This application claims the benefit of Taiwan application Serial No. 97125556, filed Jul. 7, 2008, the subject matter of which is incorporated herein by reference.
The invention relates in general to a signal detecting method and a receiver using the same.
In the field of communication, many communication systems including multiple input multiple output (MIMO) system and orthogonal frequency division multiplexing-code division multiple access (OFDM-CDMA) system are subject to the problem of signal detection. That is, the transmitting signal vector x is detected according to the receiving signal vector y=Hx+n.
Wherein, the channel matrix H and the statistic characteristics (such as Gaussian white noise) of the noise vector n are given, and the elements of the transmitting signal vector x are symbols obtained by way of quadrature amplitude modulation (QAM). In theory, the detecting method which produces minimum error rate is maximum likelihood detection (ML detection), but the complexity increases exponentially along with the dimension of the transmitting signal vector x and the magnitude of modulation, and becomes impractical to actual signal telecommunication system.
To the contrary, despite some detecting methods such as zero forcing method and Lenstra-Lenstra-Lovász (LLL) method have low complexity but the detecting performance is too poor to be accepted in actual systems.
Let the LLL method which is developed from the field of mathematics be taken for example. The contents and object of the LLL method are disclosed below.
Let the channel matrix be expressed as H=└h1, . . . ,hn
However, the detecting method still has many technical difficulties:
1. The complexity of the detecting method is related to the basis {h1, . . . ,hn
2. The detecting method aims at outputting another set of basis {{tilde over (h)}1, . . . ,{tilde over (h)}n
Thus, how to develop a signal detecting method which is efficient and can be realized and can improve the performance of error rate in detecting signals when applied to a signal telecommunication system has become an imminent issue to be resolved.
The invention is directed to a signal detecting method and a receiver using the same.
An exemplary embodiment in accordance with the invention, a method includes the following steps. A receiving signal vector y is received through a number of channels, wherein the receiving signal vector y corresponds to a transmitting signal vector x transmitted by at least one of the channels. A channel matrix H is determined, wherein the channel matrix H represents at least one of the channels. A factorization matrix D is chosen, wherein D is invertible to make the channel matrix factorized as H={tilde over (H)}D, and {tilde over (H)} is a corresponding channel matrix. The factorization matrix D is determined to make an expected value of the signal estimate error become smaller. The receiving signal vector y is detected to estimate the transmitting signal vector x according to the corresponding channel matrix {tilde over (H)} and the factorization matrix D.
In another exemplary embodiment accordance with the invention, a receiver including a receiving unit, a channel estimating unit, a processing unit, and a signal estimating unit is provided. The receiving unit is used for receiving a receiving signal vector y through a number of channels, wherein the receiving signal vector y corresponds to a transmitting signal vector x transmitted by at least one of the channels. The channel estimating unit is used for determining a channel matrix H, wherein the channel matrix H represents at least one of the channels. The processing unit is used for choosing a factorization matrix D, wherein D is invertible to make the channel matrix H expressed as H={tilde over (H)}D, wherein {tilde over (H)} is a corresponding channel matrix. The factorization matrix D is determined to make an expected value of the signal estimate error become smaller. The signal estimating unit is used for detecting the receiving signal vector y to estimate the transmitting signal vector x according to the corresponding channel matrix {tilde over (H)} and the factorization matrix D.
The invention will become apparent from the following detailed description of the preferred but non-limiting embodiments. The following description is made with reference to the accompanying drawings.
Referring to
Generally speaking, maximum likelihood detecting method is a preferred method for detecting the receiving signal vector x. That is,
That is, the estimate {circumflex over (x)} obtained by estimating the transmitting signal vector x according to the detection of the receiving signal vector y is a QAM symbol of the set Ω which minimizes the square of the distance y−Hx.
Wherein j=√{square root over (−1)}, |Ω|=M (the set Ω has M symbols, and M is a positive integer), and equation (1) is exemplified by M-ary QAM. According to equation (1), the complexity of the maximum likelihood detecting method increases exponentially as x or Ω increases.
Referring to
As indicated in
Referring to
Referring to
Furthermore, provided a channel factorization is given as H={tilde over (H)}D, the abovementioned signal model y=Hx+n can be expressed as:
y=Hx+n={tilde over (H)}Dx+n={tilde over (H)}z+n (2)
Wherein, a intermediate signal vector is defined as z=Dx.
Then, a detection matrix G is determined, a to-be-detected signal vector {tilde over (z)} is determined according to the operation Gy of the detection matrix C and the receiving signal vector y, and the to-be-detected signal vector {tilde over (z)} is used for determining the estimate {circumflex over (z)} of the intermediate signal vector z to estimate the estimate {circumflex over (x)} of the transmitting signal vector x, wherein the expected value of the signal estimate error is correlated with an expected value of an error between the to-be-detected signal vector {tilde over (z)} and the intermediate signal vector z.
For example, after the operation Gy=G{tilde over (H)}z+Gn is obtained according to equation (2), if the detection matrix G is equal to {tilde over (H)}−1, then Gy=z+Gn is easily obtained. Thus, of the many QAM symbols, the symbol closest to Gy is used as the estimate {circumflex over (z)} of the intermediate signal vector z so as to obtain the estimate {circumflex over (x)} of the transmitting signal vector x according to x=D−1z. However, when the detection matrix G is equal to {tilde over (H)}−1, the influence of noise will increase (that is, the value of G−1n being too large), so it is hard to obtain a correct estimate {circumflex over (z)}. Therefore, by selecting a more suitable detection matrix G, the influence of noise during estimation is reduced.
Let the abovementioned to-be-detected signal vector {tilde over (z)} be equal to Gy, and the abovementioned expected value of the signal estimate error be equal to the expected value of ∥Gy−z∥2(=∥{tilde over (z)}−z∥2). Moreover, let the detection matrix G be a minimum mean square error (MMSE) detection matrix GMMSE for obtaining the expected value of the signal estimate error to estimate the intermediate signal vector z.
According to the current communication theory, the abovementioned detection matrix GMMSE is expressed as:
Wherein
HH is a Hermitian matrix of the channel matrix H, σn2 is the variance of noise vector n, σx2 is the variance of the transmitting signal vector x, nt is the number of elements of the transmitting signal vector x, and In
Then, the covariance matrix ΦMMSE of the error vector GMMSEy−z which occurs when the detection matrix GMMSE is obtained according to:
Wherein the first intermediate matrix
is a Hermitian matrix for example.
As indicated in equation (4), the mean square error (MSE) of the elements of the intermediate signal vector z are the diagonal elements of the corresponding covariance matrix ΦMMSE and are related with the factorization matrix D. That is, the factorization of the channel matrix H will determine the MSE of the intermediate signal vector z. As the detection of the intermediate signal vector z is implemented by simple rounding operation, given that the minimal distance (that is, 1) is fixed during detection, it is a key object to improve the error rate performance by minimizing the MSE during detection.
In the abovementioned step of determining the factorization matrix D, the factorization matrix D is changed by making the sum of diagonal elements of the covariance matrix ΦMMSE=DADH smaller. The first intermediate matrix A is not limited to the first intermediate matrix A defined by equation (4). Any first intermediate matrix A correlated with the channel matrix H and a signal to noise ratio will do. Meanwhile, the expected value of the signal estimate error is the diagonal elements of the covariance matrix ΦMMSE.
How to search for the optimal factorization matrix D to minimize the MSE E└∥Gy−z∥2┘ is exemplified by two embodiments below.
According to the above disclosure, provided that the channel matrix H, σx2 and σn2 are given (that is, A is given), the invention obtains a suitable factorization matrix D. Suppose the channel factorization result is H={tilde over (H)}D, the invention aims at making the sum of the diagonal elements of the covariance matrix ΦMMSE=DADH smaller, that is, making the tr(ΦMMSE) smaller, wherein tr() denotes the sum of the diagonal elements of the matrix. As the factorization matrix D is a transformation bridge between the intermediate signal vector z and the output signal x (that is, z=Dx), the factorization matrix D must be invertible. Also, in order to simplify the complexity of the intermediate signal vector z, the present embodiment of the invention is preferably implemented by simple rounding operation, and preferably, the elements of the factorization matrix D are integers.
For example, the present embodiment of the invention sets the factorization matrix as D=In
After a cycle of iteration is completed, a decision regarding whether to enter the next iteration or to terminate the detecting method is made according to stopping criterion. The detecting procedures in a cycle of iteration and the stopping criterion are disclosed below.
Let the updating of the k-th row be taken as an example, and other rows can be updated in the same way. Let equation (4) be expressed as:
The k-th diagonal element of covariance matrix ΦMMSE represents the MSE msek when detecting the k-th element zk of the intermediate signal vector z, wherein k=1, . . . ,nt, and msek is expressed as:
mse
k
=d
[k,:]
Ad
[k,:]
H
,k=1, . . . ,nt (6)
Wherein d[k,:] denotes the vector formed by the elements of the k -th row of the factorization matrix D. As indicated in equation (6), provided the matrix A is given, the MSE when detecting the k-th element zk of the intermediate signal vector z is only correlated with the elements of the k-th row (that is, d[k,:]) of the factorization matrix D, so the updating of d[k,:] affects msek only, and no msek′,k′≠k will be affected.
The method of updating the factorization matrix D can be achieved by way of changing one row vector of the factorization matrix D as the sum of a linear combination of other row vectors and the abovementioned one row vector. For example, the updating of the factorization matrix D is expressed as:
d
[k,:]
new=α1*d[1,:]+ . . . +αk−1*d[k−1,:]+d[k,:]+αk+1*d[k+1,:]+ . . . +αn
Wherein α1, . . . ,αk−1,αk+1, . . . , αn
Wherein α1=[α1 . . . αk−1]T, α2=[αk+1 . . . αn
Let
then the matrix B can be partitioned as:
So, equation (8) can also be expressed as:
Wherein
Next, the equation (10) is differentiated with respect to α, and the solution obtained by setting the equation equal to 0 is expressed as:
As the αopt obtained from the equation (11) may not be an integer, the elements of the d[k,:]new of the equation (12) may not be integers either. To simplify calculation, local integer points search can be applied to the neighborhood of the obtained αopt to make the elements of the factorization matrix D be integers.
Lastly, the αupd that really needs to be updated can be obtained according to the following equation (13):
Wherein Ψα,j is a set of many integer points closest to the └αopt┘j, and └αopt┘j, j=1, . . . ,nt−1 denotes the j-th element of the vector αopt.
The numeric result shows that after the number of elements in the set Ψα,j is over 2, the improvement in performance is insignificant, so the balance between performance and complexity is obtained by setting
Lastly, if msekupd<msek, then d[k,:] is updated (that is, d[k,:]=d[k,:]upd); otherwise, d[k,:] is not updated.
According to the above disclosure, a cycle of iteration is completed by sequentially updating all rows. The abovementioned stopping criterion is:
If none row is updated in a cycle of iteration; or
The number of iteration already achieves a predetermined number N1, then the detecting method terminates, and the factorization matrix D lastly updated is outputted.
Referring to Table 1, a part of the pseudo code of the abovementioned detecting method is disclosed below:
Referring to
Firstly, the method begins at step S101, the matrix A is set as:
Next, the method proceeds to step S102, whether i is smaller than N1 is determined (that is, whether the number of iteration is smaller than N1 is determined). If i is not larger than N1, then the method proceeds to step S103; if i is smaller than N1, then the method proceeds to step S104.
In step S103, as the number of iteration already achieves N1, the factorization matrix D is outputted.
In step S104, the updated row number is set as k=1, and the determination factor is updated as n=0.
Then, the method proceeds to step S105, whether k is smaller than or equal to n, is determined (that is, whether the last row is updated is determined). If k is smaller than or equal to nt, then the method proceeds to step S106; if k is larger than nt, then the method proceeds to step S108.
In step S106, the k-th row of the factorization matrix D is updated.
As indicated in
In step S1061, it is set that
and it is obtained that d[k,:]upd=└(α1upd)H 1 (α2upd)H┘D and msekupd=d[k,:]updA(d[k,:]upd)H.
In step S1062, whether msekupd is smaller than msek is determined. If msekupd is smaller than msek, then the method proceeds to step S1063; if msekupd is not smaller than msek, then the method proceeds to step S107.
In step S1063, it is set that d[k,:]=d[k,:]upd, msek=msekupd, n=1.
In step S107, the row number k is increased by 1.
Then, the method returns to step S105, whether k is smaller than or equal to nt is determined.
In step S105, if k is larger than nt, then the method proceeds to step S108.
In step S108, whether the updating determination factor n is 0 is determined. If the updating determination factor n is not 0, then the method proceeds to step S109; if the updating determination factor n is 0, then the method proceeds to step S110.
In step S109, i is set as i=i+1.
In step S110, i is set as i=N1.
Then, the method returns to step S102.
In the above example, the factorization matrix D is changed though recursive operation by way of changing one row at a time. However the present embodiment of the invention can also obtain the factorization matrix D by a searching method to make the expected value of the signal estimate error become smaller
The effects of the present embodiment of the invention are exemplified in the following experimental diagrams. Referring to
Besides, during the operation of the conventional LLL method, the aggregate MSE does not necessarily become smaller despite the MSE shows an overall decreasing tendency (for the channel after average).
Referring to
Referring to
The present embodiment of the invention effectively overcomes the difficulties occurring to conventional technologies, and further provides a detecting method which adjusts the complexity of operation according to actual needs of hardware. Moreover, the operation of the detecting method of the present embodiment of the invention is based on the performance index which reflects improvement in the performance of the error rate of the signal detection, that is, the MSE during detection is minimized.
The present embodiment of the invention provides another method for searching for an optimum factorization matrix D in step (d) so as to minimize the sum of the diagonal elements of the covariance matrix ΦMMSE=DADH.
Furthermore, provided a channel factorization is given as H={tilde over (H)}D, the abovementioned signal model y=Hx+n can be expressed as:
y=Hx+n={tilde over (H)}DX+n={tilde over (H)}z+n (16)
Next, the minimum mean square error (MMSE) is used as a criterion in detecting the intermediate signal vector z. Firstly, let the detection matrix GMMSE be expressed as:
Then, the covariance matrix ΦMMSE of the error vector GMMSEy−z which occurs when the detection matrix GMMSE is obtained according to:
Wherein the first intermediate matrix
is a Hermitian matrix for example.
Provided the result of the channel factorization is H={tilde over (H)}D, the following method is adopted to make the sum of the diagonal elements of the covariance matrix ΦMMSE=DADH smaller (that is, to make
as smaller as possible).
After the covariance matrix ΦMMSE=DADH is determined, the first intermediate matrix A is factorized as A=RHR, wherein R is a second intermediate matrix. Then, the second intermediate matrix R is factorized according to the LLL method, and the factorization matrix D is changed accordingly, wherein A can be other intermediate matrix correlated with the channel matrix H and a signal to noise ratio, and the second intermediate matrix R is factorized many times in the LLL method. The factorization matrix D is directly obtained from the factorization result R={tilde over (R)}T of the second intermediate matrix R finally obtained so as to achieve the object of the invention without having to recursively change the factorization matrix D for several times.
In greater details, if the matrix A is transformed as A=RHR, then tr(ΦMMSE) can be expressed as:
That is, nt reasonable short lattice points {Rd[k,:]H}k=1n
Therefore, the present embodiment of the invention adopts LLL method, and R is expressed {tilde over (R)}T so as to obtain the following equations:
The factorization matrix D is obtained according to equation (25).
Referring to
Firstly, the method begins at step S201, the matrix A is set as
Next, the method proceeds to step S202, the matrix A is factorized as A=RHR.
Then, the method proceeds to step S203, the matrix R is factorized as R={tilde over (R)}T according to the Lenstra-Lenstra-Lovász (LLL) method, and the solution of the factorization matrix D is obtained according to
D=(T−1)H=[R−1{tilde over (R)}]H.
Then, the method proceeds to step S204, the factorization matrix D is outputted.
While the invention has been described by way of example and in terms of a preferred embodiment, it is to be understood that the invention is not limited thereto. On the contrary, it is intended to cover various modifications and similar arrangements and procedures, and the scope of the appended claims therefore should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements and procedures.
Number | Date | Country | Kind |
---|---|---|---|
97125556 | Jul 2008 | TW | national |