Patent Publication Number: US-9887869-B2

Title: Method of compensating carrier frequency offset in receivers

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit under 35 USC §119(a) of Indian Patent Application No. 201641015928 filed on May 6, 2016, in the Indian Patent Office and Korean Patent Application No. 10-2017-0007241 filed on Jan. 16, 2017, in the Korean Intellectual Property Office, the entire disclosures of which are incorporated herein by reference for all purposes. 
     BACKGROUND 
     1. Field 
     The following description relates to a wireless communication system, and more particularly, to a mechanism for compensating a carrier frequency offset (CFO) in a receiver. 
     2. Description of Related Art 
     In a wireless communication system, signal demodulation in a receiver may be affected by a carrier frequency offset (CFO). In general, a CFO may result from a mismatch between a local oscillator (LO) frequency in a transmitter configured to up-convert a baseband signal to a passband signal and an LO frequency in a receiver configured to down-convert the received passband signal to a baseband signal. For example, such a CFO may occur in a wireless sensor network in addition to a multi-user multi-antenna communication system. 
     Such a mismatch may result in a deviation of values in received samples, and thus demodulation of a given symbol may result in an erroneous decision. Further, the CFO may result in each of samples multiplied by a complex phasor. The CFO may be accumulated in symbol demodulation, and more adversely affect the demodulation as a data length increases. In addition to an influence on demodulation of baseband data, the mismatch may also change desired properties of a received preamble. Thus, synchronization and other operations based on the received preamble may be affected. 
     Thus, there is a desire for a simple and robust mechanism for estimating a CFO and compensating the CFO in a receiver. 
     SUMMARY 
     This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is this Summary intended to be used as an aid in determining the scope of the claimed subject matter. 
     In one general aspect, a method to compensate a carrier frequency offset (CFO) in a receiver, the method includes receiving discrete time samples, obtaining a sample vector from the received discrete time samples, obtaining tentative CFO estimates based on the sample vector, selecting a CFO having a greatest compensation coefficient from the tentative CFO estimates, and compensating the CFO in the received discrete time samples. Each of the tentative CFO estimates is associated with a compensation coefficient. 
     The obtaining of the sample vector may include obtaining sliding vectors from the received discrete time samples, dividing each of the sliding vectors into at least four sub-blocks, obtaining a correlation coefficient for each of the sliding vectors based on the at least four sub-blocks, and selecting, as the sample vector, a sliding vector having a greatest correlation coefficient from the sliding vectors. 
     The correlation coefficient for each of the sliding vectors may be obtained by adding an absolute value of a dot product between a first sub-block and a fourth sub-block and an absolute value of a dot product between a second sub-block and a third sub-block. 
     The obtaining of the tentative CFO estimates may include obtaining angles corresponding respectively to the tentative CFO estimates, and dividing a corresponding angle by a denominator term. 
     The method may further include obtaining a sub-period by identifying similar elements and elements of an opposite polarity being separated by a time duration equal to the sub-period, and obtaining, as the denominator term, a product of a multiplication of a numerical value 2, pi (π), and the sub-period. 
     The obtaining of the angles may include obtaining, as a first angle of the angles, an angle between elements of the sample vector, obtaining, as a second angle of the angles, a negative value opposite to the first angle, and obtaining a remaining angle of the angles excluding the first angle and the second angle based on a preamble angle. The preamble angle refers to an angle between samples separated from one another by a time duration equal to a preamble period. 
     The selecting of the CFO may include obtaining corresponding compensated vectors by compensating the sample vector using the tentative CFO estimates, obtaining compensation coefficients corresponding respectively to the corresponding compensated vectors, and selecting, as the CFO, one of the tentative CFO estimates for a corresponding compensated vector having a greatest compensation coefficient from the tentative CFO estimates. 
     The obtaining of the compensation coefficients may include obtaining first half compensation coefficients corresponding to a first half of the compensation coefficients by correlating first half compensated vectors with an oversampled preamble sequence, and obtaining second half compensation coefficients corresponding to a second half of the compensation coefficients by correlating second half compensated vectors with a cyclic shift sequence. 
     The oversampled preamble sequence may be obtained by repeating each element of a preamble oversampling ratio (OSR). 
     The cyclic shift sequence may be obtained as a cyclic shift of the oversampled preamble sequence by a half-period number of samples. Here, a half-period may be equal to a product of a multiplication of the OSR and a half of a preamble length. 
     A non-transitory computer-readable storage medium may store instructions to cause computing hardware to perform the method. 
     In another general aspect, a receiver comprises a processor configured to compensate a CFO, receive discrete time samples and obtain a sample vector from the received discrete time samples, obtain tentative CFO estimates based on the sample vector and select a CFO having a greatest compensation coefficient from the tentative CFO estimates, and compensate the CFO in the received discrete time samples. Here, each of the tentative CFO estimates is associated with a compensation coefficient. 
     The receiver may further comprise a block detector configured to receive the discrete time samples and obtain the sample vector from the received discrete time samples; a CFO estimator configured to obtain the tentative CFO estimates based on the sample vector and select the CFO having the greatest compensation coefficient from the tentative CFO estimates, wherein each of the tentative CFO estimates is associated with the compensation coefficient; and a CFO compensator configured to compensate the CFO in the received discrete time samples. 
     The processor may obtain sliding vectors from the received discrete time samples and divide each of the sliding vectors into at least four sub-blocks, obtain a correlation coefficient for each of the sliding vectors based on the at least four sub-blocks, and select, as the sample vector, at least one sliding vector having a greatest correlation coefficient from the sliding vectors. 
     The correlation coefficient may be obtained by adding an absolute value of a dot product between a first sub-block and a fourth sub-block and an absolute value of a dot product between a second sub-block and a third sub-block. 
     The CFO estimator may obtain the tentative CFO estimates by obtaining angles corresponding respectively to the tentative CFO estimates and dividing a corresponding angle by a denominator term. 
     The processor may obtain a sub-period by identifying similar elements and elements of an opposite polarity being separated by a time duration equal to the sub-period, and obtain, as the denominator term, a product of a multiplication of a numerical value 2, pi (π), and the sub-period. 
     The processor may obtain an angle between elements of the sample vector as a first angle of the angles and a negative value opposite to the first angle as a second angle of the angles, and obtain a remaining angle of the angles excluding the first angle and the second angle based on a preamble angle. The preamble angle refers to an angle between samples separated from one another by a time duration equal to a preamble period. 
     The processor may obtain corresponding compensated vectors by compensating the sample vector using the tentative CFO estimates, obtain compensation coefficients corresponding respectively to the corresponding compensated vectors, select, as the CFO, one of the tentative CFO estimates for a corresponding compensated vector having a greatest compensation coefficient, and compensate for the received discrete time samples using the CFO. 
     The processor may obtain first half compensation coefficients of the compensation coefficients by correlating first half compensated vectors with an oversampled preamble sequence, and second half compensation coefficients of the compensation coefficients by correlating second half compensated vectors with a cyclic shift sequence. 
     The oversampled preamble sequence may be obtained by repeating each element of a preamble OSR. 
     The cyclic shift sequence may be obtained as a cyclic shift of a distributed preamble sequence by a half-period number of samples. 
     The receiver may be in compliance with an Institute of Electrical and Electronics Engineers (IEEE) 802.15.4q standard. 
     The receiver may further include a memory storing instructions to implement the processor to receive the discrete time samples and obtain the sample vector from the received discrete time samples; obtain the tentative CFO estimates based on the sample vector and select the CFO having the greatest compensation coefficient from the tentative CFO estimates, wherein each of the tentative CFO estimates is associated with the compensation coefficient; and compensate the CFO in the received discrete time samples. 
     Other features and aspects will be apparent from the following detailed description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram illustrating an example of a configuration of a receiver configured to compensate a carrier frequency offset (CFO). 
         FIG. 2  is a flowchart illustrating an example of a method of compensating a CFO. 
         FIG. 3  is a flowchart illustrating an example of a method of obtaining a sample vector using a block detector. 
         FIG. 4  is a flowchart illustrating an example of a method of estimating a CFO using a CFO estimator. 
         FIG. 5  is a diagram illustrating an example of a computing environment to embody a method of compensating a CFO in a receiver. 
     
    
    
     Throughout the drawings and the detailed description, the same reference numerals refer to the same elements. The drawings may not be to scale, and the relative size, proportions, and depiction of elements in the drawings may be exaggerated for clarity, illustration, and convenience. 
     DETAILED DESCRIPTION 
     The following detailed description is provided to assist the reader in gaining a comprehensive understanding of the methods, apparatuses, and/or systems described herein. However, various changes, modifications, and equivalents of the methods, apparatuses, and/or systems described herein will be apparent after an understanding of the disclosure of this application. For example, the sequences of operations described herein are merely examples, and are not limited to those set forth herein, but may be changed as will be apparent after an understanding of the disclosure of this application, with the exception of operations necessarily occurring in a certain order. Also, descriptions of features that are known in the art may be omitted for increased clarity and conciseness. 
     The features described herein may be embodied in different forms, and are not to be construed as being limited to the examples described herein. Rather, the examples described herein have been provided merely to illustrate some of the many possible ways of implementing the methods, apparatuses, and/or systems described herein that will be apparent after an understanding of the disclosure of this application. 
     Terms such as first, second, A, B, (a), (b), and the like may be used herein to describe components. Each of these terminologies is not used to define an essence, order, or sequence of a corresponding component but used merely to distinguish the corresponding component from other component(s). For example, a first component may be referred to as a second component, and similarly the second component may also be referred to as the first component. 
     It should be noted that if it is described in the specification that one component is “connected,” “coupled,” or “joined” to another component, a third component may be “connected,” “coupled,” and “joined” between the first and second components, although the first component may be directly connected, coupled or joined to the second component. In addition, it should be noted that if it is described in the specification that one component is “directly connected” or “directly joined” to another component, a third component may not be present therebetween. Likewise, expressions, for example, “between” and “immediately between” and “adjacent to” and “immediately adjacent to” may also be construed as described in the foregoing. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used herein, the singular forms “a,” “an,” and “the,” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises,” “comprising,” “includes,” and/or “including,” when used herein, specify the presence of stated features, integers, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, operations, elements, components, and/or groups thereof. 
     Unless otherwise defined, all terms, including technical and scientific terms, used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure pertains based on an understanding of the present disclosure. Terms, such as those defined in commonly used dictionaries, are to be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and the present disclosure, and are not to be interpreted in an idealized or overly formal sense unless expressly so defined herein. 
     In an example, a baseband signal x(t) may be up-converted to a higher frequency carrier signal in a transmitter, e.g., the transmitter may up-convert the baseband signal x(t) to the higher frequency carrier signal, and transmit the higher frequency carrier signal. That is, a baseband signal may be up-converted to a desired carrier frequency, for example, f c . The transmitted signal may be represented by Equation 1.
 
 x ′( t )= Re{x ( t ) e   j2πf     c     t }  [Equation 1]
 
     A signal received by a receiver may be down-converted by a carrier frequency f c ′, and represented by Equation 2.
 
 y ( t )=( x ′( t )+ j HT{x ′( t )}) e   −j2πf     c     ′t   [Equation 2]
 
     In Equation 2, HT{x′(t)} denotes a Hilbert transform of a signal x′(t). In addition, f c ′=f c +δf, in which δf denotes a carrier frequency offset (CFO). That is, δf denotes a difference between f c ′ and f c . y(t) may also be represented by Equation 3.
 
 y ( t )= x ( t ) e   −j2πδft   [Equation 3]
 
     In Equation 3, x(t) is multiplied by a phasor, for example, a complex number e −j2πδft , of which an angle increases in proportion to a time or an index of samples in an output of an analog-to-digital converter (ADC). The multiplying of x(t) by the phasor may affect demodulation of data symbols. 
     A CFO may be estimated using a periodicity of a preamble, e.g., a period T may be obtained from a preamble. The period T may be identified from received discrete time samples, and the following operations may be performed to obtain a CFO estimate. 
     Equation 4 may be first calculated.
 
 y ( t )· y *( t−Y )= x ( t ) e   −j2πδft   x *( t−T ) e   j2πδf(t−T)   [Equation 4]
 
     Equation 4 may be represented by Equation 5.
 
 y ( t )· y *( t−T )=| x ( t )| 2   ·e   −2πδfT   [Equation 5]
 
     In Equation 5, an angle of y(t)·y*(t−T) is equal to 2πδfT. Thus, the angle of y(t)·y*(t−T) may be represented by Equation 6.
 
∠{ y ( t )· y *( t−T )}=−2πδ fT   [Equation 6]
 
     Thus, the CFO δf may be represented by Equation 7. 
     
       
         
           
             
               
                 
                   
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     The angle of y(t)·y*(t−T) may have the following range.
 
∠{ y ( t )· y *( t−T )} is from −π to π.
 
     An estimative CFO range may be obtained by substituting a value of T for Equation 7, e.g., in a case of an Institute of Electrical and Electronics Engineers (IEEE) 802.15.4q standard receiver of which a preamble period, or a period of a preamble, is 32, a duration of the preamble may be 32 microseconds (μs). Here, by substituting a value of T, for example, 32 μs (T=32 μs), for Equation 7 and applying a range of the angle of y(t)·y*(t−T), the CFO range may be from −15.625 kilohertz (kHz) to 15.625 kHz (−½T, ½T). 
     However, an IEEE 802.15.4q standard requires that all CFOs be within a ±40 parts-per million (ppm) range, which indicates a range from −192 kHz to +192 kHz. That is, the receiver having the estimated CFO range of −15.625 kHz to 15.625 kHz may not comply with the IEEE 802.15.4q standard. Thus, a conventional method of estimating a CFO using Equation 7 above may not be effective in a case in which the CFO range does not satisfy a requirement of the standard or other similar cases or situations. In addition, although a conventional compensation method using Equation 7 uses continuous time signals, CFO estimation and compensation may be actually performed on discrete time samples of the signals. 
     Examples to be described hereinafter relate to a method of compensating a CFO in a receiver. Although described hereinafter, the examples may be used to overcome a limitation in a range, for example, −15.625 kHz to +15.625 kHz, and expand such a CFO range to a range of −250 kHz to +250 kHz. Thus, the receiver may be compatible with the IEEE 802.15.4q standard. The method may include receiving, by the receiver, discrete time samples. The received discrete time samples may include discrete time samples of continuous time baseband signals. The method may also include obtaining, by the receiver, a sample vector from the received discrete time samples. In addition, the method may also include obtaining, by the receiver, tentative CFO estimates based on the sample vector. Here, each of the tentative CFO estimates may be associated with a compensation coefficient. Further, the method may also include selecting, by the receiver, a CFO having a greatest compensation coefficient from the tentative CFO estimates. Further, the method may also include compensating, by the receiver, the selected CFO in the received discrete time samples. 
     In one example, the obtaining of the sample vector from the received discrete time samples may include obtaining sliding vectors from the received discrete time samples. The method may also include dividing each of the sliding vectors into at least four sub-blocks. Further, the method may also include obtaining at least one correlation coefficient for each of the sliding vectors based on the four sub-blocks. Further, the method may also include selecting, as the sample vector, a sliding vector having a greatest correlation coefficient from the sliding vectors. 
     The correlation coefficient may be obtained by adding an absolute value of a dot product between a first sub-block and a fourth sub-block, and an absolute value of a dot product between a second sub-block and a third sub-block. Each of the tentative CFO estimates may be obtained by dividing a corresponding angle by a product of a multiplication of a numerical value 2, a constant pi (π), and a sub-period. Although described hereinafter, the tentative CFO estimates may be obtained based on 
     
       
         
           
             
               
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     In one example, the selecting of the CFO from the tentative CFO estimates may include the following operations:
         The operations may include compensating the sample vector using the tentative CFO estimates to obtain a plurality of corresponding compensated vectors.   The operations may also include obtaining at least one compensation coefficient corresponding to each of the corresponding compensated vectors.   The operations may also include selecting, from the tentative CFO estimates, one of the tentative CFO estimates as the CFO for a corresponding compensated vector having a greatest compensation coefficient.   The operations may also include compensating for the received discrete time samples using the CFO.       

     In one example, the obtaining of the compensation coefficients may include obtaining a plurality of first half compensation coefficient of the compensation coefficients by correlating a plurality of first half compensated vectors with an oversampled preamble sequence, and obtaining a plurality of second half compensation coefficients of the compensation coefficients by correlating a plurality of second half compensated vectors with a cyclic shift sequence. Although described hereinafter, the receiver may obtain the first half compensation coefficients based on 
                 corr     δ   ⁢           ⁢     f   est         =       ∑     n   =   1     N     ⁢         r     δ   ⁢           ⁢     f     a   x     h         a   x       ⁡     (   n   )       ·       x   p   OS     ⁡     (   n   )             ,       δ   ⁢           ⁢     f   est       ∈     {       δ   ⁢           ⁢     f     a   x     h       ,     h   ∈     [     1   ,   3     ]         }       ,         
and the second half compensation coefficients based on
 
     
       
         
           
             
               
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     In one example, the oversampled preamble sequence may be obtained by repeating each element of a preamble oversampling ratio (OSR). 
     The cyclic shift sequence may be obtained as a cyclic shift of the oversampled preamble sequence by a half-period number of samples. The half-period number may be equal to a product of a multiplication of an OSR and a half of a preamble length. 
     Dissimilar to a conventional mechanism, such a mechanism proposed herein may be used to estimate a CFO and compensate the CFO by overcoming a limitation in a range, for example, −15.625 kHz to +15.625 kHz. 
       FIG. 1  is a diagram illustrating an example of a configuration of a receiver configured to compensate a CFO. A receiver  100  illustrated in  FIG. 1  may be a coherent receiver complying with an IEEE 802.15.4q standard. However, the receiver  100  is not limited to the example receiver, and may be a coherent receiver in a wireless communication system. 
     Referring to  FIG. 1 , the receiver  100  includes a block detector  104 , a CFO estimator  106 , a CFO compensator  108 , a communicator  110 , and a storage  112 . The block detector  104 , the CFO estimator  106 , the CFO compensator  108 , the communicator  110 , and the storage  112  may be coupled to a controller  102 . 
     The block detector  104  coupled to the controller  102  may receive a plurality of discrete time samples. The discrete time samples may include a signal received from a transmitter, for example, a baseband signal. In addition, the block detector  104  may obtain a sample vector from the received discrete time samples. 
     For convenience of description, the following description is based on a preamble sequence of the IEEE 802.15.4q standard. Such a sequence has a 32-bit length, as follows: 
     X p ={1 0−1 0 0−1 0−1 1 0 1 0 0−1 0 1 1 0 1 0 0−1 0 1−1 0 1 0 0 1 0 1} 
     The sequence X p  may be divided into a plurality of blocks, for example, four blocks a x , b x , c x , and d x , as follows: 
     a x ={1 0−1 0 0−1 0−1} 
     b x ={1 0 1 0 0−1 0 1} 
     c x ={1 0 1 0 0−1 0 1} 
     d x ={−1 0 1 0 0 1 0 1} 
     The block a x  may be obtained from first eight elements, for example, a first element through an eighth element. The block b x  may be obtained from next eight elements, for example, a ninth element through 16th element. The block c x  may be obtained from next eight elements, for example, a 17th element through a 24th element. The block d x  may be obtained from next eight elements, for example, a 25th element through a 32th element. Here, a x =−d x , and b x =d x . 
     Such a fact that a x =−d x  and b x =c x  may be used to identify a starting point of the block a x  or the block c x  in a stream of samples including the sequence X p . Although described hereinafter, the block detector  104  may use the fact that a x =−d x  and b x =c x  to obtain the sample vector from the received discrete time samples. 
     Here, in the receiver  100 , an OSR is assumed to be 1. The OSR refers to a ratio between a bit duration or, a symbol duration and a sampling period. The OSR is defined as a ratio between a sampling frequency and a bit rate. The OSR of 1 in the receiver  100  may indicate that a single sample is obtained per received bit. Hereinafter, a term “bit” and a term “symbol” are used interchangeably throughout the present disclosure. In addition, the received discrete time samples are denoted as a vector r. Thus, the received discrete time samples may correspond to a collection of samples r(1), r(2), buff size . Here, buff size  denotes a buffer size of the receiver  100 . 
     In a case that r slide (n) denotes a sliding vector, for example, a plurality of sliding vectors, in a sample n, r slide (n) may be obtained by collecting the received discrete time samples, for example, r(n) through r(n+OSR−1). Under the assumption of the OSR being 1, r slide (n) may include 32 samples. By dividing r slide (n) into at least four sub-blocks, sub-blocks r a (n), r b (n), r c (n), and r d (n) may be obtained. Here, a length of each of the sub-blocks r a (n), r b (n), r c (n), and r d (n) may be 8×OSR. The sub-block r a (n) may be obtained by collecting the first 8×OSR number of samples of r slide (n). The sub-block r b (n) may be obtained by collecting the second 8×OSR number of samples of r slide (n). The sub-block r c (n) may be obtained by collecting the third 8×OSR number of samples of r slide (n). The sub-block r d (n) may be obtained by collecting the fourth 8×OSR number of samples of r slide (n). The sub-blocks r a (n), r b (n), r c (n), and r d (n) may correspond to a x , b x , c x , and d x , respectively. 
     The block detector  104  coupled to the controller  102  may detect a block by performing a sliding window correlation, as represented by Equations 8 and 9.
 
 bd   index =argmax n {σ( n )}  [Equation 8]
 
σ( n )=abs( r   a ( n )· r   d *( n ))+abs( r   b ( n )· r   c *( n ))  [Equation 9]
 
     In Equation 9, r a (n)·r d *(n) denotes an inner product, or a dot product, between r a (n) and r d *(n). σ(n) denotes a correlation coefficient for r slide (n). 
     The sample vector may be obtained as r slide (bd index ) corresponding to σ(bd index ). The sample vector r slide (bd index ) may have a greatest value among all correlation coefficients, for example, σ(n) in which n=1, 2, . . . . Here, r a (bd index )=r d *(bd index ) and r b (bd index )=r c *(bd index ), and thus σ(bd index ) may be greatest among all the correlation coefficients, for example, σ(n) in which n=1, 2, . . . . 
     The sample vector may be used to estimate a CFO. Here, for notational convenience, the sample vector r slide (bd index ) is denoted as r samp , and r samp (k) denotes a kth element of the sample vector. 
     A period of a 32 preamble length in the IEEE 802.15.4q standard is 32 μs. The period may limit a maximum range of the CFO estimates to ±15.625 kHz. However, a range required by the IEEE 802.15.4q standard is ±192 kHz. The maximum range of the CFO estimates not meeting such a standard requirement may be overcome by identifying a sub-period in a preamble, for example, X p . To meet the standard requirement, a 2 μs sub-period may need to be identified in the preamble. The receiver  100  may identify a sub-period, for example, T s  of 2 μs (T s =2 μs), of two bits in; by identifying specific elements of the preamble sequence X p . The identifying of a sub-period will be described in detail hereinafter. 
     The preamble sequence X p  may be as follows: 
     X={1 0−1 0 0−1 0−1 1 0 1 0 0−1 0 1 1 0 1 0 0−1 0 1−1 0 1 0 0 1 0 1} 
     A portion of the elements of X p  may be multiplied by an element, for example, an alternate element, positioned two shifts on a right side of the portion of the elements to generate a positive product. For example, a product of X p (6)× X p (8) is a positive value. For example, −1×−1=+1. In addition, a product of each of X p (9)×X p (11), X p (17)×X p (19), and X p (30)×X p (32) is a positive value. Here, the receiver  100  may obtain a set including an index of each of X p (6), X p (9), X p (17), and X p (30), for example, S p ={6, 9, 17, 30}. 
     In addition, a portion of the elements of X p  may be multiplied by an element, for example, an alternate element, positioned two shifts on a right side of the portion of the elements to generate a negative product, e.g., a product of X p (1)×X p (3) is a negative value. For example, 1×−1=−1. In addition, a product of each of X p (14)×X p (16), X p (22)×X p (24), and X p (25)×X p (27) is a negative value. The receiver  100  may obtain a set including an index of each of X p (1), X p (14), X p (22), and X p (25), for example, S m ={1, 14, 22, 25}. 
     An element corresponding an index included in the set S p  may be separated from an alternate element by T s =2 μs, and an element corresponding to an index included in the set S m  may be separated from an alternate element by T s =2 μs. Thus, the receiver  100  may identify a sub-period of 2 μs between some of the elements in the preamble of the period 32 μs. 
     When each element corresponding to an index included in each of the two sets S p  and S m  is multiplied by each alternate element, eight products may be generated. Although described hereinafter, such eight products may be used to estimate a CFO from the received discrete time samples. 
     When an angle between elements, or samples, having the sub-period of 2 μs is (φ 2 , an estimative CFO range may be expended based on the angle φ 2 . Hereinafter, calculating the angle φ 2  will be described. 
     Referring to Equation 7 above, a CFO associated with φ 2  may be represented by Equation 11. 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ~ 
                     
                     ⁢ 
                     f 
                   
                   = 
                   
                     
                       
                         - 
                         angle 
                       
                       ⁢ 
                       
                         { 
                         
                           φ 
                           2 
                         
                         } 
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 11, φ 2  may be determined based on Equation 12.
 
φ 2 =φ plus +φ minus   [Equation 12]
 
     In Equation 12, φ plus  and φ minus  denote positive products and negative products, respectively. That is, φ plus  denotes products of multiplications between the elements corresponding to the indices included in the set S p  and corresponding alternate elements of the elements corresponding to the indices, and φ minus  denotes products of multiplications between the elements corresponding to the indices included in the set S m  and corresponding alternate elements of the elements corresponding to the indices. φ plus  may be determined based on Equation 13, and φ minus  may be determined based on Equation 14.
 
φ plus =¼Σ nεS     p     r   samp ( n )· r   samp *( n+ 2)  [Equation 13]
 
φ minus =¼Σ nεS     m     r   samp ( n )· r   samp *( n+ 2)  [Equation 14]
 
     In one example, in the receiver  100  using an OSR not being 1, Equation 13 may be represented by Equation 15, and Equation 14 may be represented by Equation 16. 
     
       
         
           
             
               
                 
                   
                     φ 
                     plus 
                   
                   = 
                   
                     
                       1 
                       
                         4 
                         ⨯ 
                         OSR 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           ∈ 
                           
                             S 
                             p 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             r 
                             samp 
                           
                           ⁡ 
                           
                             ( 
                             
                               k 
                               n 
                             
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             samp 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 k 
                                 n 
                               
                               + 
                               
                                 2 
                                 ⨯ 
                                 OSR 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     15 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     φ 
                     minus 
                   
                   = 
                   
                     
                       1 
                       
                         4 
                         ⨯ 
                         OSR 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           ∈ 
                           
                             S 
                             m 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             r 
                             samp 
                           
                           ⁡ 
                           
                             ( 
                             
                               k 
                               n 
                             
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             samp 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 k 
                                 n 
                               
                               + 
                               
                                 2 
                                 ⨯ 
                                 OSR 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ] 
                 
               
             
           
         
       
     
     In one example, an index k n  may be represented by Equation 17.
 
 k   n =1+( n− 1)×OSR  [Equation 17]
 
     In one example, the block detector  104  may operate as follows; however, the operation of the block detector  104  is not limited to the following description. 
     The block detector  104  may obtain the sliding vectors based on the received discrete time samples. The block detector  104  may divide each of the sliding vectors into a plurality of sub-blocks, e.g., the block detector  104  may divide a sliding vector r slide (n) into r a (n), r b (n), r c (n), and r d (n). The block detector  104  may calculate a correlation coefficient for each of the sliding vectors, e.g., the block detector  104  may calculate a correlation coefficient σ(n) for the sliding vector r slide (n) by adding an absolute value of a dot product between r a (n) and r d *(n) (or r d (n)) and an absolute value of a dot product between r b (n) and r c *(n) (or r c (n)). Similarly, the block detector  104  may calculate a correlation coefficient for a remaining sliding vector. Thus, a plurality of correlation coefficients may be calculated. The block detector  104  may determine a maximum value among the correlation coefficients. The maximum value may be, for example, σ(bd index ). In such an example, the block detector  104  may determine r slide (bd index ) to be a sample vector r samp . 
     The block detector  104  may identify similar elements in the sample vector r samp  and elements at an opposite polarity that are separated from the similar elements by a certain time duration in order to identify a sub-period. Here, a product of a multiplication of each of the similar elements and each of the elements at the opposite polarity corresponding to each of the similar elements that is separated from a corresponding similar element by the time duration may be a positive value or a negative value. For example, when r samp  is {1 0−1 0 0−1 0−1 1 0 1 0 0−1 0 1 1 0 1 0 0−1 0 1−1 0 1 0 0 1 0 1}, the block detector  104  may verify that a positive value, for example, 1, may be obtained when each of 6th, 9th, 17th, and 30th elements is multiplied by a corresponding element separate on a right side of each of the 6th, 9th, 17th, and 30th elements by a distance of 2. Also, the block detector  104  may verify that a negative value, for example, −1, may be obtained when each of 1st, 14th, 22th, and 25th elements is multiplied by a corresponding element separate on a right side of each of the 1st, 14th, 22th, and 25th elements by a distance of 2. The block detector  104  may determine the 6th, 9th, 17th, and 30th elements, and the 1st, 14th, 22th, and 25th elements to be the similar elements. Also, the block detector  104  may determine, to be the elements at the opposite polarity, the elements separate from the similar elements on a right side of each of the similar elements by a distance of 2. When the similar elements and the elements at the opposite polarity are identified, the block detector  104  may determine, to be the sub-period, 2 μs corresponding to the distance 2 between each of the similar elements and each of the corresponding elements at the opposite polarity or corresponding to a time duration corresponding to the distance 2. 
     The block detector  104  may determine the sets S p  and S m , e.g., the block detector  104  may determine the set S p  including respective indices of the 6th, 9th, 17th, and 30th elements, S p ={6, 9, 17, 30}, and the set S m  including respective indices of the 1st, 14th, 22th, and 25th elements, S m ={1, 14, 22, 25}. 
     The block detector  104  may calculate φ plus  based on the sample vector and the set S p , and φ minus  based on the sample vector and the set S m , e.g., the block detector  104  may calculate φ plus  based on Equation 13 or 15, and φ minus  based on Equation 14 or 16. The block detector  104  may calculate φ 2  based on φ plus  and φ minus . Hereinafter, the CFO estimator  106  will be described. 
     In one example, the CFO estimator  106  coupled to the controller  102  may obtain a plurality of tentative CFO estimates based on the sample vector. Each of the CFO estimates may be associated with a compensation coefficient. The CFO estimator  106  may select, as a CFO, a tentative CFO estimate having a greatest compensation coefficient from the tentative CFO estimates. 
     A CFO estimate obtained based on Equation 11 using the angle φ 2  may not be sufficiently reliable in, for example, a low signal-to-noise ratio (SNR) regime. However, a reliability of the CFO estimate may be improved by using a greater angle, for example, φ T =φ 32 , between samples separated by a time duration equal to the preamble period. Here, φ 32  may be calculated using the full preamble period and may not need timing information dissimilar to the calculating of φ 2 , and thus the angle φ 32  between the samples separated by the time duration equal to the preamble period, for example, 32 μs, may have a greater reliability compared to φ 2 . 
     In one example, φ 2  may be adjusted based on φ 32 . Although described hereinafter, φ 2  may be adjusted to be φ 2,a     x     h  and φ 2,c     x     h , in which hε[1, 3]. That is, φ 2  may be adjusted to be six angles. Hereinafter, the adjusting of φ 2  will be described in detail. 
     First, φ 32  may be determined based on Equation 18.
 
φ 32   =∠{r   samp ( k   n )· r   samp *( k   n +32×OSR)}  [Equation 18]
 
     In a communication system, the preamble sequence X p  may be repeated by multiple times as specified in the standard, and thus Equation 18 may be maintained. 
     A relationship between φ 2  and φ 32  may be represented by Equation 19.
 
φ 32 +2 nπ=R   a φ 2   [Equation 19]
 
     In Equation 19, 
               R   a     =       T     T   S       =         32   ⁢           ⁢   μ   ⁢           ⁢   S       2   ⁢   μ   ⁢           ⁢   S       =   16.             
In Equation 19, n denotes an integer. To adjust φ 2 , the CFO estimator  106  and/or the controller  102  may calculate values corresponding to n in Equation 19, for example, n 1   a     x   , n 2   a     x    and n 3   a     x   . For example, the CFO estimator  106  and/or the controller  102  may calculate n 1   a     x    based on Equation 20, n 2   a     x    based on Equation 21, and n 3   a     x    based on Equation 22.
 
 n   1   a     x   =round( n   a     x   )−1  [Equation 20]
 
 n   2   a     x   =round( n   a     x   )  [Equation 21]
 
 n   3   a     x   =round( n   a     x   )+1  [Equation 22]
 
     In Equations 20, 21, and 22, n ax  may be determined based on Equation 23. 
     
       
         
           
             
               
                 
                   
                     n 
                     
                       a 
                       x 
                     
                   
                   = 
                   
                     
                       
                         16 
                         ⁢ 
                         
                           φ 
                           2 
                           
                             a 
                             x 
                           
                         
                       
                       - 
                       
                         φ 
                         32 
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     23 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 23, φ 2   a     x   =φ 2 . That is, a first angle φ 2   a     x    may be φ 2 . 
     In addition, to adjust φ 2 , the CFO estimator  106  and/or the controller  102  may calculate values corresponding to n in Equation 19, for example, n 1   c     x   , n 2   c     x   , and n 3   c     x   . For example, the CFO estimator  106  and/or the controller  102  may calculate n 1   c     x    based on Equation 24, n 2   c     x    based on Equation 25, and n 3   c     x    based on Equation 26.
 
 n   1   c     x   =round( n   c     x   )−1  [Equation 24]
 
 n   2   c     x   =round( n   c     x   )  [Equation 25]
 
 n   3   c     x   =round( n   c     x   )+1  [Equation 26]
 
     In Equations 24, 25, and 26, n cx  may be determined based on Equation 27. 
     
       
         
           
             
               
                 
                   
                     n 
                     
                       c 
                       x 
                     
                   
                   = 
                   
                     
                       
                         16 
                         ⁢ 
                         
                           φ 
                           2 
                           
                             c 
                             x 
                           
                         
                       
                       - 
                       
                         φ 
                         32 
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     27 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 27, φ 2   c     x   =φ 2   a     x   . That is, a second angle φ 2   c     x    may be a negative value opposite to the first angle φ 2   a     x   . 
     In one example, Equations 20 through 23 may correspond to the block a x  or φ 2   a     x    described along with the block detector  104 , and Equations 25 through 27 may correspond to the block c x  or φ 2   c     x    described along with the block detector  104 . 
     The tentative CFO estimates may be obtained based on Equation 28. 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         
                           a 
                           x 
                         
                         h 
                       
                     
                     = 
                     
                       
                         - 
                         
                           φ 
                           
                             2 
                             , 
                             
                               a 
                               x 
                             
                           
                           h 
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           s 
                         
                       
                     
                   
                   , 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         
                           c 
                           x 
                         
                         h 
                       
                     
                     = 
                     
                       
                         - 
                         
                           φ 
                           
                             2 
                             , 
                             
                               c 
                               x 
                             
                           
                           h 
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     28 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 28, φ 2,a     x     h  and φ 2,c     x     h  may be determined based on Equation 29 and 30, respectively. 
     
       
         
           
             
               
                 
                   
                     
                       φ 
                       
                         2 
                         , 
                         
                           a 
                           x 
                         
                       
                       h 
                     
                     = 
                     
                       
                         
                           φ 
                           32 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             n 
                             h 
                             
                               a 
                               x 
                             
                           
                           ⁢ 
                           π 
                         
                       
                       16 
                     
                   
                   , 
                   
                     h 
                     ∈ 
                     
                       
                         [ 
                         
                           1 
                           , 
                           3 
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     29 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       
                         2 
                         , 
                         
                           c 
                           x 
                         
                       
                       h 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         
                           φ 
                           32 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             n 
                             h 
                             
                               c 
                               x 
                             
                           
                           ⁢ 
                           π 
                         
                       
                       16 
                     
                   
                   , 
                   
                     h 
                     ∈ 
                     
                       
                         [ 
                         
                           1 
                           , 
                           3 
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     30 
                   
                   ] 
                 
               
             
           
         
       
     
     When n 1   a     x   , n 2   a     x   , n 3   a     x   , n 1   c     x   , n 2   c     x   , and n 3   c     x    are calculated, φ 2  may be adjusted to be the angles, for example, φ 2,a     x     h  and φ 2,c     x     h , based on Equations 29 and 30. In addition, the tentative CFO estimates may be calculated based on the angles. 
     The CFO estimator  106  may operate as follows. However, the operation of the CFO estimator  106  is not limited to the following description. 
     The CFO estimator  106  may obtain, as the first angle, an angle between elements of the sample vector and obtain, as the second angle, a negative value opposite to the first angle, e.g., the CFO estimator  106  may determine the angle φ 2  to be the first angle φ 2   a     x   , and the negative value opposite to the first angle φ 2   a     x    to be the second angle φ 2   c     x   . 
     The CFO estimator  106 , or the controller  102 , may calculate n ax  based on the first angle and the preamble angle φ 32 . An example of n ax  is as follows: 
     
       
         
           
             
               n 
               
                 a 
                 x 
               
             
             = 
             
               
                 
                   16 
                   ⁢ 
                   
                     φ 
                     2 
                     
                       a 
                       x 
                     
                   
                 
                 - 
                 
                   φ 
                   32 
                 
               
               
                 2 
                 ⁢ 
                 π 
               
             
           
         
       
     
     The CFO estimator  106 , or the controller  102 , may calculate a plurality of first integers based on n ax . Examples of the first integers are as follows: n 1   a     x   =round(n a     x   )−1, n 2   a     x   =round(n a     x   ), and n 3   a     x   =round(n a     x   )+1 
     The CFO estimator  106 , or the controller  102 , may calculate n cx  based on the second angle and the preamble angle φ 32 . An example of n cx  is as follows: 
     
       
         
           
             
               n 
               
                 c 
                 x 
               
             
             = 
             
               
                 
                   16 
                   ⁢ 
                   
                     φ 
                     2 
                     
                       c 
                       x 
                     
                   
                 
                 - 
                 
                   φ 
                   32 
                 
               
               
                 2 
                 ⁢ 
                 π 
               
             
           
         
       
     
     The CFO estimator  106 , or the controller  102 , may calculate a plurality of second integers based on n cx . Examples of the second integers are as follows: n 1   c     x   =round(n c     x   )−1, n 2   c     x   =round(n c     x   ), and n 3   c     x   =round(n c     x   )+1 
     The CFO estimator  106  may determine the angles based on the preamble angle φ 32 . Examples of the angles are as follows: 
     
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       a 
                       x 
                     
                   
                   1 
                 
                 = 
                 
                   
                     
                       φ 
                       32 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         n 
                         1 
                         
                           a 
                           x 
                         
                       
                       ⁢ 
                       π 
                     
                   
                   16 
                 
               
               , 
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       a 
                       x 
                     
                   
                   2 
                 
                 = 
                 
                   
                     
                       φ 
                       32 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         n 
                         2 
                         
                           a 
                           x 
                         
                       
                       ⁢ 
                       π 
                     
                   
                   16 
                 
               
               , 
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       a 
                       x 
                     
                   
                   3 
                 
                 = 
                 
                   
                     
                       φ 
                       32 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         n 
                         3 
                         
                           a 
                           x 
                         
                       
                       ⁢ 
                       π 
                     
                   
                   16 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       c 
                       x 
                     
                   
                   1 
                 
                 = 
                 
                   
                     
                       φ 
                       32 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         n 
                         1 
                         
                           c 
                           x 
                         
                       
                       ⁢ 
                       π 
                     
                   
                   16 
                 
               
               , 
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       c 
                       x 
                     
                   
                   2 
                 
                 = 
                 
                   
                     
                       φ 
                       32 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         n 
                         2 
                         
                           c 
                           x 
                         
                       
                       ⁢ 
                       π 
                     
                   
                   16 
                 
               
               , 
               
                 
 
               
               ⁢ 
               and 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               φ 
               
                 2 
                 , 
                 
                   c 
                   x 
                 
               
               3 
             
             = 
             
               
                 
                   
                     φ 
                     32 
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       n 
                       3 
                       
                         c 
                         x 
                       
                     
                     ⁢ 
                     π 
                   
                 
                 16 
               
               . 
             
           
         
       
     
     The CFO estimator  106  may determine the tentative CFO estimates based on the obtained angles and the sub-period, e.g., the CFO estimator  106  may obtain the tentative CFO estimates by dividing each of φ 2,a     x     1 , φ 2,a     x     2 , φ 2,a     x     3 , φ 2,c     x     1 , φ 2,c     x     2 , and φ 2,c     x     3  by 2πT s . 
     Examples of the tentative CFO estimates are as follows: 
     
       
         
           
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     a 
                     x 
                   
                   1 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         a 
                         x 
                       
                     
                     1 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     s 
                   
                 
               
             
             , 
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     a 
                     x 
                   
                   2 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         a 
                         x 
                       
                     
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     
                       s 
                       ⁢ 
                       
                           
                       
                     
                   
                 
               
             
             , 
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     a 
                     x 
                   
                   3 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         a 
                         x 
                       
                     
                     3 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     
                       s 
                       ⁢ 
                       
                           
                       
                     
                   
                 
               
             
             , 
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     c 
                     x 
                   
                   1 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         c 
                         x 
                       
                     
                     1 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     s 
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     c 
                     x 
                   
                   2 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         c 
                         x 
                       
                     
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     s 
                   
                 
               
             
             , 
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   
                     c 
                     x 
                   
                   3 
                 
               
               = 
               
                 
                   - 
                   
                     φ 
                     
                       2 
                       , 
                       
                         c 
                         x 
                       
                     
                     3 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     s 
                   
                 
               
             
           
         
       
     
     Hereinafter, the CFO compensator  108  will be described in detail. 
     In one example, the CFO compensator  108  and/or the controller  102  may compensate the sample vector using the tentative CFO estimates, or at least one CFO estimate, to obtain at least one compensated vector. 
     Compensated vectors, for example, 
                 r     δ   ⁢           ⁢     f     a   x     h         a   x       ⁢           ⁢   and   ⁢           ⁢     r     δ   ⁢           ⁢     f     a   x     h         c   x         ,         
may be obtained from Equations 29 through 32, in which hε[1, 3]. Here, a compensated vector of a kth element is assumed to be
 
                 r     δ   ⁢           ⁢     f     a   x     h         a   x       ⁡     (   k   )       .         
The compensated vector
 
             r     δ   ⁢           ⁢     f     a   x     h         a   x           
may be obtained based on Equation 31, and the compensated vector
 
             r     δ   ⁢           ⁢     f     a   x     h         c   x           
may be obtained based on Equation 32.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         r 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               a 
                               x 
                             
                             h 
                           
                         
                         
                           a 
                           x 
                         
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           r 
                           samp 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⁢ 
                       
                         e 
                         
                           
                             - 
                             j 
                           
                           × 
                           2 
                           × 
                           π 
                           × 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         
                           a 
                           x 
                         
                         h 
                       
                       × 
                       n 
                     
                   
                   , 
                   
                     h 
                     ∈ 
                     
                       [ 
                       
                         1 
                         , 
                         3 
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     31 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     
                       
                         r 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               c 
                               x 
                             
                             h 
                           
                         
                         
                           c 
                           x 
                         
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           r 
                           samp 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⁢ 
                       
                         e 
                         
                           
                             - 
                             j 
                           
                           × 
                           2 
                           × 
                           π 
                           × 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         
                           c 
                           x 
                         
                         h 
                       
                       × 
                       n 
                     
                   
                   , 
                   
                     h 
                     ∈ 
                     
                       [ 
                       
                         1 
                         , 
                         3 
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     32 
                   
                   ] 
                 
               
             
           
         
       
     
     In one example, the CFO estimator  106  and/or the controller  102  may calculate a compensation coefficient corresponding to each of the compensated vectors. The CFO estimator  106  and/or the controller  102  may calculate the compensation coefficient based on Equations 33 and 34. 
     
       
         
           
             
               
                 
                   
                     
                       corr 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             r 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 
                                   a 
                                   x 
                                 
                                 h 
                               
                             
                             
                               a 
                               x 
                             
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         · 
                         
                           
                             x 
                             p 
                             os 
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         est 
                       
                     
                     ∈ 
                     
                       { 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               a 
                               x 
                             
                             h 
                           
                         
                         , 
                         
                           h 
                           ∈ 
                           
                             [ 
                             
                               1 
                               , 
                               3 
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     33 
                   
                   ] 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         corr 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             est 
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             
                               r 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   f 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       c 
                                       x 
                                     
                                   
                                   h 
                                 
                               
                               
                                 c 
                                 x 
                               
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                           · 
                           
                             
                               x 
                               
                                 p 
                                 + 
                                 
                                   16 
                                   ⁢ 
                                   OSR 
                                 
                               
                               os 
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                       ∈ 
                       
                         { 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               
                                 c 
                                 x 
                               
                               h 
                             
                           
                           , 
                           
                             h 
                             ∈ 
                             
                               [ 
                               
                                 1 
                                 , 
                                 3 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     34 
                   
                   ] 
                 
               
             
           
         
       
     
     A length of a vector corr δf     est    may be equal to the number of the tentative CFO estimates. In Equations 33 and 34, x p   os (n) denotes an oversampled preamble sequence, and x p+16OSR   OS  denotes an oversampled preamble that is circularly or cyclically shifted by half a length of the oversampled preamble sequence. 
     In addition, the CFO estimator  106  and/or the controller  102  may select, as the CFO, one of the tentative CFO estimates. For example, the CFO estimator  106  and/or the controller  102  may select, as the CFO, a tentative CFO estimate for a corresponding compensated vector having a greatest compensation coefficient from the tentative CFO estimates. 
     The selected CFO, δf estimate , may be represented by Equation 35. 
     
       
         
           
             
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 f 
                 estimate 
               
             
             = 
             
               
                 argmax 
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       est 
                     
                   
                   ∈ 
                   
                     { 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           
                             a 
                             x 
                           
                           h 
                         
                       
                       , 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               c 
                               x 
                             
                             h 
                           
                           ⁢ 
                           h 
                         
                         ∈ 
                         
                           [ 
                           
                             1 
                             , 
                             3 
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
               ⁢ 
               
                 { 
                 
                   corr 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       est 
                     
                   
                 
                 } 
               
             
           
         
       
     
     In one example, the controller  102  coupled to the CFO compensator  108  may compensate the CFO in the received discrete time samples. However, examples are not limited to the example described in the foregoing, and the CFO compensator  108  may compensate the CFO in the received discrete time samples. The range hε[1,3] may be expanded for a greater reliability. The range hε[1,3] may be used solely for the purpose of description, and thus the range is not limited to the example described in the foregoing. For example, in Equations 20 through 23, n 1   a     x   =round(n a     x   −2, n 2   a     x   =round(n a     x   )−1, n 3   a     x   =round(n a     x   ), n 4   a     x   =round(n a     x   )+1, and =n 5   a     x   =round(n a     x   )+2 may be used in place of n 1   a     x   , n 2   a     x   , and n 3   a     x   . Similarly, in a case that hε[1,5], n 1   c     x   , n 2   c     x   , n 3   c     x   , n 4   c     x   , and n 5   c     x    may be used. In the case that hε[1,3], six tentative CFO estimates may be determined. In the case that the hε[1,5], ten tentative CFO estimates may be determined. In addition, h may include an odd number of elements, for example, h={1, 2, 3}, h={1, 2, 3, 4, 5}, and h={1, 2, 3, 4, 5, 6, 7}. 
     When the CFO δf estimate  is obtained, the CFO may be compensated based on r comp (n)=r(n)e −j×2×π×δf     estimate     ×n . Here, r comp (n) denotes an nth element of the compensated received discrete time samples r comp . 
     In one example, the communicator  110  of the receiver  100  may provide a communication interface among the controller  102 , the block detector  104 , the CFO estimator  106 , and the CFO compensator  108 , and the storage  112 . 
     The storage  112  may include at least one computer-readable storage medium. The storage  112  may include nonvolatile storage elements. 
       FIG. 2  is a flowchart illustrating an example of a method  200  of compensating a CFO. Referring to  FIG. 2 , in operation  202 , the method  200  includes receiving discrete time samples. In one example, in the method  200 , the controller  102  and/or the communicator  110  of the receiver  100  may receive the discrete time samples. 
     In operation  204 , the method  200  includes obtaining a sample vector from the received discrete time samples. In one example, in the method  200 , the controller  102  and/or the block detector  104  of the receiver  100  may obtain the sample vector from the received discrete time samples. 
     In operation  206 , the method  200  includes obtaining a plurality of tentative CFO estimates based on the sample vector. Each of the tentative CFO estimates may be associated with a compensation coefficient. In one example, in the method  200 , the controller  102  and/or the CFO estimator  106  may obtain the tentative CFO estimates based on the sample vector. 
     In operation  208 , the method  200  includes selecting a CFO having a greatest compensation coefficient from the tentative CFO estimates. In one example, in the method  200 , the controller  102  and/or the CFO estimator  106  may select the CFO having the greatest compensation coefficient from the tentative CFO estimates. 
     In operation  210 , the method  200  includes compensating the selected CFO in the received discrete time samples. In one example, in the method  200 , the controller  102  and/or the CFO compensator  108  may compensate the selected CFO in the received discrete time samples. 
     The descriptions provided with reference to  FIG. 1  may be applicable to the operations described with reference to  FIG. 2 , and thus a more detailed and repeated description will be omitted here for brevity. 
       FIG. 3  is a flowchart illustrating an example of a method  300  of obtaining a sample vector using a block detector. Referring to  FIG. 3 , in operation  302 , the method  300  includes assigning  1  to n (n=1) with respect to received discrete time samples in a detection window associated with the block detector  104  of the receiver  100 . In one example, in the method  300 , the controller  102  and/or the block detector  104  of the receiver  100  may assign  1  to n (n=1) with respect to the received discrete time samples in the detection window. 
     In operation  304 , the method  300  includes obtaining r slide (n) from the received discrete time samples. In one example, in the method  300 , the controller  102  and/or the block detector  104  may obtain r slide (n) from the received discrete time samples. 
     In operation  306 , the method  300  includes obtaining at least four sub-blocks r a (n), r b (n), r c (n), and r d (n) by dividing each sliding vector into at least four sub-blocks. That is, the method  300  may include dividing r slide (n) and obtaining r a (n), r b (n), r c (n), and r d (n). In an example, in the method  300 , the controller  102  and/or the block detector  104  may obtain the four sub-blocks r a (n), r b (n), r c (n), and r d (n). 
     In operation  308 , the method  300  includes calculating a correlation σ(n)=abs(r a  (n)·r d *(n))+abs(r b  (n)·r c *(n)). In one example, in the method  300 , the controller  102  and/or the block detector  104  may calculate the correlation σ(n)=abs r a ·r d *(n))+abs(r b  (n)·r c *(n)). 
     In operation  310 , the method  300  includes determining whether n is greater than the detection window (n&gt;detection window). In operation  312 , in response to n being greater than the detection window, the method  300  includes assigning n+1 to n (n=n+1). 
     In operation  314 , in response to n not being greater than the detection window, the method  300  includes obtaining bd index =argmax n {σ(n)}. 
     In operation  316 , the method  300  includes obtaining a sample vector r slide (bd index ). 
       FIG. 4  is a flowchart illustrating an example of a method  400  of estimating a CFO using a CFO estimator. The method  400  of estimating a CFO to be described hereinafter may be performed by the controller  102 . However, examples may not be limited to the illustrated example, and thus the method  400  may be performed by the CFO estimator  106  and/or the controller  102 . 
     Referring to  FIG. 4 , in operation  402 , the method  400  includes calculating 
     
       
         
           
             
               
                 φ 
                 
                   2 
                   , 
                   
                     a 
                     x 
                   
                 
                 h 
               
               = 
               
                 
                   
                     φ 
                     32 
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       n 
                       h 
                       
                         a 
                         x 
                       
                     
                     ⁢ 
                     π 
                   
                 
                 16 
               
             
             , 
             
               
                 φ 
                 
                   2 
                   , 
                   
                     c 
                     x 
                   
                 
                 h 
               
               = 
               
                 
                   
                     φ 
                     32 
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       n 
                       h 
                       
                         c 
                         x 
                       
                     
                     ⁢ 
                     π 
                   
                 
                 16 
               
             
             , 
             
               h 
               ∈ 
               
                 
                   [ 
                   
                     1 
                     , 
                     3 
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     Here, φ 2,a     x     h  denotes angles corresponding to a sub-block a x  or φ 2   a     x    in a case that hε[1,3], and φ 2,c     x     h  denotes angles corresponding to a sub-block c x  or φ 2   c     x    in the case that hε[1,3]. 
     In operation  404 , the method  400  includes obtaining compensated signals 
                 r     δ   ⁢           ⁢     f     a   x     h         a   x       ⁡     (   n   )       ⁢           ⁢   and   ⁢           ⁢         r     δ   ⁢           ⁢     f     a   x     h         c   x       ⁡     (   n   )       .           
Here, hε[1,3].
 
     In operation  406 , the method  400  includes performing the following two sets of correlation. 
     
       
         
           
             
               
                 
                   
                     
                       
                         corr 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             est 
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             
                               r 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   f 
                                   
                                     a 
                                     x 
                                   
                                   h 
                                 
                               
                               
                                 a 
                                 x 
                               
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                           · 
                           
                             
                               x 
                               p 
                               os 
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                       ∈ 
                       
                         { 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               
                                 a 
                                 x 
                               
                               h 
                             
                           
                           , 
                           
                             h 
                             ∈ 
                             
                               [ 
                               
                                 1 
                                 , 
                                 3 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
             
             
               
                 
                   
                     
                       corr 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             r 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     c 
                                     x 
                                   
                                 
                                 h 
                               
                             
                             
                               c 
                               x 
                             
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         · 
                         
                           
                             x 
                             
                               p 
                               + 
                               
                                 16 
                                 ⁢ 
                                 OSR 
                               
                             
                             os 
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         est 
                       
                     
                     ∈ 
                     
                       { 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               c 
                               x 
                             
                             h 
                           
                         
                         , 
                         
                           h 
                           ∈ 
                           
                             [ 
                             
                               1 
                               , 
                               3 
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     In a case that OSR is 1 (OSR=1), the following two sets of correlation may be performed. 
     
       
         
           
             
               
                 
                   
                     
                       
                         corr 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             est 
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             
                               r 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   f 
                                   
                                     a 
                                     x 
                                   
                                   h 
                                 
                               
                               
                                 a 
                                 x 
                               
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                           · 
                           
                             
                               x 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                       ∈ 
                       
                         { 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               
                                 a 
                                 x 
                               
                               h 
                             
                           
                           , 
                           
                             h 
                             ∈ 
                             
                               [ 
                               
                                 1 
                                 , 
                                 3 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
             
             
               
                 
                   
                     
                       corr 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           est 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             r 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     c 
                                     x 
                                   
                                 
                                 h 
                               
                             
                             
                               c 
                               x 
                             
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         · 
                         
                           
                             x 
                             
                               p 
                               + 
                               16 
                             
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         est 
                       
                     
                     ∈ 
                     
                       { 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               c 
                               x 
                             
                             h 
                           
                         
                         , 
                         
                           h 
                           ∈ 
                           
                             [ 
                             
                               1 
                               , 
                               3 
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     In operation  408 , the method  400  includes obtaining a CFO estimate 
     
       
         
           
             
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 f 
                 estimate 
               
             
             = 
             
               
                 argmax 
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       est 
                     
                   
                   ∈ 
                   
                     { 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           f 
                           
                             a 
                             x 
                           
                           h 
                         
                       
                       , 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             
                               c 
                               x 
                             
                             h 
                           
                           ⁢ 
                           h 
                         
                         ∈ 
                         
                           [ 
                           
                             1 
                             , 
                             3 
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
               ⁢ 
               
                 
                   { 
                   
                     corr 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         est 
                       
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
       FIG. 5  is a diagram illustrating an example of a computing environment  502  to embody a method of compensating a CFO in a receiver. Referring to  FIG. 5 , the computing environment  502  includes at least one processing unit (PU) including a controller  504  and an arithmetic logic unit (ALU)  506 , a memory  510 , a storage  512 , a plurality of networking devices  516 , and a plurality of input and output (I/O) devices  514 . The PU  508  may process instructions for the method. The PU  508  may receive commands from the controller  504  to perform processing. In addition, any logical and arithmetical operations involved in execution of the instructions may be computed or calculated with a help of the ALU  506 . 
     The overall computing environment  502  may include a plurality of homogeneous or heterogeneous cores, different types of central processing units (CPUs), special media, and other accelerators. The PU  508  may process the instructions for the method. In addition, a plurality of PUs may be positioned on a single bit or over multiple bits. 
     A method including codes and instructions needed for implementation or execution may be stored in either the memory  510  or the storage  512 , or both the memory  510  and the storage  512 . At a time of the execution, a necessary instruction may be retrieved from a corresponding memory or storage, and executed by the PU  508 . 
     The controller, the block detector, the CFO estimator, the CFO compensator, the communicator, the computing environment, and the processor illustrated in  FIGS. 1 and 5  that perform the operations described herein with respect to  FIGS. 2, 3, and 4  are implemented by hardware components. Examples of hardware components include controllers, sensors, generators, drivers, and any other electronic components known to one of ordinary skill in the art. In one example, the hardware components are implemented by one or more processors or computers. A processor or computer is implemented by one or more processing elements, such as an array of logic gates, a controller and an arithmetic logic unit, a digital signal processor, a microcomputer, a programmable logic controller, a field-programmable gate array, a programmable logic array, a microprocessor, or any other device or combination of devices known to one of ordinary skill in the art that is capable of responding to and executing instructions in a defined manner to achieve a desired result. In one example, a processor or computer includes, or is connected to, one or more memories storing instructions or software that are executed by the processor or computer. Hardware components implemented by a processor or computer execute instructions or software, such as an operating system (OS) and one or more software applications that run on the OS, to perform the operations described herein with respect to  FIGS. 2, 3, and 4 . The hardware components also access, manipulate, process, create, and store data in response to execution of the instructions or software. For simplicity, the singular term “processor” or “computer” may be used in the description of the examples described herein, but in other examples multiple processors or computers are used, or a processor or computer includes multiple processing elements, or multiple types of processing elements, or both. In one example, a hardware component includes multiple processors, and in another example, a hardware component includes a processor and a controller. A hardware component has any one or more of different processing configurations, examples of which include a single processor, independent processors, parallel processors, single-instruction single-data (SISD) multiprocessing, single-instruction multiple-data (SIMD) multiprocessing, multiple-instruction single-data (MISD) multiprocessing, and multiple-instruction multiple-data (MIMD) multiprocessing. 
     The methods illustrated in  FIGS. 2, 3, and 4  that perform the operations described in this application are performed by computing hardware, for example, by one or more processors or computers, implemented as described above executing instructions or software to perform the operations described in this application that are performed by the methods. For example, a single operation or two or more operations may be performed by a single processor, or two or more processors, or a processor and a controller. One or more operations may be performed by one or more processors, or a processor and a controller, and one or more other operations may be performed by one or more other processors, or another processor and another controller. One or more processors, or a processor and a controller, may perform a single operation, or two or more operations. 
     Instructions or software to control computing hardware, for example, one or more processors or computers, to implement the hardware components and perform the methods as described above may be written as computer programs, code segments, instructions or any combination thereof, for individually or collectively instructing or configuring the one or more processors or computers to operate as a machine or special-purpose computer to perform the operations that are performed by the hardware components and the methods as described above. In one example, the instructions or software include machine code that is directly executed by the one or more processors or computers, such as machine code produced by a compiler. In another example, the instructions or software includes higher-level code that is executed by the one or more processors or computer using an interpreter. The instructions or software may be written using any programming language based on the block diagrams and the flow charts illustrated in the drawings and the corresponding descriptions in the specification, which disclose algorithms for performing the operations that are performed by the hardware components and the methods as described above. 
     The instructions or software to control a processor or computer to implement the hardware components and perform the methods as described above, and any associated data, data files, and data structures, are recorded, stored, or fixed in or on one or more non-transitory computer-readable storage media. Examples of a non-transitory computer-readable storage medium include read-only memory (ROM), random-access memory (RAM), flash memory, CD-ROMs, CD-Rs, CD+Rs, CD-RWs, CD+RWs, DVD-ROMs, DVD-Rs, DVD+Rs, DVD-RWs, DVD+RWs, DVD-RAMs, BD-ROMs, BD-Rs, BD-R LTHs, BD-REs, magnetic tapes, floppy disks, magneto-optical data storage devices, optical data storage devices, hard disks, solid-state disks, and any device known to one of ordinary skill in the art that is capable of storing the instructions or software and any associated data, data files, and data structures in a non-transitory manner and providing the instructions or software and any associated data, data files, and data structures to a processor or computer so that the processor or computer can execute the instructions. In one example, the instructions or software and any associated data, data files, and data structures are distributed over network-coupled computer systems so that the instructions and software and any associated data, data files, and data structures are stored, accessed, and executed in a distributed fashion by the processor or computer. 
     While this disclosure includes specific examples, it will be apparent to one of ordinary skill in the art that various changes in form and details may be made in these examples without departing from the spirit and scope of the claims and their equivalents. The examples described herein are to be considered in a descriptive sense only, and not for purposes of limitation. Descriptions of features or aspects in each example are to be considered as being applicable to similar features or aspects in other examples. Suitable results may be achieved if the described techniques are performed in a different order, and/or if components in a described system, architecture, device, or circuit are combined in a different manner, and/or replaced or supplemented by other components or their equivalents. Therefore, the scope of the disclosure is defined not by the detailed description, but by the claims and their equivalents, and all variations within the scope of the claims and their equivalents are to be construed as being included in the disclosure.