Patent Publication Number: US-7899107-B1

Title: Preamble detection using low-complexity cross-correlation

Description:
This application claims the benefit of U.S. Provisional Application No. 60/792,508, filed on Apr. 17, 2006, and U.S. Provisional Application No. 60/809,733, filed on May 31, 2006. The disclosures of the above applications are incorporated herein by reference in their entirety. 
    
    
     FIELD 
     The present disclosure relates to communication systems, and more particularly to detecting preamble sequences in systems using orthogonal frequency domain multiplexing (OFDM). 
     BACKGROUND 
     The background description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventors, to the extent it is described in this background section, as well as aspects of the description which may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure. 
     Referring now to  FIG. 1 , a typical communication system  10  comprises an information source  12 , a transmitter  13 , a communication channel  20 , a receiver  27 , and a destination  28 . The transmitter  13  comprises a source encoder  14 , a channel encoder  16 , and a modulator  18 . The receiver  27  comprises a demodulator  22 , a channel decoder  24 , and a source decoder  26 . 
     The information source  12  may be an analog source such as a sensor that outputs information as continuous waveforms or a digital source such as a computer that outputs information in a digital form. The source encoder  14  converts the output of the information source  12  into a sequence of binary digits (bits) called an information sequence u. The channel encoder  16  converts the information sequence u into a discrete encoded sequence v called a codeword. The modulator  18  transforms the codeword into a waveform of duration T seconds that is suitable for transmission. 
     The waveform output by the modulator  18  is transmitted via the communication channel  20 . Typical examples of the communication channel  20  are telephone lines, wireless communication channels, optical fiber cables, etc. Noise, such as electromagnetic interference, inter-channel crosstalk, etc., may corrupt the waveform. 
     The demodulator  22  receives the waveform. The demodulator  22  processes each waveform and generates a received sequence r that is either a discrete (quantized) or a continuous output. The channel decoder  24  converts the received sequence r into a binary sequence u′ called an estimated information sequence. The source decoder  26  converts u′ into an estimate of the output of the information source  12  and delivers the estimate to the destination  28 . The estimate may be a faithful reproduction of the output of the information source  12  when u′ resembles u despite decoding errors that may be caused by the noise. 
     Communication systems use different modulation schemes to modulate and transmit data. For example, a radio frequency (RF) carrier may be modulated using techniques such as frequency modulation, phase modulation, etc. In wireline communication systems, a transmitted signal generally travels along a path in a transmission line between a transmitter and a receiver. 
     In wireless communication systems, however, a transmitted signal may travel along multiple paths. This is because the transmitted signal may be reflected and deflected by objects such as buildings, towers, airplanes, cars, etc., before the transmitted signal reaches a receiver. Each path may be of different length. Thus, the receiver may receive multiple versions of the transmitted signal. The multiple versions may interfere with each other causing inter symbol interference (ISI). Thus, retrieving original data from the transmitted signal may be difficult. 
     To alleviate this problem, wireless communication systems often use a modulation scheme called orthogonal frequency division multiplexing (OFDM). In OFDM, a wideband carrier signal is converted into a series of independent narrowband sub-carrier signals that are adjacent to each other in frequency domain. Data to be transmitted is split into multiple parallel data streams. Each data stream is modulated using a sub-carrier. A channel over which the modulated data is transmitted comprises a sum of the narrowband sub-carrier signals, which may overlap. 
     When each sub-carrier closely resembles a rectangular pulse, modulation can be easily performed by Inverse Discrete Fourier Transform (IDFT), which can be efficiently implemented as an Inverse Fast Fourier Transform (IFFT). When IFFT is used, the spacing of sub-carriers in the frequency domain is such that when the receiver processes a received signal at a particular frequency, all other signals are nearly zero at that frequency, and ISI is avoided. This property is called orthogonality, and hence the modulation scheme is called orthogonal frequency division multiplexing (OFDM). 
     Referring now to  FIGS. 2A-2C , a wireless communication system  50  may comprise base stations BS 1 , BS 2 , and BS 3  (collectively BS) and one or more mobile stations (MS). Each BS may comprise a processor  30 , a medium access controller (MAC)  32 , a physical layer (PHY) module  34 , and an antenna  36  as shown in  FIG. 2B . Similarly, each MS may comprise a processor  40 , a medium access controller (MAC)  42 , a physical layer (PHY) module  44 , and an antenna  46  as shown in  FIG. 2C . The PHY modules  34  and  44  may comprise radio frequency (RF) transceivers (not shown) that transmit and receive data via antennas  36  and  46 , respectively. Each BS and MS may transmit and receive data while the MS moves relative to the BS. 
     Specifically, each BS may transmit data using orthogonal frequency division multiplexing access (OFDMA) system. Each BS may transmit data typically in three segments: SEG 1 , SEG 2 , and SEG 3 . The MS, which moves relative to each BS, may receive data from one or more base stations depending on the location of the MS relative to each BS. For example, the MS may receive data from SEG  3  of BS 1  and SEG  2  of BS 2  when the MS is located as shown in  FIG. 2A . 
     Relative motion between MS and BS may cause Doppler shifts in signals received by the MS. This can be problematic since systems using OFDM are inherently sensitive to carrier frequency offsets (CFO). Therefore, pilot tones are generally used for channel estimation refinement. For example, some of the sub-carriers may be designated as pilot tones for correcting residual frequency offset errors. 
     Additionally, the PHY module  34  of each BS typically adds a preamble to a data frame that is to be transmitted. Specifically, the PHY module  34  modulates and encodes the data frame comprising the preamble at a data rate specified by the MAC  34  and transmits the data frame. When the PHY module  44  of the MS receives the data frame, the PHY module  44  uses the preamble in the data frame to detect a beginning of packet transmission and to synchronize to a transmitter clock of the BS. 
     According to the I.E.E.E. standard 802.16e, which is incorporated herein by reference in its entirety, a first symbol in the data frame transmitted by the BS is a preamble symbol from a preamble sequence. The preamble sequence typically contains an identifier called IDcell, which is a cell ID of the BS, and segment information. The BS selects the preamble sequence based on the IDcell and the segment number of the BS. Each BS may select different preamble sequences. Additionally, each BS may select preamble sequences that are distinct among the segments of that BS. 
     The BS modulates multiple sub-carriers with the selected preamble sequence. Thereafter, the BS performs IFFT, adds a cyclic prefix, and transmits a data frame. The MS uses the cyclic prefix to perform symbol timing and fractional carrier frequency synchronization. Unless the MS knows the preamble sequence, however, the MS cannot associate itself to a particular segment of a particular BS. 
     SUMMARY 
     A system comprises a differential demodulation module and a cross-correlation module. The differential demodulation module differentially demodulates modulated signals and generates differentially demodulated signals. The cross-correlation module generates cross-correlation values by cross-correlating states of X symbols in the differentially demodulated signals with corresponding states of X predetermined symbols in each of Y preamble sequences, where X is an integer greater than 1, and Y is an integer greater than 0. The cross-correlation module determines whether one of the Y preamble sequences is present in the modulated signals based on the cross-correlation values. 
     In another feature, the correlation module comprises a state detection module and a summing module. The state detection module detects the states of the X symbols in the differentially demodulated signals, where each of the X symbols has one of a first state and a second state. The summing module receives the Y preamble sequences and generates X sums for each of the Y preamble sequences by adding each of the states of the X symbols with the corresponding states of the X predetermined symbols in each of the Y preamble sequences. The X sums are modulo-2 sums. 
     In another feature, the modulated signals include sub-carriers that are modulated using orthogonal frequency domain multiplexing (OFDM). Every P th  one of the sub-carriers is modulated with a preamble symbol from one of the Y preamble sequences, where P is an integer greater than or equal to 1. 
     In another feature, every P th  one of the sub-carriers has one of substantially the same channel phase and substantially the same differential channel phase. 
     In another feature, the differential demodulation module generates the differentially demodulated signals by multiplying a Q th  one of the modulated signals by a complex conjugate of a (Q+P) th  one of the modulated signals, where Q is an integer greater than or equal to 1. 
     In another feature, the states of the X symbols in the differentially demodulated signals are states of real parts of the differentially demodulated signals when imaginary parts of the differentially demodulated signals are approximately equal to zero. 
     In another feature, the states of the X predetermined symbols in the Y preamble sequences are generated by inverting states of derived symbols in derived preamble sequences that are derived from the Y preamble sequences, and wherein each of the Y preamble sequences is different from others of the Y preamble sequences. 
     In another feature, each of the derived symbols of one of the derived preamble sequences has a first binary value when a corresponding symbol and a symbol adjacent to the corresponding symbol in a corresponding one of the Y preamble sequences have opposite binary values, and the each of the derived symbols has a second binary value that is opposite to the first binary value when the corresponding symbol and the symbol adjacent to the corresponding symbol have the same binary value. 
     In another feature, the derived preamble sequences have a cross-correlation that is less than or equal to a predetermined cross-correlation threshold. The predetermined cross-correlation threshold is less than approximately 0.2 for an orthogonal frequency domain multiplexing (OFDM) system using a 1024 fast Fourier transform (FFT) mode. 
     In another feature, the summing module generates the cross-correlation values by adding the X sums generated for each of the Y preamble sequences. 
     In another feature, the cross-correlation module further comprises a control module that communicates with the differential demodulation module and the summing module, that selects a largest one of the cross-correlation values, and that detects one of the Y preamble sequences in the modulated signals when the largest one of the cross-correlation values is greater than or equal to a predetermined threshold. The predetermined threshold is based on the signal strength of the modulated signals. 
     In another feature, the control module identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the control module selects the largest one of the cross-correlation values based on the signal strength of the modulated signals. 
     In another feature, the system further comprises a correlation module that communicates with the differential demodulation module and the control module and that generates correlation values by correlating at least one of the differentially demodulated signals and at least one of derived preamble sequences that is derived from the Y preamble sequences. At least one of the derived preamble sequences generates at least two of the cross-correlation values that are greater than the predetermined threshold and that are substantially the same. 
     In another feature, the control module selects one of the correlation values having a largest real part, detects one of the Y preamble sequences in the modulated signals based on the largest real part, and identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the control module selects one of the correlation values having a largest magnitude, detects one of the Y preamble sequences in the modulated signals based on the largest magnitude, and identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequence detected in the modulated signals. 
     In another feature, the modulated signals include a fractional carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the modulated signals include an integer carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the control module generates an estimate for an integer carrier frequency offset (CFO) present in the modulated signals when the control module selects the largest one of the cross-correlation values. 
     In another feature, a physical layer (PHY) module comprises the system and further comprises a transceiver module that communicates with the differential demodulation module and that receives the differentially modulated signals. 
     In another feature, a network device comprises the PHY module and further comprises at least one antenna that communicates with the transceiver module. 
     In still other features, a method comprises generating differentially demodulated signals by differentially demodulating modulated signals that include sub-carriers that are modulated using orthogonal frequency domain multiplexing (OFDM). The method further comprises generating cross-correlation values by cross-correlating states of X symbols in the differentially demodulated signals with corresponding states of X predetermined symbols in each of Y preamble sequences, where X is an integer greater than 1, and Y is an integer greater than 0. The method further comprises determining whether one of the Y preamble sequences is present in the modulated signals based on the cross-correlation values. 
     In another feature, the method further comprises detecting the states of the X symbols in the differentially demodulated signals, where each of the X symbols has one of a first state and a second state. The method further comprises generating X sums for each of the Y preamble sequences by adding each of the states of the X symbols with the corresponding states of the X predetermined symbols of each of the Y preamble sequences, where the X sums are modulo-2 sums. 
     In another feature, every P th  one of the sub-carriers is modulated with a preamble symbol from one of the Y preamble sequences, where P is an integer greater than or equal to 1. Every P th  one of the sub-carriers has one of substantially the same channel phase and substantially the same differential channel phase. 
     In another feature, the method further comprises generating the differentially demodulated signals by multiplying a Q th  one of the modulated signals by a complex conjugate of a (Q+P) th  one of the modulated signals, where Q is an integer greater than or equal to 1. 
     In another feature, the method further comprises determining the states of the X symbols in the differentially demodulated signals as states of real parts of the differentially demodulated signals when imaginary parts of the differentially demodulated signals are approximately equal to zero. 
     In another feature, the method further comprises generating the states of the X predetermined symbols in the Y preamble sequences by inverting states of derived symbols in derived preamble sequences that are derived from the Y preamble sequences. 
     In another feature, the method further comprises generating each of the derived symbols of one of the derived preamble sequences having a first binary value when a corresponding symbol and a symbol adjacent to the corresponding symbol in a corresponding one of the Y preamble sequences have opposite binary values, and having a second binary value that is opposite to the first binary value when the corresponding symbol and the symbol adjacent to the corresponding symbol have the same binary value. 
     In another feature, each of the Y preamble sequences is different from others of the Y preamble sequences, and wherein the derived preamble sequences have a cross-correlation that is less than or equal to a predetermined cross-correlation threshold. The predetermined cross-correlation threshold is less than approximately 0.2 for an orthogonal frequency domain multiplexing (OFDM) method using a 1024 fast Fourier transform (FFT) mode. 
     In another feature, the method further comprises generating the cross-correlation values by adding the X sums generated for each of the Y preamble sequences. 
     In another feature, the method further comprises selecting a largest one of the cross-correlation values and detecting one of the Y preamble sequences in the modulated signals when the largest one of the cross-correlation values is greater than or equal to a predetermined threshold. The predetermined threshold is based on the signal strength of the modulated signals. 
     In another feature, the method further comprises identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the method further comprises selecting the largest one of the cross-correlation values based on the signal strength of the modulated signals. 
     In another feature, the method further comprises generating correlation values by correlating at least one of the differentially demodulated signals and at least one of derived preamble sequences that is derived from the Y preamble sequences. At least one of derived preamble sequences generates at least two of the cross-correlation values that are greater than the predetermined threshold and that are substantially the same. 
     In another feature, the method further comprises selecting one of the correlation values having a largest real part, detecting one of the Y preamble sequences in the modulated signals based on the largest real part, and identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the method further comprises selecting one of the correlation values having a largest magnitude, detecting one of the Y preamble sequences in the modulated signals based on the largest magnitude, and identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequence detected in the modulated signals. 
     In another feature, the modulated signals include a fractional carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the modulated signals include an integer carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the method further comprises generating an estimate for an integer carrier frequency offset (CFO) present in the modulated signals when selecting the largest one of the cross-correlation values. 
     In still other features, a system comprises differential demodulation means for differentially demodulating modulated signals and generating differentially demodulated signals. The system further comprises cross-correlation means for generating cross-correlation values by cross-correlating states of X symbols in the differentially demodulated signals with corresponding states of X predetermined symbols in each of Y preamble sequences and determining whether one of the Y preamble sequences is present in the modulated signals based on the cross-correlation values, where X is an integer greater than 1, and Y is an integer greater than 0. 
     In another feature, the cross-correlation means comprises state detection means for detecting the states of the X symbols in the differentially demodulated signals, where each of the X symbols has one of a first state and a second state. The cross-correlation means further comprises summing means for receiving the Y preamble sequences and generating X sums for each of the Y preamble sequences by adding each of the states of the X symbols with the corresponding states of the X predetermined symbols in each of the Y preamble sequences. The X sums are modulo-2 sums. 
     In another feature, the modulated signals include sub-carriers that are modulated using orthogonal frequency domain multiplexing (OFDM). Every P th  one of the sub-carriers is modulated with a preamble symbol from one of the Y preamble sequences, where P is an integer greater than or equal to 1. 
     In another feature, every P th  one of the sub-carriers has one of substantially the same channel phase and substantially the same differential channel phase. 
     In another feature, the differential demodulation means generates the differentially demodulated signals by multiplying a Q th  one of the modulated signals by a complex conjugate of a (Q+P) th  one of the modulated signals, where Q is an integer greater than or equal to 1. 
     In another feature, the states of the X symbols in the differentially demodulated signals are states of real parts of the differentially demodulated signals when imaginary parts of the differentially demodulated signals are approximately equal to zero. 
     In another feature, the states of the X predetermined symbols in the Y preamble sequences are generated by inverting states of derived symbols in derived preamble sequences that are derived from the Y preamble sequences, and wherein each of the Y preamble sequences is different from others of the Y preamble sequences. 
     In another feature, each of the derived symbols of one of the derived preamble sequences has a first binary value when a corresponding symbol and a symbol adjacent to the corresponding symbol in a corresponding one of the Y preamble sequences have opposite binary values, and the each of the derived symbols has a second binary value that is opposite to the first binary value when the corresponding symbol and the symbol adjacent to the corresponding symbol have the same binary value. 
     In another feature, the derived preamble sequences have a cross-correlation that is less than or equal to a predetermined cross-correlation threshold. The predetermined cross-correlation threshold is less than approximately 0.2 for an orthogonal frequency domain multiplexing (OFDM) system using a 1024 fast Fourier transform (FFT) mode. 
     In another feature, the summing means generates the cross-correlation values by adding the X sums generated for each of the Y preamble sequences. 
     In another feature, the cross-correlation means further comprises control means for communicating with the differential demodulation means and the summing means, selecting a largest one of the cross-correlation values, and detecting one of the Y preamble sequences in the modulated signals when the largest one of the cross-correlation values is greater than or equal to a predetermined threshold. The predetermined threshold is based on the signal strength of the modulated signals. 
     In another feature, the control means identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the control means selects the largest one of the cross-correlation values based on the signal strength of the modulated signals. 
     In another feature, the system further comprises correlation means for communicating with the differential demodulation means and the control means and generating correlation values by correlating at least one of the differentially demodulated signals and at least one of derived preamble sequences that is derived from the Y preamble sequences. At least one of the derived preamble sequences generates at least two of the cross-correlation values that are greater than the predetermined threshold and that are substantially the same. 
     In another feature, the control means selects one of the correlation values having a largest real part, detects one of the Y preamble sequences in the modulated signals based on the largest real part, and identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the control means selects one of the correlation values having a largest magnitude, detects one of the Y preamble sequences in the modulated signals based on the largest magnitude, and identifies a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequence detected in the modulated signals. 
     In another feature, the modulated signals include a fractional carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the modulated signals include an integer carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the control means generates an estimate for an integer carrier frequency offset (CFO) present in the modulated signals when the control means selects the largest one of the cross-correlation values. 
     In another feature, a physical layer means (PHY) for communicating with a communication medium comprises the system and further comprises transceiver means for communicating with the differential demodulation means and receiving the differentially modulated signals. 
     In another feature, a network device comprises the PHY means and further comprises at least one antenna means for communicating with the transceiver means. 
     In still other features, a computer program executed by a processor comprises generating differentially demodulated signals by differentially demodulating modulated signals that include sub-carriers that are modulated using orthogonal frequency domain multiplexing (OFDM). The computer program further comprises generating cross-correlation values by cross-correlating states of X symbols in the differentially demodulated signals with corresponding states of X predetermined symbols in each of Y preamble sequences, where X is an integer greater than 1, and Y is an integer greater than 0. The computer program further comprises determining whether one of the Y preamble sequences is present in the modulated signals based on the cross-correlation values. 
     In another feature, the computer program further comprises detecting the states of the X symbols in the differentially demodulated signals, where each of the X symbols has one of a first state and a second state. The computer program further comprises generating X sums for each of the Y preamble sequences by adding each of the states of the X symbols with the corresponding states of the X predetermined symbols of each of the Y preamble sequences, where the X sums are modulo-2 sums. 
     In another feature, every P th  one of the sub-carriers is modulated with a preamble symbol from one of the Y preamble sequences, where P is an integer greater than or equal to 1. Every P th  one of the sub-carriers has one of substantially the same channel phase and substantially the same differential channel phase. 
     In another feature, the computer program further comprises generating the differentially demodulated signals by multiplying a Q th  one of the modulated signals by a complex conjugate of a (Q+P) th  one of the modulated signals, where Q is an integer greater than or equal to 1. 
     In another feature, the computer program further comprises determining the states of the X symbols in the differentially demodulated signals as states of real parts of the differentially demodulated signals when imaginary parts of the differentially demodulated signals are approximately equal to zero. 
     In another feature, the computer program further comprises generating the states of the X predetermined symbols in the Y preamble sequences by inverting states of derived symbols in derived preamble sequences that are derived from the Y preamble sequences. 
     In another feature, the computer program further comprises generating each of the derived symbols of one of the derived preamble sequences having a first binary value when a corresponding symbol and a symbol adjacent to the corresponding symbol in a corresponding one of the Y preamble sequences have opposite binary values, and having a second binary value that is opposite to the first binary value when the corresponding symbol and the symbol adjacent to the corresponding symbol have the same binary value. 
     In another feature, each of the Y preamble sequences is different from others of the Y preamble sequences, and wherein the derived preamble sequences have a cross-correlation that is less than or equal to a predetermined cross-correlation threshold. The predetermined cross-correlation threshold is less than approximately 0.2 for an orthogonal frequency domain multiplexing (OFDM) computer program using a 1024 fast Fourier transform (FFT) mode. 
     In another feature, the computer program further comprises generating the cross-correlation values by adding the X sums generated for each of the Y preamble sequences. 
     In another feature, the computer program further comprises selecting a largest one of the cross-correlation values and detecting one of the Y preamble sequences in the modulated signals when the largest one of the cross-correlation values is greater than or equal to a predetermined threshold. The predetermined threshold is based on the signal strength of the modulated signals. 
     In another feature, the computer program further comprises identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the computer program further comprises selecting the largest one of the cross-correlation values based on the signal strength of the modulated signals. 
     In another feature, the computer program further comprises generating correlation values by correlating at least one of the differentially demodulated signals and at least one of derived preamble sequences that is derived from the Y preamble sequences. At least one of derived preamble sequences generates at least two of the cross-correlation values that are greater than the predetermined threshold and that are substantially the same. 
     In another feature, the computer program further comprises selecting one of the correlation values having a largest real part, detecting one of the Y preamble sequences in the modulated signals based on the largest real part, and identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequences detected in the modulated signals. 
     In another feature, the computer program further comprises selecting one of the correlation values having a largest magnitude, detecting one of the Y preamble sequences in the modulated signals based on the largest magnitude, and identifying a segment of a base station that transmitted the modulated signals based on the one of the Y preamble sequence detected in the modulated signals. 
     In another feature, the modulated signals include a fractional carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the modulated signals include an integer carrier frequency offset (CFO) that generates a phase error that is substantially the same in each one of the modulated signals. 
     In another feature, the computer program further comprises generating an estimate for an integer carrier frequency offset (CFO) present in the modulated signals when selecting the largest one of the cross-correlation values. 
     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. 
     Further areas of applicability of the present disclosure 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 disclosure, are intended for purposes of illustration only and are not intended to limit the scope of the disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure will become more fully understood from the detailed description and the accompanying drawings, wherein: 
         FIG. 1  is a functional block diagram of an exemplary communication system according to the prior art; 
         FIG. 2A  is a schematic representation of an exemplary wireless communication system comprising three base stations and a mobile station according to the prior art; 
         FIG. 2B  is a functional block diagram of an exemplary base station utilized in the system of  FIG. 2A ; 
         FIG. 2C  is a functional block diagram of an exemplary mobile station utilized in the system of  FIG. 2A ; 
         FIG. 3  is a schematic representation of an exemplary wireless communication system comprising three base stations and a mobile station; 
         FIG. 4  is a table showing preamble sequences used by base stations of  FIG. 3  to transmit data; 
         FIG. 5A  is a functional block diagram of an exemplary preamble detection system according to the present disclosure; 
         FIG. 5B  is a graph showing normalized cross-correlation between preamble sequences derived from the preamble sequences shown in  FIG. 4 ; 
         FIG. 5C  is a functional block diagram of an exemplary segment selection system according to the present disclosure; 
         FIG. 5D  is a graph showing effect of integer carrier frequency offset on normalized maximum cross-correlation according to the present disclosure; 
         FIG. 6  is a flowchart of an exemplary method for preamble detection when sub-carriers have substantially similar channel phase; 
         FIG. 7A  is a functional block diagram of a vehicle control system; and 
         FIG. 7B  is a functional block diagram of a cellular phone; and 
         FIG. 7C  is a functional block diagram of a mobile device. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is in no way intended to limit the disclosure, 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 disclosure. 
     Referring now to  FIG. 3 , a wireless communication system  100  may comprise base stations BS 1 , BS 2 , and BS 3  (collectively BS) and one or more mobile stations (MS). Generally, one MS may communicate with up to three adjacent base stations. Each BS may transmit data that is modulated using an orthogonal frequency division multiplexing access (OFDMA) system. 
     Specifically, each BS may transmit data in three segments: SEG 1 , SEG 2 , and SEG 3 . The MS, which may move relative to each BS, may receive data from one or more base stations depending on the location of the MS relative to each BS. For example, the MS may receive data from SEG  3  of BS 1 , SEG  2  of BS 2 , and/or SEG  1  of BS 3  when the MS is located as shown. 
     When a receiver in the MS is turned on (i.e., when the MS is powered up), the MS may associate with an appropriate segment of a corresponding BS depending on the location of the MS. The MS, however, can process data in a frame transmitted by a BS only if the MS can correctly detect a preamble sequence in the frame. Specifically, the MS can perform frame synchronization and retrieval of a cell ID (IDcell) and a segment number of the BS from the frame if the MS can detect the preamble sequence in the frame. 
     Referring now to  FIG. 4 , OFDMA systems may use 1024 and 512 sub-carriers to modulate and transmit data. OFDMA systems using 1024 and 512 sub-carriers are generally referred to as OFDMA systems having 1024 and 512 FFT modes, respectively. Additionally, I.E.E.E. 802.16e supports 128 FFT and 2048 FFT modes. 
     A total of 114 preamble sequences exist for OFDMA systems that use fast Fourier transforms (FFT) to modulate 1024 and 512 sub-carriers. Each preamble sequence is unique. That is, each preamble sequence is distinct from another preamble sequence and is identified by an index number. The index number may be referred to as preamble sequence index. Each preamble sequence is 284 and 143 bits (symbols) long for 1024 and 512 FFT modes, respectively. 
     Since one MS may typically communicate with up to three base stations, each BS modulates every third sub-carrier. That is, each BS modulates one of every three sub-carriers. Additionally, each BS uses only one bit of the total bits in a preamble sequence when modulating every third sub-carrier. For example, in 1024 FFT mode, the BS may use bit numbers  1 ,  2 ,  3 , . . . , etc., of the 284 bits in a preamble sequence to modulate sub-carrier numbers  1 ,  4 ,  7 , . . . , etc., of the 1024 sub-carriers, respectively. 
     Each BS may use the same set of sub-carriers. Each segment in a BS, however, uses distinct sub-carriers at least for preamble purposes. For example, for each BS, segment  1  (SEG 1 ) may use sub-carriers  0 ,  3 ,  6 ,  9 , . . . , etc.; segment  2  (SEG 2 ) may use sub-carriers  1 ,  4 ,  7 ,  10 , . . . , etc.; and segment  3  (SEG 3 ) may use sub-carriers  2 ,  5 ,  8 ,  11 , . . . , etc. 
     Consequently, the MS receives distinct signals from each BS. For example, the MS may receive signals from SEG 2  of BS 2  on sub-carriers  1 ,  4 ,  7 ,  10 , . . . , etc., from SEG 1  of BS 3  on sub-carriers  0 ,  3 ,  6 ,  9 , . . . , etc., and from SEG  3  of BS 1  on sub-carriers  2 ,  5 ,  8 ,  11 , . . . , etc. Thus, the signals received by the MS may not interfere with each other since their sub-carriers are distinct. 
     A set of sub-carriers for segment n may be mathematically expressed as follows.
 
PreambleGarrierSet n   =n+ 3 k  
 
where 0≦k≦283 for 1024 FFT mode and 0≦k≦142 for 512 FFT mode. Additionally, there may be 86 guard sub-carriers on the left and right ends of the spectrum in 1024 FFT mode. In the 512 FFT mode, there may be 42 guard sub-carriers on the left end and 41 guard sub-carriers on the right end.
 
     Typically, when the receiver in the MS is turned on, the MS initially performs symbol timing and carrier frequency synchronization before the MS can detect a preamble sequence. The MS may perform these tasks using a cyclic prefix in the data frame. Thereafter, the MS determines whether a first symbol in the frame is a preamble symbol. If the first symbol is a preamble symbol, then the MS determines which preamble sequence is present in the frame. Once the MS determines the preamble sequence, the MS can associate with a corresponding segment of an appropriate BS. 
     Symbols in preamble sequences (i.e., preamble symbols) typically have higher energy than data symbols. For example, the energy of the preamble symbols is typically 8/3 times (i.e., 4.26 dB higher than) the energy of data symbols. This is useful in distinguishing preamble symbols from data symbols. 
     Additionally, the preamble sequences are almost orthogonal. That is, a cross-correlation between any two preamble sequences is very small. For example, the cross-correlation is typically less than 0.2. This is useful in distinguishing individual preamble sequences from one another. As shown in the table in  FIG. 4 , if the MS detects a preamble sequence having an index 0, then the MS associates with segment  0  of BS having cell ID 0, and so on. 
     Referring now to  FIGS. 5A-5D , a system  160  for detecting a preamble sequence in a mobile station (MS) may be implemented in a physical layer (PHY) module  162  of the MS. The system  160  shown in  FIG. 5A  comprises a differential demodulation module  164  and a cross-correlation module  163 . The cross-correlation module  163  comprises a state detection module  165 , a summing module  166 , and a control module  168  as shown in  FIG. 5A . The differential demodulation module  164  receives an input signal transmitted by a base station (BS). The input signal may be mathematically expressed as follows.
 
 Y[k]=H[k]X   i   [k]+Z[k] 
 
where k is sub-carrier index, i is preamble sequence index, Y[k] is received input signal, H[k] is channel gain, X i [k] is transmit signal, and Z[k] is noise.
 
     When a preamble bit (i.e., a preamble symbol) in a preamble sequence is a binary 0, the corresponding transmit signal X i [k] is +1 or positive. When a preamble bit in a preamble sequence is a binary 1, the corresponding transmit signal X i [k] is −1 or negative. That is, when a preamble bit in a preamble sequence is a 1, the channel phase of the sub-carrier in the transmit signal X i [k] is shifted by π relative to the channel phase of the sub-carrier when a preamble bit in a preamble sequence is 0. 
     Adjacent modulated sub-carriers (i.e., sub-carriers  1 ,  4 ,  7 , etc.) may have similar channel phase or an unknown differential channel phase that is common to all k sub-carriers. The unknown differential channel phase may be caused by presence of a symbol timing offset, which in turn may be caused by improper symbol timing synchronization. When adjacent modulated sub-carriers have similar channel phase, the channel phase difference between adjacent modulated sub-carriers is nearly zero. 
     On the other hand, when adjacent modulated sub-carriers have an unknown differential channel phase that is common to all k sub-carriers, the channel phase difference between adjacent modulated sub-carriers may be non-zero. When adjacent modulated sub-carriers have similar channel phase or unknown differential channel phase common to all sub-carriers, the sub-carriers are generally referred to as “moderately frequency selective channels.” 
     The differential demodulation module  164  performs a differential demodulation operation on the input signal and generates a differentially demodulated signal. Specifically, the differential demodulation module  164  multiplies a modulated sub-carrier by a complex conjugate of an adjacent modulated sub-carrier located three sub-carriers apart. 
     When the adjacent modulated sub-carriers have similar channel phase, the differentially demodulated signal can be mathematically expressed as follows.
 
 M[k]=Y*[k− 3 ]Y[k]=H[k]H*[k− 3 ]D   i   [k]+{tilde over (Z)}[k]≈|H[k]H*[k− 3 ]|D   i   [k]+{tilde over (Z)}[k] 
 
where Y*[k−3] denotes a complex conjugate of a modulated sub-carrier that is three sub-carriers apart from the modulated sub-carrier Y[k]. The complex conjugate is indicated by asterisk or “*”. Since the adjacent modulated sub-carriers have similar channel phase, the channel phase difference between the adjacent modulated sub-carriers is nearly zero. That is,
 
∠( H[k]H*[k− 3])≈0
 
     The control module  168  may store XOR&#39;ed versions of preamble sequences in memory. XOR&#39;ed versions of preamble sequences may also be referred to as derived preamble sequences. An XOR&#39;ed or derived preamble sequence is generated by XORing adjacent bits in the preamble sequence. For example, if one of the 114 preamble sequences in 1024 FFT mode includes bits B 1 , B 2 , B 3 , . . . , B 284 , then an XOR&#39;ed version of that preamble sequence includes bits X 1 , X 2 , X 3 , . . . , X 284 , where X 1 =B 1 ⊕B 2 , X 2 =B 2 ⊕B 3 , etc., where ⊕ denotes an XOR operation. The derived preamble sequences may be mathematically expressed as follows.
 
 D   i   [k]=X   i   [k]X   i   *[k− 3]
 
where the asterisk (i.e., “*”) denotes a complex conjugate. Actually performing complex conjugate operations to generate complex conjugates, however, is unnecessary since complex conjugates of 1 and −1 are 1 and −1, respectively. That is, 1*=1, and (−1)*=−1.
 
     A cross-correlation between the derived preamble sequences is given by the following formula. 
                 max     i   ,     j   ≠   i         ⁢     {         ∑     k   ∈     P     s   ⁡     (   i   )             ⁢         D   i     ⁡     [   k   ]       ⁢       D   j   *     ⁡     [   k   ]               ∑     k   ∈     P     s   ⁡     (   i   )             ⁢              D   i     ⁡     [   k   ]            2         }       ≈     0.1731   ⁢           ⁢   for   ⁢           ⁢   1024   ⁢           ⁢   FFT           
where Ps=set of pilot sub-carriers for segment s except left most pilot sub-carrier, and s(i)=segment number for a preamble sequence index i.  FIG. 6B  shows cross-correlation values normalized by
 
               ∑     k   ∈     P     s   ⁡     (   0   )             ⁢              D   0     ⁡     [   k   ]            2           
for 1024 FFT mode.
 
     For moderately frequency selective channels with high signal-to-noise ratio (SNR), the noise may be very low. That is,
 
 {tilde over (Z)}[k]&lt;&lt;|H[k]H*[k− 3 ]|D   i   [k] 
 
Disregarding the noise, the differentially demodulated signal M[k] may be represented by the following equation.
 
 M[k]=H[k]H*[k− 3 ]D   i   [k]+{tilde over (Z)}[k]≈|H[k]H*[k− 3 ]|D   i   [k] 
 
The sign of the real part of M[k] is the same as the sign of D i [k] in absence of noise and when ∠(H[k]H*[k−3])≈0. That is, when D i [k]=1, M[k] is positive or +1, and when D i [k]=0, M[k] is negative or −1. This may be mathematically expressed as follows.
 
 u ( Re{M[k ]})= u ( Re{D   i   [k ]})
 
where u represents the sign and Re{M[k]} and Re{D i [k]} represent real parts of M[k] and D i [k], respectively. Thus,
 
               u   ⁡     (   x   )       =     {         1           for   ⁢           ⁢   x     ≥   0             0           for   ⁢           ⁢   x     &lt;   0                   
That is, positive values of real part (e.g., x=+1) represent binary 1s, and negative values of real part (e.g., x=−1) represent binary 0s. Hereinafter, the word “sign” may be used interchangeably with (i.e., synonymously as) “state” and/or “polarity.”
 
     Signs or states of M[k] and D i [k] may be used to simplify implementation of preamble detection as follows. Instead of performing a complex operation of calculating correlation values between M[k] and D i [k], a preamble sequence may be detected by performing XOR operations on (i.e., modulo 2 sums of) signs or states of real parts of M[k] and D i [k]. The XOR operations generate values that are equivalent to correlation values that may be generated by performing correlation of M[k] and D i [k] using a correlation module. 
     Since XOR operations are simpler to implement than implementing correlation operations, the XOR operations will be hereinafter referred to as low-complexity cross-correlation. Consequently, values generated by the XOR operations will be hereinafter referred to as low-complexity cross-correlation values or cross-correlation values. 
     The state detection module  165  detects the state or sign of M[k]. The summing module  166  generates the cross-correlation values for each preamble sequence as follows. 
                 C   ~     j     =       ∑     k   ∈     P     s   ⁡     (   i   )             ⁢       u   (     Re   ⁢     {     M   ⁡     [   k   ]       }       )     ⊕     (     1   -     u   ⁡     (       D   j     ⁡     [   k   ]       )         )               
where Ps=set of pilot sub-carriers for segment s except left most pilot sub-carrier, and s(j)=segment number for a preamble sequence index j. u(Re{M[k]}) is the sign or state of the real part of M[k], and u(D j [k]) are the signs or states of the derived preamble sequences. Since the states of M[k] and D j [k] are identical, the XOR of states of M[k] and D j [k] would be zero. Therefore, instead of using states of (D j [k]), the summing module  166  uses states (1−u(D j [k])), which are opposite of states of (D j [k]). That is, (1−u(D j [k])) is essentially the same as inverted u(D j [k]).
 
     For example, if sequence D j [k] is 10110, then (1−u(D j [k])) is 01001. Thus, if received sequence in M[k] is the same as the transmitted sequence, the value of the sum C j {tilde over ( )} (i.e., the cross-correlation value) would be 5. On the other hand, if the received sequence is different from 10110, the value of the sum C j {tilde over ( )} would be less than 5. Stated generally, the cross-correlation value will be largest when a received sequence in the differentially demodulated signal M[k] matches one of differentially demodulated preamble sequences D j [k]. 
     For simplicity, (1−u(D i [k])) may be hereinafter referred to as predetermined states of preamble sequences. The control module  168  may store the states (1−u(D i [k])) for all preamble sequences. Thus, the state detection module  165  does not have to detect the predetermined states of preamble sequences, and the summing module  166  may read the predetermined states of preamble sequences directly from the control module  168  when performing cross-correlation. 
     The control module  168  estimates a preamble sequence index i from the cross-correlation values generated by the summing module  166  as follows. The control module  168  selects the preamble sequence index for which the cross-correlation value with the differentially demodulated signal is largest. This may be mathematically expressed as follows. 
               i   ^     =         arg   ⁢           ⁢   max     j     ⁢     {       C   ^     j     }             
The control module  168  determines that a preamble sequence is detected if the magnitude of the largest cross-correlation value is greater than or equal to a predetermined threshold. The predetermined threshold may be based on the strength of the input signal.
 
     Although the system  160  utilizes low-complexity cross-correlation by performing XOR operations instead of actual correlation operations to detect a preamble sequence, the system  160  may reliably detect the preamble sequence. That is, the system  160  may not mis-detect or fail to detect a preamble sequence. This is because of two reasons: First, the pilot sub-carriers for preamble sequences have higher SNRs than regular data sub-carriers since the pilot sub-carriers have approximately 9 dB higher energy than regular data sub-carriers. This helps in distinguishing a preamble symbol from a data symbol and detecting the preamble symbol thereby reducing probability of mis-detection or detection failure. 
     Second, the preamble sequences have high redundancy, which may help in increasing the accuracy with which preamble sequences may be detected despite using XOR operations instead of actual correlation operations. Specifically, in 1024 FFT mode, less than 7 bits would be sufficient to represent 144 preamble sequences (2 7 =128, which is more than 114). Yet each preamble sequence is 284-bits long wherein 7 bits are encoded to 284 bits thereby leaving many bits redundant. 
     The control module  168  determines which segment transmitted the detected preamble sequence implicitly when the control module  168  detects the preamble sequence. This is because each preamble sequence is unique as identified by a unique preamble sequence index number, and each segment transmits a unique preamble sequence using distinct sub-carriers. 
     Thus, when the control module  168  detects the preamble sequence by selecting the largest cross-correlation value, the control module  168  implicitly selects the segment having the largest channel gain (i.e., the best segment). Thus, when the control module  168  detects the preamble sequence, the control module  168  implicitly detects which segment transmitted the preamble sequence. 
     Occasionally, two or more preamble sequences may generate cross-correlation values that are larger than the rest and approximately the same. For example, when a mobile station is adjacent to a cell boundary or a segment boundary, the mobile station may receive signals including preamble sequences from adjacent base stations. 
     The input signal received by the mobile station at cell or segment boundaries may be a sum of up to three signals that may be transmitted by three different segments of three different base stations. This is mathematically expressed as follows. 
               Y   ⁡     [   k   ]       =         ∑     s   =   0     2     ⁢       H   ⁡     [     k   ,   s     ]       ⁢       X     i   ⁡     (   s   )         ⁡     [   k   ]           +     Z   ⁡     [   k   ]               
where
 
i(s)=preamble sequence index used for segment s,
 
H[k,s]=preamble OFDMA symbol of segment s, and
 
X i(s) [k]=channel gain corresponding to the segment s.
 
     In that case, the summing module  166  may generate cross-correlation values that are similar and larger that the rest due to the presence of two or more preamble sequences in the received signals. Subsequently, the control module  168  may detect two or more cross-correlation values that are similar and that are larger than the rest of the cross-correlation values. 
     Consequently, when using low-complexity cross-correlation, the best segment determined by the control module  168  may or may not in fact be the best segment depending on the channel gain and noise. For example, when channel gain H[k] is very high, say 10, and noise Z[k] is very low, say 0.1, the probability may be very high that the sign of Y[k] may be positive for a binary 0 and negative for a binary 1. In that case, the best segment determined by the control module  168  may in fact be the best segment. 
     On the other hand, when H[k] is very low, say only 1, the signs of Y[k] may still be positive and negative for 0s and 1s, respectively. However, when the channel gain is very low and the noise level is high, the signs of Y[k] may not be positive and negative for 1s and 0s, respectively. That is, high noise levels may invert the sign of Y[k] when channel gain is very low. Sign inversions due to low channel gain and/or high noise may affect the ability of the control module  168  to accurately select the segment since the low-complexity cross-correlation is based on signs of M[k] and D[k]. 
     When two or more cross-correlation values are similar and larger than the rest, the control module  168  may select the correct segment in three ways. In one way, the control module  168  may randomly select one of the two or more cross-correlation values. Thus, the control module  168  selects the segment that corresponds to the preamble sequence that yielded the randomly selected largest cross-correlation value. 
     In another way, the control module  168  may select one of the two or more larger cross-correlation values based on the strength of the input signal. That is, the control module  168  may select the larger cross-correlation value that corresponds to the strongest input signal. This may be mathematically expressed as follows. 
               s   ^     =         arg   ⁢           ⁢   max     s     ⁢     {       ∑     k   ∈     P   s         ⁢          Y   ⁡     [   k   ]              }             
The control module  168  may accurately select the segment based on input signal strength when the noise level is approximately the same for all input signals received from different segments.
 
     In yet another way, the system  160  may include a correlation module  170  that correlates the differentially demodulated input signals and the derived preamble sequences that generated the two or more larger and similar cross-correlation values. An example of such a system  161  implemented in a PHY module  163  of a mobile station is shown in  FIG. 5C . Hereinafter, the differentially demodulated input signals and the derived preamble sequences that generated the two or more larger and similar cross-correlation values will be called selected input signals and selected preamble sequences, respectively. 
     The correlation module  170  correlates the selected input signals with the selected preamble sequences, generates correlation values, and outputs the correlation values to the control module  168 . The correlation values are complex numbers, which have real and imaginary parts. The imaginary parts of the correlation values represent noise in the input signal. 
     When ∠(H[k]H*[k−3])≈0, i.e., when adjacent modulated sub-carriers have similar channel phase, the control module  168  disregards the imaginary parts and selects a correlation value having a largest real part. The control module  168  selects the segment that corresponds to the preamble sequence that generated the correlation value having the largest real part. 
     On the other hand, when the adjacent modulated sub-carriers have an unknown differential channel phase θ common to all k sub-carriers, the control module  168  calculates the magnitude of the largest correlation value instead of selecting the real part of the largest correlation value. That is, the control module  168  does not disregard the imaginary part of the largest correlation value. The control module  168  selects the segment that corresponds to the preamble sequence that generated the correlation value having the largest magnitude. 
     Occasionally, the input signal may comprise a carrier frequency offset (CFO). The CFO may be fractional or integer. An input signal comprising fractional CFO may be mathematically expressed as follows.
 
 Y[k]=C (ε,0) H[k]X[k]+I (ε, k )+ Z[k] 
 
where ε=ΔfNT represents normalized CFO. Fractional CFO introduces Inter-carrier interference (ICI) as given by the following equation.
 
               I   ⁡     (     ɛ   ,   k     )       =       ∑     r   =   1       N   -   1       ⁢       C   ⁡     (     ɛ   ,   r     )       ⁢     H   ⁡     [       (     (     k   -   r     )     )     N     ]       ⁢     X   ⁡     [       (     (     k   -   r     )     )     N     ]                 
where N=total number of sub-carriers in an FFT mode, and
 
     
       
         
           
             
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     Additionally, the fractional CFO decreases signal to noise ratio (SNR) of the input signal. This is mathematically expressed as follows. 
               E   ⁡     [     SNR   ⁡     (     ɛ   ,   k     )       ]       ≈                C   ⁡     (     ɛ   ,   0     )            2     ⁢     SNR   0             (     1   -            C   ⁡     (     ɛ   ,   0     )            2       )     ⁢           ⁢     SNR   0       +   1             
where SNR 0  represents average SNR in absence of CFO. SNR decreases as CFO increases.
 
     Since fractional CFO introduces ICI and attenuates the input signal, the fractional CFO adversely affects preamble sequence detection in system  150 . The fractional CFO, however, does not affect the preamble sequence detection significantly. This is because the fractional CFO adds a phase error that is common to all sub-carriers, which does not change the frequency selectivity of the channels. 
     On the other hand, when the CFO is an integer 1, a phase error θ introduced by the integer CFO may be common to all k sub-carriers. In that case, the input signal may be mathematically expressed as follows.
 
 Y[k]=e   jθ   H [(( k−l )) N   ]X   i [(( k−l )) N   ]+Z [(( k−l )) N ]
 
     Specifically, the integer CFO causes a cyclic shift of the input signal in the frequency domain. In other words, the integer CFO rotates the input signal in the frequency domain. Thus, a transmitted sub-carrier k may be received as a sub-carrier (k−l). θ, however, cancels out since the differential demodulation module  164  generates M[k] by differentially demodulating Y[k]. 
     The summing module  166  generates cross-correlation values using shifted version of input signals as given by the following equation. 
                 C   ~       j   ,   m       =       ∑     k   ∈     P     s   ⁡     (   j   )             ⁢       u   ⁡     (     Re   ⁢     {     M   [       (     (     k   -   m     )     )     N     ]     }       )       ⊕     (     1   -     u   ⁡     (       D   j     ⁡     [   k   ]       )         )               
Subsequently, the control module  168  detects a preamble sequence by selecting a maximum among the cross-correlation values according to the following equation.
 
               (       i   ^     ,     l   ^       )     =         arg   ⁢           ⁢   max       (     j   ,   m     )       ⁢     {          C     j   ,   m            }             
Simultaneously, the control module  168  calculates î, which is an estimated value for the integer CFO.  FIG. 5D  shows maximum cross-correlation values normalized by
 
               ∑     k   ∈     P     s   ⁡     (   0   )             ⁢                D   0     ⁡     [   k   ]            2     .           
The maximum cross-correlation values increase, but not significantly, as integer CFO increases.
 
     Occasionally, the input signal may have a small symbol timing offset due to improper symbol timing synchronization, which is performed when the MS is powered up. The symbol timing offset may cause inter-symbol interference (ISI). Additionally, the symbol timing offset may cause an inter-carrier interference (ICI). The input signal having a symbol timing offset can be mathematically expressed as follows. 
               Y   ⁡     [   k   ]       =         exp   ⁡     (       j2π   ⁢           ⁢   τ   ⁢           ⁢   k     N     )       ⁢     H   ⁡     [   k   ]       ⁢       X   i     ⁡     [   k   ]         +     I   ⁡     (     τ   ,   k     )       +     Z   ⁡     [   k   ]               
where τ represents symbol timing offset and I(τ,k) represents ISI and ICI.
 
     Specifically, the symbol timing offset introduces an extra phase offset among the sub-carriers. The phase offset may increase linearly as the sub-carrier index k increases. Additionally, the extra phase offset introduced by the symbol timing offset appears in the differentially demodulated signal generated by the differential demodulation module  164 . 
     The differentially demodulated signal with the extra phase offset is mathematically expressed as follows. 
               M   ⁡     [   k   ]       =         exp   ⁡     (       j   ⁢           ⁢   6   ⁢   π   ⁢           ⁢   τ     N     )       ⁢     H   ⁡     [   k   ]       ⁢   H   *     [     k   -   3     ]     ⁢       D   i     ⁡     [   k   ]         +       Z   ′     ⁡     [   k   ]               
where 6π=2π*P (P=3 is the spacing between modulated sub-carriers). The accuracy with which the control module  168  detects a preamble sequence and selects a segment is inversely proportional to the linearly increasing phase offset.
 
     Referring now to  FIG. 6 , a method  250  for detecting a preamble sequence when moderately frequency selective channels have substantially the same channel phase begins at step  252 . A differential demodulation module  164  receives an input signal having signals modulated using orthogonal frequency division multiplexing (OFDM) in step  254 . The differential demodulation module  164  differentially demodulates the signals in step  256 . A state detection module  165  detects states (i.e., signs) of the differentially demodulated signals in step  257 . 
     A control module  168  determines in step  258  if an integer carrier frequency offset (CFO) is present in the input signal. If the integer CFO is absent, a summing module  166  correlates (XORs) the states of differentially demodulated signals with predetermined states of preamble sequences in step  260 . If the integer CFO is present, however, the summing module  166  correlates in step  264  the predetermined states of preamble sequences with the states of differentially demodulated signals that include the shift caused by the integer CFO. 
     The control module  168  selects in step  266  a largest cross-correlation value generated by the summing module  166 . The control module  168  checks if the largest cross-correlation value is greater than or equal to a predetermined threshold in step  268 . If false, the control module  168  determines in step  270  that no preamble sequence is detected in the input signal, and the method  250  ends in step  272 . 
     If true, however, the control module  168  checks if two or more cross-correlation values are similar and larger than the rest in step  274 . If false, the control module  168  determines that a preamble sequence is detected in the input signal in step  276 . Since each preamble sequence is unique and since each segment of each base station transmits using distinct sub-carriers, the control module  168  implicitly determines in step  278  which segment of a base station transmitted the detected preamble sequence. The method  250  ends in step  272 . 
     If the result of step  274  is true, then in step  280 , a correlation module  170  correlates the differential input signals and the derived preamble sequences that generated the two or more similar and large cross-correlation values. In step  282 , the control module  168  selects either the real part or the magnitude of the largest of the correlation values generated by the correlation module  170 . The control module  168  implicitly determines in step  278  which segment of a base station transmitted the detected preamble sequence. The method  250  ends in step  272 . 
     Referring now to  FIGS. 7A-7C , various exemplary implementations incorporating the teachings of the present disclosure are shown. For example, the teachings of the present disclosure may be implemented in communication systems based on WiMAX standards, which provide mobile wireless connectivity without the need for a direct line-of-sight with a base station. 
     Referring now to  FIG. 7A , the teachings of the disclosure may be implemented in a WiMAX interface  452  of a vehicle  446 . The vehicle  446  may include a vehicle control system  447 , a power supply  448 , memory  449 , a storage device  450 , and the WiMAX interface  452  and associated antenna  453 . The vehicle control system  447  may be a powertrain control system, a body control system, an entertainment control system, an anti-lock braking system (ABS), a navigation system, a telematics system, a lane departure system, an adaptive cruise control system, etc. 
     The vehicle control system  447  may communicate with one or more sensors  454  and generate one or more output signals  456 . The sensors  454  may include temperature sensors, acceleration sensors, pressure sensors, rotational sensors, airflow sensors, etc. The output signals  456  may control engine operating parameters, transmission operating parameters, suspension parameters, etc. 
     The power supply  448  provides power to the components of the vehicle  446 . The vehicle control system  447  may store data in memory  449  and/or the storage device  450 . Memory  449  may include random access memory (RAM) and/or nonvolatile memory such as flash memory, phase change memory, or multi-state memory, in which each memory cell has more than two states. The storage device  450  may include an optical storage drive, such as a DVD drive, and/or a hard disk drive (HDD). The vehicle control system  447  may communicate externally using the WiMAX interface  452 . 
     Referring now to  FIG. 7B , the teachings of the disclosure can be implemented in a WiMAX interface  468  of a cellular phone  458 . The cellular phone  458  includes a phone control module  460 , a power supply  462 , memory  464 , a storage device  466 , and a cellular network interface  467 . The cellular phone  458  may include the WiMAX interface  468  and associated antenna  469 , a microphone  470 , an audio output  472  such as a speaker and/or output jack, a display  474 , and a user input device  476  such as a keypad and/or pointing device. 
     The phone control module  460  may receive input signals from the cellular network interface  467 , the WiMAX interface  468 , the microphone  470 , and/or the user input device  476 . The phone control module  460  may process signals, including encoding, decoding, filtering, and/or formatting, and generate output signals. The output signals may be communicated to one or more of memory  464 , the storage device  466 , the cellular network interface  467 , the WiMAX interface  468 , and the audio output  472 . 
     Memory  464  may include random access memory (RAM) and/or nonvolatile memory such as flash memory, phase change memory, or multi-state memory, in which each memory cell has more than two states. The storage device  466  may include an optical storage drive, such as a DVD drive, and/or a hard disk drive (HDD). The power supply  462  provides power to the components of the cellular phone  458 . 
     Referring now to  FIG. 7C , the teachings of the disclosure can be implemented in a network interface  494  of a mobile device  489 . The mobile device  489  may include a mobile device control module  490 , a power supply  491 , memory  492 , a storage device  493 , the network interface  494 , and an external interface  499 . The network interface  494  includes a WiMAX interface and an antenna (not shown). 
     The mobile device control module  490  may receive input signals from the network interface  494  and/or the external interface  499 . The external interface  499  may include USB, infrared, and/or Ethernet. The input signals may include compressed audio and/or video, and may be compliant with the MP3 format. Additionally, the mobile device control module  490  may receive input from a user input  496  such as a keypad, touchpad, or individual buttons. The mobile device control module  490  may process input signals, including encoding, decoding, filtering, and/or formatting, and generate output signals. 
     The mobile device control module  490  may output audio signals to an audio output  497  and video signals to a display  498 . The audio output  497  may include a speaker and/or an output jack. The display  498  may present a graphical user interface, which may include menus, icons, etc. The power supply  491  provides power to the components of the mobile device  489 . Memory  492  may include random access memory (RAM) and/or nonvolatile memory such as flash memory, phase change memory, or multi-state memory, in which each memory cell has more than two states. The storage device  493  may include an optical storage drive, such as a DVD drive, and/or a hard disk drive (HDD). The mobile device may include a personal digital assistant, a media player, a laptop computer, a gaming console, or other mobile computing device. 
     Those skilled in the art can now appreciate from the foregoing description that the broad teachings of the disclosure can be implemented in a variety of forms. Therefore, while this disclosure includes particular examples, the true scope of the disclosure 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.