CN106385394B - A method of inhibiting OFDM signal peak-to-average force ratio - Google Patents
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Abstract
The present invention discloses a kind of method for inhibiting OFDM signal peak-to-average force ratio, and specific steps, which include: (1), becomes parallel signal for the serial data signal of input in signal transmitting terminal, obtains parallel signal sequences；(2) parallel signal sequences are multiplied with the pre-coding matrix of construction, the signal after the precoding for the peak-to-average force ratio that is inhibited；(3) cyclic prefix is added to the signal after precoding, companding processing, the peak-to-average force ratio of the signal after further suppressing companding is carried out by Segment Index companding function；(4) signal sequence after companding is subjected to parallel serial conversion in signal transmitting terminal, obtains OFDM signal.The present invention has the advantages that complexity is low, and the peak-to-average force ratio of signal is significantly reduced in the case where not losing and receiving signal bit error rate, improves the performance of communication system.
Description
Technical field
The invention belongs to fields of communication technology, further relate to one of wireless communication technology field and inhibit orthogonal frequency
Divide the method for multiplexing OFDM (orthogonal frequency division multiplexing) signal peak-to-average ratio.Orthogonal frequency
Point multiplexing OFDM a major defect be signal peak-to-average power average power ratio PAPR (Peak-to-Average Power Ratio,
Abbreviation peak-to-average force ratio) it is excessive, the present invention combines precoding technique and Segment Index companding technology, can inhibit orthogonal frequency division multiplex OFDM
The peak-to-average force ratio of signal.
Background technique
Orthogonal frequency division multiplexing (Orthogonal Frequency Division Multiplexing) technology is because of its high frequency
Spectrum efficiency, outstanding ability of anti-multipath and the advantages such as implementation complexity is low, be widely used in current various Digital Transmissions and
In communication.Application scenarios include 4G wireless communication system, and power line carrier is logical, WLAN (Wireless Local Area
Network) etc..However, a major defect of orthogonal frequency division multiplex OFDM technology is exactly signal peak-to-average power power ratio (Peak-to-
Average Power Ratio) it is excessive.Excessively high signal peak-to-average ratio can make signal peak be more than linearity of amplifier working range,
To cause non-linear distortion, these non-linear distortions can reduce the bit error rate performance of system, cause transmitting signal spectrum extension,
Seriously affect the transmission quality of system.Therefore, it is necessary to handle OFDM signal, its signal peak is reduced
Average power ratio.
Paper " the Hadamard Transform Combined With that Jiangnan Xiao et al. is delivered at it
Companding Transform Technique for PAPR Reduction in an Optical Direct-
It is proposed in Detection OFDM System " (VOL.4, NO.10/OCTOBER 2012/J.OPT.COMMUN.NETW.709)
It is a kind of based on Hadamard transform and companding technology in conjunction with method it is come the peak for reducing orthogonal frequency division multiplex OFDM system signal equal
The method of ratio.The implementation steps of this method are: first, parallel signal is mapped；Second, the signal after mapping is first breathed out
Up to Hadamard transform, Inverse Fast Fourier Transforms are then done again and are modulated；Third does companding processing to resulting modulated signal, most
Emitted after adding cyclic prefix and digital-to-analogue conversion operation afterwards.This method reduces orthogonal frequency division multiplexing to a certain extent
The complexity of the peak-to-average force ratio of ofdm system signal, realization is relatively low.But the shortcoming that this method still has is first,
Due to having carried out non-linear companding processing, the performance of BER loss of system is larger；Second, it is outer that nonlinear operation results in band
Spread spectrum is more serious.Third, due to carry out Hadamard transform and Fast Fourier Transform (FFT) respectively to signal, this method
Computational complexity is higher.
Paper " the Parameter-adjustable piecewise exponential that Hu Meixia et al. is delivered at it
companding scheme for peak-to-average power ratio reduction in orthogonal
frequency division multiplexing systems”(IET Commun.,2014,Vol.8,Iss.4,pp.530–
536) a kind of Segment Index companding technology based on variable element is proposed in reduce the peak-to-average force ratio of ofdm system transmission signal
Method.The implementation steps of this method are: first, serial to parallel conversion is carried out to data-signal and obtains parallel signal；Second, it is parallel to believe
Number modulating in OFDM signal is obtained by Inverse Fast Fourier Transforms；Third believes orthogonal frequency division multiplex OFDM
Number carry out companding segment processing, wherein to small size value signal without processing, to substantially value signal carry out the processing of index companding.The
Four, transmitting operation is carried out to the signal after companding.This method has very well the peak-to-average force ratio inhibition of OFDM signal
Performance boost.But the shortcoming that this method still has is first, since Segment Index companding technology is a kind of non-
Linear treatment process, therefore will cause the out-of-band distortion and spectral leakage of system signal；Second, due to drawing for nonlinear operation
Enter, the meeting of the performance of BER of system reduces, especially when system channel is rayleigh fading channel.
Summary of the invention
It is an object of the invention to overcome the shortcomings of above-mentioned prior art, mentioned for orthogonal frequency division multiplex OFDM communication system
The new method for having gone out a kind of Segment Index companding technology for combining precoding technique and variable element, not only can significantly drop
The peak-to-average force ratio of low signal and the computational complexity of realization, and can be reduced while obtaining preferable peak-to-average force ratio inhibitory effect
The bit-error rate penalty of system.
The thinking achieved the object of the present invention is to carry out precoding processing, precoding process to OFDM original signal first
It is realized by butterfly computation, can substantially reduce the algorithm complexity of precoding, precoding can inhibit ofdm signal to a certain degree
Peak-to-average force ratio, and promote ofdm signal and showed in the bit error rate of multipath frequency-selective channel；Then precoded signal is carried out
Segment Index companding further suppresses peak-to-average force ratio.
The present invention realizes above-mentioned purpose, and specific step is as follows:
(1) signal sequence is obtained:
In the signal transmitting terminal of orthogonal frequency division multiplex OFDM system, the serial data signal of input is become into parallel letter
Number, obtain parallel signal sequences；
(2) precoding:
(2a) constructs the Inverse Fast Fourier Transforms IDFT matrix and N rank Wo Shi-hada that a N rank row bit reverses
Ma Walsh-Hadamard matrix, the even power that N value is 2；
(2b) according to the following formula, calculates the pre-coding matrix of parallel signal sequences:
Wherein, T indicates that the pre-coding matrix of parallel signal sequences, F indicate in quick inverse Fu that N rank row bit reverses
Leaf transformation IDFT matrix, W indicate N rank Wo Shi-Hadamard Walsh-Hadamard matrix, the even power that N value is 2；
(2c) utilizes butterfly computation, the precoded signal for the peak-to-average force ratio that is inhibited；
(3) companding precoded signal:
(3a) adds cyclic prefix before the precoded signal for inhibiting peak-to-average force ratio；
(3b) carries out companding processing to the precoded signal after addition cyclic prefix, by Segment Index companding function, obtains
To the companding signal for further suppressing peak-to-average force ratio；
(4) transmission signal is obtained:
Companding signal is become serially believing by (4a) in the signal transmitting terminal of orthogonal frequency division multiplex OFDM system from parallel signal
Number, it obtains orthogonal frequency division multiplex OFDM and transmits signal；
The signal transmitting terminal of (4b) orthogonal frequency division multiplex OFDM system, transmitting orthogonal frequency division multiplex OFDM transmit signal.
Compared with the prior art, the present invention has the following advantages:
First, since present invention employs the precodings of signal sequence and companding to handle, the precoding bring bit error rate
Promotion counteracts the negative effect caused by error rate of system performance of Segment Index companding method, overcomes of the existing technology
Performance of BER loses larger disadvantage, so that the present invention has good performance of BER.
Second, due to realizing the precoding of precoded signal sequence, butterfly computation energy present invention employs butterfly computation
Greatly reduce the number of calculations of addition and multiplication in precoding process, to substantially reduce the computational complexity of system, overcomes
The higher disadvantage of computational complexity of the existing technology, so that the present invention has the advantages that computational complexity is low.
Detailed description of the invention
Fig. 1 is flow chart of the invention；
Fig. 2 is peak-to-average force ratio performance simulation effect contrast figure of the present invention；
Fig. 3 is that the present invention receives signal bit error rate simulated effect comparison diagram；
Fig. 4 is inventive algorithm implementation complexity simulated effect comparison diagram.
Specific embodiment
The present invention will be further described with reference to the accompanying drawing.
Referring to attached drawing 1, specific steps of the invention are described further.
Step 1, signal sequence is obtained.
In the signal transmitting terminal of orthogonal frequency division multiplex OFDM system, the serial data signal of input is become into parallel letter
Number, obtain parallel signal sequences.
Step 2, precoding.
Construct the Inverse Fast Fourier Transforms IDFT matrix and N rank Wo Shi-Hadamard that a N rank row bit reverses
Walsh-Hadamard matrix W [N], N value be 2 even power, the present invention in N value be 256.
Specific step is as follows for the Inverse Fast Fourier Transforms IDFT matrix that construction N rank row bit reverses:
Step 1 constructs the Inverse Fast Fourier Transforms IDFT matrix of N rank according to the following formula:
Wherein, Fr[N] indicates that the Inverse Fast Fourier Transforms IDFT matrix of N rank, the even power that N value is 2, e indicate certainly
The truth of a matter of right logarithm, j indicate imaginary unit, and π indicates pi.
Step 2 indicates K to the L of the Inverse Fast Fourier Transforms IDFT matrix of N rank with binary bits, obtain two into
The line number that bit processed indicates, wherein N indicates that Inverse Fast Fourier Transforms IDFT order of matrix number, K indicate that Inverse Fast Fourier becomes
The line number of IDFT matrix the first row is changed, L indicates the line number of Inverse Fast Fourier Transforms IDFT matrix last line, K=0, L=N-
1。
Step 3 is carried out backward processing to the line number of the Inverse Fast Fourier Transforms IDFT matrix indicated with bit, is gone
The Inverse Fast Fourier Transforms IDFT matrix of bit backward.
Step 4, by the row of the Inverse Fast Fourier Transforms IDFT matrix of row bit backward, according to the ascending progress of line number
Sequence, obtains the Inverse Fast Fourier Transforms IDFT matrix that N rank row bit reverses.
N rank Wo Shi-Hadamard Walsh-Hadamard matrix of construction is as follows:
Wherein, W [N] indicates that N rank irrigates assorted Hadamard Walsh-Hadamard matrix, the even power that N value is 2.
According to the following formula, the pre-coding matrix of parallel signal sequences is calculated:
Wherein, T indicates that the pre-coding matrix of parallel signal sequences, F indicate in quick inverse Fu that N rank row bit reverses
Leaf transformation IDFT matrix, W indicate N rank Wo Shi-Hadamard Walsh-Hadamard matrix, the even power that N value is 2.
The pre-coding matrix of parallel signal sequences is as follows:
Wherein, T [N] indicates that N rank pre-coding matrix, the even power that N value is 2, A [2] indicate that 4 rank row bits reverse
Inverse Fast Fourier Transforms IDFT matrix in the submatrix that constitutes of first two columns element in front two row, it is assorted that W [2] indicates that 2 ranks are irrigated
Hadamard matrix, after B [2] is indicated in the Inverse Fast Fourier Transforms IDFT matrix that reverses of 4 rank row bits in two rows before
The submatrix that two column elements are constituted, B [4] indicate in the Inverse Fast Fourier Transforms IDFT matrix that reverses of 8 rank row bits after 4
The submatrix that preceding 4 column element in row is constituted, W [4] indicate that 4 ranks irrigate assorted hadamard matrix,It indicatesRank row bit is inverse
After in the Inverse Fast Fourier Transforms IDFT matrix of sequence arrangementBefore in rowThe submatrix that column element is constituted,Table
ShowRank irrigates assorted hadamard matrix,In the Inverse Fast Fourier Transforms IDFT matrix that expression N rank row bit reverses
AfterwardsBefore in rowThe submatrix that column element is constituted,It indicatesRank irrigates assorted hadamard matrix.
Utilize butterfly computation, the precoded signal for the peak-to-average force ratio that is inhibited.
Specific step is as follows for butterfly computation:
Parallel signal sequences are sorted according to row bit backward, every a line of signal matrix after sequence are made respectively by step 1
For butterfly computation input signal.
Step 2 arbitrarily chooses an input signal from the input signal of butterfly computation.
Step 3 calculates the twiddle factor of butterfly computation according to the following formula:
Wherein, wnIndicate that n-th of twiddle factor of butterfly computation, e indicate natural logrithm operation, j indicates imaginary unit's symbol
Number, π indicates that pi, N indicate the number of butterfly computation input.
Step 4 carries out the mathematical operation of butterfly computation to selected input signal according to the following formula:
o1=i1-(i1-i2)wn
o2=i2+(i1-i2)wn
Wherein, i1Indicate first input signal of single butterfly computation, i2Indicate second input of single butterfly computation
Signal, o1Indicate first output signal, o2Indicate second output signal.
Step 4 judges whether all input signals for having selected butterfly computation, if so, executing step 5, otherwise, executes the
2 steps.
All input signals are obtained output signal as the precoding letter for inhibiting peak-to-average force ratio by butterfly computation by step 5
Number.
Step 3, companding signal.
Cyclic prefix is added to the signal after precoding.
To addition cyclic prefix after precoded signal, by Segment Index companding function carry out companding processing, obtain into
One step inhibits the companding signal of peak-to-average force ratio, and the Segment Index companding function of construction is as follows:
Wherein, h (x) indicates Segment Index companding function, and x expression is added to the signal after the precoding of cyclic prefix, |
| indicate modulus value, AiIndicate that the waypoint for being added to the amplitude of the signal after the precoding of cyclic prefix, sgn () indicate symbol
Number function, AcIndicate that the cut off for being added to the amplitude of the signal after the precoding of cyclic prefix, d indicate the companding of Segment Index
Degree, d are the positive greater than 0, ()dExpression takes d power to operate, and e indicates the truth of a matter of natural logrithm, σ2Expression, which is added to, to follow
The variance of signal after the precoding of ring prefix,Expression takesPower operation, the effect of the bigger companding of d numerical value is better, Ac
It can be set according to design requirement, AcSmaller peak-to-average force ratio inhibitory effect is better, once AcIt determines, selectes d, AiIt can be asked by following formula
Solution:
A in present examplecValue isD value is 2.
Step 4, transmission signal is obtained.
In the signal transmitting terminal of orthogonal frequency division multiplex OFDM system, companding signal is become into serial signal from parallel signal,
It obtains orthogonal frequency division multiplex OFDM and transmits signal.
The signal transmitting terminal of orthogonal frequency division multiplex OFDM system, transmitting orthogonal frequency division multiplex OFDM transmit signal.
Effect of the invention can be further illustrated by following emulation:
1. simulated conditions:
Emulation of the invention uses Matlab R2014b simulation software, used in the setting and embodiment of system parameter
Parameter it is consistent, digital modulation mode be orthogonal PSK QPSK, number of sub carrier wave 256, circulating prefix-length 32, imitate
Transmission channel in very is ITU-R M.1225 Rayleigh channel.
2. the content and its interpretation of result of emulation:
Modeling and simulating is carried out using Matlab R2014b simulation software in emulation experiment of the invention, obtains three curves:
The peak-to-average force ratio performance curve of system, receiving end signal performance of BER curve, realize the performance curve of algorithm complexity.It will
The property of three curves and orthogonal frequency division multiplex OFDM performance curve in the prior art, Hu Meixia proposition method that the present invention obtains
Energy curve, Jiangnan Xiao propose that the performance curve of method compares and analyzes.
Fig. 2 is the peak-to-average force ratio performance curve and orthogonal frequency division multiplexing in the prior art for the proposition method of the present invention that emulation obtains
It is proposed that the peak-to-average force ratio performance curve of method, Jiangnan Xiao propose the peak of method with OFDM peak-to-average ratio performance curve, Hu Meixia
Than the comparison diagram of performance curve.Wherein, horizontal axis indicates the peak-to-average force ratio threshold value of orthogonal frequency-division multiplex singal, and unit dB is indulged
Axis indicates complementary Cumulative Distribution Function.Solid line indicates the peak-to-average force ratio emulated using orthogonal frequency division multiplex OFDM technology in Fig. 2
Performance curve, dotted line indicate to propose that the peak-to-average force ratio performance curve that method emulates, dotted line are indicated using this hair using Hu Meixia
The peak-to-average force ratio performance curve that bright proposition method emulates, chain-dotted line indicate to propose that method emulates using Jiangnan Xiao
The peak-to-average force ratio performance curve arrived.
By the simulation result diagram of Fig. 2 as it can be seen that the present invention complementary Cumulative Distribution Function be 10-4When, with orthogonal frequency division multiplexing
Ofdm signal is compared, and has the peak-to-average force ratio gain of nearly 9dB；Compared with the method that Jiangnan Xiao is proposed, 0.5dB is had more than
Peak-to-average force ratio gain；Compared with the method that Hu Meixia is proposed, there is the peak-to-average force ratio gain of 0.1dB.As it can be seen that the present invention can significantly drop
The peak-to-average force ratio of low orthogonal frequency division multiplex OFDM system transmission signal.
Fig. 3 is the performance of BER curve and the prior art of the receiving end signal for the proposition method of the present invention that emulation obtains
In the performance of BER curve of orthogonal frequency division multiplex OFDM receiving end signal, Hu Meixia propose the receiving end signal of method
Performance of BER curve, Jiangnan Xiao propose the comparison diagram of the performance of BER curve of the receiving end signal of method.
Wherein, horizontal axis indicates that signal-to-noise ratio, unit dB, the longitudinal axis indicate the bit error rate of receiving end signal.The song identified in Fig. 2 with asterisk
Line indicates the performance of BER of the receiving end signal emulated using orthogonal frequency division multiplex OFDM technology, is identified with circle
Curve indicates the performance of BER for the receiving end signal for proposing that method is emulated using Hu Meixia, the song identified with triangle
Line indicates the performance of BER of the receiving end signal emulated using proposition method of the present invention, with the curve of square mark
Indicate the performance of BER for the receiving end signal for proposing that method is emulated using Jiangnan Xiao.
By the simulation result diagram of Fig. 3 as it can be seen that compared with the method that Jiangnan Xiao is proposed, method proposed by the present invention
System performance of BER it is more preferable；It requires to be 10 when the performance of BER of system-4When, the side of the present invention and Hu Meixia proposition
Method is compared, and has the snr gain of 13dB.As it can be seen that compared with prior art, the present invention signal mistake ratio can be received not losing
The peak-to-average force ratio of signal is significantly reduced in the case where special rate.
Fig. 4 is the algorithm complexity curve and orthogonal frequency division multiplexing in the prior art for the proposition method of the present invention that emulation obtains
With the algorithm complexity curve of OFDM, based on the algorithm complexity curve comparison figure for irrigating assorted Hadamard method for precoding.Wherein, horizontal
Axis indicates system subcarrier number, arithmetical operation number needed for longitudinal axis expression system.Solid line indicates to use orthogonal frequency in Fig. 4
The algorithm complexity curve that multiplexing OFDM technology emulates, dotted line indicate the algorithm emulated using proposition method of the present invention
Complexity curve, chain-dotted line are indicated using based on the algorithm complexity curve irrigating assorted Hadamard method for precoding and emulating.
By the simulation result diagram of Fig. 4 as it can be seen that the present invention is compared with the algorithm complexity performance of orthogonal frequency division multiplex OFDM, property
It can be more excellent；The present invention significantly reduces the algorithm complexity of system with based on compared with irrigating assorted Hadamard method for precoding.As it can be seen that
The present invention greatly reduces the algorithm complexity of system compared with existing advanced technology.
Claims (3)
1. a kind of method for inhibiting OFDM signal peak-to-average force ratio, includes the following steps:
(1) signal sequence is obtained:
In the signal transmitting terminal of orthogonal frequency division multiplex OFDM system, the serial data signal of input is become into parallel signal, is obtained
Obtain parallel signal sequences；
(2) precoding:
(2a) constructs the Inverse Fast Fourier Transforms IDFT matrix and N rank Wo Shi-Hadamard that a N rank row bit reverses
Walsh-Hadamard matrix, the even power that N value is 2；
Specific step is as follows for the Inverse Fast Fourier Transforms IDFT matrix that described one N rank row bit of construction reverses:
Step 1 constructs the Inverse Fast Fourier Transforms IDFT matrix of N rank according to the following formula:
Wherein, Fr[N] indicates that the Inverse Fast Fourier Transforms IDFT matrix of N rank, the even power that N value is 2, e indicate natural logrithm
Operation, j indicate that imaginary unit's symbol, π indicate pi；
K to L in the Inverse Fast Fourier Transforms IDFT matrix of N rank is indicated with binary bits, obtains binary system by step 2
The line number that bit indicates, wherein N indicates that Inverse Fast Fourier Transforms IDFT order of matrix number, K indicate Inverse Fast Fourier Transforms
The line number of IDFT matrix the first row, L indicate the line number of Inverse Fast Fourier Transforms IDFT matrix last line, K=0, L=N-1；
Step 3 is carried out backward processing to the line number of the Inverse Fast Fourier Transforms IDFT matrix indicated with bit, obtains row bit
The Inverse Fast Fourier Transforms IDFT matrix of backward；
Step 4 is arranged the row of the Inverse Fast Fourier Transforms IDFT matrix of row bit backward according to line number is ascending
Sequence obtains the Inverse Fast Fourier Transforms IDFT matrix that N rank row bit reverses；
(2b) according to the following formula, calculates the pre-coding matrix of parallel signal sequences:
Wherein, T indicates that the pre-coding matrix of parallel signal sequences, F indicate that the Inverse Fast Fourier that N rank row bit reverses becomes
IDFT matrix is changed, W indicates N rank Wo Shi-Hadamard Walsh-Hadamard matrix, the even power that N value is 2；
(2c) utilizes butterfly computation, the precoded signal for the peak-to-average force ratio that is inhibited；
Specific step is as follows for the butterfly computation:
Step 1 sorts parallel signal sequences according to row bit backward, using the every a line of the signal matrix after sequence as butterfly
Shape algorithm input signal；
Step 2 arbitrarily chooses an input signal from the input signal of butterfly computation；
Step 3 calculates the twiddle factor of butterfly computation according to the following formula:
Wherein, wnIndicate that n-th of twiddle factor of butterfly computation, e indicate natural logrithm operation, j indicates imaginary unit's symbol, π table
Show pi, N1Indicate the number of butterfly computation input；
Step 4 carries out the mathematical operation of butterfly computation to selected input signal according to the following formula:
o1=i1-(i1-i2)wn
o2=i2+(i1-i2)wn
Wherein, i1Indicate first input signal of single butterfly computation, i2Indicate second input letter of single butterfly computation
Number, o1Indicate first output signal, o2Indicate second output signal；
Step 4 judges whether all input signals for having selected butterfly computation, if so, executing step 5, otherwise, executes step 2；
All input signals are obtained output signal as the precoded signal for inhibiting peak-to-average force ratio by butterfly computation by step 5；
(3) companding precoded signal:
(3a) adds cyclic prefix before the precoded signal for inhibiting peak-to-average force ratio；
(3b) to addition cyclic prefix after precoded signal, by Segment Index companding function carry out companding processing, obtain into
The companding signal of one step inhibition peak-to-average force ratio；
The Segment Index companding function constructs according to the following formula:
Wherein, h (x) indicates that Segment Index companding function, x indicate the precoded signal for being added to cyclic prefix, | | expression takes
Modulus value operation, AiIndicate that the waypoint for being added to the amplitude of the precoded signal of cyclic prefix, sgn () indicate sign function, Ac
Indicate that the cut off for being added to the amplitude of the precoded signal of cyclic prefix, d indicate Segment Index companding degree, d is greater than 0
Positive, ()dExpression takes d power to operate, and e indicates the truth of a matter of natural logrithm, σ2Indicate the precoding letter for being added to cyclic prefix
Number variance,Expression takesPower operation；
(4) transmission signal is obtained:
Companding signal is become serial signal from parallel signal in the signal transmitting terminal of orthogonal frequency division multiplex OFDM system by (4a),
It obtains orthogonal frequency division multiplex OFDM and transmits signal；
The signal transmitting terminal of (4b) orthogonal frequency division multiplex OFDM system, transmitting orthogonal frequency division multiplex OFDM transmit signal.
2. a kind of method for inhibiting OFDM signal peak-to-average force ratio according to claim 1, which is characterized in that
N rank Wo Shi described in step (2a)-Hadamard Walsh-Hadamard matrix is as follows:
Wherein, W [N] indicates that N rank irrigates assorted Hadamard Walsh-Hadamard matrix, the even power that N value is 2.
3. a kind of method for inhibiting OFDM signal peak-to-average force ratio according to claim 1, which is characterized in that
The pre-coding matrix of parallel signal sequences described in step (2b) is as follows:
Wherein, T [N] indicates N rank pre-coding matrix, the even power that N value is 2, and it is fast that A [2] indicates that 4 rank row bits reverse
The submatrix that first two columns element in fast inverse Fourier transform IDFT matrix in front two row is constituted, W [2] indicate that 2 ranks irrigate assorted hada
Ma matrix, the first two columns after B [2] is indicated in the Inverse Fast Fourier Transforms IDFT matrix that reverses of 4 rank row bits in two rows
The submatrix that element is constituted, after B [4] is indicated in the Inverse Fast Fourier Transforms IDFT matrix that reverses of 8 rank row bits in 4 rows
Preceding 4 column element constitute submatrix, W [4] indicate 4 ranks irrigate assorted hadamard matrix,It indicatesRank row bit backward row
After in the Inverse Fast Fourier Transforms IDFT matrix of columnBefore in rowThe submatrix that column element is constituted,It indicates
Rank irrigates assorted hadamard matrix,After in the Inverse Fast Fourier Transforms IDFT matrix that expression N rank row bit reverses
Before in rowThe submatrix that column element is constituted,It indicatesRank irrigates assorted hadamard matrix.
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