switch  410 . In amplitude modulation path  106  training mode, switch  410  couples input signal  424  (e.g., AM training waveform  420 ) to detector  431 . Accordingly, in amplitude modulation path  106  training mode, AM training waveform  424  bypasses FM-to-AM converter  210  and is coupled to input node b of switch  410 . Switch  410  may couple either signal  428  (e.g., AM training waveform  212 ) or signal  424  (either amplitude signal  104  or AM training waveform  110 ) to detector  431 . In one embodiment, detector  431  may be implemented as envelope detection module  112 , for example. Detector  431  receives AM training waveform  430 , detects the RF carrier envelope of AM training waveform  430 , and extracts I component  432  and Q component  434  from the detected envelope. In one embodiment, detector  431  may be implemented as a synchronous detector. I component  432  and Q component  434  are provided to processor  436  to correlate and determine amplitude modulation and phase modulation path delays τ α , τ φ , respectively. In one embodiment, processor  436  may be implemented as correlation and delay estimation module  116 , for example. 
     If system  400  is in phase modulation path  126  training mode, training signal  412  is frequency training waveform  202  and is coupled to phase modulation path  126  via node b of switch  404 . In one embodiment, frequency training waveform  202  may be coupled to the input node of PA  124  where it is frequency modulated with a RF carrier waveform to produce FM training waveform  420  at the output of PA  124 . In one embodiment, PA  124  may be bypassed via switch  406 . For example, frequency training waveform  202  may be coupled through switch  406  and through switch  408  directly to the input node of FM-to-AM converter  210 . More specifically, in one embodiment, PA  124  may be bypassed by coupling signal  206  to node b of switch  406 . Signal  422  at output node c of switch  406  is coupled to node b of switch  408 . In other embodiments, either signal  422  (frequency training waveform  202 ) or FM training waveform  420  may be coupled to FM-to-AM converter  210  via node c of switch  408 . In phase modulation path  126  training mode, however, switch  408  couples FM training waveform  420  to the input node of FM-to-AM converter  210 . At the output node of FM-to-AM converter  210 , AM training waveform  428  (converted AM training waveform  212 ) is coupled to input node a of switch  410 . To bypass FM-to-AM converter  210 , signal  424  is coupled from output node c of switch  408  to input node b of switch  410 . Accordingly, switch  410  may be employed to couple either AM waveform  428  or signal  424  (either frequency training waveform  206  or FM training waveform  208 ) to detector  431 . Detector  431  receives AM training waveform  430 , detects the envelope, and extracts I component  432  and Q component  434 . I component  432  and Q component  434  are coupled to processor  436  to correlate and determine amplitude modulation and phase modulation path delays τ α , τ φ , respectively. In one embodiment, processor  436  may be implemented as correlation and delay estimation module  218 , for example. 
       FIG. 5  illustrates one embodiment of detector  500  (e.g., detector  431  as illustrated in  FIG. 4 ) adapted for an amplitude and phase modulation path synchronization system. In one embodiment, detector  500  may be a synchronous detector. In one embodiment, envelope detection module  112 , for example, may be implemented as detector  431 . Detector  431  comprises a first I processing path  501 - 1  and a second Q processing path  501 - 2 . First and second input signals  502 ,  504  are coupled to detector  431 . In amplitude modulation path  106  training mode, first input signal  502  may be AM training waveform  110  from the output node of PA  124  or may be AM training waveform  212  from the output node of FM-to-AM converter  210 . In phase modulation path  126  training mode, second input signal  504  is a phase modulator output from phase modulation path  126 . Second input signal  504  is provided by the output of a phase modulator, a component which may be located in the phase path of a polar transmitter, for example. First input signal  502  is coupled to the signal input nodes of first and second mixers  506 - 1 ,  506 - 2 . Second input signal  504  is coupled to the oscillator input node of first mixer  506 - 1  and is coupled to phase shifter  508 . Phase shifter  508  produces third signal  510 , which is 90° out-of-phase with second input signal  504 . Third signal  510  is coupled to the oscillator input node of second mixer  506 - 2 . First mixer  506 - 1  down-converts first input signal  502  with second input signal  504  to produce the in-phase time-varying I component of the baseband signal. First analog low pass filter  510 - 1  extracts the low frequency I component of the baseband signal from the down-converted signal. I component is amplified and any DC-offset is removed by first DC-offset and amplifier module  512 - 1 . The output of first DC-offset and amplifier module  512 - 1  is converted to a digital version of time-varying I component  516 - 1  by first ADC  514 - 1  at a predetermined sampling rate. The output of first ADC  514 - 1  is a digital representation of the time varying I component of Cartesian I/Q baseband signals from which the polar form magnitude (R) and phase angle (θ) signals were derived. The digital I component  516 - 1  may be referred to as the detected I sine wave. Digital I component  516 - 1  is coupled to processor  518  to correlate, normalize, and/or slide-and-correlate the detected digital I component  516 - 1  in order to determine the amplitude-phase modulation path delays. Processor  518  may comprise various implementations of correlation delay and estimation modules  116 ,  218 , and processor  436 , for example. 
     As previously described, first input signal  502  is coupled to the signal input node of second mixer  506 - 2 . Second signal  504  is phase shifted by 90° by phase shifter  508  to produce third signal  510 , which is coupled to the oscillator input node of second mixer  506 - 2 . Second mixer  506 - 1  down-converts input signal  502  with third signal  510  to produce the 90° out-of-phase quadrature time-varying Q component of the baseband signal. Second analog low pass filter  510 - 2  extracts the low frequency baseband Q component of the down-converted signal. The Q component is amplified and any DC-offset removed by second DC-offset and amplifier module  512 - 2 . The output of second DC-offset and amplifier module  512 - 2  is converted to a digitized Q component  516 - 2  version of the time varying Q component by second ADC  514 - 2  at a predetermined sampling rate. Accordingly, the output of second ADC  514 - 2  is a digital representation of the time varying Q component of the Cartesian I/Q baseband signal from which the polar form magnitude (R) and phase angle (θ) signals were derived. The digital Q component  516 - 2  may be referred to as the detected Q sine wave. Digital Q component  516 - 2  is provided to processor  518  to correlate, normalize, and/or slide-and-correlate the detected digital Q component  516 - 2  in order to determine the amplitude-phase modulation path delays. Processor  518  may comprise various implementations of correlation and delay estimation modules  116 ,  218 , and processor  436 , for example. 
     The accuracy of the analog-to-digital conversion in detector  431  is related to the selected sampling rate for ADCs  512 - 1 ,  512 - 2  (ADC  512 ). The analog-to-digital conversion may be conducted in one-step or multiple-steps. Although ADC  512  is shown as a single module or block, ADC  512  may comprise a single ADC or multiple staged ADCs. In a one-step direct analog-to-digital conversion process, performance considerations may dictate a sufficiently high sampling rate, where a higher sampling rate yields better performance. ADC  512  may be adapted to operate at a sampling rate to achieve a predetermined level of performance. In a two-step analog-to-digital conversion process, a first conversion may be performed at a lower-sampling-rate and a second conversion may be performed at a higher-sampling rate. In one embodiment, in a first step, the method is to conduct the analog-to-digital conversion process at a moderate sampling rate. In a second step, the analog-to-digital conversion process is digitally up-sampled (e.g., the sampling rate of a signal is increased). Although, the two-step conversion process may not be as accurate as a high-sampling rate one-step direct sampling method, it may be better than the one-step direct conversion process at a lower sampling rate. The quantization limits of ADC  512  also should be considered for performance purposes. For example, the bit-width of the ADC  512  should be selected to satisfy a worst-case quantization error that does not result-in or correspond-to a timing error that is larger than the required accuracy of the specific implementation. 
     FM-to-AM converter  210  may have a non-linear response and may generate higher order frequency harmonics. Accordingly, performance of systems  200  and  400  depends in part on the non-linearity and the elimination of higher order harmonics generated by FM-to-AM converter  210 . In one embodiment, FM-to-AM converter  210  may be implemented as a low-pass filter (LPF). The LPF may have a non-linear FM-to-AM response. For example, the AM response a(t) of FM-to-AM converter  210  to an FM modulated carrier signal (e.g., a sine wave tone sin(ω tone t)) may be written as follows:
 
 a ( t )= c   0   +c   1 (Δ FM ·sin(ω tone t))+ c   2 (Δ FM ·sin (ω tone t)) 2   +c   3 (Δ FM ·sin(ω tone t)) 3 +  (1)
 
     Where F tone =2πω tone  is the FM modulated carrier signal frequency (e.g., sine wave frequency). Accordingly, the non-linearity will generate higher order harmonics (i.e., at k*F tone ; k=2,3, . . . ), which can be suppressed via digital filtering. 
     Quantization and circuit noise suppression in FM-to-AM converter  210  also should be considered. In detector  431 , for, example, quantization noise as a function of frequency N Q (f) may be represented as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                           F 
                           s 
                         
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   Where 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     = 
                     
                       
                         Quantization 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Step 
                       
                       = 
                       
                         
                           V 
                           max 
                         
                         
                           ( 
                           
                             
                               2 
                               
                                 ADC 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 BitWidth 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       F 
                       s 
                     
                     = 
                     
                       Sampling 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Rate 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       of 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ADC 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       512 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The circuit noise in detector  431  may be represented as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         N 
                         C 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     = 
                     
                       
                         P 
                         N 
                       
                       
                         F 
                         s 
                       
                     
                   
                   ; 
                   
                     
                       - 
                       
                         
                           F 
                           s 
                         
                         2 
                       
                     
                     ≤ 
                     f 
                     ≤ 
                     
                       
                         F 
                         s 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Where P N =Noise Power and F S =Sampling Rate. 
     In one embodiment, filtering (e.g., LPF or band-pass-filtering (BPF) with corner/center at F tone ) may suppress most of the noise which is spread flat across the F 2  range to detect a single tone located at a FM modulated carrier frequency F tone . 
     It is worthwhile noting that PA  124  bias may be such that the amplitude-modulation/phase-modulation (AM/PM) will rotate the I/Q components in a manner that makes the I or Q signal in one path stronger than the other. To balance out the I and Q components in I and Q paths  501 - 1 ,  501 - 2 , the DC-offset and gain of the I and Q components may be independently adjusted by the first and second DC-offset and amplifier modules  512 - 1 ,  512 - 2 . Once the DC-offset is removed and the gain is properly adjusted, the signals in both I and Q paths  501 - 1 ,  501 - 2  may cover the entire dynamic range of respective ADCs  514 - 1 ,  514 - 2 . Otherwise, the weaker path signal may suffer excessive quantization if the weaker path signal is combined with the stronger path signal at processor  518  thus affecting the accuracy of the amplitude-phase time delay estimate. 
       FIG. 6  graphically illustrates one embodiment of an AM waveform  600 . The modulation frequency for AM waveform  600  is FM modulated carrier frequency F tone . In one embodiment, AM waveform  600  is normalized and any DC-offset or bias removed by DC-Offset and amplifier modules  512 - 1 ,  512 - 2  in each I and Q component paths  501 - 1 ,  501 - 2 , respectively, prior to analog-to-digital conversion. Multiple techniques may be used to normalize and remove the bias in AM waveform  600 . In one embodiment, AM waveform  600  may be represented as a signal having a predetermined waveform shape and frequency. In the illustrative embodiments described below, AM waveform  600  may be defined as sine wave tone sin(ω tone t) at a frequency of ω tone =2πF tone , for example. Normalization and bias removal techniques may comprise max-min, mean, and mean-square methods. In max-min method max is defined the positive peak and min is defined as the negative peak of AM waveform  600 . In accordance with the max-min method, the mean and amplitude of AM waveform  600  may be calculated as: 
     
       
         
           
             
               
                 
                   
                     mean 
                     = 
                     
                       
                         ( 
                         
                           Max 
                           + 
                           Min 
                         
                         ) 
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   amplitude 
                   = 
                   
                     
                       
                         ( 
                         
                           Max 
                           - 
                           Min 
                         
                         ) 
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In accordance with the mean and mean-square methods, if the sampling rate is an integer multiple of the sine wave frequency ω tone =2πF tone , then the mean and amplitude of AM waveform  600  may be calculated as: 
     
       
         
           
             
               
                 
                   
                     mean 
                     = 
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           s 
                           n 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     amplitude 
                     = 
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           s 
                           n 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   Where 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   N 
                   = 
                   
                     
                       Number 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       of 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       samples 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       per 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cycles 
                     
                     = 
                     
                       
                         ⌊ 
                         
                           
                             F 
                             s 
                           
                           
                             F 
                             tone 
                           
                         
                         ⌋ 
                       
                       = 
                       
                         
                           
                             F 
                             s 
                           
                           
                             F 
                             tone 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     s 
                     n 
                   
                   = 
                   
                     amplitude 
                     · 
                     
                       ( 
                       
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   F 
                                   tone 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     n 
                                     / 
                                     
                                       F 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         mean 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     If the sampling rate is not equal to an integer multiple of the AM waveform  600  carrier frequency F tone , the summations above may not exactly equal to the mean and amplitude of the sine wave, and there may be a finite amount of bias error in the calculation. 
       FIG. 7  illustrates one embodiment of a combination amplitude and phase modulation path synchronization system  700 . In one embodiment, system  700  may be implemented in a RF polar modulation transmitter topology. In one embodiment, system  700  comprises similar components previously described with reference to  FIGS. 1 ,  2 ,  4 , and  5 , the details of which will not be restated hereinbelow. Detector  702  is substantially similar to detector  431  previously discussed with reference to  FIG. 5 . Detector  702  includes I-path  701 - 1  and Q-path  701 - 2  comprising respective digital filters  704 - 1  and  704 - 2 . Each digital filter  704 - 1 ,  704 - 2  may perform digital filtering processes with or without prior up-sampling. Output signals of digital filters  704 - 1 ,  704 - 2  comprise filtered digitized I and Q components  706 - 1  and  706 - 2 , respectively. Digital I and Q components  706 - 1  and  706 - 2  are coupled to processor  708  for processing and determination of amplitude-phase modulation path delays. Processor  708  may comprise various embodiments of correlation delay and estimation modules  116 ,  218 , and processor  436 ,  518 , for example. 
     Although, the following functions are described as being performed by processor  708 , these functions may be performed by any one of the previously discussed correlation delay and estimation modules  116 ,  218 , and processors  436 ,  518  for example. Digital I and Q components  706 - 1 ,  706 - 2  may be combined and processed at the output of synchronous detector  702  by processor  708  to calculate amplitude-phase modulation path delays. Digital I and Q components  706 - 1 ,  706 - 2  may be represented as follows: 
                     r   ⁡     (   t   )       =       I   ⁡     (   t   )       +     j   ⁢           ⁢     Q   ⁡     (   t   )                   (   11   )                 I   ⁡     (   t   )       =       (         cos   ⁡     (     ϕ     AM   ⁢     /     ⁢   PM       )       ·     sin   ⁡     (       ω   tone     ⁡     (     t   -   τ     )       )         +       n   I     ⁡     (   t   )         )       ︸     I   ⁢           ⁢   rail                 (   12   )                 Q   ⁢         (   t   )     =     (         sin   ⁡     (     ϕ     AM   ⁢     /     ⁢   PM       )       ·     sin   ⁡     (       ω   tone     ⁡     (     t   -   τ     )       )         +       n   Q     ⁡     (   t   )         )       ︸         Q   ⁢           ⁢   rail             (   13   )                 r ( t )=sin(ω tone ( t−τ))·[cos(φ   AM/PM )+ j sin(φ AM/PM )]+[ n   l ( t )+ jn   Q ( t )]  (14) φ AM/PM   =PA AM/PM  rotation (distortion)  (15)   n   l   +jn   Q =(Circuit+Quantization) noise  (16) τ=Amplitude(τ α ) or Phase(τ φ )Modulation Pathdelay to be estimated  (17) 
     Several techniques may be employed to combine and process digital I and Q components  706 - 1 ,  706 - 2 , represented herein as sine waves for illustration purposes only, for example, to determine τ, the amplitude (τ α ) or phase (τ φ ) modulation path delay. The techniques may comprise, for example, selecting the I or Q path  701 - 1 ,  7012  based on signal strength, averaging the signal strengths in the I and Q paths  701 - 1 ,  7012 , and determining the magnitude of the envelope of the signals in the I and Q paths  701 - 1 ,  701 - 2  as:
 
| R|=√ {square root over (I 2   +Q   2 )}  (18)
 
     These methods are by no means exhaustive and other methods may be employed without departing from the scope. 
     Accordingly, in one embodiment, either I-path  701 - 1  or Q path  701 - 2  may be selected for processing based on which path has the stronger signal. This technique processes only the stronger of the two I or Q paths  701 - 1 ,  701 - 2 . Differences in signal strengths may be attributed to bias imbalances in PA  124  due to AM/PM rotation of the IQ component in a way that makes one path (either I-path  701 - 1  or Q-path  701 - 2 ) substantially stronger than the other. Instead of balancing I-path  701 - 1  or Q-path  701 - 2  signals, in one embodiment, the path with stronger signal is selected for processing. This technique may be desirable if the DC-offset and gain stages in each I and Q paths  701 - 1 ,  701 - 2  cannot be independently adjusted to cover the full dynamic range of the ADCs  514 - 1 ,  514 - 2 . Otherwise, if combined with the stronger path signal, the weaker path signal may be subject to excessive quantization error which may affect the accuracy of the amplitude-phase delay timing estimate (i.e., the weaker signal may act more like noise than useful information). Only one clean-up LPF and one normalization and bias-removal calculation is employed in I path  701 - 1  or Q path  701 - 1 , whichever has the stronger signal. 
     If the signal power in I, Q paths  701 - 1 ,  701 - 2  cannot be independently adjusted for DC-offset and gain, the signal power in each I, Q path  701 - 1 ,  701 - 2  may be averaged. Also, if PA  124  bias is such that the AM/PM rotation will result in balanced I and Q components  706 - 1 ,  706 - 2  in respective I, Q paths  701 - 1 ,  701 - 2  having comparable strengths, then it may be possible to combine the I and Q components  706 - 1 ,  706 - 2  by calculating their average after normalization and bias-removal. This approach requires processing in both I and Q paths  701 - 1 ,  701 - 2 . For example, two clean-up LPFs and two normalization and bias-removal calculations, and then the averaging operation in each I and Q path  701 - 1 ,  701 - 2 . 
     In the envelope √{square root over (I 2 +Q 2 )} determination method, the envelope of the digital I and Q components  706 - 1 ,  706 - 2  is determined, normalized, and the bias is removed. This method requires a CORDIC (or a square root function calculation), one clean-up LPF, and one normalization and bias-removal calculation. 
     Once digital I and Q components  706 - 1 ,  706 - 2  are combined, using any of the processes previously described, a maximum likelihood estimation technique is performed by processor  708 . For maximum-likelihood estimation, digital I and Q components  706 - 1 ,  706 - 2  may be written as:
 
 r ( t )= I ( t )+ jQ ( t )  (11)
 
 r ( t )=( a+jb )·sin(ω tone ( kT   s −τ))+ n   k,Circ   +n   k,Quan   (19)
 
 n   k,Circ   =n   k,Circ,l   +jn   k,Circ,Q =Circuit noise (Additive Gaussian noise)  (20)
 
 n   k,uan   =n   k,Quan,l   +jn   k,Quan,Q =Quantization noise  (21)
 
     If ADCs  514 - 1  and  514 - 2  have sufficient bit-widths (sufficient number of output bits) to resolve digital I and Q components  706 - 1 ,  706 - 2  the solution is one of detecting a tone in the presence of additive Gaussian noise (AGN), a process which may be computationally complex. However, in the present case, processor  708  is to determine an estimate of the phase (or timing) of the tone and not an estimate of the magnitude or frequency of the tone. Therefore, the solution may be more easily processed. For a discussion on techniques for coarse frequency acquisition through the discrete Fourier transform (DFT) computation for samples of a received waveform, reference is made to Walid K. M. Ahmed and P. J. McLane, “A Method for Coarse Frequency Acquisition for Nyquist Filtered MPSK”, IEEE Transactions on Vehicular Technology, Nov. 1996, vol. 45, number 4, pp. 720-731, which is incorporated herein by reference in its entirety. 
     For an AGN channel, the maximum likelihood estimation technique attempts to find a time-delay that minimizes the log-likelihood metric (LLM) as follows:
 
 L   ML (τ)=( S   r   −s   t (τ))· C   Noise   −1 ·( S   r   −S   t (τ)) T   (22)
 
where
 
 C   Noise   −1 =Coveriance Matrix of the  AGN   (23)
 
 s   r=[s   l,r   s   2,r    . . . s   M,r   ]; s   m,r   =c·sin(ω   tone ( mT   s−τ))+   n   m   (24)
 
 n   m =Complex−valued Additive Gaussian Noise Random Variable  (25)
 
 s   t   =[s   l,t   s   2,t    . . . s   M,t   ];s   m,t =sin(ω tone ( mT   s −τ))  (26)
 
     The timing estimate is the one that satisfies the condition: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         τ 
                         ^ 
                       
                       ⁢ 
                       
                         : 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           L 
                           ML 
                         
                         ⁡ 
                         
                           ( 
                           
                             τ 
                             ^ 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         min 
                         τ 
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             L 
                             ML 
                           
                           ⁡ 
                           
                             ( 
                             τ 
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     This operation may be conducted through an exhaustive search by scanning all possible delay values, which may be computationally complex. In the special case of additive white Gaussian noise (AWGN), the LLM is equivalent to the least-mean-square (LMS) method. The LMS method may be preferred over the LLM method because it is less computationally complex than the exhaustive search method. It can be shown that the maximum likelihood estimated frequency, amplitude, and delay time τ (i.e., tone phase) may be computed via the DFT of the received digital I and Q component  706 - 1 ,  706 - 2  samples. The maximum likelihood estimated frequency, amplitude, and delay time τ also may be computed using the fast Fourier transform (FFT) if M=2l, for example. 
     The calculation becomes simpler because the delay estimate τ or the tone phase (timing) is all that needs to be extracted by processor  708 . Accordingly, it can be shown that the maximum likelihood estimated phase delay τ of the detected tone is: 
     
       
         
           
             
               
                 
                   
                     θ 
                     ^ 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           Angle 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   S 
                                   r 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 | 
                                 
                                   ω 
                                   = 
                                   
                                     ω 
                                     tone 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           Angle 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   j 
                                 
                                 · 
                                 
                                   
                                     S 
                                     r 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     ω 
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 | 
                                 
                                   ω 
                                   = 
                                   
                                     ω 
                                     tone 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       τ 
                       ^ 
                     
                     = 
                     
                       
                         θ 
                         ^ 
                       
                       / 
                       
                         ( 
                         
                           ω 
                           tone 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   Where 
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       S 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     DFT 
                     ⁡ 
                     
                       ( 
                       
                         s 
                         
                           m 
                           , 
                           r 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       S 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       M 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           s 
                           
                             m 
                             , 
                             r 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               mT 
                               s 
                             
                             ⁢ 
                             ω 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Where T s  is the sampling rate of ADCs  514 - 1 ,  514 - 2 . 
     Therefore, the DFT (or the FFT) is not computed for the entire frequency range, but only at two points:
 
f=±f tone   (32)
 
     To implement these maximum likelihood estimation techniques, the operating point of the various components of system  700  may be selected according to the following criteria. For example, selecting the proper operating point of PA  124 , converting a FM training waveform to an AM training waveform with FM-to-AM converter  210 , and performing a synchronous detection of the AM training waveform with synchronous detector  702  such that the signal-to-noise ratio (SNR) at the inputs of ADCs  514 - 1 ,  514 - 2  is high enough for other techniques to be reliable and well performing. A sufficient dynamic range at the inputs of respective ADCs  514 - 1 ,  514 - 2  should be maintained to ensure sufficient bit-widths and minimum possible quantization noise in digital I and Q components  706 - 1 ,  706 - 2 . After ADCs  514 - 1 ,  514 - 2  digital filters  704 - 1 ,  704 - 2  reduce the noise power and thus increase the SNR. If these maximum likelihood estimation techniques cannot be implemented, other techniques may be employed. 
     For example, if the aforementioned maximum likelihood estimation techniques are computationally complex with respect to given implementation limitations, other techniques may be employed to estimate the phase delay τ of the detected tone. For example, if the SNR in system  700  is relatively high, these other techniques may asymptotically approach the performance level of an optimal maximum likelihood estimation technique, provided they are not biased estimators. If the SNR in system  700  is relatively low, however, these other techniques may yield considerably inferior performance than the optimal maximum likelihood estimation technique. The following embodiments describe various techniques to determine amplitude-phase delay τ using a process other than the maximum likelihood estimation technique described above. 
     One amplitude and phase path delay τ estimation technique that may be employed is a one-step inverse-sine approach via look-up-table (LUT), for example. A delay estimate {circumflex over (τ)} InvSin  may be formed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       τ 
                       ^ 
                     
                     
                       Inv 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Sin 
                     
                   
                   = 
                   
                     
                       1 
                       M 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               sin 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 s 
                                 
                                   r 
                                   , 
                                   m 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             mT 
                             s 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     In equation (33), the phase wrapping and quadrant determination have been properly accounted for prior to its application. 
     One embodiment of an amplitude and phase path delay τ estimation technique is a one-step slide-and-correlate estimation technique that employs high-resolution sampling and high-resolution slide (HRS-HRS) techniques. A least-mean-square delay estimate {circumflex over (τ)} 1Stp-C&amp;S   (LMS)  or a least-mean-absolute delay estimate {circumflex over (τ)} 1Stp-C&amp;S   (LMA)  may be formed as follows: 
     Least-Mean-Square (LMS): 
     
       
         
           
             
               
                 
                   
                     
                       τ 
                       ^ 
                     
                     
                       
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Stp 
                           ⁢ 
                           
                             - 
                           
                           ⁢ 
                           C 
                         
                         &amp; 
                       
                       ⁢ 
                       S 
                     
                     
                       ( 
                       LMS 
                       ) 
                     
                   
                   = 
                   
                     Arg 
                     ⁢ 
                     
                       { 
                       
                         
                           max 
                           
                             
                               
                                 
                                   τ 
                                   = 
                                   
                                     kT 
                                     sample 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     k 
                                     = 
                                     
                                       - 
                                       
                                         ( 
                                         
                                           M 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   
                                     - 
                                     1 
                                   
                                   , 
                                   0 
                                   , 
                                   1 
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   
                                     L 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         M 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               ∑ 
                               m 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     s 
                                     
                                       r 
                                       , 
                                       m 
                                     
                                   
                                   - 
                                   
                                     s 
                                     
                                       t 
                                       , 
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           k 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Or Least-Mean-Absolute (LMA): 
     
       
         
           
             
               
                 
                   
                     
                       τ 
                       ^ 
                     
                     
                       
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Stp 
                           ⁢ 
                           
                             - 
                           
                           ⁢ 
                           C 
                         
                         &amp; 
                       
                       ⁢ 
                       S 
                     
                     
                       ( 
                       LMA 
                       ) 
                     
                   
                   = 
                   
                     Arg 
                     ⁢ 
                     
                       { 
                       
                         
                           max 
                           
                             
                               
                                 
                                   τ 
                                   = 
                                   
                                     kT 
                                     sample 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     k 
                                     = 
                                     
                                       - 
                                       
                                         ( 
                                         
                                           M 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   
                                     - 
                                     1 
                                   
                                   , 
                                   0 
                                   , 
                                   1 
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   
                                     L 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         M 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               ∑ 
                               m 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   s 
                                   
                                     r 
                                     , 
                                     m 
                                   
                                 
                                 - 
                                 
                                   s 
                                   
                                     t 
                                     , 
                                     
                                       ( 
                                       
                                         m 
                                         - 
                                         k 
                                       
                                       ) 
                                     
                                   
                                 
                               
                                
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     These techniques may employ high sampling-rate ADCs  514 - 1 ,  514 - 2  and multiple correlation computations. For a high-resolution estimate (e.g., 2 ns for WCDMA) T s  should be as small as, or smaller than, the resolution required for the delay estimate τ. 
     One embodiment of an amplitude and phase path delay τ estimation technique is a one-step slide-and-correlate technique that employs low-resolution sampling and high-resolution slide (LRS-HRS). A least-mean-square delay estimate {circumflex over (τ)} 1Stp-C&amp;S   (LMS)  or a least-mean-absolute delay estimate {circumflex over (τ)} 1Stp-C&amp;S   (LMA)  may be formed as follows: 
     Least-Mean-Square (LMS): 
     
       
         
           
             
               
                 
                   
                     
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                       LMS 
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                   = 
                   
                     Arg 
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                               max 
                               
                                 
                                   
                                     
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                                       = 
                                       
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                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     Or Least-Mean-Absolute (LMA): 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   37 
                   ) 
                 
               
             
             
               
                 
                   l 
                   = 
                   
                     
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                       sample 
                     
                     / 
                     
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                       slide 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     This technique may not employ high sampling rate ADCs  514 - 1 ,  514 - 2 . This technique may employ multiple correlation computations to scan all possible time delays at a high-resolution slide. Performance, however, may be expected to be somewhat inferior to that of the HRS-HRS method previously described. 
     Some techniques may require multiple steps. For example, an amplitude and phase path delay τ estimation technique is a two-step slide and correlate technique. During a first step, I and Q components  703 - 1 ,  703 - 2  at the respective inputs of ADCs  514 - 1 ,  514 - 2  are sampled at a low rate. A slide-and-correlate is conducted at the lower sampling rate using low-resolution-sampling and low-resolution slide (LRS-LRS). A first course timing estimate τ course  is determined. In a second step, I and Q components  703 - 1 ,  703 - 2  at the respective inputs of ADCs  514 - 1 ,  514 - 2  are digitally up-sampled at a higher sampling rate. A slide-and-correlate is conducted at the higher sampling rate using low-resolution-sampling and high-resolution slide (LRS-HRS) around the first course timing estimate τ course  and a second fine timing estimate τ fine  is determined. 
     A two-step inverse-sine slide and correlate approach also may be employed as an amplitude and phase path delay τ estimation technique. In a first step, I and Q components  703 - 1 ,  703 - 2  at the respective inputs of ADCs  514 - 1 ,  514 - 2  are sampled at a low rate, or may be sampled at a low rate and digitally up-sampled. A first coarse timing estimate τ coarse  may be calculated using the one-step inverse sine computation previously described with reference to equation ( 33 ) to solve for {circumflex over (τ)} InvSin . In a second step, I and Q components  703 - 1 ,  703 - 2  at the inputs of respective ADCs  514 - 1 ,  514 - 2  are sampled at the same low rate, or may be further digitally up-sampled. A slide and correlate is conducted using LRS-HRS around the course timing estimate τ coarse  and a second fine timing estimate τ fine  is then determined. 
       FIG. 8  illustrates one embodiment of a combination amplitude and phase modulation path synchronization system  800 . In one embodiment, system  800  may be implemented in a RF polar modulation transmitter topology. System  800  comprises down converter  804  to mix-down output signal  802  with an un-modulated oscillator signal  803  from LO  806 . Training waveform generator  402  generates training waveform  412 . Training waveform  412  is passed to both amplitude modulation path  106  and phase modulation path  126  simultaneously. Amplitude training waveform  824  is coupled to the amplitude modulation node of PA  124  and is amplitude modulated with a carrier waveform. Frequency training waveform  826  is coupled to the input node of PA  124  and is frequency modulated with the carrier waveform. Output signal  802  of PA  124  is coupled to down converter  804 . LO  806  is coupled to the oscillator node of down converter  804 . LO  806  generates oscillator signal  803  to mix with output signal  802 . In one embodiment, down converter  804  may be implemented as one embodiment of detectors  431 ,  702  (e.g., a synchronous detector and the like) with the oscillator node coupled to LO  806 . Down converted output signal  808  is coupled to a CORDIC module  810  to extract amplitude signal  812  and phase signal  814 . Amplitude signal  812  is coupled to amplitude timing extraction module  816  to determine the amplitude delay τ α  and phase signal  814  is coupled to phase timing extraction module  818  to determine the phase delay τ φ . 
     If training signal  412  is a sine wave, the analysis discussed with respect to  FIG. 7  may be applied to system  800  to determine the amplitude delay τ α  and the phase delay τ φ . Down-converted signal  808  is coupled to a CORDIC module  810  to extract amplitude and phase signals  812 ,  814 . Relative to systems  100 ,  200 ,  400 ,  600 , and  700 , however, system  800  may provide a much larger dynamic range of the extracted amplitude and phase signals  812 ,  814  relative to any associated DC offsets. This may relax the circuit noise tolerance of the feedback path RF components, for example. System  800  saves acquisition time by sending both the amplitude and phase training waveform  412  patterns simultaneously to both amplitude and phase modulation paths  106 ,  126  and then uses CORDIC module  810  to separate amplitude and phase signals  812 ,  814 . Once the amplitude and phase signals  812 ,  814  are separated, the amplitude delay τ α  and the phase delay τ φ  are determined by the respective amplitude timing extraction module  816  and phase timing extraction module  818 . In this embodiment it is assumed that PA  124  can be biased at a point where the AM/PM error is small such that no distortion, manifested as an additive noise quantity, takes place relative to phase signal  814 . Otherwise, training sessions for amplitude modulation path  106  and phase modulation path  126  should be conducted separately as in the FM-to-AM approaches discussed with reference to  FIGS. 1-7  above. 
       FIG. 9  illustrates one embodiment of an amplitude and phase modulation path synchronization system  900 . In one embodiment, system  900  may be implemented in a RF polar modulation transmitter topology. System  900  is a conceptual level functional illustration of synchronization system  900  and is not an illustration of actual circuit connections. Training waveform generator  402  generates training waveform  412 , which is passed to both amplitude modulation path  106  and phase modulation path  126  simultaneously. Amplitude path training waveform  902  is passed directly to detector  912 . Phase path training waveform  904  is coupled to cascaded FM-to-AM converter  210  and to detector  912 . In one embodiment, phase modulation path  126  may comprise a voltage controlled oscillator (VCO) coupled to FM-to-AM converter  210 . In one embodiment, FM-to-AM converter  210  may be implemented as a 450 MHz LPF (for a 900 MHz and 1800 MHz carrier frequency, for example), for example. During a synchronization training period, either PA  124  or the antenna is switched-off (or both) to effectively bypass PA  124  and to essentially disable input node  906 , amplitude modulation node  908 , and output node  910 . This is conceptually illustrated by the ground symbols at the input node  906 , amplitude modulation node  908 , and output node  910 . Amplitude modulation path  106  may comprise an anti-alias filter (AAF) coupled to the amplitude modulation node  908  of PA  124 . 
     Detector  912  comprises first and second paths  914 - 1 ,  914 - 2 . Either path  914 - 1  or path  914 - 2  may be employed to process amplitude training waveform  902  or phase training waveform  904 . In the illustrated embodiment, the input node  907  of phase shifter  508  is disabled and mixer  506 - 2 , analog LPF  510 - 2  and DC-offset and amplifier  512 - 2  are bypassed. In one embodiment, amplitude path training waveform  902  may be tapped at amplitude modulation node  908  of PA  124 , i.e., at the AAF output, and it is routed “directly” to ADC  514 - 2  in second path  914 - 2  of detector  912  via switch  920 . Phase training waveform  904  is coupled to the oscillator input node of mixer  506 - 1 . AM training waveform  916  at the output node of FM-to-AM converter  210  is coupled to the input node of mixer  506 - 1 . Accordingly, phase training waveform  904  and AM training waveform  916  are mixed at mixer  516 - 1 . Mixer  506 - 1  produces signal  917 , which is processed along path  914 - 1 . 
     Amplitude training waveform  902  and phase training waveform  904  are processed by detector  912 . Detected signals  922 - 1  and  922 - 2  at the output node of each ADC  514 - 1 ,  514 - 2  are processed independently by respective digital filters  704 - 1 ,  704 - 2 . Accordingly, detected signals  924 - 1 ,  924 - 2  may be processed independently by processor  708  to extract the timing information associated with each signal, e.g., amplitude training waveform  902  and phase training waveform  904 . Accordingly, amplitude modulation path delay τ α  and phase modulation path delay τ φ  may be obtained simultaneously. 
       FIG. 10  is a flow diagram  1000  of one embodiment of a synchronization process to determine a delay in an amplitude modulation path of an RF transmitter in a polar modulation topology. As previously discussed, the synchronization process may be conducted by estimating the amplitude modulation path delay and the phase modulation path delay either individually as shown with reference to systems  100 ,  400 ,  700  shown in respective  FIGS. 1 ,  4 , and  7 , for example, or simultaneously as shown with reference to systems  800 ,  900  shown in  FIGS. 8 and 9 , for example. In this embodiment, the synchronization process is described with reference to system  100  shown in  FIG. 1 , although a similar synchronization process may be applied to systems  400  and  700  in  FIGS. 4 and 7 . Synchronization is conducted by determining the amplitude modulation path  106  delay τ α  separately from the phase modulation path  126  delay τ φ . Baseband amplitude waveform generator  102  generates baseband amplitude training waveform  104  and provides amplitude training waveform  104  in the amplitude modulation path  106 . Baseband amplitude waveform generator  102  passes ( 1002 ) baseband amplitude training waveform  104  through the amplitude modulation path  106  of a polar transmitter. At PA  124  amplitude modulation node, baseband amplitude training waveform  104  is amplitude modulated with an RF carrier waveform to produce AM training waveform  110 . Envelope detection module  112  detects ( 1004 ) the RF carrier envelope of AM training waveform  110  at an output node of PA  124 . Envelope detection module  112  and/or processor  116  processes ( 1006 ), e.g., normalize and slide-and-correlate AM training waveform  110 . Processor  116  determines ( 1008 ) the amplitude path delay τ α  between detected envelope waveform  114  and baseband amplitude training waveform  104 . The best-estimate time delay is the one that corresponds to the best correlation factor between detected envelope waveform  114  and baseband amplitude training waveform  104 . Processor  116  records ( 1010 ) the amplitude path delay τ α  and determines ( 1012 ) the time delay mismatch between the amplitude and phase paths τ α −τ φ , where τ φ  is the phase path time delay. 
       FIG. 11  is a flow diagram  1100  of one embodiment of a synchronization process to determine a delay in a phase modulation path of an RF transmitter in a polar modulation topology. As previously discussed, the synchronization process may be conducted by estimating the amplitude modulation path delay and the phase modulation path delay either individually as shown with reference to systems  100 ,  400 ,  700  shown in respective  FIGS. 1 ,  4 , and  7 , for example, or simultaneously as shown with reference to systems  800 ,  900  shown in  FIGS. 8 and 9 , for example. In this embodiment, the synchronization process is described with reference to system  200  shown in  FIG. 2 , although a similar synchronization process may be applied to systems  400  and  700  in  FIGS. 4 and 7 . Synchronization is conducted by determining the phase modulation path  126  delay τ φ  separately from the amplitude modulation path  106  delay τ α . Baseband frequency waveform generator  118  generates baseband frequency training waveform  202  and provides frequency training waveform  202  in the phase modulation path  126 . Baseband frequency training waveform  202  may be integrated by integrator  204 . Integrated baseband frequency training waveform  206  is passed ( 1102 ) through the phase modulation  126  path of the polar transmitter. At the input node of PA  124 , integrated baseband frequency waveform  206  is frequency modulated with the carrier waveform to produce FM training waveform  208 . FM-to-AM converter  210  converts ( 1104 ) FM training waveform  208  to AM training waveform  212 . As previously discussed, in one embodiment, FM-to-AM converter  210  may be a LPF, a tuning circuit, or the like. Envelope detection module detects ( 1106 ) the envelope waveform of the RF carrier of AM training waveform  212 . Envelope detection module  112  or processor  218  processes ( 1108 ), e.g., normalize and slide-and-correlate AM training waveform  212 . Processor  218  determines ( 1110 ) the phase path delay τ φ  between detected envelope training waveform  214  and baseband frequency training waveform  202 . The best-estimate time delay is the one that corresponds to the best correlation factor between detected envelope waveform  214  and baseband frequency training waveform  202 . Processor  218  records ( 1112 ) the amplitude path delay τ φ . Processor  218  determines ( 1114 ) the time delay mismatch between the amplitude and phase paths τ α −τ 100  , where τ α is the amplitude path time delay. 
       FIG. 12  is a flow diagram  1200  of one embodiment of a process to estimate delay in an amplitude path and a phase path simultaneously. The amplitude modulation path delay τ α  and the phase modulation path delay τ φ may be calculated simultaneously. Reference is made to synchronization system  800  illustrated in  FIG. 8 , for example. Training waveform generator  402  generates baseband amplitude and phase training waveform  412  and passes ( 1202 ) baseband amplitude training waveform  412  through the amplitude modulation path  106  and phase modulation path  126  of a polar transmitter simultaneously. At an amplitude modulation input node of PA  124 , baseband amplitude modulation training waveform  824  is amplitude modulated with a carrier waveform. At an input node of PA  124 , baseband frequency waveform  826  is frequency modulated with the carrier waveform. Combination AM/FM modulation training waveform  802  is produced at an output node of PA  124 . Down converter  804  mixes ( 1204 ) combination AM/FM modulation training waveform  802  oscillator signal  803 . Down converted output signal  808  is detected ( 1206 ) by CORDIC module  810 . CORDIC module  810  extracts ( 1208 ) amplitude signal  812  and phase signal from down converted output signal  808 . Amplitude timing extraction module  816  extracts ( 1210 ) amplitude modulation path  106  delay τ α . Phase timing extraction module  818  extracts ( 1212 ) phase modulation path  126  delay τ φ . Once the amplitude modulation path  106  delay τ 60   and the phase modulation path  126  delay τ φ  are extracted a processor may be employed to record ( 1214 ) the amplitude modulation path  106  delay τ α  and the phase modulation path  126  delay τ φ  and determine ( 1216 ) the time delay mismatch between the amplitude and phase paths τ α −τ φ , where τ α  is the amplitude path time delay. 
       FIG. 13  is a flow diagram  1300  of one embodiment of a process to estimate delay in an amplitude path and a phase path simultaneously. The amplitude modulation path delay τ α  and the phase modulation path delay τ 100   may be calculated simultaneously. Reference is made to synchronization system  900  illustrated in  FIG. 9 , for example. Training waveform generator  402  generates baseband amplitude and phase training waveform  412  and passes ( 1302 ) amplitude training waveform  412  to separate amplitude modulation path  106  and phase modulation path  126 . Amplitude path training waveform  902  is passed ( 1304 ) directly to detector  912 . Detector  912  detects ( 1306 ) the amplitude of amplitude path training waveform  902 . Processor  708  determines ( 1306 ) the amplitude path delay τ α . Phase path training waveform  904  is coupled to cascaded FM-to-AM converter  210  and to detector  912 . FM-to-AM converter  210  converts ( 1308 ) phase path training waveform  904  to AM training waveform  916 . Mixer  506 - 1  mixes ( 1310 ) AM training waveform  916  with phase path training waveform  904  and produces signal  917 . Signal  917  is processed in path  914 - 1 . Detector  912  detects ( 1316 ) the phase of phase path training waveform  904 . Processor  708  determines ( 1312 ) the phase path delay τ φ . Once the amplitude modulation path  106  delay τ α  and the phase modulation path  126  delay τ φ  are extracted, processor  708  determines ( 1314 ) the time delay mismatch between the amplitude and phase paths τ α −τ φ , where τ α  is the amplitude path time delay. 
     Once the time delay mismatch is determined, then the timing mismatch between phase and amplitude paths can be corrected in polar transmitter in accordance with conventional techniques. Accordingly, the time delay mismatch may be used to synchronize the phase and amplitude paths. For example, the time delay mismatch may be applied to correct the timing misalignment between the phase and amplitude path signals because the timing misalignment may be detrimental to the reconstructed I/Q components, for example. Accordingly, polar transmitters amplitude R and phase θ signals may be synchronized employing the time delay mismatch to correct for the timing misalignments due to different delays encountered by these signals in the separate amplitude R and phase θ processing paths. 
     In various embodiments, the systems described herein may be illustrated and described as comprising several separate functional elements, such as modules and/or blocks. Although certain modules and/or blocks may be described by way of example, it can be appreciated that additional or fewer modules and/or blocks may be used and still fall within the scope of the embodiments. Further, although various embodiments may be described in terms of modules and/or blocks to facilitate description, such modules and/or blocks may be implemented by one or more hardware components (e.g., processors, DSPs, PLDs, ASICs, circuits, registers), software components (e.g., programs, subroutines, logic) and/or combination thereof. 
     Numerous specific details have been set forth herein to provide a thorough understanding of the embodiments. It will be understood by those skilled in the art, however, that the embodiments may be practiced without these specific details. In other instances, well-known operations, components and circuits have not been described in detail so as not to obscure the embodiments. It can be appreciated that the specific structural and functional details disclosed herein may be representative and do not necessarily limit the scope of the embodiments. 
     It is also worthy to note that any reference to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment. 
     Some embodiments may be implemented using an architecture that may vary in accordance with any number of factors, such as desired speed, power levels, heat tolerances, semiconductor manufacturing processing, input rates, output rates, memory resources, and other performance constraints. 
     Some embodiments may be described using the expression “coupled” along with their derivatives. It should be understood that the term “coupled” may be used to indicate that two or more elements are in direct physical or electrical contact. The term “coupled,” however, also may mean that two or more elements are not in direct contact with each other, but yet still co-operate or interact with each other. The embodiments are not limited in this context. 
     While certain features of the embodiments have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is therefore to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.