JP3454407B2 - Transmission / reception method and device - Google Patents

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, which simultaneously realizes high quality of a communication line and reduction of peak power of a composite signal generated by multiplexing a plurality of modulated signals.

[0002]

2. Description of the Related Art In future information communication, services that provide images, sounds, databases represented by multimedia, etc. in real time are conceivable. In order to realize such a service, it is indispensable for the information transmission means to realize a high quality of the line and a high information transmission speed.

As a method for improving the quality of a line, there are a method of increasing signal energy on a transmission line, a method of applying transmission line coding, a method of applying diversity reception and the like. As a method for realizing a higher information transmission speed, there are a method of applying a higher code transmission speed and a method of applying a high efficiency modulation.

There are several methods of information transmission that simultaneously satisfy the high quality of the line and the high speed of the information transmission speed, and the parallel transmission means using error correction is effective from the viewpoint of circuit technology. By using error correction to reduce the required Eb / N0 (ratio of the power per information bit to the noise power per 1Hz), it may be possible to realize economical equipment and increase the line capacity. There is a possibility that the current circuit technology can reduce the code transmission rate per single carrier by performing parallel transmission.

As an example of error correction, an automatic retransmission technique (Aut
omatic Repeat reQuest: ARQ), error correction (Forward Error
Correction: FEC), etc. As an example of parallel transmission, multi-carrier transmission, code division multiple access (Code
Division Multiple Acess (CDMA), Orthogonal Frequency Division Multiplex (OFDM), and the like.

[0006]

The parallel transmission means using error correction has the following problems. First, the use of error correction leads to an increase in the signal bandwidth on the transmission line or a decrease in the line throughput, which hinders the effective use of the communication line. Second, it is a peak due to the multiplexing of multiple modulated signals. The power is to increase. Therefore, in order to solve the first problem and the second problem, research has been conducted individually.

The first problem is actively studied as an error correction technique. The second problem is being studied as a method of reducing peak power. Until now, the method 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; B-II, N
o.11, pp.663-671, Nov.1995), searching for possible combinations of signals in advance and taking countermeasures (Tokuheihei 6-50
4175, Peak-to-average power ratio reduction method in QAM communication system), which detects peak power and multiplexes signals to reduce peaks (Shigeru Tomisato, Hiroshi Suzuki; Envelope smoothing parallel modulation / demodulation method; 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), etc. have been proposed. These techniques have problems such as an increase in the number of carriers transmitted in parallel and the inapplicability to arbitrary input signals.

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 an input signal or without increasing channels in parallel transmission. Is to provide. A second object of the present invention is to provide a transmission / reception method and a transmission / reception apparatus using the same, which can improve the quality of communication lines in parallel transmission.

[0009]

According to a first aspect of the present invention, a transmitter comprises a modulation means for respectively modulating parallel digital signals of N channels and N is an arbitrary integer of 2 or more. And an orthonormal transform means for performing an orthonormal transform on the N-channel modulated signal output from the modulating means so as to reduce the cross-correlation between the transform outputs, and outputting the transformed N-channel signal. ,including.

According to a second aspect of the present invention, a receiving device receives N-channel signals from N transmission channels and performs a conversion opposite to the orthonormal transformation on the received N-channel signals. An inverse orthonormal transform means for outputting a corresponding N channel modulation signal, and a demodulation means for demodulating the N channel modulation signal output from the inverse orthonormal transform means to output an N channel digital signal. Including.

According to a third aspect of the present invention, the transmitting / receiving method includes the following steps: (a) At the transmitting side, for N-channel modulated signals,
Orthogonal transformation is performed so that the cross-correlation between the output signals is reduced, N-channel converted signals are generated, N-transmission channels are transmitted, and (b) N-channel reception signals are obtained on the receiving side. The received signals are subjected to the inverse transform from the orthonormal transform to obtain an N channel modulated signal, and (c) the N channel modulated signal is demodulated to obtain an N channel digital signal.

[0012]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 shows a basic configuration of a conventional transmitter to which the present invention is not applied, in order to explain the principle of the present invention according to the first aspect. In this example, the input signal of each channel is a bit string, and the modulator 11 _i outputs a QP signal.
The case where SK modulation is performed to form a complex baseband signal v (Ia, Qa) is shown. The input signal of the N channel is modulated by the modulator 11
Are modulated by N modulators 11 _{0 to} 11 _N-1 to form a modulated signal U _i (Ia, Qa) of the i-th channel,
These N-channel modulated signals are combined in the complex adder 12 to obtain a complex combined signal U _t (Ia, Qa). The N-channel input signals may be signals independent of each other, 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 N = 4 in FIG. 1, here, a complex baseband signal (Ia, Qa) and a complex adder 12 The complex composite signal Ut (It, Qt) at the output of is shown in the complex plane. The phase of each modulation signal is either π / 4, 3π / 4, 5π / 4, or 7π / 4, and they all have the same amplitude, but the complex composite signal Ut (It, Qt The amplitude value of) depends on the combination of the complex baseband signals of the modulator outputs, and the theoretical maximum amplitude is 4 and the minimum amplitude is zero.
FIG. 3 shows the amplitude value of the complex composite signal obtained by combining the complex baseband signals of all modulator outputs when N = 4. From this 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-synthesizing an N-channel modulated signal, which is a complex baseband signal in this example, is provided.
The difference from the prior art of FIG. 1 is that a complex orthonormal transform unit 13 is provided between the modulators 11 _{0 to} 11 _N-1 and the complex addition unit 12. The complex orthonormal transform unit 13 outputs N
The channel modulation signals u ₀ (t) to u _N-1 (t) are input and output so that the cross-correlation between N channel conversion outputs v ₀ (t) to v _N-1 (t) is reduced. . As a result, the peak power of the complex composite signal of the complex adder 12 is reduced.

FIG. 5A shows the converted signal v _i (Ib, Qb) output from the complex orthonormal transform unit 13 at N = 4 in FIG. 4 on the complex plane. As a result of the complex orthonormal transformation of the modulated signal, the amplitude is not always constant and the phase is (2n + 1)
It is not always π / 4. Therefore, the amplitude value of the converted signal is different from that of the complex baseband signal of FIG. A complex composite signal of the converted signals is shown in FIG. 5B. 5B is the same as FIG.
The conventional complex composite signal Ut (It, Qt) and the complex composite signal locus Vt (It, Qt) based on the principle of the present invention are shown. 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 the orthonormal basis used in the complex orthonormal transform unit 13, the Walsh-Hadamard transform (Walsh-Ha
damard Transform; WHT) basis and the Karhunen-Loeve Transform (KLT) basis, which is an approximation of Discrete Cosin Transfom (DCT) basis. The results of obtaining the amplitude value of the complex composite signal by combining the complex baseband signals of the modulator output are shown in FIGS. 6 and 7, respectively. FIG. 6 shows an example of DCT basis, and FIG. 7 shows WHT.
It is an example of the base. In the WHT basis example, 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. Depending on the orthonormal base actually used from the example of the DCT base, the amplitude value of the complex composite signal may vary. This is because the DCT basis approximates the KLT basis, and thus the amplitude value fluctuates.

In the signal processing of the present invention, as in the case of FIG. 7, the peak power generated by multiplexing a plurality of modulated waves becomes the same as 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, multi-carrier transmission is assumed for convenience of explanation. In the past, independent information was transmitted for each transmission channel, whereas the N-channel modulated signals u ₀ (t) to u _N-1 (t) output from the modulator 11 are normalized according to the present invention. The signal v _i (t) of each transmission channel after orthonormal transformation by the orthogonal transformation unit
, The N-channel modulation signals u ₀ (t) to u _N−1 (t) before conversion are superimposed on each other, 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 any one or more of N-channel transmission paths, the signals in the remaining transmission channels are used to perform the inverse orthonormal transform. The transmission signal can be restored to some extent by the unit 22. This effect is the same as improving the transmission characteristics by using a general error correction code. Therefore, the distributed transmission of the present invention has a kind of error correction effect.

The frequency diversity effect obtained as a result of the distributed transmission of the orthonormal transform unit will be described. By largely separating the frequencies of the carriers of the modulators of a plurality of channels from each other, the number of channels missing due to frequency selective fading can be reduced, so that the remaining channels can be restored as described above. Therefore, it is possible to prevent the deterioration of the transmission quality due to the frequency selective fading of the transmission path.

Next, the above qualitative explanation will be quantitatively explained using mathematical expressions. FIG. 9A shows an embodiment of the present invention applied to the transmitter 100. There is one or a plurality of modulators 11. The orthonormal transform unit 13 has the same input signal number N and output signal number N as the number of multiplexed modulated signals. The operation of the orthonormal transform unit 13 is mathematically expressed as an orthonormal transform matrix. In the present invention, in order to keep the average power of the transmission signal before orthonormal transform and after orthonormal transform constant, an orthonormal transform unit using an orthonormal basis is used. Let u be the input signal vector to the orthonormal transform unit at time t.
(t), u _i (t) (i = 0,1, ..., N-1) is the complex baseband signal in the i-th transmission channel, and the N × N orthonormal transform matrix realized by the orthonormal transform unit is A, the element of the complex number
(i, j) is a _ij , the orthonormal basis vector is a _i , the orthogonal transform unit output vector is v (t), and the complex output signal in the i-th transmission channel is v _i (t) _{, and} T is 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 of 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 takes 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 transformed into the output signal vector v (t) by the 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 the orthonormal transformation 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 is shown. 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 the correlation matrix of the input signal vector u (t). R _uu, which is generally a non-negative definite real symmetric matrix
(t) can be transformed 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 modulated signals to be multiplexed 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 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) and adders 1A ₀ , ..., 1A _{N -1} . The adder 1A _i of the i-th channel has coefficients a _{i, 0} , ..., a
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 that are elements of the matrix
Orthonormal transformation is performed by appropriately selecting a _0,0 , ..., a _{N-1, N-1} . Such numerical calculation can be realized by a computer. It is also possible to perform the matrix calculation mathematically by a computer.

The output signal v (t) which has been orthonormally transformed is
It is multiplexed into one carrier by the multiplexing unit 14 and sent to the receiving apparatus 200 of FIG. 8B via the transmission path. Noise is generally added to the transmitted multiplexed signal on the transmission line. 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 device 200
The received signal vector y (t) input to is expressed by the following equation.

Y (t) = v (t) + n (t) = Au (t) + n (t) (12) The receiving device includes a detection unit 27 and an inverse normal transform corresponding to the orthonormal transform unit 13 in FIG. 9A. Orthogonal transformation unit 22 and demodulation unit 2
3 and 3. The inverse orthonormal transform unit 22 performs the transform represented by the inverse matrix of Expression (2). If the output signal vector of the inverse orthonormal transform unit 22 is z (t) and its elements are complex numbers z _n (t), then 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) Is. The inverse orthonormal transform unit 22 can obtain the modulation signal vector u (t) from the equation (14).

FIG. 10B shows a circuit configuration for realizing the N × N inverse orthonormal transform matrix realized by the inverse orthonormal transform unit 22. As shown in the figure, the inverse orthonormal transformation matrix is a multiplier 2M _{i, j} (i = 0,1, ..., N-1; j that multiplies the coefficient a _ij of the inverse orthonormal basis.
= 0, 1, ..., N-1) and adders 2A ₀ , ..., 2A _N-1 .
The adder 2A _i of the i-th channel adds the weighted input signals u ₀ (t) to u _N-1 (t) with the coefficients a ^* _{i, 0} , ..., a ^* _{i, N-1} , and outputs the output signal. Get v _i (t). The coefficient a ^* _{i, j} is the complex conjugate of the coefficient a _{i, j} . Coefficients a ^* _0,0 are elements of the matrix, ..., by appropriately selecting the ^{_{a * N-1, N -1}} , performs inverse orthonormal transform. Such numerical calculation can be realized by a computer. It is also possible to perform the matrix calculation mathematically by a computer.

The noise vector of equation (14) is converted into a vector obtained by multiplying the inverse matrix 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 nth transmission channel. The variance of the noise vector subjected to the inverse orthonormal transform of Expression (14) is the same as the variance of the noise vector before the transform at time t. Therefore, in the ideal static transmission path, the deterioration of the transmission characteristics due to the inverse orthonormal transform means does not occur. The reception signal z _i (t) of each channel that has undergone the inverse orthonormal transformation is demodulated by the demodulation unit 23, and a transmitted digital signal is obtained.

9A and 9B, the multiplexing unit 14 and the detection unit 21 are omitted in some cases, and the modulated signal output from the orthonormal transform unit 13 is transmitted to the N-channel transmission path as it is in the base band, and the reverse signal is transmitted. It may be received by the orthonormal transform unit 22. Further, as shown by the dotted line in FIGS. 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 the transmission filter is used will be described. 11A has a configuration in which transmission filters 15 ₀ to 15 _N-1 for preventing interference between channels are provided on the output sides of the modulators 11 _{0 to} 11 _N-1 of the transmission device in FIG. 9A. Each transmission filter 15 _i performs a predetermined transmission band limitation. Each transmission filter 15
Mth-order complex impulse response h of _i , input vector U of transmission filter 15 _i of channel i dating back to time M before
_{Let i} (t) and the output of the transmission filter be 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 the 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. Transmission filter output vector u _'i of formula (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.

[0032] v (t) = AU '( t) (20) when attention is paid to the i-th _{channel, v i (t) = Σ} n = 0 N-1 a in u' n (t) (21) Equation ( _U'n (t) in (21) is band-limited by Eq. (19). Further, the orthonormal matrix A set in the orthonormal transform unit 13
The element a _{in of} is a coefficient that does not include the frequency in the variable. Therefore, the output v _i of the i channel of the orthonormal transform unit 13 of the equation (21)
band (t) is the transmission filter 12 _0-12 set of all outputs of the _{N-1 u 'n (t} ) (n = 0, ..., N-1) is the same as. Therefore, even if the signal is converted by the orthonormal transform unit 13 according to the present invention, the disadvantage of expanding and contracting the transmission band does not occur.

The receiving apparatus shown in FIG. 11B is similar to the receiving apparatus shown in FIG. 9B in that the detecting section 21, the inverse orthonormal section 22, and the demodulating section 2 are included.
3 (demodulators 23 ₀ to 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 Orthogonal Transform Unit 22 for i-th Channel
The output signal z _i (t) of is

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 receive filter output 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 _i (t) (26) The output signal of the receive filter is calculated by the equation (26). z ′ _i (t) is band-limited by the impulse response h of the reception filter.

As described above, by disposing the transmission filter before the input signal of the orthonormal transform unit of the present invention and by arranging the reception filter after the inverse orthonormal transform unit, the transmission filter and the reception filter are root roll-off filters. Can be used as. Further, the orthonormal transform matrix of the present invention only needs to satisfy equations (5), (6), (7), and (9). Therefore, a plurality of orthonormal transformation matrices can be used. If this orthonormal transformation matrix is kept secret from other users, the confidentiality of communication will be improved.

Next, a specific example of the characteristic of the orthonormal transform unit will be shown. The peak power reduction effect and the transmission characteristic for a static communication channel when a DCT base and a WHT base are used as an orthonormal transform unit are shown. FIG. 12 is a block diagram of the transmitter of the present invention in which the peak power is examined by computer simulation. The input digital signal is converted into an N-channel parallel digital signal by the serial / parallel converter 15, and then the modulator 11
QPSK modulation is performed from _{0 to} 11 _N−1 , orthonormal transform unit 13 performs orthonormal transform, and the converted output is added by 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 the orthonormal transform is performed using the WHT basis. The horizontal axis represents the number of carriers (multiplexing number) N, and the vertical axis represents the peak-to-average power ratio. As is clear from the figure, by using the DCT basis and the WHT basis, the peak-to-average power ratio (PAPR) can be set to 4.5 dB when 8 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 N-channel baseband signal is directly transmitted / received without multiplexing, the multiplexing unit and the detection unit of the transmission / reception system are omitted. 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. The noise given in each transmission line is simulated by adding the noise generated from the Gaussian noise sources 7 _{0 to} 7 _N-1 by the adders 6 _{0 to} 6 _N-1 . From this result, it can be seen that the transmission characteristics are not deteriorated by using the DCT base and the WHT base. This transmission characteristic is the same even when the multiplexer and the detector are used. The transmission / reception system of FIG. 14 can be applied to N-channel wired transmission, for example. In the receiving device 200, the demodulation unit 2
A parallel / serial conversion unit 24 is added to the subsequent stage of 3.

In the transmission / reception system of FIG. 14, the transmission unit
With 100, one series of digital signals is input and
N channels (4 channels here) by the parallel conversion unit 9
After being converted into digital signals of the above and modulated respectively, the signals are orthonormally converted by the orthonormal converter 13 and the converted outputs of 4 channels are sent out to the transmission channels of 4 channels.
In the receiving device 200, the inverse orthonormal transform unit 22 performs an orthonormal transform on the N-channel received signal and demodulates the output thereof. A parallel / serial conversion unit 24 is provided in the subsequent stage of the demodulation unit 23,
The demodulated N-channel baseband digital signal is converted into a 1-channel digital signal.

Next, an example of a multicarrier transmission system according to the present invention will be described. 16 and 17 are examples of a transmitting device 100 and a receiving device 200 to which the present invention is applied to multicarrier transmission. The transmitter 100 includes a modulator 11 _i (i = 0, ..., N-1) to which digital signals # 1, ..., # N are input.
And an orthonormal transform unit 13, a transmission filter 15 _i , a digital-analog converter 16 _{i, and the} like. Modulator 1
Each of the modulators 11 _i constituting 1 inputs a different digital signal and performs amplitude or phase or frequency modulation. The plurality of modulator outputs are input to the orthonormal transform unit 13. The orthonormal transform unit 13 performs signal processing using a predetermined matrix so as to reduce the cross-correlation coefficient between the transform output signals v ₀ (t) to v _N−1 (t), and outputs the processed signal. Each output signal v _i (t) is converted into an analog signal by the digital-analog converter 16 _i . The carrier wave of the analog signal is mixed with the carrier wave from the oscillator OSCi in the frequency converter MIX _i to be a radio frequency signal, and the power combiner 14 ′ combines the N channel radio frequency signals and the transmission power amplifier 17 to the antenna. Sent from ANT _T. The receiving device 200 (FIG. 17) includes an antenna ANT _R ,
The radio frequency signal received through the low-noise amplifier 26 is power-distributed to N channels by the power distributor 27, is mixed with the carrier waves from the carrier oscillator OSC _i by the mixer MIX _i in each channel, and is then passed through the bandpass filter 28. It is converted to a baseband signal via _i . The baseband reception signal is converted into a digital signal by the AD converter 29 _i , and the reception filter 25
_It is given to the inverse orthonormal transform unit 22 via _i and undergoes the inverse orthonormal transform. The inverse transform outputs are demodulated by the demodulators 23 _i to obtain data signals. Thus, in the case of multicarrier transmission, the number of carriers is the number of rows and 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. The 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 each element of u (t). Therefore, the equation (27) is generally expressed 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) of is defined as follows. γ _ij (t) = u _i (t) u ^* _j (t) / {| u _i (t) ‖u ^* _j (t) |} (29) Using equation (29), the overall correlation coefficient Γ (t) is defined as follows.

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

The horizontal axis of FIG. 18 represents the correlation coefficient of equation (30),
The vertical axis shows the peak power. As you can see 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 4.0 [W], that is, the sum of the average power of each channel. Like this, 4
In the case of channel QPSK, the peak power depends on the correlation of the modulating waves to be multiplexed.

The horizontal and vertical axes of FIG. 19 are the same as those of FIG. FIG. 19 is 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 4.0W. In this way, in the example using the WHT basis of the present invention, the sum of the peak power and the average power of the input signal matches. Also, the correlation coefficient does not change. This is because the orthonormal transform unit 13 orthogonalizes the input signal.

The horizontal and vertical axes of FIG. 20 are the same as those of FIG. FIG. 20 is 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 peak power and the correlation coefficient between −0.33 and 0.35. From FIG. 20, the peak power can be reduced by the present invention. Comparing FIG. 18 and FIG. 20, the maximum peak power is reduced from 16 [W] to 8 [W].

As described above, by using the present invention for reducing the correlation between the multiplexed modulated signals, the peak power of the modulated signals generated after the multiplexing can be reduced. 10A and 10B, the transmission filter 15 and the reception filter 25 are inserted in the front stage of the orthonormal transform unit 13 and the rear stage of the inverse orthonormal transform unit 22, respectively. The transmission filter 15 and the reception filter 25 may be inserted in the subsequent stage of the orthonormal transform unit 13 and in the preceding stage of the inverse orthonormal transform unit 22 as shown in FIGS. The case of FIGS. 16 and 17 will be described below.

The Mth-order complex impulse response of the transmission filter 15 is h, the output vector of the orthonormal transform unit 13 of the present invention is v (t), and the output signal vector of the orthonormal transform unit 13 of the channel i traced back to the time M before. _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) (34) In equation (34), the transmission filter output w ^(tx) _i (t) for the i-th channel is band-limited by h.

Similarly to the transmission filter, the Mth-order complex impulse response of the reception filter 25 is set to the mixer MI of the channel i.
Let Z _i (t) be the output signal vector of the detector output from X _i, which traces back to time M before, and w _rx (t) be the output of the reception filter. 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 Expression (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, some transmission lines of the transmission / reception system are not particularly clarified, but in general, when a radio frequency carrier is used, the transmission lines are three.
It may be a dimensional space or a metallic cable. Also, an optical fiber cable can be used if electro-optical conversion is performed. On the other hand, when transmitting a baseband signal, a metallic cable is used, but if electro-optical conversion is used, an optical fiber cable can also be used. An example of a typical transmission / reception system using an optical fiber cable as a transmission path of a transmission / reception system to which the present invention is applied and a case of using a metallic cable will be shown below.

FIG. 21 shows an embodiment of the parallel transmission system using the optical fiber cable 32. The transmitter 100 includes a modulator 11, a transmission filter 15, a serial / parallel converter 19, an orthonormal converter 13, a transmitter 10, and an electrical / optical converter E /.
Composed of O. The transmitter 10 has a signal multiplexer 14 and a frequency converter MIX. The receiver 200 includes an optical / electrical converter O / E, a receiver 20, and an orthonormal inverse transformer 22.
And a parallel-serial converter 24, a reception filter 25, and a demodulator 23. The reception unit 20 is a frequency conversion unit MI
It has an X and a demultiplexing unit 27. The demultiplexing unit 27 that demultiplexes the multiplexed signal is configured by a band limiting filter for the signal multiplexed by the frequency, and by a despreader for the signal multiplexed by the spreading code, A filter for wavelength detection is used for signals multiplexed by wavelength.

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

In the receiving device 200, the received optical signal
Convert to electrical converter O / E. The receiving unit 20 separates signals multiplexed by frequency, code, wavelength and 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 transformation unit 24 as a digital signal sequence. The reconstructed signal is waveform-shaped by the reception filter 25 and demodulated by the demodulator 23.

In the embodiment of FIG. 21, an electric / optical converter
E / O and opto-electrical converter O / E are removed, and a metallic cable is used instead of the optical fiber cable 31 to transmit the signal.
The output of 0 and the input of the receiver 20 may be connected. According to this embodiment, parallel transmission using orthonormal transform processing can be performed. FIG. 22 shows an example of baseband transmission using a metallic cable. The transmitter 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 receiving device 200 includes a switch SW2, an orthonormal inverse transforming unit 22, a parallel / serial transforming unit 24, a receiving 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 optimum waveform shaping when transmitting to the transmission line 32 of the metallic cable. The serial-parallel converter 9 parallel-converts the digital signal sequence output from the transmission filter 15. The parallel-converted digital signal sequence is simultaneously input to the orthonormal transform unit 13. The orthonormal transform unit 13 outputs a parallel digital signal sequence whose cross-correlation is 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 path 32. Signal processing suitable for transmission by the metallic cable 32 is performed and the signal is transmitted to the transmission path 32.

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

In the embodiment of FIG. 22, as shown in FIG. 23, the electric / optical converter E /
By providing O and an optical / electrical converter O / E at the input of the switch SW2 on the receiving side, and connecting these converters E / O and O / E with 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. Gaussian noise from the Gaussian noise sources 7 _{0 to} 7 _N-1 is given to the _N- channel transmission line by adders 6 _{0 to} 6 _N-1 . The simulation was set up as follows. CNR in the transmission path of one channel, for example channel i = 0, is fixed to 0, 4, 6, and 8 dB sequentially, and each is fixed.
Figure 1 shows the CNR of the remaining channels i = 1, 2, 3 with respect to the CNR value.
As shown on the horizontal axis of 5, the noise sources 7 ₁ , 7 ₂ , 7 _{3 are} added to adders 6 ₁ , 6 ₂ , so that 0, 2, 4, ..., 12 dB are obtained.
The Gaussian noise level given to 6 ₃ was controlled. Like this
Under the condition of CNR, the baseband digital signal output in the receiver when the orthonormal transform in the orthonormal transform unit 13 used in the present invention is performed using the WHT standard and when the DCT standard is used. 24, the symbol error probabilities are indicated by circles and triangles, respectively, and theoretical values (solid lines) in the case where the present invention is not applied are also shown in comparison with the broken line connecting them. It should be noted that this condition assumes that the line quality of some transmission lines is significantly deteriorated. Theoretical value P
Was _calculated 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 probability of the remaining 3 channels.

It should be noted that the fixed CNR of channel i = 0
Is 4 dB, for example, until the CNRs of the other three channels become 4 dB, which is the same as channel i = 0, the symbols of the output baseband digital signal of the parallel / serial converter 24 with and without the present invention are applied. The error probabilities vary almost the same, but the CNR of channels i = 1, 2, 3 is 4
When the value exceeds dB, the error probability when the present invention is not applied is not so much improved, whereas the error probability when the present invention is applied is further improved.
The same can be said when the CNR of channel i = 0 is 6 dB and 8 dB. This is because N
When a channel modulated signal is orthonormally transformed and transmitted through an N-channel transmission line, the ability to improve the symbol error probability of a received baseband digital signal even if the transmission line characteristic of one transmission line deteriorates, It is shown to be superior to the conventional one without orthonormal transform. That is, it can be said that a certain kind of error correction effect is obtained by the present invention.

[0060]

According to the conventional technique, the number of carriers to be transmitted in parallel is increased, and it cannot be applied to arbitrary input signals.
There were problems such as. According to the present invention, these problems are solved as in (a) and (b) below. (a) Increasing number of carriers to be transmitted in parallel:
As shown in (5), the number of channels in the transmission line does not obviously increase even if the orthonormal transform unit is used. The peak power reduction effect according to the present invention is shown in FIG. As described above, the present invention can reduce the peak-to-average power ratio without increasing the number of transmission channels.

(B) Dependence on the input signal: In the process of deriving the principle, the present invention does not specify the statistical property of the input signal, as shown in the equation (1). Therefore, the present invention does not rely on the statistical nature of the input signal,
The peak-to-average power ratio caused by multiplexing a plurality of modulated signals can be reduced. Other effects are as follows.

(C) Since the information to be transmitted in parallel by the orthogonal transformation unit is converted into the format defined by the orthonormal transformation matrix, the information originally transmitted in one channel is 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 method that is resistant to variations in the quality of the communication line.

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

[Brief description of drawings]

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

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

3 is a graph showing the magnitude of a complex composite signal of the 4-channel QPSK signal of FIG.

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

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

FIG. 6 is a block diagram of the transmitter of FIG.
The graph which shows the magnitude | size of the complex synthetic | combination signal of the signal which carried out the orthogonal orthogonal transformation of the signal by DCT basis.

FIG. 7 is a block diagram of a 4-channel QPSK in the transmitter of FIG.
The graph which shows the magnitude | size of the complex synthetic signal of the signal which carried out the orthogonal orthogonal transformation of the signal by WHT basis.

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

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

10A is a block diagram showing a configuration of an orthonormal transform unit 13 in FIG. 9A, and B is a block diagram showing a configuration of an inverse orthonormal 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 B 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.

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.

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

FIG. 16 is a block diagram showing a configuration when the transmitter of FIG. 8A is applied to a multicarrier transmission system.

FIG. 17 is a block diagram showing a configuration when the receiving device of FIG. 8B is applied to a multicarrier transmission system.

FIG. 18 is a graph showing the relationship between the peak power of a signal obtained by multiplexing a 4-channel QPSK signal and the correlation coefficient between the signals of the respective channels.

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

FIG. 20 is a graph showing the relationship between the peak power of a multiplexed signal after orthogonally transforming a 4-channel QPSK signal using a DCT basis and the correlation coefficient between the converted signals of each channel.

FIG. 21 is a block diagram of a transmission / reception system in the case where the present invention is applied to multicarrier transmission when the 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 line is a metallic cable.

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

FIG. 24 is a graph for explaining an effect when the present invention is applied to 4-channel multicarrier transmission by QPSK.

Continuation of front page (56) Reference JP-A-7-264162 (JP, A) JP-A-8-274748 (JP, A) JP-A-8-340361 (JP, A) JP-A-9-98146 (JP , A) JP 9-98147 (JP, A) JP 9-107345 (JP, A) JP 9-116251 (JP, A) JP 9-149090 (JP, A) 6-504175 (JP, A) Yasuyoshi Suzuki, Ken Kumagai, Toshio Nojima, Reduction of peak-to-average power ratio using direct rectification, Proceedings of the 1997 IEICE Communications Society Conference, Japan, The Institute of Electronics, Information and Communication Engineers, August 13, 1997, 1, p. 409 (58) Fields investigated (Int.Cl. ⁷ , DB name) H04J 11/00

Claims (21)

(57) [Claims] 1. Modulating means for respectively modulating N-channel parallel digital signals, and N is an integer of 2 or more, and a conversion output for the N-channel modulated signal output from the modulating means. Orthonormal transform means for performing orthonormal transform so as to reduce the cross-correlation between them and outputting the transformed N-channel signal. 2. The transmitting apparatus according to claim 1, further comprising: a multiplexing unit that multiplexes outputs of the orthonormal transforming unit and sends out a multiplexed signal. 3. The transmission device according to claim 1 or 2, wherein a transmission filter for band limitation is inserted on the input side or output side of the orthonormal transform means. 4. The transmission device according to claim 1, wherein the input side of said modulation means is a serial / parallel conversion means for converting a serial digital input signal into said N-channel parallel digital signal and giving it to said modulation means. It is provided. 5. The transmitting device according to claim 1, wherein the N-channel output signals of the orthonormal transform means are sequentially selected and output in series, and the serial signal from the switch means is converted into an optical signal. And an electrical / optical conversion means for transmitting the optical fiber cable to the optical fiber cable. 6. The transmission device according to claim 1, further comprising a switching means for sequentially selecting the N-channel output signals of the orthonormal transformation means and outputting them in series to a metallic cable. 7. The transmitting apparatus according to claim 1, wherein the orthonormal transform means is a transform means based on a WHT basis. 8. The transmitter according to claim 1, wherein the orthonormal transform means is a transform means based on a DCT basis. 9. The transmitting apparatus according to claim 1, wherein the input signal vector to the orthonormal transform means at time t is u (t), and the complex baseband signal in the i-th transmission channel is u _i (t). (i = 0,1, ..., N-1), A is an N × N orthonormal transform matrix realized by the above orthonormal transform means, a complex element (i, j) is a _ij , orthonormal basis vector Is a _i , the orthogonal transformation unit output vector is v (t), and the complex output signal in the i-th transmission channel is v _i (t) _{, and} T is a transpose, u (t) = [u ₀ (t ) u ₁ (t) ... u _N-1 (t)] ^T [Equation 1] 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) Then, the orthonormal transformation matrix A satisfies Σ _{n = 0} ^N-1 a _in a ^* _jn = δ _ij , where δ _ij is 1 when i = j and 0 when i ≠ j, * Is a complex conjugate. 10. Orthonormal transform is performed in the transmitter,
A receiving device for receiving N-channel signals transmitted via N transmission channels, including: receiving N-channel signals from N transmission channels and receiving them
An inverse orthonormal transform means for performing the inverse transform of the orthonormal transform on the channel signal to output a corresponding N-channel modulated signal, and an N-channel modulated signal output from the inverse orthonormal transform means Demodulation means for demodulating and outputting an N-channel digital signal. 11. The receiving device according to claim 10, wherein a detection means for detecting a signal from the N transmission channel, outputting a baseband signal of the N channel, and providing the signal as an input signal to the inverse orthonormal transform means is provided. It is provided. 12. The receiving device according to claim 10 or 11, wherein a receiving filter for band limitation is inserted on the input side or output side of the inverse orthonormal transform means. 13. The receiving apparatus according to claim 10 or 11, wherein parallel signals for converting N-channel parallel digital signals into one serial digital signal are provided on the output side of the demodulating means.
A serial conversion means is provided. 14. A transmission / reception method of transmitting an N-channel signal through an N-transmission channel and demodulating an N-channel signal received by a receiving side, including the steps of: (a) For the modulated signal,
Orthogonal transformation is performed so that the cross-correlation between the output signals is reduced, N-channel converted signals are generated, N-transmission channels are transmitted, and (b) N-channel reception signals are obtained on the receiving side. The received signals are subjected to the inverse transform of the orthonormal transform to obtain an N channel modulated signal, and (c) the N channel modulated signal is demodulated to obtain an N channel digital signal. 15. A transmission / reception method of transmitting an N-channel signal through an N-transmission channel and demodulating an N-channel signal received by a receiving side, including the steps of: (a) For the modulated signal,
Orthogonal transform is performed so that the cross-correlation between the converted and output signals is reduced to generate an N-channel converted signal, (b) the N-channel converted signal is multiplexed and transmitted, and (c) a receiving side. At N input signals are obtained from the input multiplex signal, and the received signals are subjected to inverse conversion from the orthonormal transformation to obtain N channel modulated signals. (D) The N channel modulated signals are obtained. It demodulates to obtain an N-channel digital signal. 16. The method according to claim 14 or 15,
The step of converting the input serial digital signal into an N-channel parallel digital signal on the transmitting side to obtain the N-channel modulated signal, and the receiving side of the demodulated N-channel parallel digital signal into one serial digital signal. And a step of converting into a signal. 17. The method according to claim 14 or 15, wherein the transmitting side sequentially selects the N-channel output signals obtained by the orthonormal transformation, outputs as a serial signal, and converts the serial signal into an optical signal. The method further includes the steps of sending to the optical fiber cable and converting the serial signal received at the receiving side into an electric signal and sequentially distributing to an N channel to obtain the N channel received signal. 18. The method according to claim 14 or 15, wherein the transmitting side sequentially selects the N-channel output signals obtained by the orthonormal transformation and outputs the N-channel output signals in series to the metallic cable, and the receiving side. The method includes the steps of sequentially dividing the received serial signals into N channels to obtain the received signals of the N channels. 19. The method according to claim 14 or 15,
The orthonormal transform is a transform based on the WHT basis. 20. The method according to claim 14 or 15,
The orthonormal transform means is a transform based on the DCT basis. 21. The method of claim 14 or 15,
The input signal vector subjected to the orthonormal transformation at time t is u (t), the complex baseband signal in the i-th transmission channel is u _i (t) (i = 0,1, ..., N-1), An N × N orthonormal transformation matrix realized by orthogonal transformation is A, its complex element (i, j) is a _ij , and an orthonormal basis vector is a.
Let _i , v (t) be the output vector of the orthogonal transformation unit, and v _i (t) be _the complex output signal in the i-th transmission channel _{, and} T be the transpose, then u (t) = [u ₀ (t) u ₁ (t)… u _N-1 (t)] ^T [Equation 2] 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) Then, the orthonormal transformation matrix A satisfies Σ _{n = 0} ^N−1 a _in a ^* _jn = δ _ij , where δ _ij is 1 when i = j, and 0 when i ≠ j, * Is a complex conjugate.