Cp reconstruction method of ofdm signal with multi-subband joint extrapolation

Abstract

The application provides a CP reconstruction method of OFDM signal by multi-subband joint extrapolation. When subcarrier mapping is performed at the transmitting end, a plurality of subbands are artificially divided. After IFFT, the signal omits the step of adding CP, and the whole period OFDM signal without redundant information and guard interval is sent out. At the receiving end, the baseband signal obtained by frequency conversion is again frequency-converted by a multi-subband filter to each subband, so that the signal bandwidth to be processed meets the requirement of extrapolation. The signal output by the multi-subband filter can be divided into two parts, one part is the unreliable part affected by ISI, and the other part is the reliable part not affected by ISI. According to the data signal of the reliable part, the data of the unreliable part affected by ISI is extrapolated, the CP characteristic is recovered, and the problem of inter-symbol interference and inter-carrier interference existing in the traditional OFDM signal without guard interval is overcome.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication technology, and in particular relates to a method for OFDM signal CP reconstruction by multi-subband joint extrapolation. Background Technology

[0002] As a multi-carrier wireless transmission method, Orthogonal Frequency Division Multiplexing (OFDM) systems improve spectral efficiency by utilizing the orthogonality between subcarriers. In actual communication, signals are affected by multipath fading in the channel, resulting in inter-symbol interference (ISI). Simultaneously, within the duration of an OFDM symbol, other delayed path components no longer satisfy the integer period characteristic, disrupting the orthogonality between subcarriers and causing inter-channel interference (ICI). To ensure reliable OFDM symbol transmission and avoid the effects of ISI and ICI, a cyclic prefix (CP) is introduced into the OFDM transmission system. CP extends the symbol duration, concentrating the ISI generated by multipath channels within the cyclic prefix. When the cyclic prefix is ​​removed, ISI is also eliminated. Furthermore, CP allows each path signal to maintain a complete symbol period within a single FFT time window, ensuring the orthogonality between subcarriers and thus avoiding ICI. The role of CP is not limited to this. At the receiver, the received signal can be regarded as the result of the linear convolution of OFDM symbols and the channel impulse response matrix, plus the superposition of noise. This will bring challenges to channel equalization. The existence of CP transforms the linearly convolved channel matrix into a circularly convolved matrix. Its diagonalization property transforms the convolution in the time domain into scalar multiplication in the frequency domain, thus simplifying the frequency domain equalization processing of the signal at the receiver.

[0003] While the introduction of CP (Cyclic Symbol Length) brings numerous benefits to OFDM signal transmission, it also consumes additional transmission time, leading to a decrease in transmission efficiency. This is especially true in scenarios requiring high-speed communication, where symbol durations under broadband conditions are becoming increasingly shorter. Furthermore, the duration of CP is related to channel delay spread, and given the current relatively fixed channel conditions, the duration of CP remains relatively constant. Therefore, CP consumes more of the overall OFDM symbol time-domain resources, reducing the transmission efficiency of the communication system and contradicting the current theme of green communication. To avoid the impact of CP redundancy on the transmission efficiency of the communication system while ensuring communication quality, extensive research has been conducted, primarily falling into the following categories: 1. Reducing the impact of channel delay spread through channel filters to decrease the length of CP; 2. Suppressing interference caused by insufficient CP length through interference suppression algorithms; 3. Adaptive CP length to cope with different channel conditions; 4. CP-free transmission systems that achieve reliable transmission by restoring CP. CP-free transmission methods completely eliminate redundancy, maximizing the system's spectral efficiency. Several CP-free transmission algorithms already exist. Among them, the CP recovery algorithm based on band-limited signal extrapolation can guarantee the bit error rate performance of the CP-OFDM system and is a technology worth paying attention to. However, this technology has strict requirements on the bandwidth of the received signal. Summary of the Invention

[0004] The purpose of this invention is to solve the problem of reduced system spectral efficiency caused by CP redundancy in traditional OFDM signals, and to propose a multi-subband joint extrapolation method for OFDM signal CP reconstruction.

[0005] This invention is achieved through the following technical solution: This invention proposes a multi-subband joint extrapolation OFDM signal CP reconstruction method, which is applied at the transmitting end and specifically includes the following steps:

[0006] Step A1: Divide the bit stream to be transmitted into several data blocks. After constellation mapping, the i-th data block is used to obtain the serial raw data a corresponding to the i-th OFDM data symbol. i :a i =[a i (0),a i (1),…,a i (N-1)] T ;

[0007] Among them, a i (0) is the first code element in the original serial data, a i (1) is the second code element in the original serial data, a i (N-1) is the Nth code element in the serial raw data, where N is the number of code elements transmitted in the i-th OFDM data symbol;

[0008] Step A2, by analyzing a i Perform M-point subcarrier mapping, where M is an integer power of 2 and M≥N, to obtain the subcarrier-mapped symbol s. i :s i =[s i (0),s i (1),…,s i (M-1)] T ;

[0009] Where s i (0) is the data to be transmitted by the first subcarrier in the OFDM signal, s i (1) is the data to be transmitted by the second subcarrier, s i (M-1) is the data to be transmitted on the Mth carrier. For carriers without a... i The corresponding data is padded with zeros;

[0010] Put s i Divide into several sub-bands, written as s i =[s i,0 ,s i,1 ,…,s i,j …] T s i,j It is represented as the set of subcarriers of the j-th subband;

[0011] Step A3, for s i Perform an M-point IFFT operation to obtain the IFFT result.

[0012] in, Given an M×M IFFT matrix, the element in the m-th row and n-th column is... The result of the IFFT operation The first code metadata in The result of the IFFT operation The second code metadata in The result of the IFFT operation The Mth code metadata in;

[0013] Step A4, for The parallel-to-serial conversion, D / A conversion, and RF I / Q modulation are performed sequentially, and then the modulation result is transmitted to the wireless channel via the RF antenna.

[0014] Furthermore, in step A2, the sub-bands are pre-divided, which can be written as s i =[s i,0 ,s i,1 ,…,s i,j …,s i,J-1T , s i,j represented as a set of subcarriers of the j-th subband, with J subbands divided in total; similarly, divide a i data in is uniformly divided into J groups, which are respectively placed into J subbands. Since the number of data in a subband is greater than the number of data in each group, the subband is filled by zero-padding at the end.

[0015] The present invention proposes a CP reconstruction method for OFDM signals based on multi-subband joint extrapolation, which is applied to a receiving end, and the method specifically comprises the following steps:

[0016] Step B1: after a signal arrives at a receiver through a wireless channel, a baseband signal is obtained through down-conversion processing;

[0017] Step B2: performing multi-subband filtering on the baseband signal, setting the upsampling multiple to U, upsampling the output result of each subband through A / D, and each subband will obtain a signal containing UP sampling points represents the symbol data of the i-th received OFDM symbol at the j-th subband;

[0018] wherein, is the 1st sampling point, is the 2nd sampling point, is the UP-th sampling point;

[0019] in a wireless channel, is the baseband impulse response of the channel, wherein L < M, then can be written as:

[0020]

[0021] wherein, is a lower triangular Toeplitz matrix of UP×UP whose first column is , which represents the linear convolution matrix of the channel impulse response, is an upper triangular Toeplitz matrix of UP×UP whose first row is , which represents the ISI caused by the previous symbol to the next symbol after passing through the multipath channel , is the upsampled signal corresponding to the data vector of the j-th subband in the i-th OFDM symbol at the transmitting end, is the noise signal sampled after passing through the subband low-pass filter, and B represents the noise bandwidth before digital sampling;

[0022] Step B3: dividing the signal into two parts, namely for ​Band-limited signal extrapolation is performed, that is, UL data points are extrapolated forward to obtain a signal with circular convolution characteristics.

[0023] in, For data to be pushed forward and outward, The unreliable data portion that experiences ISI due to multipath channel interference is... The first UL data; The reliable part that is not affected by ISI is The last UP-UL data; and reliable data Together, they can recover a signal with the same characteristics as CP, which is located within an FFT window;

[0024]

[0025] in This is the first action The UP×UP cyclic matrix, [·] ∪ Indicates the extrapolation result;

[0026] Step B4, for the signal Perform serial-to-parallel conversion to obtain the converted signal; set the decimation interval to U, decimate the converted signal at equal intervals, and then perform a P-point FFT operation on the decimated signal to obtain the data.

[0027]

[0028] Wherein, the FFT matrix F is of size P×P P The matrix element in the m-th row and n-th column is h=[h(0),h(1),…,h(L-1),0,0,…] T Therefore, f s =P / T b The channel impulse response after rate sampling, N B It is the extracted noise vector;

[0029] Step B5, for Perform equalization and decision-making to obtain the demodulated data of the j-th subband.

[0030] Step B6: Repeat steps B3 to B5 until all sub-band data has been demodulated, then... Subcarrier inverse mapping is performed to obtain the final demodulated signal.

[0031] Furthermore, for noise in the channel, the multi-subband low-pass filter limits the bandwidth of the noise to the bandwidth of a set subband, after passing through a sampling rate of f. s =UP / T b After upsampling, where T b The duration of the transmitted OFDM symbol is determined by upsampling the signal. Spectral analysis yielded a bandwidth of B for a single subband. OFDM =P / T b The bandwidth is in the same range as the noise bandwidth.

[0032] Furthermore, the aforementioned The specific process of extrapolating band-limited signals is as follows:

[0033] Step B31: For Zero-padding is performed to obtain the initial extrapolated signal.

[0034] Step B32: For Perform an FFT transformation on the up and down points to obtain

[0035] Step B33: For For frequency domain filtering, only consider bandwidth B. OFDM =P / T b The value is set to 0 for all other values, and the result is obtained.

[0036] Step B34: [The rest of the text appears to be incomplete and requires further context.] Perform an IFFT to obtain the extrapolated value.

[0037] Step B35: Put and The splicing process is as follows:

[0038] Step B36: Repeat steps B32 to B35 until the condition is met. Iteration stops when the value is less than the threshold. As a signal Output;

[0039] in, This is the splicing result obtained in the (n+1)th iteration. This is the splicing result obtained in the nth iteration.

[0040] Furthermore, in step B4, the signal is first... Perform serial-to-parallel conversion, then perform an up-down FFT operation on the converted signal, and finally extract the low-frequency data from the FFT signal.

[0041] The beneficial effects of this invention are as follows:

[0042] This invention artificially divides the subcarrier mapping at the transmitter into multiple subbands. The signal after IFFT eliminates the need for the CP (Programmable Component Interference) step, transmitting a full-cycle OFDM signal without redundancy or guard intervals. At the receiver, the down-converted baseband signal is further down-converted through a multi-subband filter to ensure the bandwidth of the processed signal meets the extrapolation requirements. The signal output from the multi-subband filter can be divided into two segments: an unreliable segment affected by ISI (Inter-Signal Interference) and a reliable segment unaffected by ISI. The data from the unreliable segment affected by ISI is extrapolated from the reliable segment, thus restoring the CP characteristic and overcoming the problems of inter-symbol interference and inter-carrier interference in traditional OFDM signals without guard intervals. The absence of CP also avoids the redundancy problem of traditional OFDM systems, reducing transmitter power and improving the spectral efficiency of the OFDM system. Attached Figure Description

[0043] Figure 1 This is a schematic diagram of OFDM signal CP reconstruction based on multi-subband joint extrapolation; the receiver takes one subband as an example.

[0044] Figure 2 Flowchart of the multi-subband filter at the receiver; baseband signal after down-conversion. Through multiplier and The signal is down-converted again to obtain several sub-band signals. A low-pass filter removes signals other than those in the current sub-band. The bandwidth of the low-pass filter is 2πs / T. b Where s is the number of subcarriers in the subband, and T b For OFDM symbol duration;

[0045] Figure 3 This is a flowchart of the signal extrapolation process in step B3 of the present invention;

[0046] Figure 4 This is a simulation diagram of the band-limited signal extrapolation effect according to step B3 of the present invention;

[0047] Figure 5 This is a simulation comparison chart of the bit error rate of the system of the present invention under Rayleigh channel. Detailed Implementation

[0048] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] Specific Implementation Method 1: Combination Figure 1 and Figure 2 This embodiment describes a method for OFDM signal CP reconstruction using multi-subband joint extrapolation, which specifically includes the following steps:

[0050] At the sending end

[0051] Step A1: Divide the bit stream to be transmitted into several data blocks. After constellation mapping, the i-th data block is used to obtain the serial raw data a corresponding to the i-th OFDM data symbol. i :a i =[a i (0),a i (1),…,a i (N-1)] T ;

[0052] Among them, a i (0) is the first code element in the original serial data, a i (1) is the second code element in the original serial data, a i (N-1) is the Nth code element in the serial raw data, where N is the number of code elements transmitted in the i-th OFDM data symbol;

[0053] Step A2, by analyzing a i Perform M-point subcarrier mapping, where M is an integer power of 2 and M≥N, to obtain the subcarrier-mapped symbol s. i :s i =[s i (0),s i (1),…,s i (M-1)] T ;

[0054] Where s i (0) is the data to be transmitted by the first subcarrier in the OFDM signal, s i (1) is the data to be transmitted by the second subcarrier, s i (M-1) is the data to be transmitted on the Mth carrier. For carriers without a... i The corresponding data is padded with zeros;

[0055] Put s i Dividing into several sub-bands can be written as s i =[s i,0 ,s i,1 ,…,s i,j …] T s i,j It represents the set of subcarriers of the j-th subband.

[0056] Step A3: for s i , perform IFFT at M points to obtain an IFFT operation result

[0057] wherein, is an M×M IFFT matrix, and the element at the m-th row and n-th column of the matrix is is the first symbol data in the IFFT operation result , is the second symbol data in the IFFT operation result , is the M-th symbol data in the IFFT operation result ;

[0058] Step A4: for , sequentially perform serial-to-parallel conversion, D / A conversion and radio frequency I / Q modulation, and then transmit the modulated result to a wireless channel via a radio frequency antenna.

[0059] The difference between the transmitter of this embodiment and the transmitter of a traditional OFDM system is that after IFFT processing of the signal, the step of adding a CP is omitted, that is, the guard interval is removed, and a CP-free OFDM signal is transmitted. And the concept of multiple sub-bands is introduced in the subcarrier mapping step.

[0060] At the receiving end

[0061] Step B1: after the signal reaches the receiver through the wireless channel, obtain a baseband signal through down-conversion processing;

[0062] Step B2: perform multi-subband filtering on the baseband signal, wherein there are a total of sub-bands, set an upsampling factor as U, upsample the output result of each subband through A / D conversion, and each subband will obtain a signal comprising UP sampling points represents the received i-th OFDM symbol and the symbol data of the j-th subband;

[0063] wherein, is the first sampling point, is the second sampling point, is the UP-th sampling point;

[0064] In the wireless channel, is the baseband impulse response of the channel, where L<M, so can be written as:

[0065]

[0066] wherein, It is the first to be listed The lower triangular Topplitz matrix, denoted as UP×UP, represents the linear convolution matrix of the channel impulse response. This is the first action The UP×UP upper triangular Topplitz matrix represents the previous symbol after passing through the multipath channel. The ISI generated for the next symbol, This refers to the upsampled signal corresponding to the data vector of the j-th subband in the i-th OFDM symbol transmitted from the transmitter. B represents the noise signal sampled after passing through the subband low-pass filter, and B represents the noise bandwidth before digital sampling.

[0067] For noise in the channel, a multi-subband low-pass filter limits the bandwidth of the noise to the bandwidth of a set subband, after passing through a sampling rate of f. s =UP / T b After upsampling, where T b The duration of the transmitted OFDM symbol. This is determined by upsampling the signal. Spectral analysis easily reveals that the bandwidth of a single subband is B. OFDM =P / T b The bandwidth is in the same range as the noise bandwidth. This facilitates signal extrapolation, which is usually difficult and non-convergent. However, after multi-subband low-pass filtering, the noise only changes the amplitude of the OFDM signal subcarrier, without making the signal non-bandlimited.

[0068] Step B3, transfer the signal Divided into two parts, namely right Band-limited signal extrapolation is performed, that is, UL data points are extrapolated forward to obtain a signal with circular convolution characteristics.

[0069] in, For data to be pushed forward and outward, The unreliable data portion that experiences ISI due to multipath channel interference is... The first UL data; The reliable part that is not affected by ISI is The last UP-UL data; and reliable data Together, they can recover a signal with the same characteristics as CP, which is located within an FFT window;

[0070]

[0071] in This is the first action The UP×UP cyclic matrix, [·] ∪ Indicates the extrapolation result;

[0072] Combination Figure 3 Explanation of The specific process of extrapolating band-limited signals is as follows:

[0073] Step B31: For Zero-padding is performed to obtain the initial extrapolated signal.

[0074] Step B32: For Perform an FFT transformation on the up and down points to obtain

[0075] Step B33: For For frequency domain filtering, only consider bandwidth B. OFDM =P / T b The value is set to 0 for all other values, and the result is obtained.

[0076] Step B34: [The rest of the text appears to be incomplete and requires further context.] Perform an IFFT to obtain the extrapolated value.

[0077] Step B35: Put and The splicing process is as follows:

[0078] Step B36: Repeat steps B32 to B35 until the condition is met. Iteration stops when the value is less than the threshold. As a signal Output;

[0079] in, This is the splicing result obtained in the (n+1)th iteration. This is the splicing result obtained in the nth iteration.

[0080] Step B4, for the signal Perform serial-to-parallel conversion to obtain the converted signal; set the decimation interval to U, decimate the converted signal at equal intervals, and then perform a P-point FFT operation on the decimated signal to obtain the data.

[0081]

[0082] Wherein, the FFT matrix F is of size P×P P The matrix element in the m-th row and n-th column is h=[h(0),h(1),…,h(L-1),0,0,…]T Therefore, f s =P / T b The channel impulse response after rate sampling, N B It is the extracted noise vector;

[0083] Step B5, for Perform equalization and decision-making to obtain the demodulated data of the j-th subband.

[0084] Step B6: Repeat steps B3 to B5 until all sub-band data has been demodulated, then... Subcarrier inverse mapping is performed to obtain the final demodulated signal.

[0085] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that, in step A2, sub-band blocks are pre-divided, which can be written as s i =[s i,0 ,s i,1 ,...,s i,j ...,s i,J-1 ] T s i,j Let a be the set of subcarriers of the j-th subband, which is divided into J subbands. Similarly, let a i The data is evenly divided into J groups and placed into J sub-bands. Since the number of data in a sub-band is greater than the number of data in each group, the sub-bands are filled with zeros at the end.

[0086] The other steps and parameters are the same as in Specific Implementation Method 1.

[0087] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One in that, in step B4, the signal is first processed... Perform serial-to-parallel conversion, then perform an up-down FFT operation on the converted signal, and finally extract the low-frequency data from the FFT signal.

[0088] The multi-subband joint extrapolation OFDM signal CP reconstruction method proposed in this invention has been described in detail above. Specific examples have been used to illustrate the principle and implementation of this invention. The description of the above embodiments is only for the purpose of helping to understand the method and core idea of ​​this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation and application scope based on the idea of ​​this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

Claims

1. A method for OFDM signal CP reconstruction by multi-subband joint extrapolation, characterized in that: The method is applied to both the sending end and the receiving end, and the method specifically includes the following steps: At the transmitting end: Step A1: Divide the bit stream to be sent into several data blocks. For the first... i After the data block is mapped by constellation, the first data block is obtained. i Serial raw data corresponding to each OFDM data symbol : ; in, It is the first code data element in the serial raw data. It is the second code data element in the original serial data. It is the first in the serial raw data N Individual code metadata N It is the first i The number of symbols transmitted per OFDM data symbol; Step A2, through the... conduct M Dot subcarrier mapping, where M It is a power of 2, and Obtain the symbols after subcarrier mapping : ; in This is the data to be transmitted by the first subcarrier in the OFDM signal. This is the data to be transmitted by the second subcarrier. It is the first M The data to be transmitted by each carrier, for those without The corresponding data is padded with zeros; Bundle Divide into several sub-bands, written as , Represented as the first j The set of subcarriers of each subband; Step A3, for conduct M Point IFFT operation, to obtain the IFFT result. ; in, for The IFFT matrix, the matrix's th m OK n The elements of the column are , The result of the IFFT operation The first code metadata in The result of the IFFT operation The second code metadata in The result of the IFFT operation The first in M Individual code metadata; Step A4, for The parallel-to-serial conversion, D / A conversion, and RF I / Q modulation are performed sequentially, and the modulation result is then transmitted to the wireless channel via the RF antenna. At the receiving end: Step B1: After the signal reaches the receiver via the wireless channel, it undergoes down-conversion processing to obtain the baseband signal; Step B2: Perform multi-subband filtering on the baseband signal, setting the upsampling factor to [value missing]. U By upsampling the output of each sub-band using an A / D converter, each sub-band will obtain a result containing... UP Signal at each sampling point : , indicating the received number of i The OFDM symbol, the first j The metadata of each child's code; in, This is the first sampling point. This is the second sampling point. It is the first One sampling point; In wireless channels, Let be the baseband impulse response of the channel, where L < M ,So It can be written as: in, It is the first to be listed of The lower triangular Topplitz matrix represents the linear convolution matrix of the channel impulse response. This is the first action of The upper triangular Topletz matrix represents the previous symbol's path through the multipath channel. The ISI generated for the next symbol, For the first time of the sending end i The first OFDM symbol j The upsampled signal corresponding to the data vector of each sub-band This is the noise signal sampled after passing through the sub-band low-pass filter. B Indicates the noise bandwidth before digital sampling; Step B3, transfer the signal Divided into two parts, namely ,right Extrapolate the band-limited signal, that is, extrapolate forward. UL From the given data, obtain a signal with circular convolution properties. : ; in, For data to be pushed forward and outward, The unreliable data portion that experiences ISI due to multipath channel interference is... The former UL One data point; The reliable part that is not affected by ISI is After UP - UL One data point; and reliable data Together, they can recover a signal with the same characteristics as CP, which is located within an FFT window; in This is the first action of Circular matrix Indicates the extrapolation result; Step B4, for the signal Perform serial-to-parallel conversion to obtain the converted signal; set the decimation interval to... U The serial-to-parallel converted signal is decimated at equal intervals, and then a P-point FFT operation is performed on the decimated signal to obtain data. ; Among them, the size is FFT matrix The Middle m OK n The matrix elements of the column are , Therefore Channel impulse response after rate sampling It is the extracted noise vector; Step B5, for Perform equilibrium and decision-making to obtain the first j Demodulation data of individual bands ; Step B6: Repeat steps B3 to B5 until all sub-band data has been demodulated, then... Subcarrier inverse mapping is performed to obtain the final demodulated signal. .

2. The method according to claim 1, characterized in that, In step A2, the sub-bands are pre-divided, which can be written as follows: , Represented as the first j The set of subcarriers of each subband is divided into . J A single belt; similarly, put The data in the middle are evenly divided into J Group, put into J Since the number of data in a sub-band is greater than the number of data in each group, the sub-band is filled with zeros at the end.

3. The method according to claim 1, characterized in that, For noise in the channel, a multi-subband low-pass filter limits the bandwidth of the noise to the bandwidth of a set subband, after passing through a sampling rate of... After upsampling, among The duration of the transmitted OFDM symbol is determined by upsampling the signal. Spectral analysis yielded the bandwidth of a single subband as follows: The bandwidth is in the same range as the noise bandwidth.

4. The method according to claim 3, characterized in that, The pair The specific process of extrapolating band-limited signals is as follows: Step B31: For Zero-padding is performed to obtain the initial extrapolated signal. ; Step B32: For Do UP Point FFT transformation to obtain ; Step B33: For For frequency domain filtering, only the bandwidth is considered. The value is set to 0 for all other values, and the result is obtained. ; Step B34: [The rest of the text appears to be incomplete and requires further context.] Perform an IFFT to obtain the extrapolated value. ; Step B35: Put and The splicing process is as follows: ; Step B36: Repeat steps B32 to B35 until the condition is met. Iteration stops when the value is less than the threshold. As a signal Output; in, For the first The splicing result obtained in the next iteration For the first The stitching result obtained in the next iteration.

5. The method according to claim 1, characterized in that, In step B4, the signal is first... Perform serial-to-parallel conversion, and then process the converted signal. UP Point-wise FFT operation, then extracting low-frequency data from the FFT signal. .

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