JPH10178411A - Transmitting and receiving method and its equipment - Google Patents

Abstract

PROBLEM TO BE SOLVED: To reduce peak power without increasing channels for parallel transmission independent of the static characteristic of an input signal by orthonormally converting N channels of modulation signals so as to reduce a mutual relation between the converted outputs of the N channels and outputting the converted N channels of the signal. SOLUTION: A modulator consists of N number of modulators modulating N channels of input signals, and a complex adding part complex-mixes the base and signals of the N channels of modulation signals. A complex orthonormal conversion part is provided between these modulation part and the complex addition part, the N channels of modulation signals U (t)=[u0 (t) to un-1 (t)]T are inputted and outputted so that the mutual relation between N channels of conversion outputs V (t)=[V0 (t) to Vn-1 (t)] may become small. Then, V (t)=AU (t). At this time, A is expressed by the expression of an N N orthonormal conversion matrix, ai=(ai0 to ain-1 ), and T expresses transposition.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a transmission / reception method and a transmission / reception apparatus using the same to simultaneously improve the quality of a communication line and reduce the peak power of a composite signal generated by multiplexing a plurality of modulated signals.

[0002]

2. Description of the Related Art In the future information communication, a service that provides images, sounds, databases, and the like represented by multimedia in real time is considered. In order to realize such a service, it is essential for the information transmission means to realize high quality of the line and high information transmission speed.

As a method of realizing high quality of a line, there are a method of increasing signal energy on a transmission path, a method of applying transmission path coding, a method of applying diversity reception, and the like. Methods for increasing the information transmission rate include a method for increasing the code transmission rate, a method for applying high-efficiency modulation, and the like.

[0004] There are several methods for simultaneously transmitting information of high quality and high information transmission speed, and a parallel transmission method using error correction is effective from the viewpoint of circuit technology. By using the error correction to reduce the required Eb / N0 (ratio of power per information bit to noise power per 1 Hz), it is possible to realize economical equipment and increased line capacity. Reducing the code transmission rate per single carrier by performing parallel transmission may be realized by current circuit technology.

As an example of error correction, an automatic retransmission technique (Aut
omatic Repeat reQuest: ARQ), error correction (Forward Error
Correction: FEC). Examples of parallel transmission include multicarrier transmission and code division multiplexing multiple access (Code
Division Multiple Access: CDMA) and Orthogonal Frequency Division Multiplex (OFDM).

[0006]

The parallel transmission means using error correction has the following problems. First, the use of error correction causes an increase in signal bandwidth on a transmission path or a reduction in line throughput, which hinders effective use of a communication line. Second, the multiplexing of a plurality of modulated signals causes a peak. The power is to increase. For this reason, studies have been individually made to solve the first problem and the second problem.

The first problem is being actively studied as an error correction technique. The second problem is being studied as a method of reducing peak power. Up to now, methods to reduce the peak power and adjust the initial phase of the carrier (Shoichi Narahashi, Toshio Nojima; Initial phase setting method to reduce the peak-to-average power ratio (PAPR) of multitone signals; IEICE B-II, N
o.11, pp.663-671, Nov.1995), search for possible signal combinations in advance and take countermeasures (Tokuhei Heihei 50
4175, Method of reducing peak-to-average power ratio in QAM communication system), multiplexing signals to detect peak power and reduce peak (Shigeru Tomisato, Hiroshi Suzuki; Envelope smoothing parallel modulation / demodulation method; IEICE) Technique RCS95-77, Sept. 1995),
A method of newly transmitting a signal corresponding to an error correction symbol as a channel for suppressing peak power (Wilkinso
n TA, Jones AE,; Minimization of the peak to me
an envelope power ratio of multicarrier transmissi
onschemes by block coding; Proc.IEEE VTS pp.825-8
29, 1995). These techniques have problems such as an increase in the number of carriers to be transmitted in parallel and an inapplicability to any input signal.

A first object of the present invention is to provide a transmission / reception method and a transmission / reception apparatus using the same, which can reduce peak power without depending on statistical properties of input signals or without increasing channels in parallel transmission. To provide. A second object of the present invention is to provide a transmission / reception method for realizing a high quality communication line in parallel transmission and a transmission / reception apparatus using the same.

[0009]

According to a first aspect of the present invention, a transmitting apparatus includes a modulating means for modulating N-channel parallel digital signals, and N is an integer of 2 or more. An orthonormal transform unit that performs an orthonormal transform on the N-channel modulated signal output from the modulating unit so as to reduce the cross-correlation between the converted outputs, and outputs a converted N-channel signal; ,including.

[0010] According to a second aspect of the present invention, a receiving apparatus receives N-channel signals from N transmission channels, and performs an inverse transform of the orthonormal transform on the received N-channel signals. And inverse demodulation means for outputting a corresponding N-channel modulated signal, and demodulation means for demodulating the N-channel modulated signal output from the inverse orthonormal transform means and outputting an N-channel digital signal. Including.

According to a third aspect of the present invention, a transmitting / receiving method includes the following steps: (a) On the transmitting side, for an N-channel modulated signal,
Perform orthonormal transform so as to reduce the cross-correlation between the converted and output signals, generate an N-channel converted signal, send out N transmission channels, (b) obtain an N-channel received signal on the receiving side, The received signals are converted in the reverse order from the above-described orthonormal transform to obtain N-channel modulated signals. (C) The N-channel modulated signals are demodulated to obtain N-channel digital signals.

[0012]

FIG. 1 shows a basic configuration of a conventional transmitting apparatus to which the present invention is not applied in order to explain the principle of the present invention from a first viewpoint. Input signals of the channels in this example is a bit string, QP by the modulator 11 _i
The figure shows a case where the complex baseband signal v (Ia, Qa) is modulated by SK modulation. The N-channel input signal is output from the modulation unit 11
Are modulated by the N modulators 11 _{0 to} 11 _N−1 forming a modulated signal U _i (Ia, Qa) of the i-th channel, respectively.
These N-channel modulated signals are combined in the complex adder 12, and a complex combined signal U _t (Ia, Qa) is obtained. The N-channel input signals may be mutually independent signals, or may be signals obtained by converting one serial signal into parallel.

FIG. 2 shows a modulation signal output from each of the modulators 11 _{0 to} 11 _{N -1} when the number of channels is N = 4 in FIG. 1, that is, a complex baseband signal (Ia, Qa) and a complex adder 12. The complex composite signal Ut (It, Qt) of the output is shown on the complex plane. The phase of each modulated signal takes one of π / 4, 3π / 4, 5π / 4, and 7π / 4, and all have the same amplitude, but a complex composite signal Ut (It, Qt ) Differs depending on the combination of the complex baseband signals output from the respective modulators. The maximum value of the theoretical amplitude is 4, and the minimum value is 0.
FIG. 3 shows the amplitude value of the complex composite signal obtained by combining the complex baseband signals of all the modulator outputs when N = 4. From FIG. 3, it can be confirmed that the maximum amplitude value of the complex composite signal is 4, and the minimum amplitude value is zero.

FIG. 4 shows a complex signal synthesizer corresponding to FIG. 1 for explaining the basic principle of the present invention according to the first aspect. Similar to FIG. 1, a modulation section 11 of N modulators 11 ₀ to 11 _N-1 for modulating an input signal of N-channel,
A complex adder 12 for complex-combining an N-channel modulated signal, in this example, a complex baseband signal, is provided.
1 in that a complex orthonormal transform unit 13 is provided between modulators 11 _{0 to} 11 _N−1 and a complex adder 12. The complex orthonormal transform unit 13 calculates N
And outputs as the cross-correlation between channels of modulated signals _{u 0 (t) ~u N-} 1 (t) to input N-channel conversion output v of _{0 (t) ~v N-1} (t) becomes smaller . As a result, the peak power of the complex composite signal of the complex adder 12 is reduced.

FIG. 5A shows a converted signal v _i (Ib, Qb) output from the complex orthonormal transform unit 13 when N = 4 in FIG. 4 on a complex plane. As a result of the complex orthonormal transform, the amplitude of the modulated signal is not always constant, and the phase is also (2n + 1)
It is not always π / 4. Accordingly, the amplitude value of the converted signal is different from that of the complex baseband signal of FIG. FIG. 5B shows a complex composite signal of the converted signal. FIG. 5B corresponds to FIGS.
2 shows a complex composite signal Ut (It, Qt) of the related art and a complex composite signal locus Vt (It, Qt) based on the principle of the present invention. The peak power of the complex composite signal according to the present invention is smaller than the peak power of the complex composite signal according to the prior art.

As an orthonormal basis used in the complex orthonormal transform unit 13, Walsh-Hadamard transform (Walsh-Ha
Using the discrete cosine transform (DCT) basis, which is an approximation of the damard transform (WHT) basis and the Karhunen-Loeve Transform (KLT) basis, all of the N = 4 cases in FIG. FIGS. 6 and 7 show the results of obtaining the amplitude value of the complex composite signal by combining the complex baseband signals output from the modulator. FIG. 6 is an example of a DCT basis, and FIG.
Here is an example of a basis. In the example of the WHT basis, the amplitude value of the complex composite signal is 2.0. In the example of the DCT basis, as shown in FIG. 6, the amplitude value of the complex composite signal is from the minimum value 1.0 to the maximum value 3.
It fluctuates to zero. The amplitude value of the complex composite signal may fluctuate due to the orthonormal basis actually used from the example of the DCT basis. This causes a fluctuation in the amplitude value because the DCT base approximates the KLT base.

In the signal processing of the present invention, as shown in FIG. 7, the peak power generated by multiplexing a plurality of modulated waves is equal to the sum of the average power of each modulated wave. That is, no signal peak occurs. Next, the principle of the error correction effect obtained by the distributed transmission of the present invention according to the second aspect will be described.
FIG. 8 shows the principle of distributed transmission according to the present invention. Here, multicarrier transmission is assumed for convenience of explanation. Conventionally, independent information is transmitted for each transmission channel, whereas the N-channel modulated signals u ₀ (t) to u _N-1 (t) of the output of the modulator 11 are normalized according to the present invention. Signal v _i (t) of each transmission channel after orthonormal transform by orthogonal transform unit
, N-channel modulated signals u ₀ (t) to u _N-1 (t) before conversion are superimposed as schematically shown in FIG. Therefore, when compared with the conventional multi-carrier transmission, even if a signal is partially lost due to fading in one or more of the N-channel transmission paths, inverse orthonormal transform is performed using the remaining transmission channel signals. The transmission signal can be restored to some extent by the unit 22. This effect is the same as improving transmission characteristics using a general error correction code. Therefore, the distributed transmission according to the present invention has a kind of error correction effect.

A description will be given of the frequency diversity effect obtained as a result of the distributed transmission of the above-described orthonormal transform unit. By making the frequencies of the carriers of the modulators of a plurality of channels far apart from each other, the number of missing channels due to frequency selective fading can be reduced, and thus the rest of the channels can be restored as described above. Therefore, it is possible to prevent the transmission quality from deteriorating due to the frequency selective fading of the transmission path.

Next, the above qualitative explanation will be explained quantitatively using mathematical expressions. FIG. 9A shows an embodiment of the present invention applied to the transmitting apparatus 100. One or more modulators 11 exist. The orthonormal transformer 13 has the same number N of input signals and the number N of output signals as the number of multiplexed modulated signals. When the operation of the orthonormal transform unit 13 is mathematically expressed, it is described by an orthonormal transform matrix. In the present invention, an orthonormal transform unit using an orthonormal basis is used in order to keep the average power of the transmission signal before the orthonormal transform and after the orthonormal transform constant. The input signal vector to the orthonormal transform unit at time t is u
(t), u _i (t) (i = 0,1,..., N−1) the complex baseband signal in the i-th transmission channel, and the N × N orthonormal transform matrix realized by the orthonormal transform unit. A, the element of the complex number
(i, j) represents a _ij , the orthonormal basis vector a _i , the orthogonal transformation unit output vector v (t), the complex output signal in the _ith transmission channel v _i (t) _{, and} T the transpose Then, u (t) = [u ₀ (t) u ₁ (t)… u _N-1 (t)] ^T (1)

[0020]

(Equation 3)

A _i = (a _i0 a _i1 ... A _iN-1 ) (3) v (t) = [v ₀ (t) v ₁ (t)... V _N-1 (t)] ^T (4) v (t) = Au (t) (5) The orthonormal transformation matrix A in the equation (2) has the following properties. Σ _{n = 0} ^N-1 a _in a ^* _jn = δ _ij (6) AA ^H = A ^H A = E (7) | v (t) | ² = Σ _{n = 0} ^N-1 ρ ² _n (t) (8) | u (t) | ² = | v (t) | ² (9) Here, δ _ij is 1 when i = j and 0 when i ≠ j. H is the complex conjugate transpose, * is the complex conjugate, E is the identity matrix,
ρ ² _n (t) is the square of the absolute value of the n-th output signal v _n (t) at time t. The input signal vector u (t) is converted into an output signal vector v (t) by an orthonormal transformation matrix A. From equation (9), the power of the signal before and after conversion is preserved. Here, the input signal u _n (t) is transformed by an orthonormal transform matrix so as to reduce the cross-correlation coefficient between the input signals to the multiplexing unit 14.

The orthogonality of the elements of the output signal vector by the complex orthonormal transform unit 13 according to the present invention will be described. When the output signal vector v the correlation matrix of the (t) and _{R vv (t), R vv} (t) = v (t) v H (t) = Au (t) u H (t) A H = AR uu ( t) A ^H (10) where R _uu (t) is a correlation matrix of the input signal vector u (t). In general, a non-negative definite real symmetric matrix R _uu
(t) can be converted into a diagonal matrix R _vv (t) having positive elements by an appropriate orthogonal transformation. That is, equation (11) is obtained.

[0023]

(Equation 4)

If the cross-correlation between the multiplexed modulated signals at a certain time is reduced, the peak power after the multiplexing can be reduced. This will be described later. Next, the orthonormal transform unit 1
FIG. 10A shows a circuit configuration for realizing the N × N orthonormal transform matrix realized in No. 3. As shown, the orthonormal transformation matrix multiplier 1M _i for multiplying the coefficients a _ij of orthogonal _{bases, j} (i
= 0,1, ..., N-1 ; j = 0,1, ..., N-1) adders 1A _0, ..., it consists of 1A _{N -1.} The adder 1A _i of the i-th channel has coefficients a _{i, 0} ,.
_The input signals u ₀ (t) to u _N−1 (t) weighted by _{i, N−1} are added to obtain an output signal v _i (t). Coefficients of matrix elements
Perform orthonormal transformation by appropriately selecting a _0,0 ,..., a _{N−1, N−1} . Such a numerical operation can be realized by a computer. The calculation of the matrix can be performed mathematically by a computer.

The output signal v (t) that has undergone the orthonormal transform is
The signal is multiplexed into one carrier by the multiplexing unit 14 and transmitted to the receiving apparatus 200 of FIG. 8B via a transmission path. Generally, noise is added to a transmitted multiplex signal on a transmission path. Assuming that the received signal vector band-limited by the low-pass filter is y (t) and the noise vector of the transmission path is n (t), the receiving apparatus 200
The received signal vector y (t) input to the signal is represented by the following equation.

Y (t) = v (t) + n (t) = Au (t) + n (t) (12) The receiving apparatus comprises a detector 27 and an inverse normal corresponding to the orthonormal transformer 13 in FIG. 9A. Orthogonal transform unit 22 and demodulation unit 2
3 is provided. The inverse orthonormal transform unit 22 performs a conversion represented by an inverse matrix of Expression (2). Assuming that the output signal vector of the inverse orthonormal transform unit 22 is z (t) and its element is a complex number z _n (t), z (t) = [z ₀ (t) z ₁ (t) z ₂ (t). z _N-1 (t)] ^T (13) z (t) = A ^H y (t) = A ^H (Au (t) + n (t)) = u (t) + A ^H n (t) (14) It is. The inverse orthonormal transform unit 22 can obtain a modulation signal vector u (t) from Expression (14).

FIG. 10B shows a circuit configuration for realizing an N × N inverse orthonormal transform matrix realized by the inverse orthonormal transform unit 22. As shown in the figure, the inverse orthonormal transform matrix is a multiplier 2M _{i, j} (i = 0,1,..., N−1; j) that multiplies the coefficient a _ij of the inverse orthonormal base.
= 0,1, ..., N-1 ) and the adder 2A _0, ..., it consists of 2A _N-1 Tokyo.
Adder 2A _i of the i channel coefficient _{^{a * i, 0, ...,}} a * i, adds the input signal weighted by the _{N-1 u 0 (t)} ~u N-1 (t), the output signal Get v _i (t). The coefficient a ^* _{i, j} is the complex conjugate of the coefficient _{ai, j} . Coefficients a ^* _0,0 are elements of the matrix, ..., by appropriately selecting the ^{_{a * N-1, N -1}} , performs inverse orthonormal transform. Such a numerical operation can be realized by a computer. The calculation of the matrix can be performed mathematically by a computer.

The noise vector of the equation (14) is converted into a vector obtained by multiplying the noise vector by the inverse orthonormal transform unit 22. The transformed noise vector is an n (t) inverse orthonormal transform vector. That is, equation (15) is obtained. ^{<[A H n (t)} ] H A H n (t)> = <n H (t) AA H n (t)> = <n H (t) n (t)> = Σ n = 0 N- ¹ σ ² _n (15) where <> is the time average, and σ ² _n is the variance of the noise vector of the n-th transmission channel. The variance of the noise vector subjected to the inverse orthonormal transform in Expression (14) is the same as the variance at time t of the noise vector before the conversion. Therefore, in the ideal static transmission path, the transmission characteristics do not deteriorate due to the inverse orthonormal transform means. The received signal z _i (t) of each channel subjected to the inverse orthonormal transform is demodulated by the demodulation unit 23 to obtain a transmitted digital signal.

In FIGS. 9A and 9B, the multiplexing unit 14 and the detection unit 21 may be deleted in some cases, and the modulated signal output from the orthonormal transform unit 13 is transmitted to the N-channel transmission path without changing the baseband. The data may be received by the orthonormal transform unit 22. 9A and 9B, the serial / parallel converter 15 may be provided on the input side of the modulator 11 and the parallel / serial converter 24 may be provided on the output side of the demodulation unit 23. is there.

The limitation of the transmission band when using the transmission filter will be described. Figure 11A is a structure in which each transmission filter 15 ₀ ~15 _N-1 for preventing interference between channels on the output side of the modulator 11 ₀ ~11 _N-1 of the transmitter of FIG. 9A. Each transmission filter 15 _i performs a predetermined transmission band limitation. Each transmission filter 15
i, the M-th complex impulse response h, the input vector U of the transmission filter 15 _i of the channel i going back to the time M
_i (t), and the output of the transmission filter is U ′ (t).

H = [h ₀ h ₁ h ₂ ... H _M-1 ] ^T (16) U _i (t) = [u _i (t), u _i (t-1), u _i (t-2) ,…, U _i (t−M + 1)] ^T (17) U '(t) = [u' 0 (t), u '1 (t), u' 2 (t), ..., u 'N-1 (t)] T (18) u ′ _i (t) = h ^H U _i (t) (19) From equation (19), the output vector u ′ _i of the transmission filter 15 _i
(t) is band-limited by the impulse response h of the transmission filter. The transmission filter output vector u ′ _{i of} equation (19)
(t) is input to the orthonormal transform unit 13 of the present invention. The output vector v (t) of the orthonormal transform unit 13 is as follows.

V (t) = AU ′ (t) (20) Focusing on the i-th channel, v _i (t) = Σ _{n = 0} ^N−1 a _in u ′ _n (t) (21) U ′ _n (t) in 21) is band-limited by equation (19). Also, the orthonormal matrix A set in the orthonormal transform unit 13
The elements a _in a coefficient which does not include a frequency in the variable. Therefore, the i-channel output v _{i of the} orthonormal transform unit 13 in equation (21)
The band of (t) is the same as the set u ′ _n (t) (n = 0,..., N−1) of all outputs of the transmission filters 12 _{0 to} 12 _N−1 . Therefore, even if the signal is converted by the orthonormal transform unit 13 according to the present invention, the inconvenience of expanding and reducing the transmission band does not occur.

The receiver shown in FIG. 11B has a detector 21, an inverse orthonormal unit 22, and a demodulator 2 similar to the receiver shown in FIG. 9B.
3 having (demodulator 23 ₀ ~23 _N-1). In FIG. 11B, reception filters 25 _{0 to} 25 _N−1 are further provided between the output of each channel of the inverse orthonormal transform unit 22 and the demodulation unit 23. Receiving filter 25 ₀ ~25 _N-1 has the same complex impulse response and the transmit filter 15 ₀ ~15 _N-1 in FIG. 10A. Output vector z of the inverse orthonormal transform unit 22
(t) is represented by the following equation: z (t) = [z ₀ (t), z ₁ (t), z ₂ (t),..., z _N−1 (t)] ^T (22) Inverse orthonormal transform unit 22 of the i-th channel
Output signal z _i (t) is as follows.

Z (t) = A ^H y (t) = u (t) + A ^H n (t) (23) z _i (t) = Σ _{n = 0} ^N−1 a ^* _in y _n (t) ( 24) Here, the output of the reception filter of the i-th channel is z ′
_i (t), when the time M before to back channels i-th receiving filter input signal vector _{Z i (t), Z i} (t) = [z i (t), z i (t-1), z _{i (t-2), ...} , z i (t-M + 1)] T (25) z 'i (t) = h H Z by _i (t) (26) equation (26), the reception filter output signal z ′ _i (t) is band-limited by the impulse response h of the reception filter.

As described above, the transmission filter is arranged before the input signal of the orthonormal transform unit of the present invention, and the reception filter is arranged after the inverse orthonormal transform unit. It can be used as Further, the orthonormal transform matrix of the present invention only needs to satisfy Expressions (5), (6), (7), and (9). Therefore, a plurality of orthonormal transform matrices can be used. When this orthonormal transform matrix is kept secret from other users, the confidentiality of communication is improved.

Next, specific examples of the characteristics of the orthonormal transform unit will be described. The peak power reduction effect when a DCT base and a WHT base are used as the orthonormal transform unit and transmission characteristics for a static communication channel are shown. FIG. 12 is a block diagram of the transmitting apparatus of the present invention in which the peak power has been studied by computer simulation. The input digital signal is converted into an N-channel parallel digital signal by a serial / parallel
_The QPSK modulation is performed on _{0 to} 11 _N−1 , the orthonormal transform is performed by the orthonormal transform unit 13, and the converted output is added by the adder 12 and transmitted. FIG. 13 shows the results of a computer simulation study of the peak power reduction effect when the orthonormal transform unit 13 performs the orthonormal transform using the DCT basis and when performing the orthonormal transform using the WHT basis. The horizontal axis indicates the number of carriers (multiplex number) N, and the vertical axis indicates the peak-to-average power ratio. As is apparent from the figure, by using the DCT base and the WHT base, it is possible to reduce the peak-to-average power ratio (PAPR) to 4.5 dB when eight carriers are multiplexed.

Next, based on the second aspect of the present invention,
As shown in FIG. 14, when the transmission system of the present invention is used when the transmission system of the present invention is used when the multiplexing unit and the detection unit of the transmission / reception system are omitted and the N-channel baseband signal is transmitted / received as it is without multiplexing. FIG. 15 shows the result of the effect obtained by computer simulation. The horizontal axis is the CNR due to Gaussian noise added in the transmission path, and the vertical axis is the symbol error probability rate. Noise given by each transmission path is simulated by adding a noise generated from the Gaussian noise source 7 ₀ ~7 _N-1 in the adder 6 ₀ ~6 _N-1. From this result, it is understood that there is no deterioration in the transmission characteristics due to the use of the DCT base and the WHT base. This transmission characteristic is the same even when a multiplexing unit and a detecting unit are used. The transmission / reception system of FIG. 14 can be applied to, for example, N-channel wired transmission. In the receiving device 200, the demodulation unit 2
3, a parallel / serial conversion unit 24 is added.

In the transmitting / receiving system shown in FIG.
In 100, a series of digital signals are input,
N channels (4 channels in this case) by the parallel converter 9
, And after being respectively modulated, the signal is subjected to orthonormal transform by the orthonormal transform unit 13, and the converted output of four channels is transmitted to the transmission path of four channels.
In the receiving apparatus 200, the N-channel received signal is inverse-orthogonal-transformed by the inverse-orthogonal-orthogonal transformation unit 22, and the output is demodulated. A parallel / serial conversion unit 24 is provided at a subsequent stage of the demodulation unit 23,
The demodulated N-channel baseband digital signal is converted into a one-channel digital signal.

Next, an embodiment of the multicarrier transmission system according to the present invention will be described. FIGS. 16 and 17 show an embodiment of a transmitting apparatus 100 and a receiving apparatus 200 in which the present invention is applied to multicarrier transmission. The transmitting apparatus 100 receives the modulator 11 _i (i = 0,..., N−1) to which the digital signals # 1,.
Orthonormal transform unit 13 and the transmission filter 15 _i a, and a digital-to-analog converter 16 _i and the like. Modulation unit 1
Each of the modulators 11 _i that constitutes 1 receives a different digital signal and modulates amplitude, phase, or frequency. The outputs of the plurality of modulators are input to the orthonormal transform unit 13. The orthonormal transform unit 13 performs signal processing so as to reduce the cross-correlation coefficient between the converted output signals v ₀ (t) to v _N−1 (t) using a predetermined matrix, and outputs the result. Each output signal v _i (t) is converted to an analog signal by the digital / analog converter 16 _i . The analog signal is mixed with the carrier from the oscillator OSCi by the frequency converter MIX _i to be converted into a radio frequency signal, and the N-channel radio frequency signal is synthesized by the power synthesis unit 14 ′. Sent from ANT _T. The receiving device 200 (FIG. 17) includes an antenna ANT _R ,
The radio frequency signal received via the low noise amplifier 26 is divided into N channels by a power distribution unit 27, and mixed with a carrier from a carrier oscillator OSC _i by a mixer MIX _i in each channel to form a band-pass filter 28. _The signal is converted to a baseband signal via _i . The baseband reception signal is converted into a digital signal by an AD converter 29 _i and received by a reception filter 25.
_{The signal} is given to the inverse orthonormal transform unit 22 via _i , and undergoes inverse orthonormal transform. Inverse transform output and each is demodulated by the demodulator 23 _i, the data signal is obtained. Thus, in the case of multicarrier transmission, the number of carriers is the number of rows and the number of columns of the orthonormal transform matrix.

Here, the relationship between the correlation coefficient and the peak power is shown only for 4-channel QPSK transmission. Input signal vector u of the orthonormal transform unit at time t according to the present invention
The correlation matrix R _uu (t) of (t) is as follows. R _uu (t) = u (t) u ^H (t) (27) Generally, orthogonality is not maintained between the elements of u (t). Therefore, equation (27) is generally represented as follows.

[0041]

(Equation 5)

R _uu (t) at time t for equation (28)
The correlation coefficient γ _ij (t) of each element (i, j) is defined as follows. γ _ij (t) = u _i (t) u ^* _j (t) / {| u _i (t) ‖u ^* _j (t) |} (29) (t) is defined as follows.

Γ (t) = {1 / (N ² −N)} Σ _{i = 0} ^N− 1Σ _{j = 0} ^N−1 γ _ij (t) (where i ≠ j) (30) 30) is used as an evaluation parameter, and all four channels are QP
Assuming that SK is used, the calculation results of the respective peak powers when the orthonormal transform is not performed as in the conventional case, when the orthonormal transform based on the WHT is performed, and when the orthonormal transform based on the DCT is used are shown. 18, FIG. 19 and FIG. In this example, it is assumed that the baseband signal is simply multiplexed.

The horizontal axis of FIG. 18 shows the correlation coefficient of equation (30).
The vertical axis indicates the peak power. As is clear from this figure,
There is clearly a proportional relationship between the correlation coefficient and the peak power. When the correlation coefficient is 0, the peak power is equal to 4.0 [W], that is, the sum of the average power of each channel. Thus, 4
In the case of channel QPSK, the peak power depends on the correlation between the modulated waves to be multiplexed.

The horizontal and vertical axes in FIG. 19 are the same as those in FIG. FIG. 19 shows an example in which a WHT basis is used as the orthonormal basis of the present invention. The input signal is the same as in FIG. From this figure, the correlation was 0 and the peak power was 4.0 W. As described above, in the example using the WHT basis of the present invention, the sum is equal to the sum of the peak power and the average power of the input signal. Further, the correlation coefficient does not change. This is because the orthogonal transform unit 13 orthogonalizes the input signal.

The horizontal and vertical axes in FIG. 20 are the same as those in FIG. FIG. 20 shows an example in which a DCT basis is used as the orthonormal basis of the present invention. The input signal is the same as in FIG. Also from this figure, a proportional relationship is established between the correlation coefficient and the peak power between -0.33 and 0.35. From FIG. 20, the present invention can reduce the peak power. 18 and 20, the maximum peak power is reduced from 16 [W] to 8 [W].

As described above, by using the present invention that reduces the correlation between modulated signals to be multiplexed, the peak power of the modulated signal generated after multiplexing can be reduced. 10A and 10B, the transmission filter 15 and the reception filter 25 are inserted before the orthonormal transform unit 13 and after the inverse orthonormal transform unit 22, respectively. These transmission filter 15 and reception filter 25 can be inserted after the orthonormal transform unit 13 and before the inverse orthonormal transform unit 22, respectively, as shown in FIGS. Hereinafter, the cases of FIGS. 16 and 17 will be described.

H is the M-order complex impulse response of the transmission filter 15, v (t) is the output vector of the orthonormal transform unit 13 of the present invention, and the output signal vector of the orthonormal transform unit 13 of the channel i which goes back before the time M. _Is V _i (t), and the output of the transmission filter is w _tx (t). w _tx (t) = [w ^(tx) ₀ (t) w ^(tx) ₁ (t) w ^(tx) ₂ (t) ... w ^(tx) _N-1 (t)] ^T (31) V _i ( _{t) = [v i (t} ) v i (t-1) v i (t-2) ... v i (t-M + 1)] T (32) h = [h 0 h 1 h 2 ... h M _-1 ] ^{T _{(33) w (tx) i}} = h H V i (t) in (34) (34), i-th transmission to the channel filter output w ^(tx) _i (t) is subjected to band limitation by h.

The M-th order complex impulse response of the reception filter 25 is converted into the mixer MI of h and channel i in the same manner as the transmission filter.
Let Z _i (t) be the output signal vector of the detector output from X _i before time M, and let w _rx (t) be the receive filter output. w _rx (t) = [w ^(rx) ₀ (t) w ^(rx) ₁ (t) w ^(rx) ₂ (t)… w ^(rx) _N-1 (t)] ^T (35) Z _i ( t) = [z _i (t) z _i (t-1) z _i (t-2)… z _i (t-M + 1)] ^T (36) w ^(rx) _i (t) = h ^H Z _i ( t) (37) In equation (35), the reception filter output w ^(rx) _i (t) for the i-th channel is band-limited by h.

In the various embodiments described above, there are some transmission lines of the transmission / reception system that are not particularly clarified. However, when a radio frequency carrier is used, the transmission line is generally 3 channels.
It may be a dimensional space or a metallic cable. Further, if electro-optical conversion is performed, an optical fiber cable can be used. On the other hand, when transmitting a baseband signal, a metallic cable is used. However, if electro-optical conversion is used, an optical fiber cable can be used. Hereinafter, examples of typical transmission / reception systems when an optical fiber cable is used as a transmission path of a transmission / reception system to which the present invention is applied and when a metallic cable is used will be described.

FIG. 21 shows an embodiment of the parallel transmission system using the optical fiber cable 32. The transmission device 100 includes a modulator 11, a transmission filter 15, a serial-parallel conversion unit 19, an orthonormal conversion unit 13, a transmission unit 10, and an electro-optical converter E /
Consists of O. The transmission section 10 has a signal multiplexing section 14 and a frequency conversion section MIX. The receiving device 200 includes an optical / electrical converter O / E, a receiving unit 20, and an orthonormal inverse converting unit 22.
, A parallel / serial conversion unit 24, a reception filter 25, and a demodulator 23. The receiving unit 20 includes a frequency conversion unit MI
X and a demultiplexing unit 27. The demultiplexing unit 27 that separates the multiplexed signal is configured by a band-limiting filter for a signal multiplexed by frequency, and configured by a despreader for a signal multiplexed by a spreading code, The signal multiplexed by the wavelength is constituted by a wavelength detection filter.

The information signal to be transmitted is modulated by the modulator 11. The transmission filter 15 is realized by digital signal processing, and performs optimal waveform shaping for transmission to a transmission path. The serial / parallel converter 9 converts the digital signal sequence output from the transmission filter 15 into a parallel signal. The parallel-converted digital signal sequence is input to the orthonormal transform unit 13 at the same time. The orthonormal transform unit 13 outputs a parallel digital signal sequence with reduced cross-correlation by digital signal processing and is input to the transmitting unit 10. The transmission unit 10 multiplexes an input signal using a frequency, a code, a wavelength, and the like. The multiplexed signal is
The signal is converted into an optical signal by the electric / optical converter E / O and sent to the receiving device 200 through the optical fiber cable 31.

The receiving apparatus 200 converts the received optical signal into an optical signal.
Convert to electrical converter O / E. The receiving unit 20 separates a signal multiplexed by frequency, code, wavelength, or the like. The separated signal is inversely transformed by the orthonormal inverse transformation unit 22 and the inverse transformation output is reconstructed by the parallel / serial conversion unit 24 into a digital signal sequence. The reconstructed signal is shaped by the reception filter 25 and demodulated by the demodulator 23.

In the embodiment shown in FIG.
E / O and optical / electrical converter O / E are eliminated, and the transmission unit 1 is replaced with a metallic cable instead of the optical fiber cable 31.
The output of 0 and the input of the receiving unit 20 may be connected. According to this embodiment, parallel transmission using orthonormal transform processing can be performed. FIG. 22 shows an embodiment of baseband transmission using a metallic cable. The transmission device 100 includes a modulator 11, a transmission filter 15, a serial-parallel converter 9,
It is composed of an orthonormal transform unit 13 and a switch SW1. The receiver 200 includes a switch SW2, an orthonormal inverse converter 22, a parallel-serial converter 24, a reception filter 25,
And a demodulator 23.

The information signal to be transmitted is modulated by the modulator 11. The transmission filter 15 is realized by digital signal processing, and performs optimal waveform shaping when transmitting to the transmission path 32 of a metallic cable. The serial / parallel converter 9 converts the digital signal sequence output from the transmission filter 15 into parallel. The parallel-converted digital signal sequence is input to the orthonormal transform unit 13 at the same time. The orthonormal transform unit 13 outputs a parallel digital signal sequence whose cross-correlation has been reduced by digital signal processing according to the present invention, and is input to the switch SW1. The switch SW1 selects the output signal of the orthonormal transform unit 13 in the order of transmission to the transmission line 32. The signal processing suitable for transmission on the metallic cable 32 is performed, and the signal is transmitted to the transmission path 32.

The signal received via the transmission line of the metallic cable 32 is converted into a digital signal by A / D conversion, and the switch SW2 reconstructs a parallel digital signal sequence formed on the transmission side. The reconstructed parallel digital signal is inversely transformed by the orthonormal inverse transform unit 22, and the output of the inverse transform is shaped by the reception filter 25 and demodulated by the demodulator 23. Through these processes, baseband transmission using orthonormal transform can be performed.

In the embodiment of FIG. 22, as shown in FIG. 23, the output of the transmission-side switch SW1 is connected to the electric / optical converter E / E.
O, an optical / electrical converter O / E is provided at the input of the receiving side switch SW2, and these converters E / O and O / E are connected by an optical fiber cable 31 instead of the metallic cable 32. Can also constitute a baseband transmission system.

FIG. 24 shows the effect of the parallel transmission system of the present invention in 4-multicarrier transmission using QPSK.
In FIG. 24, the horizontal axis is the carrier-to-noise ratio CNR due to Gaussian noise added in the transmission path, and the vertical axis is the symbol error probability. The adder 6 ₀ to 6 _N-1 in the transmission path of the N-channel providing the Gaussian noise from the Gaussian noise source 7 ₀ ~7 _N-1. The simulation was set as follows. The CNR in one channel, for example, the transmission path of channel i = 0, is fixed to 0, 4, 6, and 8 dB sequentially, and each fixed
The CNR of the remaining channels i = 1, 2, and 3 is shown in Fig. 1 for the CNR value.
As indicated on the horizontal axis of 5 0, 2, 4, ..., each of the noise source ₇₁ so as to 12dB, 7 _2, 7 ₃ adders 6 ₁ from 6 _2,
It was controlled Gaussian noise level given to 6 _3. Like this
Under the condition of CNR, the baseband digital signal output in the receiving apparatus when the orthonormal transform in the orthonormal transform unit 13 used in the present invention is performed using the WHT rule and when the orthonormal transform is performed using the DCT rule. The symbol error probabilities are indicated by circles and triangles in FIG. 24, respectively, and the theoretical values (solid lines) when the present invention is not applied are also shown in comparison with the broken lines connecting them. Note that this condition is based on the assumption that the line quality of some transmission paths is significantly deteriorated. Theoretical value P
Was determined by P = (P _fix + P ₁ + P ₂ + P ₃ ) / 4. P _fix indicates the symbol error probability of the channel whose channel quality is fixed, and P ₁ , P ₂ , and P ₃ indicate the symbol error probabilities of the remaining three channels.

It should be noted that the fixed CNR of channel i = 0
Is 4 dB, for example, until the CNR of the other three channels becomes 4 dB, which is the same as that of the channel i = 0. The error probabilities vary approximately the same, but the CNR of channel i = 1, 2, 3 is 4
When the value exceeds dB, the error probability in the case where the present invention is not applied is not so improved, whereas the error probability in the case where the present invention is applied is further improved.
The same can be said for the case where the CNR of channel i = 0 is 6 dB and 8 dB. This means that the invention
When a channel modulated signal is orthonormally transformed and transmitted through an N-channel transmission path, even if the transmission path characteristics of one transmission path deteriorate, the ability to improve the symbol error probability of the received baseband digital signal is improved. This shows that it is superior to the conventional one that does not perform orthonormal transform. That is, according to the present invention, it can be said that a certain kind of error correction effect is obtained.

[0060]

According to the conventional technique, the number of carriers to be transmitted in parallel increases, that the technique cannot be applied to an arbitrary input signal,
And so on. According to the present invention, these problems are solved as follows (a) and (b). (a) Increasing the number of carriers transmitted in parallel: In the present invention, the expression
As shown in (5), the number of channels in the transmission path does not increase clearly even if the orthonormal transform unit is used. The peak power reduction effect according to the present invention is shown in FIG. Thus, the present invention can reduce the peak-to-average power ratio without increasing the number of transmission channels.

(B) Dependence on input signal: In the process of deriving the principle, the present invention does not specify the statistical properties of the input signal as shown in equation (1). Thus, the present invention does not rely on the statistical properties of the input signal,
The peak-to-average power ratio generated by multiplexing a plurality of modulation signals can be reduced. Other effects include the following.

(C) Since the information to be transmitted in parallel by the orthogonal transformation unit is converted into a format defined by the orthonormal transformation matrix, and the information originally transmitted in one channel is apparently distributed to all the channels transmitted in parallel. Even if one channel is lost, it can be restored to some extent from the information of the remaining channels. Therefore, it is possible to provide a transmission system that is resistant to fluctuations in communication line quality.

(D) Since the orthonormal transformation matrix is realized by digital signal processing, the circuit can be simplified. (e) Since the coefficients of the orthonormal transformation matrix can be changed as needed, it is possible to flexibly cope with systems of various sizes. (f) The secrecy of communication can be obtained by not revealing the orthonormal transform matrix to others. (g) Diversity effect can be obtained by distributed transmission by the orthonormal transform unit.

[Brief description of the drawings]

FIG. 1 is a block diagram of a conventional transmission device that multiplexes N-channel QPSK signals.

FIG. 2 is a diagram showing an output signal of each modulator and a synthesized signal on a complex plane when N = 4 in FIG. 1;

FIG. 3 is a graph showing the magnitude of a complex composite signal of the four-channel QPSK signal of FIG. 1;

FIG. 4 is a block diagram of a transmission device when the modulator 11 performs QPSK modulation.

5A is a diagram showing an output signal of a complex orthonormal transform unit 13 of FIG. 4 on a complex plane, and FIG. 5B is a diagram showing a complex signal obtained by combining the complex orthonormal transform output signal of FIG. The figure shown on the plane.

FIG. 6 shows a 4-channel QPSK in the transmitting apparatus of FIG.
9 is a graph showing the magnitude of a complex composite signal of a signal obtained by orthonormally transforming a signal using a DCT basis.

FIG. 7 shows a 4-channel QPSK in the transmitting apparatus of FIG.
9 is a graph showing the magnitude of a complex composite signal of a signal obtained by orthonormally transforming a signal using a WHT basis.

FIG. 8 is a diagram for explaining distributed transmission by the orthonormal transform unit.

FIG. 9A is a block diagram showing an embodiment of a transmitting apparatus according to the present invention, and FIG. 9B is a block diagram showing an embodiment of a receiving apparatus according to the present invention.

10A is a block diagram illustrating a configuration of the orthonormal orthogonal transform unit 13 in FIG. 9A, and FIG. 10B is a block diagram illustrating a configuration of the inverse orthonormal orthogonal transform unit 22 in FIG. 9B.

11A is a block diagram when a transmission filter is provided in the transmission device of FIG. 9A, and FIG. 11B is a block diagram when a reception filter is provided in the reception device of FIG. 9B.

FIG. 12 is a block diagram showing another embodiment of the present invention.

FIG. 13 is a graph showing the relationship between the peak-to-average power ratio and the number of carriers in the embodiment of FIG.

FIG. 14 is a block diagram showing an embodiment of the present invention when performing parallel transmission.

FIG. 15 is a graph showing a relationship between a symbol error rate and a carrier to noise power ratio in the embodiment of FIG. 14;

FIG. 16 is a block diagram showing a configuration when the transmitting apparatus of FIG. 8A is applied to a multicarrier transmission scheme.

FIG. 17 is a block diagram showing a configuration in a case where the receiving apparatus of FIG. 8B is applied to a multicarrier transmission scheme.

FIG. 18 is a graph showing a relationship between a peak power of a signal obtained by multiplexing four-channel QPSK signals and a correlation coefficient between signals of each channel.

FIG. 19 is a graph showing the relationship between the peak power of a multiplexed signal and the correlation coefficient between the converted signals of the respective channels after orthonormal transform of the 4-channel QPSK signal by the WHT basis.

FIG. 20 is a graph showing the relationship between the peak power of a multiplexed signal and the correlation coefficient between the converted signals of the respective channels after orthonormal transform of the 4-channel QPSK signal by DCT basis.

FIG. 21 is a block diagram of a transmission / reception system in a case where the present invention is applied to multicarrier transmission in a case where a transmission path is an optical fiber cable.

FIG. 22 is a block diagram of a transmission / reception system when the present invention is applied to multicarrier transmission when a transmission path is a metallic cable.

FIG. 23 is a block diagram in a case where the present invention is applied to baseband transmission in a case where the transmission path is an optical fiber cable.

FIG. 24 is a graph for explaining an effect when the present invention is applied to four-channel multicarrier transmission by QPSK.

Claims (21)

[Claims] 1. A modulating means for modulating N-channel parallel digital signals, and N is an integer of 2 or more, and a conversion output for an N-channel modulated signal output from said modulating means. A normal orthogonal transform unit that performs an orthogonal transform so as to reduce the cross-correlation between the signals and outputs a converted N-channel signal. 2. The transmitting apparatus according to claim 1, further comprising multiplexing means for multiplexing an output of said orthonormal transform means and transmitting a multiplexed signal. 3. The transmission device according to claim 1, wherein a transmission filter for band limitation is inserted on an input side or an output side of the orthonormal transform unit. 4. The transmitting apparatus according to claim 1, wherein a serial / parallel converter for converting a serial digital input signal into the N-channel parallel digital signal and supplying the converted signal to the modulator is provided on an input side of the modulator. Is provided. 5. The transmitting apparatus according to claim 1, wherein the switching means for sequentially selecting the N-channel output signals of the orthonormal transform means and outputting the signals in series, and converting the serial signal from the switching means into an optical signal. And an electric / optical conversion means for transmitting the data to an optical fiber cable. 6. The transmitting apparatus according to claim 1, further comprising switching means for sequentially selecting N-channel output signals of said orthonormal transform means and outputting the signals in series to a metallic cable. 7. The transmitting apparatus according to claim 1, wherein the orthonormal transform unit is a transform unit based on a WHT basis. 8. The transmitting apparatus according to claim 1, wherein said orthonormal transform unit is a transform unit based on a DCT basis. 9. The transmitting apparatus according to claim 1, wherein an input signal vector to said orthonormal transform means at time t is u (t), and a complex baseband signal in an i-th transmission channel is u _i (t). (i = 0, 1,..., N−1), A is the N × N orthonormal transformation matrix realized by the above orthonormal transformation means, a _{ij is the} complex element (i, j), and orthonormal basis vector Let a _{i denote} the output vector of the orthogonal transform unit, v (t), the complex output signal of the i-th transmission channel v _i (t) _{, and} T the transpose, u (t) = [u ₀ (t ) u ₁ (t)… u _N-1 (t)] ^T a _i = (a _i0 a _i1 ... a _iN-1 ) v (t) = [v ₀ (t) v ₁ (t) ... v _N-1 (t)] ^T v (t) = Au (t) There, the orthonormal transform matrix a satisfies ^{_{Σ n = 0 n-1 a}} in a * jn = δ ij, where, [delta] _ij is zero when the 1, i ≠ j when i = j, * Is a complex conjugate. 10. The transmitting apparatus performs orthonormal transform,
A receiving device for receiving N-channel signals transmitted via N transmission channels, including: receiving N-channel signals from N transmission channels, and receiving the received N signals;
Inverse orthonormal transform means for performing the inverse transform of the orthonormal transform on the channel signal and outputting a corresponding N-channel modulated signal; and converting the N-channel modulated signal output from the inverse orthonormal orthogonal transform means. Demodulation means for demodulating and outputting N-channel digital signals. 11. The receiving apparatus according to claim 10, wherein said detecting means detects a signal from said N transmission channel, outputs an N-channel baseband signal, and provides the signal as an input signal to said inverse orthonormal transform means. Is provided. 12. The receiving apparatus according to claim 10, wherein a receiving filter for band limitation is inserted on an input side or an output side of said inverse orthonormal transform means. 13. The receiving apparatus according to claim 10, wherein an output side of said demodulation means is a parallel / digital converter for converting N-channel parallel digital signals into one serial digital signal.
Serial conversion means is provided. 14. A transmission / reception method for transmitting an N-channel signal through an N transmission channel and demodulating an N-channel signal received at a receiving side, including the following steps: (a) transmitting a N-channel signal at a transmitting side; For the modulated signal,
Perform orthonormal transform so that the cross-correlation between the converted and output signals is reduced, generate a converted signal of N channels, transmit N transmission channels, (b) obtain a received signal of N channels on the receiving side, The received signals are converted in the inverse of the above-described orthonormal transform to obtain N-channel modulated signals. (C) The N-channel modulated signals are demodulated to obtain N-channel digital signals. 15. A transmission / reception method for transmitting an N-channel signal through an N transmission channel and demodulating an N-channel signal received at a receiving side, including the following steps: For the modulated signal,
Orthonormal transform is performed so that the cross-correlation between the converted and output signals is reduced, an N-channel converted signal is generated, (b) the N-channel converted signal is multiplexed and transmitted, and (c) a receiving side. And N-channel received signals are obtained from the input multiplexed signal, an N-channel modulated signal is obtained from the received signals by performing an inverse transform of the orthonormal transform, and (d) the N-channel modulated signal is Demodulation is performed to obtain an N-channel digital signal. 16. The method of claim 14 or claim 15, wherein
A step of obtaining an N-channel modulated signal by converting an input serial digital signal into an N-channel parallel digital signal on a transmitting side; and converting the demodulated N-channel parallel digital signal into one serial digital signal on a receiving side. Converting to a signal. 17. The method according to claim 14 or 15, wherein an N-channel output signal obtained by the orthonormal transform is sequentially selected on the transmitting side, output as a serial signal, and the serial signal is converted into an optical signal. The method further includes a step of transmitting the signal to the optical fiber cable and a step of converting the serial signal received on the receiving side into an electric signal, and sequentially distributing the electric signal to N channels to obtain the N-channel received signal. 18. The method according to claim 14, wherein the transmitting side sequentially selects the N-channel output signals obtained by the orthonormal transform and outputs the N-channel output signals in series to a metallic cable. A step of sequentially distributing the received serial signals into N channels to obtain the N-channel received signals. 19. The method of claim 14 or claim 15, wherein
The orthonormal transform is a transform based on the WHT basis. 20. The method according to claim 14, wherein
The orthonormal transform means is a transform based on a DCT basis. 21. The method according to claim 14, wherein
The input signal vector that undergoes the above orthogonal transformation at time t is u (t), the complex baseband signal in the _ith transmission channel is u _i (t) (i = 0, 1,..., N−1), A denotes an N × N orthonormal transform matrix realized by orthogonal transform, a _ij denotes a complex element (i, j), and a denotes an orthonormal basis vector.
Let _i , the output vector of the orthogonal transform unit be v (t), the complex output signal in the i-th transmission channel be v _i (t) _{, and} T be the transpose, u (t) = [u ₀ (t) u ₁ (t)… u _N-1 (t)] ^T a _i = (a _i0 a _i1 ... a _iN-1 ) v (t) = [v ₀ (t) v ₁ (t) ... v _N-1 (t)] ^T v (t) = Au (t) There, the orthonormal transform matrix a satisfies ^{_{Σ n = 0 n-1 a}} in a * jn = δ ij, where, [delta] _ij is zero when the 1, i ≠ j when i = j, * Is a complex conjugate.