Patent Publication Number: US-7720179-B2

Title: Method for timing detection

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of U.S. Provisional Application No. 60/685,152, filed on May 27, 2005. The disclosure of the above application is incorporated herein by reference in its entirety. 

   FIELD OF THE INVENTION 
   The present invention relates to communication systems, and more particularly to systems and methods for demodulating data in wireless communication systems. 
   BACKGROUND OF THE INVENTION 
   Personal Handy-phone System (PHS) is a mobile telephone system that operates in the 1.88-1.93 GHz frequency band. PHS is a cordless telephone system with capability to handover signals from one cell to another. PHS cells are smaller than cells of cellular phone systems that use Global System for Mobile communication (GSM). 
   Typically, PHS has a transmission power of 500 mW and a range of 10-100 meters. PHS provides service with minimal congestion in areas of heavy call-traffic such as business districts, downtown, etc. This is accomplished by installing cell stations at a radial distance of every 100-200 meters. Thus, PHS is particularly suitable for use in urban areas. 
   PHS-based phones can be used in homes, offices, and outdoors. PHS offers a cost-effective alternative to conventional phone systems that use ground lines. Additionally, PHS-based phones can interface with conventional phone systems. Thus, where ground lines of conventional phone systems cannot reach a physical location of a subscriber, the subscriber can use PHS to establish communication with the conventional phone system and reach other subscribers served by the conventional phone system. 
   PHS uses Time division multiple access (TDMA) as radio interface and adaptive differential pulse code modulation (ADPCM) as voice coder-decoder (codec). A codec includes an analog-to-digital converter (ADC) and a digital-to-analog converter (DAC) that translate signals between analog and digital formats. 
   TDMA is a digital signal transmission scheme that allows multiple users to access a single radio-frequency (RF) channel. Interference between channels is avoided by allocating unique time slots to each user within each channel. For example, a PHS frame comprises four channels: one control channel and three traffic channels. Each channel is divided into two time slots. The control channel assigns each caller one time slot for uplink or transmission and one time slot for downlink or reception. 
   Unlike PCM codecs that quantize speech signals directly, ADPCM codecs quantize a difference between a speech signal and a prediction made of the speech signal. If the prediction is accurate, the difference between actual and predicted speech may have a variance that is lower than the variance in actual speech. Additionally, the difference may be accurately quantized with fewer bits than the number of bits that would be needed to quantize the actual speech. While decoding, a quantized difference signal is added to a predicted signal to reconstruct an original speech signal. The performance of the codec is aided by using adaptive prediction and quantization so that a predictor and a difference quantizer adapt to changing characteristics of speech being coded. 
   Referring now to  FIG. 1 , an exemplary PHS phone  10  comprises an antenna  12 , a signal processing module  16  comprising a transmit module  18  and a receive module  20 , memory  22 , a power supply  24 , and an I/O module  26 . The I/O module  26  may comprise various user-interfaces such as a microphone  26 - 1 , a speaker  26 - 2 , a display  26 - 3 , a keypad  26 - 4 , a camera  26 - 5 , etc. 
   The transmit module  18  converts user input from the microphone  26 - 1 , the camera  26 - 5 , etc., into PHS-compatible signals. The receive module  20  converts data received from the antenna  12  into a user-recognizable format and outputs the same via speaker  26 - 2 , camera  26 - 5 , etc. The signal processing module  16  uses memory  22  to process data transmitted to and received from the antenna  12 . The power supply  24  provides power to the phone  10 . 
   Digital data is typically represented by zeros and ones, which are called bits. Data is generally transmitted by modulating amplitude, frequency, or phase of a carrier signal with a baseband information-bearing signal. Quadrature phase shift keying (QPSK) is a form of phase modulation generally used in communication systems. In QPSK, information bits are grouped in pairs called dibits. Thus, QPSK uses four symbols that represent dibit values 00, 01, 10, and 11. QPSK maps the four symbols to four fixed phase angles. For example, symbol 00 may be mapped to (+3π/4). On the other hand, π/4-DQPSK uses differential encoding wherein mapping between symbols and phase angle varies. Additionally, π/4-DQPSK maps each of the four symbols to a real and an imaginary phase angle resulting in an eight-point constellation. 
   Referring now to  FIGS. 2A-2B , the transmit module  18  comprises an ADPCM module  50 , a framer module  52 , a serial-to-parallel converter module  54 , a DQPSK mapper module  56 , a square-root raised cosine (SRRC) filter module  58 , and an upsample module  60 . The receive module  20  comprises a downsample module  70 , an automatic gain control (AGC) module  72 , a demodulator  75  comprising a carrier acquisition module  74  and an equalization module  76 , a de-mapper and parallel-to-serial converter module  78 , a de-framer module  80 , and an ADPCM module  82 . 
   When transmitting data from the phone  10  on a channel, the ADPCM module  50  converts audio and/or video signal into bits of digital data. The framer module  52  partitions the digital data into frames. The serial-to-parallel converter module  54  converts the bits in the frames into symbols. The DQPSK mapper module  56 , which may utilize a modulation scheme such as π/4-DQPSK modulation, maps four real and four imaginary values of four symbols in each frame to a total of eight phase angles and generates a complex baseband signal. 
   The SRRC filter module  58 , which is essentially a Nyquist pulse-shaping filter, limits the bandwidth of the signal. Additionally, the SRRC filter module  58  removes mixer products from the complex baseband signal. The upsample module  60  comprises a quadrature carrier oscillator that is used to convert the phase-modulated baseband signal into a phase-modulated carrier signal. The upsample module  60  transmits the phase-modulated carrier signal on the channel at a sampling frequency that is greater than twice the Nyquist frequency. 
   When the phone  10  receives a signal from the antenna  12 , the downsample module  70  downsamples the signal using an asynchronous oscillator. The downsample module  70  down-converts the signal from the phase-modulated carrier signal to the phase modulated baseband signal. The AGC module  72  maintains the gain of the signal relatively constant despite variation in input signal strength due to transmission losses, noise, interference, etc. 
   The carrier acquisition module  74  demodulates the signal, retrieves carrier phase information, and decodes symbol values from the signal. The equalization module  76  corrects any distortion present in the signal. The de-mapper and parallel-to-serial converter module  78  de-maps and converts the demodulated signal into a serial bit-stream. The de-framer module  80  de-partitions the frames into digital data bits. The ADPCM module  82  converts the digital data bits into audio and/or video data and outputs the data to the speaker  26 - 2  and/or the display  26 - 3  of the phone  10 . 
   SUMMARY OF THE INVENTION 
   A wireless communication system comprises a sampling module that samples a first portion and a second portion of a channel using a sampling time. A correlator module selectively correlates the first portion and the second portion, generates correlation samples, and calculates an offset based on the correlation samples. A timing module selectively adjusts the sampling time based on the offset. 
   In other features, the first portion is a preamble (PR) and the second portion is a unique word (UW). The correlator module selectively correlates the first portion and the second portion when the channel is a traffic channel. The timing module adjusts the sampling time based on the offset when the channel is a traffic channel. The timing module adjusts the sampling time based on the first portion when the channel is a control channel. 
   In other features, the correlation module generates a correlation waveform using the correlation samples, determines a peak of the correlation waveform, and calculates the offset based on a distance of the peak from one of the correlation samples. 
   In other features, the correlation module generates a correlation waveform using the correlation samples and determines a peak of the correlation waveform using parabolic curve fitting, and calculates the offset based on a distance of the peak from one of the correlation samples. 
   In other features, the correlator module correlates the first portion and the second portion for each time slot of the channel. The correlator module correlates using training symbols in the channel. The channel comprises a plurality of time slots and wherein the timing module calculates the offset for each of the time slots. The correlator module estimates a location of the second portion in a traffic channel based on the sampling time of a control channel and based on a burst detected in the control channel. 
   In other features, a burst detector module communicates with the sampling module, detects a burst in a control channel, and generates a burst-detect signal when a moving average of one of N phases is less than a predetermined threshold. The one of N phases is an array of Nth sample of every symbol in a signal sampled at a sampling rate of N samples per symbol, and N is an integer greater than 1. The burst-detect signal activates the timing module. A carrier offset estimator module communicates with the burst detector module and generates a carrier offset for a control channel when activated by the burst-detect signal. 
   In other features, a carrier recovery module communicates with the sampling module and uses a phase-locked loop (PLL) to recover a carrier signal from a sample output by the sampling module. The PLL is initialized with the carrier offset and wherein the carrier offset is update when the PLL locks for each time slot in the channel. A differential decoder module communicates with the carrier recovery module and decodes a symbol from the sample. The sampling module uses one of a cubic interpolator and a parabolic interpolator. An output sample rate of the sampling module is equal to a symbol rate and wherein an input sample rate of the sampling module is an integer multiple of the symbol rate. A personal handy-phone system (PHS) receiver comprises the wireless communication system. 
   A wireless communication system comprises sampling means for sampling a first portion and a second portion of a channel using a sampling time. Correlator means selectively correlates the first portion and the second portion, generates correlation samples, and calculates an offset based on the correlation samples. Timing means selectively adjusts the sampling time based on the offset. 
   In other features, the first portion is a preamble (PR) and the second portion is a unique word (UW). The correlator means selectively correlates the first portion and the second portion when the channel is a traffic channel. The timing means adjusts the sampling time based on the offset when the channel is a traffic channel. The timing means adjusts the sampling time based on the first portion when the channel is a control channel. The correlation means generates a correlation waveform using the correlation samples, determines a peak of the correlation waveform, and calculates the offset based on a distance of the peak from one of the correlation samples. The correlation means generates a correlation waveform using the correlation samples and determines a peak of the correlation waveform using parabolic curve fitting, and calculates the offset based on a distance of the peak from one of the correlation samples. 
   In other features, the correlator means correlates the first portion and the second portion for each time slot of the channel. The correlator means correlates using training symbols in the channel. The channel comprises a plurality of time slots and wherein the timing means calculates the offset for each of the time slots. 
   In yet other features, the correlator means estimates a location of the second portion in a traffic channel based on the sampling time of a control channel and based on a burst detected in the control channel. Burst detector means communicates with the sampling means, detects a burst in a control channel, and generates a burst-detect signal when a moving average of one of N phases is less than a predetermined threshold, wherein the one of N phases is an array of Nth sample of every symbol in a signal sampled at a sampling rate of N samples per symbol, and N is an integer greater than 1. The burst-detect signal activates the timing means. Carrier offset estimator means communicates with the burst detector means and generates a carrier offset for a control channel when activated by the burst-detect signal. Carrier recovery means communicates with the sampling means and uses phase-locked loop (PLL) means for recovering a carrier signal from a sample output by the sampling means. The PLL is initialized with the carrier offset. The carrier offset is updated when the PLL locks for each time slot in the channel. 
   In other features, differential decoder means communicates with the carrier recovery means and decodes a symbol from the sample. The sampling means uses one of a cubic interpolator and a parabolic interpolator. An output sample rate of the sampling means is equal to a symbol rate and wherein an input sample rate of the sampling means is an integer multiple of the symbol rate. A personal handy-phone system (PHS) receiver comprising the wireless communication system. 
   A computer program executed by a processor for operating wireless communication system comprises sampling a first portion and a second portion of a channel using a sampling time; selectively correlating the first portion and the second portion; generating correlation samples; calculating an offset based on the correlation samples; selectively adjusting the sampling time based on the offset. 
   In other features, the first portion is a preamble (PR) and the second portion is a unique word (UW). The computer program selectively correlates the first portion and the second portion when the channel is a traffic channel. The computer program includes adjusting the sampling time based on the offset when the channel is a traffic channel. The computer program includes adjusting the sampling time based on the first portion when the channel is a control channel. The computer program includes generating a correlation waveform using the correlation samples; determining a peak of the correlation waveform; and calculating the offset based on a distance of the peak from one of the correlation samples. 
   In other features, the computer program includes generating a correlation waveform using the correlation samples; determining a peak of the correlation waveform using parabolic curve fitting, and calculating the offset based on a distance of the peak from one of the correlation samples. The computer program includes correlating the first portion and the second portion for each time slot of the channel. The computer program includes correlating using training symbols in the channel. 
   In other features, the channel comprises a plurality of time slots and further comprising calculating the offset for each of the time slots. The computer program includes estimating a location of the second portion in a traffic channel based on the sampling time of a control channel and based on a burst detected in the control channel. The computer program includes detecting a burst in a control channel; and generating a burst-detect signal when a moving average of one of N phases is less than a predetermined threshold. The one of N phases is an array of Nth sample of every symbol in a signal sampled at a sampling rate of N samples per symbol, and N is an integer greater than 1. 
   In other features, the computer program includes adjusting the sampling time when the burst-detect signal occurs. The computer program includes generating a carrier offset for a control channel when the burst-detect signal occurs. The computer program includes using a phase-locked loop to recover a carrier signal from a sample output. The computer program includes initializing the PLL with the carrier offset; and updating the carrier offset when the PLL locks for each time slot in the channel. 
   In other features, the computer program includes decoding a symbol from the sample. The computer program includes using one of a cubic interpolator and a parabolic interpolator. The computer program includes setting an output sample rate equal to a symbol rate; and setting an input sample rate equal to an integer multiple of the symbol rate. 
   In still other features, the systems and methods described above are implemented by a computer program executed by one or more processors. The computer program can reside on a computer readable medium such as but not limited to memory, non-volatile data storage and/or other suitable tangible storage mediums. 
   A computer method comprises sampling a first portion and a second portion of a channel using a sampling time; selectively correlating the first portion and the second portion; generating correlation samples; calculating an offset based on the correlation samples; selectively adjusting the sampling time based on the offset. 
   In other features, the first portion is a preamble (PR) and the second portion is a unique word (UW). The computer method selectively correlates the first portion and the second portion when the channel is a traffic channel. The computer method includes adjusting the sampling time based on the offset when the channel is a traffic channel. The computer method includes adjusting the sampling time based on the first portion when the channel is a control channel. The computer method includes generating a correlation waveform using the correlation samples; determining a peak of the correlation waveform; and calculating the offset based on a distance of the peak from one of the correlation samples. 
   In other features, the computer method includes generating a correlation waveform using the correlation samples; determining a peak of the correlation waveform using parabolic curve fitting, and calculating the offset based on a distance of the peak from one of the correlation samples. The computer method includes correlating the first portion and the second portion for each time slot of the channel. The computer method includes correlating using training symbols in the channel. 
   In other features, the channel comprises a plurality of time slots and further comprising calculating the offset for each of the time slots. The computer method includes estimating a location of the second portion in a traffic channel based on the sampling time of a control channel and based on a burst detected in the control channel. The computer method includes detecting a burst in a control channel; and generating a burst-detect signal when a moving average of one of N phases is less than a predetermined threshold. The one of N phases is an array of Nth sample of every symbol in a signal sampled at a sampling rate of N samples per symbol, and N is an integer greater than 1. 
   In other features, the computer method includes adjusting the sampling time when the burst-detect signal occurs. The computer method includes generating a carrier offset for a control channel when the burst-detect signal occurs. The computer method includes using a phase-locked loop to recover a carrier signal from a sample output. The computer method includes initializing the PLL with the carrier offset; and updating the carrier offset when the PLL locks for each time slot in the channel. 
   In other features, the computer method includes decoding a symbol from the sample. The computer method includes using one of a cubic interpolator and a parabolic interpolator. The computer method includes setting an output sample rate equal to a symbol rate; and setting an input sample rate equal to an integer multiple of the symbol rate. 
   In still other features, the systems and methods described above are implemented by a computer method executed by one or more processors. The computer method can reside on a computer readable medium such as but not limited to memory, non-volatile data storage and/or other suitable tangible storage mediums. 
   Further areas of applicability of the present invention will become apparent from the detailed description provided hereinafter. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will become more fully understood from the detailed description and the accompanying drawings, wherein: 
       FIG. 1  is a functional block diagram of an exemplary personal handy-phone system (PHS) phone according to the prior art; 
       FIG. 2A  is a functional block diagram of an exemplary transmitter used in a PHS phone of  FIG. 1  according to the prior art; 
       FIG. 2B  is a functional block diagram of an exemplary receiver used in a PHS phone of  FIG. 1  according to the prior art; 
       FIG. 3  is a functional block diagram of an exemplary demodulator used in a receiver of a PHS phone according to the present invention; 
       FIG. 4  is a functional block diagram of an exemplary burst detector used in the demodulator of  FIG. 3  according to the present invention; 
       FIG. 5  is a graph showing a correlation curve with parabolic curve-fitting according to the present invention; 
       FIG. 6  is a functional block diagram of an exemplary carrier recovery module having automatic frequency control (AFC) that is used in the demodulator of  FIG. 3  according to the present invention; 
       FIG. 7  is a pi-chart of exemplary simulation test results showing performance of the demodulator of  FIG. 3  according to the present invention; and 
       FIG. 8  is a flowchart for a method used by the demodulator of  FIG. 3  to demodulate symbols according to the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The following description of the preferred embodiment(s) is merely exemplary in nature and is in no way intended to limit the invention, its application, or uses. For purposes of clarity, the same reference numbers will be used in the drawings to identify similar elements. As used herein, the term module, circuit and/or device refers to an Application Specific Integrated Circuit (ASIC), an electronic circuit, a processor (shared, dedicated, or group) and memory that execute one or more software or firmware programs, a combinational logic circuit, and/or other suitable components that provide the described functionality. As used herein, the phrase at least one of A, B, and C should be construed to mean a logical (A or B or C), using a non-exclusive logical or. It should be understood that steps within a method may be executed in different order without altering the principles of the present invention. 
   The present disclosure is applicable to communications systems. For example, the present disclosure is applicable to wireless communications systems. The present disclosure is also applicable to time division multiple access (TDMA) systems. In the foregoing description, the present disclosure discusses a personal handy-phone system (PHS). However, the present disclosure is not meant to be limited to PHS or TDMA systems. 
   A receiver in a personal handy-phone system (PHS) phone estimates carrier phase (and frequency), recovers symbol timing, and estimates a most likely value of a received symbol. The receiver utilizes a symbol timing recovery scheme to estimate timing of symbols in a received signal. The receiver obtains correct values of symbols if the received signal is sampled at proper times. 
   A timing signal determines a time at which the received signal may be sampled to retrieve correct values of symbols. After the receiver identifies correct symbol timing, the receiver samples the received signal at the correct symbol timing. The receiver retrieves carrier and symbols from samples generated by sampling the received signal at the correct symbol timing and estimates most likely values of the symbols. Thereafter, the receiver converts the estimates into dibits. 
   Some symbol timing recovery schemes may use a phase difference between a current sample and a previous sample to determine symbol timing. If a phase estimate is wrong, however, the symbol values may be wrong since the receiver may be effectively using a different symbol mapping than a symbol mapping used by a transmitter while transmitting the symbols. 
   Some transmitters insert a fixed synchronization pattern into modulation when transmitting signals. The receiver searches for the pattern and correctly recovers symbols. This scheme, however, consumes bandwidth that can otherwise be used to carry data. Alternatively, the receiver can accurately recover symbols by using additional hardware. Adding hardware, however, may increase system cost and decrease system marketability. 
   In the present disclosure, a PHS receiver samples and recovers symbols from received signals with substantial accuracy by using a demodulation scheme that includes phase correlation, parabolic curve fitting, and interpolation using correct symbol timing. A signal received by the PHS receiver is downsampled at a sampling rate equal to three times a symbol rate. Thus, each symbol has three samples. The demodulation scheme identifies best of the three samples to demodulate. The best sample is decoded to obtain a correct value for each symbol. 
   Generally, a PHS signal comprises a series of time division multiple access (TDMA) frames. Each TDMA frame may be 5 milliseconds (mS) in duration with 2.5 mS for uplink or transmission and 2.5 mS for downlink or reception. Each TDMA frame may comprise four channels: one control channel and three traffic channels. Each channel has two time slots: one time slot for uplink and one time slot for downlink. Thus, each TDMA frame has a total of eight time slots. 
   The control channel uses a carrier frequency that is different from the carrier frequency (or frequencies) used by the traffic channels. The traffic channels can use same or different carrier frequencies. 
   Each TDMA frame comprises a preamble (PR) and a unique word (UW) for each channel. A UW includes identifying information for the PHS. The PHS uses UW as a security feature to authenticate access by subscribers to the PHS. 
   Typically, a TDMA signal is transmitted in bursts. A burst generally comprises an initial increase in amplitude from zero to a normal value followed by modulated data and a decrease in amplitude to zero. The burst is detected in the control channel, and bit timing is recovered for the control channel by utilizing a periodic pattern in the PR. The burst comprises a sequence of eight training symbols. The sequence is generally inserted in the middle of each time slot and is called a midamble. An equalizer in the PHS receiver uses the sequence to reduce inter-symbol interference. 
   Additionally, the demodulation scheme uses the training symbols for correlation in traffic channels. Based on the burst detection in the control channel and the bit timing of the control channel, an approximate position of the UW in traffic channels is determined. Substantially accurate bit timing for traffic channels is calculated by performing a phase correlation of PR and UW for each traffic channel. The phase correlation of PR and UW is performed using training symbols, and correlation samples are generated for each time slot. A correlation curve is generated using correlation samples for each time slot. The correlation curve is fitted onto a parabola, and a peak of the parabola is calculated. 
   The peak of the parabola approximately equals a peak of the correlation curve. The peak of the correlation curve corresponds to the best sample from which a symbol may be correctly decoded. An x-coordinate of the peak of the parabola corresponds to a best time to sample a symbol to get the correct value of the symbol. 
   Additionally, a distance between the peak of the parabola and an adjacent point on the correlation curve corresponds to a distance between the best sample and a sample adjacent to the best sample. A timing offset is calculated based on the distance between the peak of the parabola and the adjacent point on the correlation curve. A sampling time for sampling the symbols is adjusted by the timing offset. An interpolator, which is essentially a sampling module, uses adjusted sampling times to sample symbols at correct times. In other words, the demodulation scheme uses the peak of the parabola to estimate best times to sample subsequent symbols. Accordingly, best subsequent samples are demodulated and correct values of symbols are obtained therefrom. 
   Additionally, a carrier offset is estimated for the control channel, and a carrier phase is offset by an estimated carrier offset value. The estimated carrier offset value is calculated based on a first-order differentiation of a downsampled signal. Once carrier phase is offset and locked using automatic frequency control (AFC), subsequent samples are demodulated at correct times and are decoded to obtain substantially accurate values of the symbols. 
   For traffic channels, the AFC is initialized with the estimated carrier offset calculated for the control channel. A frequency step size of the AFC is decreased until the AFC locks. The estimated carrier offset is updated for each time slot. Samples are demodulated and decoded to obtain substantially accurate values of the symbols. 
   Referring now to  FIG. 3 , a demodulation system  75 - 1  of a PHS phone receiver comprises an arctangent module A  100 , an arctangent module B  100 - 1 , a single differentiator module  102 , a double differentiator module  104 , a burst detector module  106 , a timing module  108 , a UW correlator module  110 , a sampling module  112 , an offset estimator module  114 , a carrier recovery module  117 , and a differential decoder module  120 . 
   The demodulation system  75 - 1  receives a signal that is downsampled at a rate of three samples per symbol (3f b ). An array of first samples of every symbol is called phase  1 . An array of second samples for every symbol is called phase  2 . An array of third samples of every symbol is called phase  3 . The arctangent module A  100  receives the signal at the 3f b  rate. The arctangent module A  100  recovers a phase angle of the signal based on in-phase (I) and quadrature (Q) components of the signal. The arctangent module A  100  outputs the phase angle information for phase  1 , phase  2 , and phase  3  (collectively phases i) to the single differentiator module  102 . This may be represented by arctangentOut(i−3), where i represents one of phases i. 
   The sampling module  112  interpolates the three samples of every symbol based on a sampling signal generated by the timing module  108 . The sampling module  112  outputs one sample called best sample per symbol. Thus, output of the sampling module  112  is at the symbol rate (f b ). 
   The arctangent module B  100 - 1  receives the output of the sampling module  112 . The arctangent module B  100 - 1  recovers a phase angle of the output of the sampling module  112  and outputs the phase angle information to the single differentiator module  102  and the carrier recovery module  117 . The output of the arctangent module B  100 - 1  may be represented by arctangentOut(i). 
   The single differentiator module  102  and the double differentiator module  104  perform differentiation for each phase separately. For example, singleDiff (i)=arctangentOut (i)−arctangentOut (i−3). An output of the double differentiator module  104  is input to the burst detector module  106 . 
   The burst detector module  106  detects a burst in the control channel. The burst detector module  106  adds two contiguous outputs of the double differentiator module  104  for phase i and calculates an absolute value of a sum of the two contiguous outputs. Moving averages of absolute values for each one of phases i are calculated and compared to a burst threshold ThB. The burst detector module  106  detects the burst if a moving average of any one of phases i is less than ThB. 
   The burst detector module  106  generates a burst-detect signal that enables the timing module  108  and the offset estimator module  114 . The timing module  108  utilizes a periodic pattern in the preamble (PR) in the control channel instead of performing a phase correlation to recover bit timing. The timing module  108  determines bit timing and generates a correct sampling time for the control channel based on the PR in the control channel. 
   Burst detection is not performed in traffic channels because the PR in traffic channels is not as long as the PR in the control channel. Additionally, the timing module  108  cannot recover bit timing for traffic channels based on PR alone since the PR in traffic channels is not of sufficient length. Therefore, approximate bit timing for traffic channels is initially estimated based on the burst detection and the bit timing of the control channel. 
   An approximate position of the UW in traffic channels is determined based on the burst detection in the control channel and bit timing of the control channel. Thereafter, a phase correlation of PR and UW is performed for each traffic channel and a timing offset calculated. Substantially correct bit timing and sampling time for traffic channels are obtained by adjusting the approximate bit timing and sampling times of the control channel by the timing offset. 
   Specifically, the UW correlator module  110  performs the phase correlation of PR and UW for each traffic channel and generates correlation samples. The UW correlator module  110  generates a correlation curve using the correlation samples for each time slot of each traffic channel in a frame. The UW correlator module  110  performs a parabolic curve-fitting. That is, the correlation curve is fitted onto a parabola and a peak of the parabola is calculated. The peak of the parabola approximately corresponds to a peak of the correlation curve. 
   The UW correlator module  110  calculates the timing offset based on a distance between the peak of the parabola and an adjacent correlation sample on the correlation curve. The timing module  108  adjusts the bit timing and the sampling times of the traffic channels by the timing offset. 
   The sampling module  112  samples symbols in best of the three samples at sampling times generated by the timing module  108 . The sampling module  112  essentially interpolates three samples comprising one symbol and samples the interpolated data at the correct sampling time generated by the timing module  108 . The sampling module  112  effectively generates one sample called the best sample from the three samples, which yields a correct value of a symbol when decoded. 
   Thus, an output sample rate of the sampling module  112  is equal to the symbol rate. The symbol rate is determined by the number of symbols used in modulation and may be expressed as number of symbols per second. Additionally, by using the correct sampling times, the sampling module  112  can estimate substantially correct sampling times to sample subsequent symbols in the frame. 
   The offset estimator module  114  estimates a carrier offset for the control channel based on an output of the single differentiator module  102 . The offset estimator module  114  stores the estimated carrier offset in a register. A carrier recovery module  117  utilizes automatic frequency control (AFC), which is essentially a phase-locked loop (PLL). The carrier recovery module  117  initializes the AFC with the estimated carrier offset. 
   Thereafter, the carrier recovery module  117  decreases a frequency step size of the AFC based on the estimated carrier offset. When the AFC is locked, the carrier recovery module  117  recovers the carrier signal from the best sample output by the sampling module  112  and generates a demodulated output comprising a correct value of the symbol. A differential decoder module  120  decodes the demodulated output and generates digital data represented by the symbol. 
   For traffic channels, the carrier recovery module  117  initializes the AFC with the estimated carrier offset calculated for the control channel. Thereafter, the carrier recovery module  117  decreases the frequency step size of the AFC based on the estimated carrier offset. Once the AFC is locked, the carrier recovery module  117  updates the estimated carrier offset for each time slot of each traffic channel. Alternatively, the frequency step size may be fixed, AFC may be performed for an entire input symbol sequence, and the estimated carrier offset may be updated at the end of each time slot. 
   When the AFC is locked, the carrier recovery module  117  recovers the carrier signal from the best sample output by the sampling module  112  and generates a demodulated output comprising the correct value of the symbol. The differential decoder module  120  decodes the demodulated output and generates digital data represented by the symbol. 
   Referring now to  FIG. 4 , the burst detector module  106  detects the burst in the control channel. Specifically, an output of the double differentiator module  104  is input to a serial-to-parallel converter module  130 , which separates phases i. Adder modules  132 - 1 ,  132 - 2 , and  132 - 3  (collectively  132 ) add two contiguous outputs of the double differentiator module  104  for phases i. Absolute function modules  134 - 1 ,  134 - 2 ,  134 - 3  (collectively  134 ) calculate absolute values of outputs of respective adder modules  132 . Moving average modules  136 - 1 ,  136 - 2 ,  136 - 3  (collectively  136 ) calculate moving averages of respective absolute values. 
   A comparator module  138  compares moving averages of phases i to a predetermined burst threshold ThB and generates the burst-detect signal if a moving average of one of the phases i is less than ThB. The burst-detect signal enables the timing module  108  and the offset estimator module  114 . 
   The phase correlation, parabolic curve fitting, and interpolation performed by the demodulation system  75 - 1  can be mathematically explained as follows. For convenience, explanation is limited to time field. A signal received by the demodulation system  75 - 1  can be expressed by the following equation.
 
 r ( t )= A ( t ) e   j2πΔft+φ(t−εT)   +n ( t )
 
where A(t) is the amplitude of the signal, Δf is the carrier offset, φ(t) is the phase information of the signal, ε is the bit timing offset, and n(t) is Gaussian white noise.
 
   For π/4-DQPSK, the phase information φ(t) can be expressed as follows.
 
φ( t )=φ( t−T )+θ( t )
 
where θ(t) is the phase angle mapped to symbols (a k , b k ) in the signal as shown in the following table.
 
   
     
       
         
             
             
             
           
             
                 
                 
             
             
                 
               (a k , b k ) 
               θ(k) 
             
             
                 
                 
             
           
          
             
                 
               (0, 0) 
                π/4 
             
             
                 
               (0, 1) 
               3π/4 
             
             
                 
               (1, 1) 
               −3π/4  
             
             
                 
               (1, 0) 
               −π/4 
             
             
                 
                 
             
          
         
       
     
   
   After single differentiation by the single differentiator module  102 , the phase information is given by the following equation.
 
phsSingleDiff( t )=phaseRx( t )−phaseRx( t−T )=2πΔ fT+φ ( t )−φ( t−T )
 
After double differentiation by the double differentiator module  104 , the phase information is given by the following equation.
 
phsDoubleDiff(t)=phsSingleDiff( t )−phsSingleDiff( t−T )=φ( t )+φ( t− 2 T )−2φ( t−T )
 
   The burst detector module  106 , the timing module  108 , and the carrier recovery module  117  use algorithms that are based on a periodicity of the preamble PR. In the burst detector module  106 , inputs to the comparator module  138  are given by following equations. 
   
     
       
         
           
             sumBurst 
             ⁢ 
             
                 
             
             ⁢ 
             1 
           
           = 
           
             
               ∑ 
               
                 m 
                 = 
                 0 
               
               
                 M 
                 - 
                 1 
               
             
             ⁢ 
             
               abs 
               ⁡ 
               
                 ( 
                 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         mT 
                       
                       ) 
                     
                   
                   + 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         mT 
                         - 
                         T 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     
       
         
           
             sumBurst 
             ⁢ 
             
                 
             
             ⁢ 
             2 
           
           = 
           
             
               ∑ 
               
                 m 
                 = 
                 0 
               
               
                 M 
                 - 
                 1 
               
             
             ⁢ 
             
               abs 
               ⁡ 
               
                 ( 
                 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         
                           T 
                           / 
                           3 
                         
                         - 
                         mT 
                       
                       ) 
                     
                   
                   + 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         
                           T 
                           / 
                           3 
                         
                         - 
                         mT 
                         - 
                         T 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     
       
         
           
             sumBurst 
             ⁢ 
             
                 
             
             ⁢ 
             3 
           
           = 
           
             
               ∑ 
               
                 m 
                 = 
                 0 
               
               
                 M 
                 - 
                 1 
               
             
             ⁢ 
             
               abs 
               ⁡ 
               
                 ( 
                 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         
                           2 
                           ⁢ 
                           
                             T 
                             / 
                             3 
                           
                         
                         - 
                         mT 
                       
                       ) 
                     
                   
                   + 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         
                           2 
                           ⁢ 
                           
                             T 
                             / 
                             3 
                           
                         
                         - 
                         mT 
                         - 
                         T 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
   
   If one of sumBurst1, sumBurst2, or sumBurst3 is less than a predetermined threshold ThB, the comparator module  138  generates a burst detect signal that enables the timing module  108  and the offset estimator module  114 . 
   The timing module  108  performs bit timing recovery for the control channel and the traffic channels as follows. For the control channel, phsDoubleDiff(t) is a signal with a period of 2T. A sampling error, if any, may be expressed as phsDoubleDiff(t−εT). Expanding phsDoubleDiff(t−εT) using Fourier series, we get 
   
     
       
         
           
             
               
                 
                   phsDoubleDiff 
                   ⁡ 
                   
                     ( 
                     
                       t 
                       - 
                       
                         ɛ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         T 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     a 
                     0 
                   
                   + 
                   
                     
                       a 
                       1 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               f 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   ɛ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   T 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       b 
                       1 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               f 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   ɛ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   T 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   + 
                 
               
             
           
           
             
               
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           2 
                         
                         ⁢ 
                         
                           
                             f 
                             0 
                           
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 ɛ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             f 
                             0 
                           
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 ɛ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 ⋯ 
               
             
           
         
       
     
     
       
         where 
       
     
     
       
         
           
             
               a 
               0 
             
             = 
             0 
           
           , 
           
             
 
           
           ⁢ 
           
             
               a 
               n 
             
             = 
             
               
                 
                   ∫ 
                   0 
                   
                     2 
                     ⁢ 
                     T 
                   
                 
                 ⁢ 
                 
                   
                     
                       phsDoubleDiff 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     · 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             π 
                             · 
                             
                               nf 
                               0 
                             
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
               ≠ 
               0 
             
           
         
       
     
     
       
         
           
             b 
             n 
           
           = 
           
             
               
                 ∫ 
                 0 
                 
                   2 
                   ⁢ 
                   T 
                 
               
               ⁢ 
               
                 
                   
                     phsDoubleDiff 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   · 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             nf 
                             0 
                           
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
             = 
             0 
           
         
       
     
     
       
         
           
             f 
             0 
           
           = 
           
             1 
             
               2 
               ⁢ 
               T 
             
           
         
       
     
     
       
         
           Therefore 
           , 
           
             
 
           
           ⁢ 
           
             
               phsDoubleDiff 
               ⁡ 
               
                 ( 
                 
                   t 
                   - 
                   
                     ɛ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     T 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   a 
                   1 
                 
                 ⁢ 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           f 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               ɛ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       
                         π 
                         · 
                         2 
                       
                       ⁢ 
                       
                         
                           f 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               ɛ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               ⋯ 
             
           
         
       
     
   
   The output of the timing module  108  for the control channel is given by the following equation. 
   
     
       
         
           ɛ 
           = 
           
             
               1 
               π 
             
             ⁢ 
             
               arctan 
               ⁡ 
               
                 ( 
                 
                   w 
                   u 
                 
                 ) 
               
             
           
         
       
     
     
       
         where 
       
     
     
       
         
           
             
               
                 u 
                 = 
                   
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     
                       2 
                       ⁢ 
                       T 
                     
                   
                   ⁢ 
                   
                     
                       
                         phsDoubleDiff 
                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               ɛ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                       · 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               π 
                               · 
                               2 
                             
                             ⁢ 
                             
                               f 
                               0 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     
                       2 
                       ⁢ 
                       T 
                     
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             a 
                             1 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         ɛ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         ɛ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         ⋯ 
                       
                       ⁢ 
                       
                           
                       
                       ] 
                     
                     · 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           0 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         a 
                         1 
                       
                       · 
                       2 
                     
                     ⁢ 
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             0 
                           
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           T 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         a 
                         1 
                       
                       · 
                       2 
                     
                     ⁢ 
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             1 
                             
                               2 
                               ⁢ 
                               T 
                             
                           
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           T 
                         
                         ) 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       a 
                       1 
                     
                     · 
                     2 
                   
                   ⁢ 
                   T 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       πɛ 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     
       
         and 
       
     
     
       
         
           
             
               
                 w 
                 = 
                   
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     
                       2 
                       ⁢ 
                       T 
                     
                   
                   ⁢ 
                   
                     
                       
                         phsDoubleDiff 
                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               ɛ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                       · 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               π 
                               · 
                               2 
                             
                             ⁢ 
                             
                               f 
                               0 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     
                       2 
                       ⁢ 
                       T 
                     
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             a 
                             1 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         ɛ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             a 
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   2 
                                 
                                 ⁢ 
                                 
                                   f 
                                   0 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       ɛ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       T 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         ⋯ 
                       
                       ] 
                     
                     · 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           0 
                         
                         ⁢ 
                         T 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         a 
                         1 
                       
                       · 
                       2 
                     
                     ⁢ 
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             0 
                           
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           T 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         a 
                         1 
                       
                       · 
                       2 
                     
                     ⁢ 
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             1 
                             
                               2 
                               ⁢ 
                               T 
                             
                           
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           T 
                         
                         ) 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       a 
                       1 
                     
                     · 
                     2 
                   
                   ⁢ 
                   T 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       πɛ 
                       ) 
                     
                   
                 
               
             
           
         
       
     
   
   On the other hand, for traffic channels, the length of the preamble PR is insufficient to perform burst detection and accurate bit timing recovery. Therefore, the UW correlator module  110  uses burst detection and bit timing information of the control channel to estimate UW position in traffic channels and performs phase correlation of PR and UW to determine the best phase. The AFC in the carrier recovery module  117  is locked using the best phase. 
   Initially, frequency offset information is removed from an output of the single differentiator module  102  as follows.
 
phsSingleDiff( t )=phaseRx( t )−phaseRx( t−T )=2πΔ fT+φ ( t )−φ( t−T )
 
Thereafter, the UW correlator module  110  performs correlation as follows.
 
   
     
       
         
           
             Corr 
             ⁡ 
             
               ( 
               t 
               ) 
             
           
           = 
           
             
               ∫ 
               
                 t 
                 - 
                 
                   12 
                   ⁢ 
                   T 
                 
               
               t 
             
             ⁢ 
             
               
                 ( 
                 
                   
                     phsSingleDiff 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     πΔ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     fT 
                   
                 
                 ) 
               
               * 
               
                 UWmapping 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               ⁢ 
               
                 ⅆ 
                 t 
               
             
           
         
       
     
   
   Referring now to  FIG. 5 , a correlation curve  150  is plotted for the correlation Corr(t). Since the peak of the correlation curve corresponds to the best phase, the correlation curve is fitted onto a parabola  152  for the purpose of finding the peak of the correlation curve. A parabola is expressed by the following equation.
 
 y=ax   2   +bx+c  
 
where a, b, and c are the coefficients of the parabola. The peak of the parabola is calculated based on the coefficients by the following formula.
 
   
     
       
         
           ɛ 
           = 
           
             - 
             
               b 
               
                 2 
                 ⁢ 
                 a 
               
             
           
         
       
     
   
   The coefficients of the parabola  152  can be calculated based on coordinates of three points on the parabola  152 : before peak  154 , peak  156 , and after peak  158 . If the coordinates of the three points are (x 1 ,y 1 )=(0,y 1 ), (x 2 ,y 2 )=(1,y 2 ), (x 2 ,y 2 )=(2,y 2 ), the coefficients are given by the following equations. 
             a   =         y   1     -     2   ⁢     y   2       +     y   3       2       ,           ⁢     b   =       y   2     -     y   1     -   a       ,           ⁢     c   =     y   1             
After the peak of the parabola  152  is calculated based on the coefficients, an x-coordinate of the peak is determined. The x-coordinate of the peak of the parabola  152  corresponds to the best time to sample a symbol to get the correct value of the symbol.
 
   The peak of the parabola  152  is used to estimate best samples. The correlation module  110  calculates a timing offset based on a distance between the peak of the parabola and an adjacent point on the correlation curve. The timing module  108  adjusts the bit timing and the sampling times of the traffic channels by the timing offset. The sampling module  112  interpolates the three samples, samples interpolated data at the sampling time adjusted by the timing module  108 , and generates one sample that comprises the correct value of the symbol. 
   If the signal is noisy, a cubic interpolator may be used instead of a parabolic interpolator. For example, a 4-point cubic interpolator is mathematically expressed by the following equation. 
   
     
       
         
           
             y 
             ⁡ 
             
               ( 
               n 
               ) 
             
           
           = 
           
             
               ∑ 
               
                 i 
                 = 
                 
                   I 
                   1 
                 
               
               
                 I 
                 2 
               
             
             ⁢ 
             
               
                 C 
                 i 
               
               ⁢ 
               
                 x 
                 ⁡ 
                 
                   ( 
                   
                     
                       I 
                       1 
                     
                     + 
                     
                       I 
                       2 
                     
                     - 
                     i 
                   
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 I 
                 1 
               
             
             = 
             
               - 
               2 
             
           
           , 
           
               
           
           ⁢ 
           
             
               I 
               2 
             
             = 
             1 
           
           , 
           
               
           
           ⁢ 
           
             
               and 
               ⁢ 
               
                   
               
               ⁢ 
               
                 C 
                 i 
               
             
             = 
             
               
                 ∏ 
                 
                   
                     j 
                     = 
                     
                       I 
                       1 
                     
                   
                   , 
                   
                     j 
                     ≠ 
                     i 
                   
                 
                 
                   I 
                   2 
                 
               
               ⁢ 
               
                 
                   t 
                   - 
                   
                     t 
                     j 
                   
                 
                 
                   
                     t 
                     i 
                   
                   - 
                   
                     t 
                     j 
                   
                 
               
             
           
         
       
     
     
       
         
           
             C 
             
               - 
               2 
             
           
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             + 
             μ 
           
         
       
     
     
       
         
           and 
           ⁢ 
           
               
           
           ⁢ 
           now 
         
       
     
     
       
         
           
             C 
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   Generally, μ is within [0,1). Therefore, if the peak ε of the parabola  152  is less than 0, then ε is modulated with osrRx so that ε is within [0,osrRx). 
   Referring now to  FIG. 6 , the carrier recovery module  117  utilizes a phase-locked loop (PLL) that performs automatic frequency control (AFC). The carrier recovery module  117  comprises a phase rotator module  117 - 10  that subtracts π/4 from a π/4-DQPSK signal and generates a DQPSK signal. The carrier offset generated for the control channel by the offset estimator module  114  is input to an accumulator module  117 - 2 . An adder module  117 - 1  adds an output of the accumulator module  117 - 2  to the DQPSK signal and outputs a sum to a phase-shift calculator module  117 - 3  and to a detector module  117 - 4 . The phase-shift calculator module  117 - 3  calculates a carrier phase-shift. The detector module  117 - 4  detects codes and provides a feedback to the phase-shift calculator module  117 - 3 . 
   The carrier phase-shift is filtered by a carrier filter module  117 - 5 . The carrier filter module  117 - 5  may utilize a first-order filter. A filtered carrier phase-shift is input to a digital fixed frequency (DFF) module  117 - 6 . An output of the DFF module  117 - 6  is input to the detector module  117 - 4  and to a differentiator module  117 - 7 . The differentiator module  117 - 7  differentiates the output of the DFF module  117 - 6 . A digital filter module  117 - 8  filters a differentiated signal output by the differentiator module  117 - 7  and provides a feedback to the carrier filter module  117 - 5 . An up-down counter module  117 - 9  counts add times and subtract times and provides an output to the accumulator module  117 - 2 . When add times and subtract times of the up-down counter module  117 - 9  are nearly equal, the AFC frequency is locked, and the carrier filter module  117 - 5  recovers the carrier signal. 
   The carrier recovery module  117  receives the carrier offset generated by the offset estimator module  114 . The offset estimator module  114  calculates the carrier offset for the control channel based on an output of the single differentiator module  102 . The output of the single differentiator module  102 , phsSingleDiff (t), is a periodic function of period  2 T and is expressed by the following equation.
 
phsSingleDiff( t )=phaseRx( t )−phaseRx( t−T )=2πΔ fT+φ ( t )−φ( t−T )
 
   The carrier offset is determined by the following equation. 
             Δ   ⁢           ⁢   f     =         ∫   0     0       +   2     ⁢   MT         ⁢       (       phsSingleDiff   ⁡     (   t   )       -     π   /   4       )     ⁢     ⅆ   t           4   ⁢   π   ⁢           ⁢     MT   2               
Offset estimation is improved by using burst detection performed by the burst detector module  106 . An improved carrier offset is given by the following equation.
 
             Δ   ⁢           ⁢   f     =         ∑     m   =   0       m   =       2   ⁢   M     -   1         ⁢     (       phsSingleDiff   ⁡     (     k   +   mT     )       -     π   /   4       )         4   ⁢   π   ⁢           ⁢   M   ⁢           ⁢   T             
The improved carrier offset increases probability of selecting best samples for carrier recovery. The carrier recovery module  117  uses best samples to recover the carrier signal.
 
   The carrier recovery module  117  decreases a frequency step size of the AFC using the following AFC algorithm. The AFC algorithm utilizes a characteristic of DQPSK. A relationship between DQPSK and π/4-DQPSK is shown in the following table. 
   
     
       
         
             
             
             
          
             
                 
                 
             
             
                 
               Phase difference 
                 
             
          
         
         
             
             
             
          
             
               Transmitted 
               π/4 − DQPSK 
               DQPSK 
             
             
               signal 
               Δθ(k) 
               Δθ(k) − π/4 
             
             
                 
             
             
               (0, 0) 
                π/4 
               0 
             
             
               (0, 1) 
               3π/4 
                π/2 
             
             
               (1, 1) 
               −3π/4  
               π 
             
             
               (1, 0) 
               −π/4 
               −π/2 
             
             
                 
             
          
         
       
     
   
   A DQPSK signal can be obtained by subtracting π/4 from π/4-DQPSK signal. Therefore, the input signal is rotated by π/4 by the phase rotator module  117 - 10 . This is mathematically expressed as follows. 
                   θ   Pk     =       θ   Rk     -     k   ·     π   /   4                     =       θ   Tk     +     ϕ   ek     +     n   ek     -     k   ·     π   /   4                     =         I   k     ·     π   /   2       +     ϕ   ek     +     n   ek                   
where I k =I k−1 +I T , φ ek  is phase change caused by frequency offset, and n ek  is phase change caused by Gaussian noise.
 
   The carrier recovery module  117  initializes the AFC with a frequency offset within a small predetermined range. Thereafter, the carrier recovery module  117  decreases the frequency offset based on the improved carrier offset provided by the offset estimator module  114 . When add times and subtract times of the up-down counter module  117 - 9  are nearly equal, the PLL is locked, and the carrier filter module  117 - 5  begins carrier recovery. 
   An equation for the carrier filter module  117 - 5  is expressed as follows. 
               ϕ   ⋒     ek     =         1   -   α       1   -     α   ⁢           ⁢     z     -   1             ⁢     ϕ   ek   ′             
where φ′ ek  is input phase error, {circumflex over (φ)} ek  is filtered phase error, α=exp(−2/Q), and Q k =10·k/N f (k&lt;N f )=10(k&gt;=N f ).
 
   The carrier recovery module  117  comprises the phase-shift calculator module  117 - 3  that uses a feedback mechanism, which may be a form of reverse modulation. A code detected by the detector module  117 - 4  is fed back to the phase-shift calculator module  117 - 3 . This eliminates a phase change caused by the phase information in the signal. This is mathematically expressed as follows.
 
φ′ ek =( I   k   −Î   k )·π/2+φ ek   +n   ek  
 
Î can be obtained by the following equation.
 
 Î=[(θ   Pk −{circumflex over (φ)} ek−1 )/(π/2)+½]
 
Thereafter, the differential decoder module  120  decodes the output of the carrier recovery module  117 . Thus, Î T =Î k −Î k−1 .
 
   Referring now to  FIG. 7 , bit timing estimated using correlation, parabolic curve fitting, and interpolation may be accurate at least 65% of the time as compared to known accurate timing. This is shown in the pi-chart by the region having timing offset=0. 
   Referring now to  FIG. 8 , a demodulation method  200  used in a PHS receiver begins at step  202 . A downsample module  70  downsamples a signal received by the PHS receiver at a rate of three samples per symbol or 3f b  in step  204 . Whether a channel is a control channel or a traffic channel is determined in step  205 . 
   If the channel is control channel, a burst detector module  106  detects a burst in step  206 . The burst detector module  106  determines if the burst is detected in step  208  based on whether moving average of one of three phases (phase  1 , phase  2 , or phase  3 , wherein phase i is an array of i-th of three samples for every symbol) is less than a predetermined burst threshold ThB. If true, the burst detector module  106  generates a burst-detect signal in step  210 . Otherwise, the burst detector module  106  continues to detect burst in step  206 . 
   The burst-detect signal enables a timing module  108  and an offset estimator module  114  in step  212 . The timing module  108  determines correct sampling time for the control channel in step  213  based on PR in the control channel. A sampling module  112  samples symbols in subsequent samples in step  224  using the sampling time generated by the timing module  108 . 
   The offset estimator module  114  estimates a carrier offset in step  226 . A carrier recovery module  117  initializes an AFC with an estimated value of carrier offset in step  228 . The AFC checks in step  230  if a PLL is locked. If false, the carrier recovery module  117  decreases a frequency step size of the AFC in step  232 . If true, the carrier recovery module  117  locks the AFC in step  234 . The demodulation method  200  determines in step  236  that the sample is on time, that is, the sample is the best sample that comprises a correct symbol value. The demodulation system  75 - 1  demodulates the sample, and a differential decoder module  120  decodes a symbol from the sample in step  238 . The method  200  restarts in step  202 . 
   On the other hand, if the channel is traffic channel, a UW correlator module  110  correlates PR and UW phases in step  214  and generates a correlation curve in step  216  using correlation samples. To correlate UW, the correlator module  110  uses the burst detection and bit timing information of the control channel to estimate UW position in traffic channels. 
   The UW correlator module  110  fits the correlation curve to a parabola in step  218 . The UW correlator module  110  calculates a peak of the parabola and a peak time based on an x-coordinate of the peak in step  220 . The peak time represents the best time to sample a symbol. The UW correlator module  110  calculates a timing offset in step  222  based on a distance between the peak of the parabola and an adjacent point on the correlation curve. 
   The timing module  108  determines correct sampling time for the traffic channel in step  223  based on the timing offset. The sampling module  112  samples symbols in subsequent samples in step  224  using the sampling time generated by the timing module  108 . 
   The offset estimator module  114  estimates the carrier offset for the control channel in step  226 . To recover carrier in traffic channel, the carrier recovery module  117  initializes AFC with the estimated value of the carrier offset in step  228 . The AFC checks in step  230  if the PLL is locked based on the carrier offset. If false, the carrier recovery module  117  decreases the frequency step size of the AFC in step  232 . If true, the carrier recovery module  117  locks the AFC in step  234  and updates the carrier offset. 
   The demodulation method  200  determines in step  236  that the sample is on time, that is, the sample is the best sample that comprises a correct symbol value. The demodulation system  75 - 1  demodulates the sample, and a differential decoder module  120  decodes a symbol from the sample in step  238 . The method  200  restarts in step  202 . 
   As can be appreciated, the systems and methods disclosed herein may be used to determine bit timing in any communication system that utilizes TDMA. For example, the systems and methods disclosed herein may be used to determine bit timing in wireless communication systems, optical communication systems, etc. 
   Those skilled in the art can now appreciate from the foregoing description that the broad teachings of the present invention can be implemented in a variety of forms. Therefore, while this invention has been described in connection with particular examples thereof, the true scope of the invention should not be so limited since other modifications will become apparent to the skilled practitioner upon a study of the drawings, the specification and the following claims.