Patent Publication Number: US-2023155624-A1

Title: Transceiver and operating method thereof

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This U.S. non-provisional patent application claims priority under 35 U.S.C. 119 to Korean Patent Application No. 10-2021-0155771 filed on Nov. 12, 2021 in the Korean Intellectual Property Office and Korean Patent Application No. 10-2022-0007892 filed on Jan. 19, 2022 in the Korean Intellectual Property Office, the disclosures of which are incorporated by reference in their entireties herein. 
     1. TECHNICAL FIELD 
     The present disclosure relates to a transceiver and an operating method thereof. 
     2. DISCUSSION OF THE RELATED ART 
     Orthogonal frequency-division multiplexing (OFDM) is a technology that may be used for next-generation communication including long-term evolution (LTE), which is used to mobile, fixed, and portable broadband access. In OFDM, multiple closely spaced orthogonal subcarrier signals with overlapping spectra are transmitted to carry data in parallel. 
     However, in OFDM, there is a large out-of-band emission (OOBE) phenomenon in which a signal flows into an adjacent channel. Accordingly, a guard band may be used to reduce an influence of interference between adjacent channels. However, the guard band uses an empty frequency band, and thus, efficiency may be decreased when using OFDM. 
     SUMMARY 
     At least one embodiment of the present disclosure provides a transceiver in which an out-of-band emission (OOBE) phenomenon is decreased. 
     At least one embodiment of the present disclosure provides an operating method of a transceiver in which an OOBE phenomenon is decreased. 
     According to an embodiment of the present inventive concept, there is provided a transceiver including, a transmitter up-converting a frequency of input data to transmit data of a first frequency, an orthogonal basis generator receiving the input data from the transmitter to generate an orthogonal basis; and a receiver receiving the orthogonal basis from the orthogonal basis generator and receiving the transmitted data for down-converting a frequency of the received data to a second frequency different from the first frequency. The orthogonal basis generator includes, a multiplier multiplying the input data by a Gaussian characteristic value obtained by extracting a Gaussian characteristic of the input data to generate an out of band emission (OOBE) value, and a polynomial block generating the orthogonal basis by using the OOBE value as an input to an Itô-Hermite polynomial. 
     According to an embodiment of the present inventive concept, there is provided a transceiver up-converting a frequency of input data to transmit data of a first frequency, and down-converting the first frequency to a second frequency different from the first frequency and receiving data having the second frequency. The transceiver includes an orthogonal basis generator receiving the input data. The orthogonal basis generator includes, a multiplier multiplying the input data by a Gaussian characteristic value obtained by extracting a Gaussian characteristic of the data to generate an OOBE value, and a polynomial block generating the orthogonal basis by using the OOBE value as an input to an Itô-Hermite polynomial. 
     According to an embodiment of the present inventive concept, there is provided an operating method of a transceiver. The method includes, up-converting a frequency of input data to transmit data of a first frequency, through a transmitter, generating an orthogonal basis based on the input data received from the transmitter through an orthogonal basis generator, and receiving the orthogonal basis from the orthogonal basis generator and the transmitted data to downscale a frequency of the transmitted data to a second frequency different from the first frequency, through a receiver. The generating of the orthogonal basis includes multiplying the input data by a Gaussian characteristic value obtained by extracting a Gaussian characteristic of the data to generate an OOBE value, through a multiplier in the orthogonal basis generator; and generating the orthogonal basis by using the OOBE value as an input to an Itô-Hermite polynomial, through a polynomial block in the orthogonal basis generator. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects and features of the present disclosure will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings, in which: 
         FIG.  1    is a diagram for describing a transceiver according to an exemplary embodiment. 
         FIG.  2    is a diagram for describing an orthogonal basis generator according to an exemplary embodiment. 
         FIGS.  3  and  4    are illustrative diagrams for describing an adaptive filter according to an exemplary embodiment. 
         FIG.  5    is illustrative graphs illustrating power spectral densities (PSDs) of the transceiver according to an exemplary embodiment. 
         FIG.  6    is illustrative graphs illustrating a mean square error (MSE) of the transceiver according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Out-of-band emission (OOBE) may be largely generated due to an orthogonal frequency-division multiplexing (OFDM) signal generating process and/or nonlinearity of a power amplifier (PA). 
     First, the OOBE generated in the OFDM signal generating process will be described. OFDM has high spectral efficiency by using orthogonality in a frequency domain, and is effective for channel fading. However, the OFDM has a large side lobe by each sub-carrier signal, which may cause the OOBE. Spectrum shaping filtering, guard band insertion, and/or precoding techniques at a transmitting end in a transceiver may be performed for generating a transmission signal to prevent OOBE generated in the OFDM signal generating process. 
     Then, the OOBE generated due to a power amplifier will be described. The nonlinearity of the power amplifier may be offset using a digital pre-distortion (DPD) technique to prevent OOBE generated due to the power amplifier. However, the digital pre-distortion technique uses the inverse of a power amplifier response, and requires calculation or calibration of the inverse according to the power amplifier connected to each transmitting end. 
     An adaptive filter may be used as a potential OOBE removal technology. A signal flowing into an adjacent channel needs to be removed up to a noise level to prevent OOPE. An adaptive filter may use an algorithm for tracking a time invariant filter coefficient. However, when OOBE is considered at a receiving end in a transceiver, nonlinearity may occur due to a difference between an up-converted frequency of a transmitting end and a down-converted frequency of the receiving end since a value may change over time. 
     In addition, a nonlinearity effect due to the power amplifier has also an influence on a signal at the receiving end, and accordingly, the effect of the power amplifier needs to be removed through modeling. 
     In a transceiver according to an embodiment of the present disclosure, which is one transceiver including a transmitting end and a receiving end having different frequency bands, a method of removing OOBE using a signal reflecting information on a difference between a frequency transmitted from the transmitting end and a frequency received from the receiving end in a transmission signal or transmission data with a basis function of an adaptive filter is proposed. 
     In particular, a transceiver according to an embodiment having an improved convergence speed and better performance at a convergence value using an Itô-Hermite polynomial having orthogonality between orders of the basis function when data transmitted from the transmitting end has a Gaussian distribution as a statistical characteristic of the data, as the basis function of the adaptive filter in the present disclosure, will be described. 
       FIG.  1    is a diagram for describing a transceiver according to an exemplary embodiment. 
     Referring to  FIG.  1   , a transceiver  10  according to an exemplary embodiment includes a transmitting end and a receiving end. The transmitting end includes a transmitter  100 , a multiplier  110 , a power amplifier (PA)  130 , an oscillator  120 , and a transmitting antenna  140 . The receiving end includes a receiver  200 , a multiplier  210 , an oscillator  220 , and a receiving antenna  240 ; and an orthogonal basis generator  300 . The multiplier  110  may be implemented by a logic circuit. 
     The transmitter  100  includes a signal generator  101  generating data x[n] and a processor  102  performing pulse shaping on the data x[n]. 
     The transmitter  100  may pulse-shape the data (x[n]) according to Equation 1. 
         x   p [ n ]=( g   t   *x )[ n ]=Σ l=0   L   g   t [ l ] x [ n− 1],  x   b ( t )=Σ n   x   p [ n ]sinc( Wt−n )  [Equation 1].
 
     In Equation 1, g t [ ] is a pulse shaping filter function, W is a bandwidth, x p  is a discrete signal of the pulse-shaped data, and x b [ ] is a continuous signal of the pulse-shaped data. 
     The data x b (t) pulse-shaped through the transmitter  100  may be transferred to the multiplier  110 . The multiplier  110  may up-sample the pulse-shaped data x b (t) using an output of the oscillator  120 , and the up-sampled data may be amplified to amplified data x tx (t) as expressed in Equation 2 through the power amplifier  130 . 
         x   tx ( t )=ƒ PA (Real( x   b ( t ) e   −j2πƒ     up     t ))  [Equation 2].
 
     In Equation 2, f PA  is an operation function of the power amplifier  130 , and an ideal power amplifier may have characteristics of a linear function with a gain of G, such as f PA (x)=Gx. f up  may be a frequency up-converted or upscaled from a frequency of data x[n]. 
     The amplified data x tx (t) may be transmitted to the receiving antenna  240  via a channel (e.g., a wireless channel) through the transmitting antenna  140 . In this case, it is assumed that the channel is wide enough to track a channel coherence time in which an influence of the channel does not significantly change for a predetermined time. The transmitter  100 , the multiplier  110 , the oscillator  120 , and the power amplifier  130  may together perform an OFDM operation. 
     The amplified data x tx (t) transmitted through the channel may include noise. Data y rx (t) received under an influence of the noise may be expressed in a convolutional form as in Equation 3. 
         y   rx ( t )=( h*x   tx )( t )+ v ( t )  [Equation 3].
 
     In Equation 3, h is a channel, and v is a white Gaussian noise signal. An average of the white Gaussian noise signals may be 0, and a variance value may be determined depending on a magnitude of a noise level. 
     To emphasize nonlinearity due to a difference between the up-converted frequency f up  and a down-converted (or downscaled) frequency, a delay channel (hd(t)=δ(t−t 0 )) may be assumed. In addition, assuming an ideal power amplifier, the received data (y rx (t)) received through the receiving antenna  240  according to Equation 3 may be converted into data y dc (t) having a down-converted frequency through the multiplier  210  and an output of the oscillator  220 . For example, a downscaling or a down-converting may be performed to convert the received data. The data y dc (t) having the down-converted frequency f dw  may be generated through Equation 4. In an embodiment, the down-converted frequency f dw  and the up-converted frequency f up  are different from each other. 
         y   dc ( t )= y   rx ( t ) e   j2πf     dw     t   [Equation 4].
 
     The data y dc (t) having the down-converted frequency f dw  may include a portion having no frequency information and a portion having a frequency twice the up-converted frequency f up , as expressed in Equation 5. In this case, in Equation 5, it is assumed that the down-converted frequency f dw  and the up-converted frequency f up  are the same as each other. 
     
       
         
           
             
               
                 
                   
                     
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     In Equation 5, x b   I (t),x b   Q (t) indicates a real number part and an imaginary number part of x b (t), respectively, and x b *(t) indicates a complex conjugate of x b (t). 
     Referring to Equation 5, the down-converted data y dc (t) is expressed in the form of a convolution of the delay channel multiplied by a constant and a transmission signal. When this signal passes through a low pass filter, the transmission signal may remain. 
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                         b 
                       
                       ( 
                       t 
                       ) 
                     
                     2 
                   
                   ⁢ 
                   
                     e 
                     
                       j 
                       ⁢ 
                       2 
                       ⁢ 
                       
                         π 
                         ⁡ 
                         ( 
                         
                           
                             f 
                             up 
                           
                           - 
                           
                             f 
                             dw 
                           
                         
                         ) 
                       
                       ⁢ 
                       t 
                     
                   
                 
                 + 
                 
                   
                     
                       
                         x 
                         b 
                         * 
                       
                       ( 
                       t 
                       ) 
                     
                     2 
                   
                   ⁢ 
                   
                     e 
                     
                       
                         - 
                         j 
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                         π 
                         ⁡ 
                         ( 
                         
                           
                             f 
                             up 
                           
                           + 
                           
                             f 
                             dw 
                           
                         
                         ) 
                       
                       ⁢ 
                       t 
                     
                   
                 
               
               ) 
             
           
         
       
     
     Through Equation 6, it may be confirmed that when the up-converted frequency f up  and the down-converted frequency f dw  are different from each other, a time variant signal by a frequency difference and a frequency sum exists. 
     The data y dc (t) having the down-converted frequency f dw  is transmitted to the receiver  200 . 
     The data y dc (t) having the down-converted frequency f dw  may be pulse-shaped through a processor  202 , and may then be transmitted to an adaptive filter  203 . 
     The adaptive filter  203  may receive an orthogonal basis generated from the orthogonal basis generator  300  to track an unknown system parameter to be described later. 
     In this case, the orthogonal basis generator  300  may receive the data x[n] generated from the transmitter  100 , more specifically, the signal generator  101 , and generate the orthogonal basis Φ m (x) from the received data x[n]. The generation of the orthogonal basis Φ m (x) will be described in detail with reference to  FIG.  2   . 
       FIG.  2    is a diagram for describing an orthogonal basis generator according to an exemplary embodiment. 
     Referring to  FIGS.  1  and  2   , it can be seen that the data x[n] approaches a Gaussian distribution as a size of fast Fourier transform (FFT) increases, through inverse fast Fourier transform (IFFT) through the transmitter  100 , as expressed in Equation 7. 
     
       
         
           
             
               
                 
                   
                     x 
                     [ 
                     n 
                     ] 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       
                         
                           
                             x 
                             F 
                           
                           [ 
                           k 
                           ] 
                         
                         ⁢ 
                         
                           
                             e 
                             
                               j 
                               ⁢ 
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 ( 
                                 
                                   n 
                                   / 
                                   N 
                                 
                                 ) 
                               
                               ⁢ 
                               k 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 7, N is a size of the FFT, and x F  is a signal in a frequency domain of the data x[n]. 
     In this case, a basis used in an Itô-Hermite polynomial may have the form of a polynomial as expressed in Equation 8. 
       Ψ m ( X )=| x|   2m-1   x   [Equation 8].
 
     In this case, when the input data x[n] for the polynomial follows a Gaussian distribution as expressed in Equation 9, the respective orders may have orthogonality as expressed in Equation 10. 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         Φ 
                         m 
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           
                             m 
                             - 
                             1 
                           
                           2 
                         
                       
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           i 
                         
                         ⁢ 
                         
                           i 
                           ! 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 i 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     m 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 i 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         x 
                         ⁢ 
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             x 
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           
                             m 
                             - 
                             1 
                             - 
                             
                               2 
                               ⁢ 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     9 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     E 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             Φ 
                             m 
                           
                           ( 
                           x 
                           ) 
                         
                         ⁢ 
                         
                           
                             Φ 
                             
                               m 
                               ⁢ 
                               ′ 
                             
                             * 
                           
                           ( 
                           y 
                           ) 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             m 
                             + 
                             1 
                           
                           2 
                         
                         ) 
                       
                       ! 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             m 
                             - 
                             1 
                           
                           2 
                         
                         ) 
                       
                       ! 
                     
                     ⁢ 
                     E 
                     ⁢ 
                     
                       { 
                       
                         xy 
                         * 
                       
                       } 
                     
                     ⁢ 
                     
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           E 
                           ⁢ 
                           
                             { 
                             
                               xy 
                               * 
                             
                             } 
                           
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                       
                         m 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         δ 
                         [ 
                         
                           m 
                           - 
                           
                             m 
                             ′ 
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     10 
                   
                   ] 
                 
               
             
           
         
       
     
     In particular, when OFDM signals are transformed into time domain signals by an inverse Fourier transform as expressed in Equation 7, the OFDM signals may be expressed as the sum of frequency domain signals. 
     In particular, when Equation 7 is considered from a statistical point of view, in a case where a value of N becomes large, the data x[n] approaches the Gaussian distribution by a central limit theorem. 
     In the present disclosure, the OFDM signals are used at the transmitting end, and when the Itô-Hermite polynomial is used, the respective orders may have near orthogonality. When the basis of the polynomial as described above is used, nonlinearity of a system including the transceiver  10  may be tracked with the adaptive filter  203 , and the OFDM signals have near orthogonality by passing through a basis function, and thus, a convergence speed and performance may be improved. 
     Specifically, the orthogonal basis generator  300  receives the data x[n], and multiplies the data x[n] by a Gaussian characteristic value g[n] obtained by extracting a Gaussian characteristic from the data x[n] through a multiplier  302 . For example, the Gaussian characteristic value g[n] may be expressed as in Equation 11. 
         g [ n ]= e   j2π(Δƒ/Bw)n   [Equation 11]
 
     In Equation 11, BW is a bandwidth. 
     The product of the generated data x[n] and the Gaussian characteristic value g[n] through the multiplier  302  may be transmitted to a polynomial block  304  to become an input to the Itô-Hermite polynomial of Equation 9. The polynomial block  304  may be implemented by a logic circuit. 
     Therefore, an orthogonal basis Φ m (x[n]*g[n]) of Equation 9 may be generated. 
     In this case, an orthogonal basis as expressed in Equation 12 (EQ12) may be generated by using an influence generated due to a difference between the up-converted frequency f up  and the down-converted frequency f dw  as an input to the Itô-Hermite polynomial. 
     
       
         
           
             
               
                 
                   
                     
                       
                         Φ 
                         m 
                       
                       ( 
                       
                         
                           x 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     f 
                                     up 
                                   
                                   - 
                                   
                                     f 
                                     dw 
                                   
                                 
                                 ) 
                               
                               BW 
                             
                             ⁢ 
                             n 
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         Φ 
                         m 
                       
                       ( 
                       
                         
                           x 
                           OOBE 
                         
                         [ 
                         n 
                         ] 
                       
                       ) 
                     
                   
                   , 
                   
 
                   
                     
                       
                         x 
                         OOBE 
                       
                       [ 
                       n 
                       ] 
                     
                     = 
                     
                       
                         x 
                         [ 
                         n 
                         ] 
                       
                       ⁢ 
                       
                         e 
                         
                           j 
                           ⁢ 
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   f 
                                   up 
                                 
                                 - 
                                 
                                   f 
                                   dw 
                                 
                               
                               ) 
                             
                             BW 
                           
                           ⁢ 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     EQ 
                     ⁢ 
                     12 
                   
                   ] 
                 
               
             
           
         
       
     
     The orthogonal basis generated by the orthogonal basis generator  300  may be used in the adaptive filter  203 . 
       FIGS.  3  and  4    are illustrative diagrams for describing an adaptive filter according to an exemplary embodiment. 
     First, referring to  FIGS.  1  to  3   , an unknown system parameter for an unknown system  2030  may be tracked through an adaptive filter  203   a . The adaptive filter of  FIG.  1    may be implemented with the adaptive filter  203   a.    
     For example, when the number of taps of the adaptive filter  203   a  is p, a vector x[n]=[x[n], x[n−1], . . . x[n−p+1]]H of input data x[n] of an n-th time and d[n](=y[n]+v[n]) in which noise or interference is added to an output signal y[n] of the unknown system  2030  may be required. 
     The adaptive filter  203   a  may output an output signal ŷ[n] through a block  2033  in which ŵ[n]=[ŵ 0 [n], ŵ 1  [n], . . . ŵ p-1 [n]] H  generated through an estimated weight filter  2035  is multiplied by orthogonal bases (Φ 1 (x[n]), Φ 2 (x[n]), to Φ m (x[n]))  2034  received from the orthogonal basis generator  300 . 
     In this case, a difference between an estimated output signal ŷ[n] of the unknown system  2030  and an actual signal d(n) of the unknown system  2030  may be defined as an error e[n]. 
     In this case, the adaptive filter  203   a  needs to decrease the square (∥e[n]∥ 2 ) of a magnitude of an error, which is a cost function (C[n]). The adaptive filter  203   a  may decrease the square (∥e[n]∥ 2 ) of the magnitude of the error through Equation 13 based on steepest descent. 
     
       
         
           
             
               
                 
                   
                     
                       e 
                       [ 
                       n 
                       ] 
                     
                     = 
                     
                       
                         
                           d 
                           [ 
                           n 
                           ] 
                         
                         - 
                         
                           
                             y 
                             ^ 
                           
                           [ 
                           n 
                           ] 
                         
                       
                       = 
                       
                         
                           d 
                           [ 
                           n 
                           ] 
                         
                         - 
                         
                           
                             
                               
                                 w 
                                 ^ 
                               
                               H 
                             
                             [ 
                             n 
                             ] 
                           
                           ⁢ 
                           
                             x 
                             [ 
                             n 
                             ] 
                           
                         
                       
                     
                   
                   ⁢ 
                   
 
                   
                     
                       C 
                       [ 
                       n 
                       ] 
                     
                     = 
                     
                       E 
                       ⁢ 
                       
                         { 
                         
                           
                              
                             
                               e 
                               [ 
                               n 
                               ] 
                             
                              
                           
                           2 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
 
                   
                     
                       
                         ∇ 
                         
                           
                             w 
                             ^ 
                           
                           H 
                         
                       
                       
                         C 
                         [ 
                         n 
                         ] 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       E 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ∇ 
                               
                                 
                                   w 
                                   ^ 
                                 
                                 H 
                               
                             
                             
                               e 
                               [ 
                               n 
                               ] 
                             
                           
                           ⁢ 
                           
                             
                               e 
                               * 
                             
                             [ 
                             n 
                             ] 
                           
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
 
                   
                     
                       
                         ∇ 
                         
                           
                             w 
                             ^ 
                           
                           H 
                         
                       
                       
                         e 
                         [ 
                         n 
                         ] 
                       
                     
                     = 
                     
                       - 
                       
                         x 
                         [ 
                         n 
                         ] 
                       
                     
                   
                   ⁢ 
                   
 
                   
                     
                       
                         
                           E 
                           ^ 
                         
                         ⁢ 
                         
                           { 
                           
                             
                               x 
                               [ 
                               n 
                               ] 
                             
                             ⁢ 
                             
                               
                                 e 
                                 * 
                               
                               [ 
                               n 
                               ] 
                             
                           
                           } 
                         
                       
                       = 
                       
                         
                           
                             1 
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             
                               
                                 x 
                                 [ 
                                 
                                   n 
                                   - 
                                   i 
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 
                                   e 
                                   * 
                                 
                                 [ 
                                 
                                   n 
                                   - 
                                   i 
                                 
                                 ] 
                               
                             
                           
                         
                         = 
                         
                           
                             x 
                             [ 
                             n 
                             ] 
                           
                           ⁢ 
                           
                             
                               e 
                               * 
                             
                             [ 
                             n 
                             ] 
                           
                         
                       
                     
                     , 
                     
 
                     
                       ( 
                       
                         N 
                         = 
                         1 
                       
                       ) 
                     
                   
                   ⁢ 
                   
 
                   
                     
                       
                         
                           w 
                           ^ 
                         
                         H 
                       
                       [ 
                       
                         n 
                         + 
                         1 
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           H 
                         
                         [ 
                         n 
                         ] 
                       
                       - 
                       
                         
                           μ 
                           2 
                         
                         ⁢ 
                         
                           
                             ∇ 
                             
                               
                                 w 
                                 ^ 
                               
                               H 
                             
                           
                           
                             C 
                             [ 
                             n 
                             ] 
                           
                         
                       
                     
                   
                   ⁢ 
                   
 
                   
                     
                       
                         
                           w 
                           ^ 
                         
                         H 
                       
                       [ 
                       
                         n 
                         + 
                         1 
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           H 
                         
                         [ 
                         n 
                         ] 
                       
                       + 
                       
                         
                           μ 
                           2 
                         
                         ⁢ 
                         
                           x 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         
                           
                             e 
                             * 
                           
                           [ 
                           n 
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     13 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 13, E{⋅} is an expected value, and μ is a step size of the steepest descent and may have an influence on a convergence speed of least mean square (LMS). 
     Equation 13 has an expected value in a gradient of the cost function, but a calculation may be performed using an empirical expected value. For example, a case where N=1, which is the simplest case of calculating an empirical expected value, may be considered. 
     As described above, by using an orthogonal basis having linearity as a basis for the least mean square, tracking efficiency for the unknown system parameter may be increased. 
     Referring to  FIGS.  1 ,  2 , and  4   , an adaptive filter  203   b  may perform parameter tracking for an unknown system  2030  including a time variant system. The adaptive filter  203  of  FIG.  1    may be implemented by the adaptive filter  203   b.    
     For example, in consideration of a non-linear time delay of the power amplifier  130  as expressed in Equation 14, when the Itô-Hermite polynomial is used, whitening (or noise whitening) may be performed through a covariance block  2036  performing a covariance matrix operation on a time sample and an estimated weight filter  2035 . 
         f   PA ( x [ n ])=Σ m=1,odd   M Σ q     1     =0   Q     1   Σ q     2     =0   Q     2     d   mq     1     q     2     x [ n−q   1 ]| x [ n−q   2 ]| m-1   [Equation 14]
 
     The adaptive filter  203   b  may use orthogonal bases (Φ 1 (x[n]), Φ 2 (x[n]), to Φ m (x[n])))  2034 _ 1  to  2034 _ p +1 for each of delayed data x[n] to x[n−p]. 
     For example, a cubic Itô-Hermite polynomial of the delayed data x[n] to x[n−p] may calculate empirical covariance using vectors in which data on delay times are accumulated as expressed in Equation 15, a whitening matrix as expressed in Equation 16 and a whitening method using the whitening matrix may be applied in order to decrease correlation according to the delayed data x[n] to x[n−p]. 
     
       
         
           
             
               
                 
                   
                     
                       Φ 
                       
                         3 
                         , 
                         P 
                       
                     
                     [ 
                     n 
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             Φ 
                             3 
                           
                           ( 
                           
                             
                               x 
                               OOBE 
                             
                             [ 
                             n 
                             ] 
                           
                           ) 
                         
                         , 
                         
 
                         
                           
                             Φ 
                             3 
                           
                           ( 
                           
                             
                               x 
                               OOBE 
                             
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ) 
                         
                         , 
                         … 
                             
                         , 
                         
                           
                             Φ 
                             3 
                           
                           ( 
                           
                             
                               x 
                               OOBE 
                             
                             [ 
                             
                               n 
                               - 
                               P 
                             
                             ] 
                           
                           ) 
                         
                       
                       ] 
                     
                     T 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     15 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ∑ 
                     3 
                   
                   
                     = 
                     
                       
                         E 
                         [ 
                         
                           
                             
                               Φ 
                               
                                 3 
                                 , 
                                 P 
                               
                             
                             [ 
                             n 
                             ] 
                           
                           ⁢ 
                           
                             
                               
                                 Φ 
                                 
                                   3 
                                   , 
                                   P 
                                 
                               
                               [ 
                               n 
                               ] 
                             
                             H 
                           
                         
                         ] 
                       
                       ≃ 
                       
                         
                           1 
                           N 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           
                             
                               
                                 Φ 
                                 
                                   3 
                                   , 
                                   P 
                                 
                               
                               [ 
                               n 
                               ] 
                             
                             ⁢ 
                             
                               
                                 
                                   Φ 
                                   
                                     3 
                                     , 
                                     P 
                                   
                                 
                                 [ 
                                 n 
                                 ] 
                               
                               H 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     Equation 
                     ⁢ 
                         
                     16 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ∑ 
                     3 
                   
                   
                     = 
                     
                       VDV 
                       H 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     16 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             S 
             = 
             
               
                 D 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
               ⁢ 
               
                 V 
                 H 
               
             
           
         
       
       
         
           
             
               
                 
                   Φ 
                   
                     3 
                     , 
                     P 
                   
                 
                 _ 
               
               [ 
               n 
               ] 
             
             = 
             
               S 
               ⁢ 
               
                 
                   Φ 
                   
                     3 
                     , 
                     P 
                   
                 
                 [ 
                 n 
                 ] 
               
             
           
         
       
     
     Φ 3,P [n] of Equation 15 is a vector of delay times having a cubic Itô-Hermite polynomial, and empirical covariance using a time of n according to a system including the transceiver  10  may be used as the covariance. 
     In addition, Equation 16 may represent a method of generating a whitening matrix S by using an eigen value decomposition of the covariance of Equation 15 and a method of generating a new basis  Φ 3,P   [n] with low correlation from the whitening matrix. 
     A correlation coefficient according to a time may be decreased with reference to the example as described above, and in particular, covariance according to each time may have the form of a block diagonal matrix using the fact that a correlation between respective orders is close to 0 when an Itô-Hermite polynomial having orthogonality between the orders is used, and the block diagonal matrix may have a complexity gain as compared with a matrix of the same size. 
       FIG.  5    includes graphs illustrating power spectral densities (PSDs) of the transceiver according to an exemplary embodiment. 
     Referring to  FIG.  5   , PSD graphs according to a frequency are illustrated. 
     It can be seen that power spectral density (PSD) characteristics are improved when the adaptive filter uses the LMS using the Itô-Hermite polynomial using the orthogonal basis described above with reference to  FIGS.  1  to  4    than when the adaptive filter uses least mean square (LMS), parallel Hammerstein model LMS (PHLMS), and orthogonal LMS (OLMS). 
       FIG.  6    includes graphs illustrating a mean square error (MSE) of the transceiver according to an exemplary embodiment. 
     Referring to  FIG.  6   , mean square error (MSE) graphs according to data samples of a transmitter are illustrated. 
     It can be seen that mean square error (MSE) characteristics are improved more when the adaptive filter uses the LMS using the Itô-Hermite polynomial using the orthogonal basis described above with reference to  FIGS.  1  to  4    than when the adaptive filter uses least mean square (LMS), parallel Hammerstein model LMS (PHLMS), and orthogonal LMS (OLMS). 
     Exemplary embodiments of the present disclosure have been described hereinabove with reference to the accompanying drawings, but the present disclosure is not limited to the above-described exemplary embodiments, and may be implemented in various different forms. For example, one of ordinary skill in the art to which the present disclosure pertains may understand that the present disclosure may be implemented in other specific forms without changing the technical spirit of the present disclosure.