Abstract:
A method for carrier tracking in AM in-band on-channel radio receivers comprises the steps of receiving an input signal, generating a local oscillator signal in response to an oscillator control signal, mixing the input signal with a local oscillator signal to produce a first signal, filtering the first signal to produce a filtered first signal at a decimated sample rate, detecting the phase error and frequency error of the filtered first signal normalized to mitigate effects of signal fades, and using an adaptive loop filter to produce the oscillator control signal in response to the phase error and frequency error of the filtered first signal. An apparatus that performs the method is also provided.

Description:
FIELD OF THE INVENTION  
       [0001]     This invention relates to radio broadcasting and, more particularly, to methods of and apparatus for tracking carrier signals in a receiver for use with an in-band on-channel digital broadcasting system.  
       BACKGROUND OF THE INVENTION  
       [0002]     An in-band on-channel (IBOC) digital broadcasting system simultaneously broadcasts analog and digital signals in a standard AM broadcasting channel. One AM IBOC system is described in U.S. Pat. No. 5,588,022. The broadcast signal includes an amplitude modulated radio frequency signal having a first frequency spectrum. The amplitude modulated radio frequency signal includes a first carrier modulated by an analog program signal. The signal also includes a plurality of digitally modulated subcarriers within a bandwidth that encompasses the first frequency spectrum. Each of the digitally modulated subcarriers is modulated by a digital signal. A first group of the digitally modulated subcarriers lies within the first frequency spectrum and is modulated in quadrature with the first carrier signal. Second and third groups of the digitally modulated subcarriers lie outside of the first frequency spectrum and are modulated both in-phase and in quadrature with the first carrier signal. The subcarriers are divided into primary, secondary and tertiary partitions. Some of the subcarriers are complementary subcarriers.  
         [0003]     The subcarriers must be acquired and tracked in the receivers prior to demodulation of the received signal. Although the performance of existing carrier tracking algorithms is assumed to be reasonably good, they are difficult to analyze or modify without extensive simulation and verification in all possible tracking modes and logic conditions within the tracking algorithms.  
         [0004]     Therefore there is a need for a simpler method of carrier tracking that operates autonomously without explicit acquisition or coarse/wideband/narrowband tracking control.  
       SUMMARY OF THE INVENTION  
       [0005]     This invention provides a method for carrier tracking in AM in-band on-channel radio receivers. The method comprises the steps of receiving an input signal, generating a local oscillator signal in response to an oscillator control signal, mixing the input signal with a local oscillator signal to produce a first signal, filtering the first signal to produce a filtered first signal at a decimated sample rate, detecting the phase error and frequency error of the filtered first signal normalized to mitigate effects of signal fades, and using an adaptive loop filter to produce the oscillator control signal in response to the phase error and frequency error of the filtered first signal. An apparatus that performs the method is also provided.  
         [0006]     In another aspect, the invention provides an apparatus for carrier tracking in AM in-band on-channel radio receivers. The apparatus comprises an input for receiving an input signal, a local oscillator for generating a local oscillator signal in response to an oscillator control signal, a mixer for mixing the input signal with a local oscillator signal to produce a first signal, a filter for filtering the first signal to produce a filtered first signal at a decimated sample rate, a detector for detecting the phase error and frequency error of the filtered first signal, wherein the filtered first signal is normalized to mitigate effects of signal fades, and an adaptive loop filter for producing the oscillator control signal in response to the phase error and frequency error of the filtered first signal. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0007]      FIG. 1  is a spectral diagram of the AM hybrid IBOC signal.  
         [0008]      FIG. 2  is a spectral diagram of the AM all-digital IBOC signal.  
         [0009]      FIG. 3  is a functional block diagram of an AM IBOC receiver.  
         [0010]      FIG. 4  is a block diagram of a modem for an AM IBOC receiver.  
         [0011]      FIG. 5  is a functional block diagram of AM carrier tracking frequency/phase-locked loop.  
         [0012]      FIG. 6  is a functional block diagram of an adaptive loop filter.  
         [0013]      FIG. 7  is a linear model of a phase-locked loop, useful in determining gain values.  
         [0014]      FIG. 8  is a schematic diagram that illustrates analog resistor-capacitor (RC) time constants.  
         [0015]      FIG. 9  is a schematic diagram that illustrates equivalent digital time constants.  
         [0016]      FIGS. 10-13  are graphs showing simulated performance of the invention. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0017]     This invention provides a method for carrier tracking for AM HD Radio™ receivers. It is intended to be suitable for all AM modes, including analog demodulation.  
         [0018]     Referring to the drawings,  FIG. 1  is a spectral diagram of an AM hybrid IBOC signal. The AM hybrid IBOC waveform  10  includes the conventional AM analog signal  12  (bandlimited to about ±5 kHz) along with a nearly 30 kHz wide digital audio broadcasting (DAB) signal  14  transmitted beneath the AM signal. The spectrum is contained within a channel  16  having a bandwidth of about 30 kHz. The channel is divided into a central frequency band  18 , and upper  20  and lower  22  frequency bands. The central frequency band is about 10 kHz wide and encompasses frequencies lying within about ±5 kHz of the center frequency f o  of the channel. The upper sideband extends from about +5 kHz from the center frequency to about +15 kHz from the center frequency. The lower sideband extends from about −5 kHz from the center frequency to about −15 kHz from the center frequency.  
         [0019]     AM hybrid IBOC DAB signal format in one embodiment of the invention comprises the analog modulated carrier signal  24  plus 162 OFDM subcarrier locations spaced at approximately 181.7 Hz, spanning the central frequency band and the upper and lower sidebands. Coded digital information, representative of the audio or data signals (program material), is transmitted on the subcarriers. The symbol rate is less than the subcarrier spacing due to a guard time between symbols.  
         [0020]     As shown in  FIG. 1 , the upper sideband is divided into a primary partition  26  and a secondary partition  28 , and the lower sideband is divided into a primary partition  30  and a secondary partition  32 . The digital signals are transmitted in the primary and secondary partitions on either side of the host analog signal, as well as underneath the host analog signal in a tertiary partition  34 . The tertiary partition  34  can be considered to include a plurality of groups of subcarriers labeled  36 ,  38 ,  40  and  42  in  FIG. 1 . Subcarriers within the tertiary partition that are positioned near the center of the channel are referred to as inner subcarriers and subcarriers within the tertiary partition that are positioned farther from the center of the channel are referred to as outer subcarriers. In this example, the power level of the inner subcarriers in groups  38  and  40  is shown to decrease linearly with frequency spacing from the center frequency. The remaining groups of subcarriers  36  and  42  in the tertiary sideband have substantially constant power levels.  
         [0021]      FIG. 1  also shows two reference subcarriers  44  and  46 , for system control, that are positioned at the first subcarrier positions immediately adjacent to the analog modulated carrier and have power levels which are fixed at a value that is different from the other sidebands.  
         [0022]     The center carrier  24 , at frequency f o , is not QAM modulated, but carries the main analog amplitude modulated carrier. The synchronization and control subcarriers  44  and  46  are modulated in quadrature to the carrier. The remaining subcarriers of the tertiary partition, positioned at locations designated as  2  through  26  and − 2  through − 26  on either side of the AM carrier, are modulated with QPSK. Representative subcarrier locations are identified by the subcarrier index shown in  FIG. 1 . Subcarriers at locations  2  through  26  and − 2  through − 26  on either side of the central frequency, are referred to as tertiary subcarriers and are transmitted in complementary pairs such that the modulated resultant DAB signal is in quadrature to the analog modulated AM signal. The use of complementary subcarrier pairs in an AM IBOC DAB system is shown in U.S. Pat. No. 5,859,876. The synchronization and control subcarriers  44  and  46  are also modulated as a complementary pair.  
         [0023]     The double sideband (DSB) analog AM signal occupies the bandwidth in the ±5 kHz region. The lower and upper tertiary partitions occupy sub-bands from about 0 to about ±5 kHz and from about 0 to about +5 kHz regions, respectively. These tertiary partitions are negative complex conjugates of each other and are characterized as complementary. This complementary property maintains an orthogonal relationship between the analog and digital tertiary signals such that they can be separated in a receiver, while existing conventional receivers can still receive the analog AM signal. The tertiary partitions must be complementary combined to extract the digital signal while canceling the analog crosstalk. The secondary partitions also have the complementary property, so they can be processed at the receiver either independently, or after complementary combining, depending on interference conditions and audio bandwidth. The primary partitions are transmitted independently.  
         [0024]      FIG. 2  is a spectral diagram of an all-digital IBOC signal  50 . The power of the central frequency band  52  subcarriers is increased, relative to the hybrid format of  FIG. 1 . Again, the two subcarriers  54  and  56  located at locations − 1  and + 1  use binary phase shift keying to transmit timing information. A core upper sideband  58  is comprised of carriers at locations  2  through  26 , and a core lower sideband  60  is comprised of subcarriers at locations − 2  through − 26 . Sidebands  58  and  60  form primary partitions. Two groups  62  and  64  of additional enhancement subcarriers occupy locations  27  through  54  and − 54  through − 27  respectively. Group  62  forms a secondary partition and group  64  forms a tertiary partition. The all-digital format of  FIG. 2  is very similar to the hybrid format except that the AM signal is replaced with a delayed and digitally encoded tuning and backup version of the program material. The central frequency band occupies approximately the same spectral location in both the hybrid and all-digital formats. In the all-digital format, there are two options for transmitting the main version of the program material in combination with the tuning and backup version. The all-digital system has been designed to be constrained within ±10 kHz of the channel central frequency, f o , where the main audio information is transmitted within ±5 kHz of f o , and the less important audio information is transmitted in the wings of the channel mask out to ±10 kHz at a lower power level. This format allows for graceful degradation of the signal while increasing coverage area. The all-digital system carries a digital time diversity tuning and backup channel within the ±5 kHz protected region (assuming the digital audio compression is capable of delivering both the main and audio backup signal within the protected ±5 kHz). The modulation characteristics of the all-digital system are based upon the AM IBOC hybrid system.  
         [0025]     The all-digital IBOC signal includes a pair of primary partitions in the ±5 kHz region, a secondary partition in the −5 kHz to −10 kHz region, and a tertiary partition in the +5 kHz to +10 kHz region. The all-digital signal has no analog component, and all partitions are transmitted independently (that is, the partitions are not complementary).  
         [0026]      FIG. 3  is a functional block diagram of an IBOC receiver  84  constructed in accordance with this invention. The IBOC signal is received on antenna  86 . A bandpass preselect filter  88  passes the frequency band of interest, including the desired signal at frequency f c , but rejects the image signal at f c − 2 f if  (for a low side lobe injection local oscillator). Low noise amplifier  90  amplifies the signal. The amplified signal is mixed in mixer  92  with a local oscillator signal f lo  supplied on line  94  by a tunable local oscillator  96 . This creates sum (f c +f lo ) and difference (f c −f lo ) signals on line  98 . Intermediate frequency filter  100  passes the intermediate frequency signal f if  and attenuates frequencies outside of the bandwidth of the modulated signal of interest. An analog-to-digital converter  102  operates using a clock signal f s  to produce digital samples on line  104  at a rate f s . Digital down converter  106  frequency shifts, filters and decimates the signal to produce lower sample rate in-phase and quadrature signals on lines  108  and  110 . A digital signal processor based demodulator  112  then provides additional signal processing to produce an output signal on line  114  for output device  116 .  
         [0027]     The receiver in  FIG. 3  includes a modem constructed in accordance with this invention.  FIG. 4  is a functional block diagram of an AM HD Radio™ modem  130  showing the functional location of the carrier tracking of this invention. An input signal on line  132  from the digital down converter is subject to carrier tracking and automatic gain control as shown in block  134 . The resulting signal on line  136  is subjected to a symbol tracking algorithm  138  that produces the BPSK signal on lines  140  and  142 , symbol vectors (in the time domain) on line  144 , and the analog modulated carrier on line  146 . BPSK processing, as shown in block  148  produces block/frame sync and mode control information  150  that is used by functions illustrated in other blocks. An OFDM demodulator  152  demodulates the time domain symbol vectors to produce frequency domain symbol vectors on line  154 .  
         [0028]     The equalizer  156  processes the frequency domain symbol vectors in combination with the BPSK and carrier signals to produce equalized signals on line  158  and channel state information on line  160 . These signals are processed to produce branch metrics  162 , deinterleaved in a deinterleaver  164 , and mapped in a deframer  166  to produce soft decision bits on line  168 . A Viterbi decoder  170  processes the soft decision bits to produce decoded program data units on line  172 .  
         [0029]     This invention relates to the carrier tracking function in block  134  of  FIG. 4 .  FIG. 5  is a functional block diagram of AM carrier tracking frequency/phased-locked loop (FPLL). The input signal sig n , comprising a stream of time domain samples on line  182 , is mixed to dc in mixer  184  using an output signal NCO n  on line  186  from a frequency and phase-locked numerically controlled oscillator (NCO)  188  to produce a signal mult n  on line  190 . The mult signal is the mixed-to-dc signal.  
         [0030]     The subscript n indicates the n th  sample index at the input sample rate. The output sigdc n  on line  192  is obtained through multiplication of mult n , by a gain control signal sgain nDF  on line  194 , in multiplier  196 . The subscript nDF indicates the sample index after the Decimation Filter (decimate by  9 ).  
         [0031]     The closed-loop processing starts with the signal mult n  and generates the NCO n  value for the next input sample of sig n . The NCO is comprised of a phase accumulator  198  and a processor  200  for computing a complex phasor e −j·theta . The NCO input values of dtheta nDF  (in radians, initialized to zero) are accumulated to produce theta n  on line  202 . The FPLL output sigdc n  is obtained after mixing the input signal to dc, then multiplying by a gain control value sgain n  (computed later in the loop, initialized to zero), which attempts to maintain the main carrier at unity magnitude. The process can be summarized as: 
        “Compute first signal variables using input signal sig, and output samples sigdc”    theta(n)=theta(n−1)+dtheta(n−1); “dtheta, theta init to  0  for first iteration”    NCO(n)=exp{−j·theta(n)}; “compute conjugate phasor value”    mult(n)=sig(n)·NCO(n); “input signal shifted to dc”    sigdc(n)=mult(n)·sgain; “compute output samples with agc (carrier mag=1)”.        
 
         [0037]     The signal mult n  is filtered and decimated by a factor of 9 (decimated sample rate approx 5168 Hz=44100*15/128) as shown in block  204 , using the 45-tap FIR filter LPFl, producing filtered samples multfilt nDF  on line  206 . Filter LPFl limits the bandwidth of multfilt nDF  to approximately ±2 kHz. This reduces the effects of interference and permits some subsequent operations to operate at the decimated rate using decimated index nDF, instead of n. The signal multfilt is input to a frequency detector  208 , a phase detector  210 , and the gain control function  212  to compute cmaghold ns  and sgain nDF . The index nS implies samples at the symbol rate, which is decimated by a factor of  30  from the first Decimation Filter rate nDF. The decimation filter coefficients can be computed as follows:  
                 ``     ⁢     Compute   ⁢           ⁢   and   ⁢           ⁢   prestore   ⁢           ⁢   coefficients   ⁢           ⁢   of   ⁢           ⁢   decimatioin   ⁢           ⁢   filter   ⁢           ⁢   LPF   ⁢           ⁢     1   ″           
         LPF   ⁢           ⁢   1   ⁢     x   ⁡     (   k   )         =     {                     1   ;             if   ⁢             ⁢             ⁢   k     =   22                                 sin   ⁡     [         2   ·   π     23     ·     (     k   -   22     )       ]             2   ·   π     23     ·     (     k   -   22     )         ·       sin   ⁡     [       π   50     ·     (     k   +   3     )       ]       2       ;             k   =     0   ⁢           ⁢   …   ⁢           ⁢   44       ,     k   ≠   22                   ⁢     
     ⁢             LPF   ⁢           ⁢   1   ⁢     (   k   )       =       LPF   ⁢           ⁢   1   ⁢     x   ⁡     (   k   )             ∑     kk   =   0     44     ⁢     LPF   ⁢           ⁢   1   ⁢     x   ⁡     (   kk   )                     ;     k   =     0   ⁢           ⁢   …   ⁢           ⁢       44   ⁢             ``     ⁢   normalize   ⁢           ⁢   for   ⁢           ⁢   unity   ⁢           ⁢   d   ⁢           ⁢   c   ⁢           ⁢     gain   ″                 ⁢         
     ⁢             ``     ⁢   Compute   ⁢           ⁢   output   ⁢           ⁢   of   ⁢           ⁢   LPF   ⁢           ⁢   1     ,           decimate   ⁢           ⁢   by   ⁢           ⁢     9   ″     ⁢     
     ⁢           ⁢     multfilt   ⁡     (   nDF   )         =       ∑     k   =   0     44     ⁢         mult   ⁡     (       9   ·   nDF     +   k     )       ·   LPF     ⁢           ⁢   1   ⁢     (   k   )           ;     ⁢         
     ⁢             ``     ⁢           ⁢   apply   ⁢           ⁢     L   ⁢   PF     ⁢           ⁢   1     ,     decimate   ⁢           ⁢   by   ⁢           ⁢       9   ″     .               
 
         [0038]     The phase detector estimates the sample phase error in radians, while the frequency detector estimates the phase difference in radians between each pair of LPFl decimated samples multfilt nDF . The phase or frequency estimates on lines  214  and  216 , of the complex samples rely on small angles being approximated by the imaginary component divided by its magnitude and passed to a loop filter  218 . The value cmaghold ns  is used instead of the instantaneous sample magnitude to allow the detectors to “flywheel” through signal fades where the magnitude is typically small with large phase noise. The phase and frequency detector estimates are computed at the decimated sample rate as follows:  
               ⁢       phdet   ⁡     (   nDF   )       =     min   ⁡     [     1   ,     max   ⁡     [       -   1     ,       Im   ⁢     {     multfilt   ⁡     (   nDF   )       }         cmaghold   ⁡     (   nDF   )           ]         ]             
           fdet   ⁡     (   nDF   )       =     min   ⁡     [     1   ,     max   ⁡     [       -   1     ,       Im   ⁢     {       multfilt   ⁡     (   nDF   )       ·       multfilt   *     ⁡     (     nDF   -   1     )         }           cmaghold   ⁡     (   nDF   )       2         ]         ]         ;       
               ⁢         where   ⁢             *     ⁢           ⁢   is   ⁢           ⁢   the   ⁢           ⁢   complex   ⁢           ⁢     conjugate   .           
 
         [0039]     The gain control variables symbolmag, cmag, and cmaghold are updated at the symbol rate, and are derived from the decimated filter output groups of 30 samples. The variable cmaghold ns  represents an average magnitude of a previous symbol and is used in the phase and frequency estimators. The variable sgain nDF  is updated at the decimated rate (index nDF) and used to scale the level of the output signal. The automatic gain control (agc) action of sgain nDF  is used to keep the main carrier at a magnitude of one. The gain control variables in this example are computed as:  
                 symbolmag   =       1   30     ·       ∑     nDF   =   0     29     ⁢          multfilt   ⁡     (   nDF   )                  ;     ⁢                                 ``     ⁢     updated   ⁢           ⁢   at   ⁢           ⁢   symbol   ⁢           ⁢     rate   ″                     cmag   =         3   4     ·   cmag     +       1   4     ·   symolmag         ;                         ``     ⁢     updated   ⁢           ⁢   at   ⁢           ⁢   symbol   ⁢             ⁢             ⁢     rate   ″                     cmaghold   =     max   ⁡     (       cmaghold   ·     255   256       ,   cmag   ,     10     -   6         )         ;                         ``     ⁢     updated   ⁢           ⁢   at   ⁢           ⁢   symbol   ⁢           ⁢     rate   ″                       sgain   ⁡     (   nDF   )       =         255   256     ·     sgain   ⁡     (   nDF   )         +     1     256   ·   cmaghold           ;                         ``     ⁢     updated   ⁢           ⁢   at   ⁢           ⁢   decimated   ⁢           ⁢       rate   ″     .                 
 
         [0040]     The carrier tracking of this invention uses an adaptive third order frequency/phase-locked loop (FPLL). The gain of the loop is adaptive in such a manner as to maintain a nearly constant damping factor over the entire range of operation from initial frequency acquisition through narrowband tracking. This feature ensures closed-loop stability while continuously maximizing tracking performance without excessive overshoot. The frequency detector is effective during acquisition to quickly bring the FPLL within phase acquisition range. Initial acquisition can acquire an initial frequency offset up to at least 2000 Hz.  
         [0041]     Carrier tracking is implemented using an algorithm designed to operate independently of symbol synchronization, as the operations are performed on a sample-by-sample basis (270 samples/symbol) at a sample rate of approximately 46,512 Hz (44100*135/128). This obviates the need for external synchronization (e.g., symbol clock, or FFT lock conditions, etc.).  
         [0042]     Details of the loop filter  218  are shown in  FIG. 6 . The loop filter adaptively adjusts its gain in a manner that maintains a nearly constant damping factor around unity, as shown in block  220 . This gain is a function of input phase bias or frequency error, and acts to facilitate fast acquisition and slow tracking when sufficiently phase-locked. The adaptive loop gain (g) is derived as a function of the internally estimated phase error bias magnitude (biasmag), which provides an appropriate metric for controlling the gain. Details and analyses of the loop filter parameters are discussed below. The inputs are the estimated frequency and phase detector values. The output on line  222  is the phase increment dtheta nDF  for the NCO. All functions are performed at the decimated rate (index nDF). All variables are initialized to zero, except biasmag nDF  is initialized to  1  after the first symbol output value of cmaghold nDF  is computed.  
         [0043]      FIG. 6  is a functional block diagram of the adaptive loop filter (using a floating point implementation). A novel phase smoothing feature produces a cancellation signal dphi nDF  (on line  224 ) used to reduce the phase noise present in the output dtheta nDF . This phase noise reduction technique also produces another variable phi nDF  (on line  226 ) used to stabilize the loop. In effect, the loop stability operates as if the noise reduction filter were not present since the noise cancelled from the dtheta nDF  output is reinserted into the loop at a point after it has no significant effect on dtheta nDF . This additional stabilization (nearly) cancels the additional loop filtering effects resulting from dphi nDF  cancellation.  
         [0044]     The adaptive loop gain parameter g is computed as a function of the frequency and phase detector values. A phase error bias is first estimated to determine if the loop is in acquisition mode (large biasmag nDF ) or tracking (small biasmag nDF ). The computed value of the phase error phdet nDF  is adjusted by phi nDF  for stability compensation (as illustrated by summation point  228 ), and is used in the following algorithm to compute the adaptive value of  
                       ``     ⁢     Compute   ⁢           ⁢   adaptive   ⁢           ⁢   loop   ⁢           ⁢   gain   ⁢           ⁢   parameter   ⁢           ⁢     g   ″                       phdetphi   ⁡     (   nDF   )       =       phdet   ⁡     (   nDF   )       -     phi   ⁡     (     nDF   -   1     )           ;                         ``     ⁢     stability   ⁢           ⁢   compensation   ⁢           ⁢   for   ⁢           ⁢     dphi   ″                     bias   =         63   ·   bias     +     fdet   ·   g     +   phdetphi     64       ;                         ``     ⁢     phase   ⁢           ⁢   bias   ⁢           ⁢   estimate   ⁢           ⁢   after   ⁢           ⁢     filtering   ″                     biasmag   =         63   ·   biasmag     +        bias          64       ;                         ``     ⁢     filter   ⁢             ⁢             ⁢   bias   ⁢           ⁢     magnitude   ″                     g   =     min   ⁡     (     1   ,     max   ⁡     (       2   ·   biasmag     ,     1   32       )         )         ;                         ``     ⁢       compute   ⁢           ⁢     1   /   32       &lt;=   g   &lt;=       1   ″     .                 
 
         [0045]     The remaining loop filter parameters are computed next. The loop filter computes the first and second filter outputs filt 1   nDF  and filt 2   nDF , on lines  230  and  232 . These signals are used to compute the next value of dtheta nDF , which determines the phase increment for the NCO samples, using the following algorithm.  
                       ``     ⁢       Compute   ⁢           ⁢   loop   ⁢           ⁢   output   ⁢           ⁢   dtheta     ,     decimated   ⁢           ⁢   sample   ⁢           ⁢   rate   ⁢           ⁢     nDF   ″                         filt   ⁢           ⁢   1     =       g   64     ·   phdet       ;                         ``     ⁢     first   ⁢     -     ⁢   order   ⁢           ⁢   loop   ⁢           ⁢     parameter   ″                     phiprev   =   phi     ;                         ``     ⁢     save   ⁢           ⁢   prev   ⁢           ⁢   value   ⁢           ⁢   for   ⁢           ⁢   differential   ⁢           ⁢     dphi   ″                     phi   =     max   ⁡     (         -   1     512     ,     min   ⁡     (       1   512     ,       filt   ⁢           ⁢   1     +     phi   ·     127   128           )         )         ;                         ``     ⁢     stability   ⁢           ⁢   compensation   ⁢           ⁢   for   ⁢           ⁢   small   ⁢           ⁢     angles   ″                     dphi   =     phiprev   -   phi       ;                         ``     ⁢     phase   ⁢           ⁢   noise   ⁢           ⁢   estimate   ⁢           ⁢       (   negative   )     ″                       filt   ⁢           ⁢   2     =     max   ⁡     (         -   5     16     ,     min   ⁡     (       5   16     ,         g   16     ·     [           g   2     32     ·   fdet     +     filt   ⁢           ⁢   1       ]       +     filt   ⁢           ⁢   2         )         )         ;                         ``     ⁢     second   ⁢     -     ⁢   order   ⁢           ⁢   loop   ⁢           ⁢     filter   ″                                   ``     ⁢       Compute   ⁢           ⁢   dtheta   ⁢           ⁢   from   ⁢           ⁢   loop   ⁢           ⁢   filter     ,                   with   +       -   1531     ⁢           ⁢   Hz   ⁢           ⁢   limit       ,               &amp;           ⁢   phase   ⁢           ⁢   noise   ⁢           ⁢   cancellation   ⁢           ⁢     dphi   ″                       dtheta   =     max   ⁡     (         -   5     16     ,     min   ⁡     (       5   16     ,       filt   ⁢           ⁢   1     +     filt   ⁢           ⁢   2     +   dphi       )         )                             ``     ⁢     phase   ⁢           ⁢   increment   ⁢           ⁢   for   ⁢           ⁢       NCO   ″     .                 
 
         [0046]     The derivation of the loop parameters is described next. The stability, damping factor and other performance parameters of the PLL are most conveniently analyzed in steady state operation using an ideal linear model approximation of the PLL. The linear model allows conventional servo control theory analysis techniques to determine appropriate design parameters, particularly for the loop filter, to control stability and performance in operation. This model, shown in  FIG. 7 , describes frequency in units of radians/sec, and signal values in volts.  
         [0047]      FIG. 7  is a simplified linear model of a PLL useful in determining gain values. A goal of the analysis is to determine the appropriate values for gain parameters a and b. For this analysis, however, we will start with derived values of a and b determined in a full analysis, and then characterize the resulting PLL performance with these assumed values. A brief summary of the PLL linear model analysis follows.  
         [0048]     Referring to  FIG. 7 , the phase detector  250 , including an amplifier  252  having a gain Kd, is designed with a gain of Kd=1 volt/rad. The NCO  254  is designed for a gain of Ko=fs rad/sec-volts. The two factors Kd and Ko can be conveniently expressed as one parameter K=fs.  
         [0049]     The closed-loop transfer function H(s) of the linear model of the PLL can be used to assess the performance and stability. The transfer function is best described using Laplace Transform techniques. That is,  
         H   ⁡     (   s   )       =       K   ·     F   ⁡     (   s   )           s   +     K   ·     F   ⁡     (   s   )                 
 
 where F(s) is the embedded loop filter transfer function. An ideal second-order loop filter has a transfer function  
         F   ⁡     (   s   )       =     -       (       1       s   ·   C   ·   R     ⁢           ⁢   1       +       R   ⁢           ⁢   2       R   ⁢           ⁢   1         )     .           
 
         [0050]     Conventional analysis of the loop filter describes important characteristics of the PLL in terms of time constants π1 and π2. These time constants refer to properties of an integrator and gain components of a loop filter implemented with RC components used in an ideal second-order PLL. The relationships between these time constants and their digital equivalents are illustrated in  FIGS. 8 and 9 . It is assumed that the sample rate of the integrator in the loop filter is fsd, which is decimated by a factor of 9 relative to the FPLL signal input/output sample rate fs.  
         [0051]      FIG. 8  is a schematic diagram that illustrates analog RC time constants. In  FIG. 8 , the time constants are:
 π1 =R 1 ·C , and π2 =R 2 ·C.   
         [0052]      FIG. 9  is a schematic diagram that illustrates equivalent digital time constants. In  FIG. 9 , the time constants are:  
         τ1   =     1     b   ·   fsd         ,       and   ⁢           ⁢   τ2     =       a     b   ·   fsd       .             
         [0053]     The resulting transfer function for the PLL can now be rewritten as  
         H   ⁡     (   s   )       =         K   ·     F   ⁡     (   s   )           s   +     K   ·     F   ⁡     (   s   )             =         K   ·       (       s   ·   τ2     +   1     )     /   τ1           s   2     +     s   ·     (     K   ·     τ2   /   τ1       )       +     K   /   τ1         .           
 
 Furthermore the transfer function can be described in servo terminology as  
         H   ⁡     (   s   )       =         2   ·   ζ   ·     ω   n     ·   s     +     ω   n   2           s   2     +     2   ·   ζ   ·     ω   n     ·   s     +     ω   n   2             
 
 where ω n , is the natural frequency and ζ is the damping factor of the PLL, and  
         ω   n     =         K     τ   ⁢           ⁢   1         =           K   ·   b   ·   fsd       ⁢           ⁢   and   ⁢           ⁢   ζ     =         τ   ⁢           ⁢     2   ·     ω   n         2     =         a   ·     ω   n         2   ·   b   ·   fsd       .               
 
         [0054]     The analysis, design and simulation performance of the PLL suggests a desired value of a=g/64, and b=g 2 /1024. These values are chosen as a function of a controlled loop gain parameter g that allows the loop to acquire quickly, then track smoothly. The square of g is used in the second-order filter in order to maintain a constant damping factor as g varies. This relationship should become clear when we examine the expression for the damping factor. The PLL natural frequency can be computed as  
         ω   n     =         fs   ·   b   ·   fsd       =         g   ·   fs       32   ·   3       ≅       484   ·   g     ⁢           ⁢   rad   ⁢     /     ⁢   sec     ≅       77   ·   g     ⁢           ⁢     Hz   .               
 
 The resulting damping factor is then  
       ζ   =         a   ·     ω   n         2   ·   b   ·   fsd       =         3   ·   a       2   ·     b         =     0.75   .             
 
         [0055]     This damping factor is set slightly above critical damping (0.7071), which results in fast acquisition times with minimal overshoot. It is particularly important to notice that the damping factor is independent of the controlled loop gain value g. This is a result of using g as the multiplying factor for gain a, while g 2  is used as the multiplying factor for gain b. The square root in the denominator of the damping factor expression allows the variable g to cancel. This feature allows the PLL to operate consistently and converge quickly over a wide range of adaptive gain control. The addition of the frequency detection on the PLL is effective only during initial frequency acquisition.  
         [0056]     Performance results of a simulation of the invention are shown in  FIGS. 10-13 .  FIG. 10  shows the calculated phase error.  FIG. 11  shows the carrier magnitude estimate.  FIG. 12  shows the frequency error.  FIG. 13  shows the adaptive gain g.  
         [0057]     Previously existing AM carrier tracking algorithms are complex and involve various modes of operation. Initial analyses, simulation performance, and field testing of the present invention indicate that the performance of this invention is as good or better with various channel impairments and outages. In addition, the final design can be adjusted by the parameter settings if needed.  
         [0058]     The functions shown in the drawings can be implemented using known circuit components, including but not limited to, one or more processors or application specific integrated circuits.  
         [0059]     While the invention has been described in terms of several examples, it will be apparent to those skilled in the art that various changes can be made to the described examples without departing from the scope of the invention as set forth in the following claims.