CN101026433A - Signal-to-noise ration estimation method for adaptive modulating-coding - Google Patents

Abstract

In the method, the received data are expressed function containing UH, where U as unitary matrix, UHu=UUH=INs*Ns, and WHSmWNm=WNmWHNm=INm*Nm. WNm is as FFT unitary matrix or orthogonal Walsh-Hadamad transformation matrix, and H denotes conjugated transposition of matrix. INm*Nm is diagonal matrix with diagonal elements being as 1. Using the received data containing UH, the method calculates SNR. The obtained SNR is a vector of received SNR corresponding to each constellation symbol. Calculating SNR of reception accurately, the invention is applicable to broad sensed frequency division multiplex system.

Description

A kind of signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding

Technical field

The present invention relates to a kind of method of accurate estimation received signal to noise ratio, relate in particular to a kind of received signal to noise ratio evaluation method that is used for the Adaptive Modulation and Coding of broad sense Frequency Division Multiplexing system and single-carrier system, simultaneously, the present invention also can be directly used in code division multiple access (CP-CDMA) system based on Cyclic Prefix.

Background technology

Adaptive Modulation and Coding (AMC) a lot of documents in the past that are used for OFDM (OFDM) and CP-SC (single carrier) system all are studied and proposed.In the Adaptive Modulation and Coding based on frame, the modulation coding mode of frame data (MCS) is constant, but according to the difference of the state information of channel, the MCS between frame and the frame changes.For ofdm system, bit error rate performance depends primarily on that subcarrier of signal to noise ratio minimum.And for the CP-SC system, the energy of each bit all is to be evenly distributed on the entire spectrum.Therefore, the serious frequency that is caused by decline property channel declines also little to the influence of CP-SC system.Why this has just adopted can be considerably beyond the same ofdm system that has adopted based on frame AMC based on the throughput of the CP-SC system of the Adaptive Modulation and Coding of frame.If each subcarrier for ofdm system adopts different MCS, the throughput of system can improve greatly, but this need increase a large amount of signaling redundancies, and is very unactual.

Broad sense Frequency Division Multiplexing system (GOFDM) proposes as the half-way house of OFDM (OFDM) and CP-SC (single carrier) system.In GOFDM, a plurality of undersized OFDM symbols carry out the GOFDM frame that a FFT length is formed in cascade in time domain.Same, Cyclic Prefix is added in the front end of each GOFDM frame.At receiving terminal, adopt frequency-domain equalizer that data are carried out equilibrium.It is 1 when the sub-carrier number of OFDM symbol that the GOFDM system can be regarded as traditional CP-SC system, also can be seen as ofdm system equals FFT when the sub-carrier number of OFDM symbol length.

Broad sense Frequency Division Multiplexing system (GOFDM) is a good candidate scheme, based on to complexity, and frequency diversity, peak-to-average power ratio is especially based on the AMC of frame.

In the prior art, for the broad sense Frequency Division Multiplexing system because this system not by practical application, therefore also nobody proposes signal-noise ratio computation method based on GOFDM.

Summary of the invention

Technical problem to be solved by this invention provides a kind of signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding, is used in accurate Calculation received signal to noise ratio in the broad sense Frequency Division Multiplexing system.

On solving the problems of the technologies described above, the present invention has adopted following technical scheme:

The signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding of the present invention is characterized in that, will receive data and be expressed as and contain U ^HFunction, wherein, U is a unitary matrice, And $U^{H}U=U{U}^H=I_{N_s\×N_s},$ $W_{N_m}=\frac{1}{\sqrt{N_m}}{\left(\begin{array}{cccc}1& W_{N_m}^{1-1}& ...& W_{N_m}^{1-N_m}\\ 1& W_{N_m}^{2-1}& ...& W_{N_m}^{2-N_m}\\ ...& ...& ...& ...\\ 1& W_{N_m}^{N_m-1}& ...& W_{N_m}^{N_m\·N_m}\\ & & & \\ & & & \end{array}\right)}_{N_m\×N_m},$ And $W_{N_m}^H{W}_{N_m}=W_{N_m}{W}_{N_m}^H=I_{N_m\×N_m};$

Wherein, W _NmBeing the FFT unitary matrice, also can be quadrature Walsh-Hadamad transformation matrix, ^HRepresenting matrix grip transposition altogether.I _{Nm * Nm}It for diagonal element 1 diagonal matrix.

With the above-mentioned U that contains ^HThe reception data be used to calculate signal to noise ratio, the signal to noise ratio of acquisition is a vector corresponding to the received signal to noise ratio of each constellation symbol.

Description of drawings

Fig. 1 is the structural representation of the dispensing device of broad sense Frequency Division Multiplexing system.

Fig. 2 is the structural representation of the receiving system of broad sense Frequency Division Multiplexing system.

Fig. 3 is the simulation performance schematic diagram of the throughput of system of multiple modulation coding mode correspondence.

Fig. 4 is the CP-SC that adopts based on the AMC of frame, the throughput of system performance schematic diagram of GOFDM and OFDM.

Embodiment

The transmission that Fig. 1 has provided the GOFDM system receives structured flowchart.At transmitting terminal, long for the data sequence of Ns is modulated onto on the constellation point, going here and there and changing length overall is that the transmission sequence d of Ns is divided into the short sequence d that the K group length is Nm ^k, Ns=Nm * k, 0≤k≤K-1.These short sequences are the IFFT that Nm is ordered respectively, then obtain s ^k, can be expressed as:

$s^k=W_{N_m}^H{d}^k---\left(1\right)$

Wherein, W _NmBe the FFT unitary matrice, $W_{N_m}=\frac{1}{\sqrt{N_m}}{\left(\begin{array}{cccc}1& W_{N_m}^{1-1}& ...& W_{N_m}^{1-N_m}\\ 1& W_{N_m}^{2-1}& ...& W_{N_m}^{2-N_m}\\ ...& ...& ...& ...\\ 1& W_{N_m}^{N_m-1}& ...& W_{N_m}^{N_m\·N_m}\\ & & & \\ & & & \end{array}\right)}_{N_m\×N_m},$ And $W_{N_m}^H{W}_{N_m}=W_{N_m}{W}_{N_m}^H=I_{N_m\×N_m}.$ ^HRepresenting matrix grip transposition altogether.I _{Nm * Nm}It for diagonal element 1 diagonal matrix.Then with s ^kCascade, also string converts sequence s to again, and s can be expressed as:

s＝U ^Hd (2)

Wherein,

And $U^{H}U=U{U}^H=I_{N_s\×N_s}.$

Final nucleotide sequence s sends after adding Cyclic Prefix.

As shown in Figure 2: at receiving terminal, at first the data that will receive obtain r after removing Cyclic Prefix behind the process channel, and r can be expressed as:

r＝Cs+n (3)

＝CU ^Hd+n

Wherein, channel matrix C is that circular matrix is expressed as:

Andh=[h ₀h ₁H _L-1] be the time domain impulse response of channel, l is a channel exponent number, n is a white Gaussian noise.Channel matrix C can also be expressed as in addition:

$C=W_{N_s}^{H}H{W}_{N_s}---\left(4\right)$

Wherein Be N _s* N _sDiagonal matrix, diagonal element is the frequency response of channel, all the other are zero.W _NsBe N _s* N _sThe FFT unitary matrice.Formula (4) substitution (3) is obtained:

$r=C{U}^{H}d+n$

$=W_{N_s}^{H}H{W}_{N_s}{U}^{H}d+n---\left(5\right)$

The received signal that obtains in time domain behind the removal CP is passed through N _sPoint FFT can transform to frequency domain,

$R=W_{N_s}r$

$=W_{N_s}(W_{N_s}^{H}H{W}_{N_s}{U}^{H}d+n)---\left(6\right)$

$=H{W}_{N_s}{U}^{H}d+W_{N_s}n$

1.1 ZF (ZF) equilibrium

Adopt the ZF equilibrium, the data sequence R after the FFT conversion that is about to order through Ns multiply by frequency channels matrix (H ^HH) ^-1H ^H:

$\hat{R}={\left(H^{H}H\right)}^{-1}{H}^{H}R$

${=\left(H^{H}H\right)}^{-1}{H}^H(H{W}_{N_s}{U}^{H}d+W_{N_s}n)---\left(7\right)$

$=W_{N_s}{U}^{H}d+H^{-1}{W}_{N_s}n$

Carrying out the IFFT conversion that Ns orders again obtains , be expressed as:

$\hat{r}=W_{N_s}^H\hat{R}$

$=W_{N_s}^H(W_{N_s}{U}^{H}d+H^{-1}{W}_{N_s}n)---\left(8\right)$

$=U^{H}d+W_{N_s}^H{H}^{-1}{W}_{N_s}n$

At last, will

Divide into groups, do the FFT conversion that Nm is ordered, obtain

, can be expressed as:

$\hat{d}=U\hat{r}$

$=U(U^{H}d+W_{N_s}^H{H}^{-1}{W}_{N_s}n)---\left(9\right)$

$=d+U{W}_{N_s}^H{H}^{-1}{W}_{N_s}n$

Because the ZF equilibrium does not have ISI, so received signal to noise ratio can be expressed as:

${\mathrm{SNR}}_{\mathrm{ZF}}=\frac{{\mathrm{\σ}}_s^2\·I}{\left({\mathrm{UW}}_{N_s}^H{H}^{-1}{W}_{N_s}n\right){\left(U{W}_{N_s}^H{H}^{-1}{W}_{N_s}n\right)}^H}$

$=\frac{{\mathrm{\σ}}_s^2\·I}{U{W}_{N_s}^H{H}^{-1}{W}_{N_s}n\·n^H{W}_{N_s}^H{H}^{-1,H}{W}_{N_s}{U}^H}---\left(10\right)$

$=\frac{{\mathrm{\σ}}_s^2\·I}{U{W}_{N_s}^H{\mathrm{\σ}}_n^2{\left(H^{H}H\right)}^{-1}{W}_{N_s}{U}^H}$

σ wherein _n ²Be noise variance, σ _s ²Be signal energy, be assumed to 1.I _{Ns * Ns}It for diagonal element 1 diagonal matrix.Because W and U are unitary matrice, so SNR _ZFIt is a vector corresponding to the received signal to noise ratio of Ns constellation symbol.

1.2 least mean-square error (MMSE) equilibrium

The MMSE equilibrium is that sequence R be multiply by (HH ^H+ I σ _n ²) ^-1H ^H, obtain:

$\hat{R}={(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}R$

$={(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H(H{W}_{N_s}{U}^{H}d+W_{N_s}n)---\left(11\right)$

$={(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^{H}d+{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n$

Same, carry out the IFFT conversion that Ns orders again and obtain , be expressed as:

$\hat{r}=W_{S_s}^H\hat{R}$

$=W_{S_s}^H({(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^{H}d+{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n)---\left(12\right)$

$=W_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^{H}d+W_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n$

At last, will

Divide into groups, do the FFT conversion that Nm is ordered, obtain

, can be expressed as:

$\hat{d}=U\hat{r}$

$=U({W_{N_s}^H(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^{H}d+W_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H^{H}W}_{N_x}n)$

$=U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^{H}d+U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n---\left(13\right)$

Wherein, first is the signal of expectation, and middle one is remaining intersymbol interference ISI, and last is an additive Gaussian noise.Remaining intersymbol interference and additive noise sum are designated as J:

$J={\mathrm{\σ}}_s^2(U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H-I_{N_s\×N_s}){(U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H-I_{N_s\×N_s})}^H$

$+\left(U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n\right){\left(U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W}_{N_s}n\right)}^H$

$=I-U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H---\left(14\right)$

Last received signal to noise ratio can be expressed as:

${\mathrm{SNR}}_{\mathrm{MMSE}}=\frac{{\mathrm{\σ}}_s^2\·I-J}{J}$

$=\frac{U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H}{I-U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H}---\left(15\right)$

Because W and U are unitary matrice, so SNR _MMSEIt is a vector corresponding to the received signal to noise ratio of Ns constellation symbol.The analytical method of this signal to noise ratio also can be directly used in other system, for example CP-CDMA.For CP-CDMA, unique difference replaces with quadrature Walsh-Hadamad transformation matrix with the U matrix exactly.

2. the received signal to noise ratio that is used for frame AMC

Among the AMC based on entire frame, the The data of one frame is with a kind of modulation coding mode, the selection of modulating-coding is according under all optional modulation coding modes, the error rate that system obtains in the white Gaussian noise environment or frame error rate performance, set signal-noise ratio threshold, utilize the received signal to noise ratio of frame data to carry out the selection of modulating-coding.The received signal to noise ratio computational methods are as follows: when adopting ZF balanced, suppose the energy through the symbol of constellation point modulation ${\mathrm{\σ}}_s^2=1,$ Received signal to noise ratio SNR according to formula (10) one frame data _{ZF_all}Be expressed as:

${\mathrm{SNR}}_{\mathrm{ZF}\_\mathrm{all}}=\mathrm{trace}\left(\frac{{\mathrm{\σ}}_s^2\·I}{U{W}_{N_s}^H{\mathrm{\σ}}_n^2{\left(H^{H}H\right)}^{-1}{W}_{N_s}{U}^H}\right)$

$=\frac{N_s}{\underset{q=0}{\overset{N_s-1}{\mathrm{\Σ}}}\frac{{{|H}_{\mathrm{qq}}|}^2}{|{H_{\mathrm{qq}}|}^2+{\mathrm{\σ}}_n^2}}---\left(16\right)$

When adopting MMSE balanced, according to formula (15), the received signal to noise ratio SNR of frame data _{MMSE_all}Be expressed as:

${\mathrm{SNR}}_{\mathrm{MMSE}\_\mathrm{all}}=\mathrm{trace}\left(\frac{U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H}{I-U{W}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^{H}H{W}_{N_s}{U}^H}\right)$

$=\frac{\underset{q=0}{\overset{N_s-1}{\mathrm{\Σ}}}\frac{{{|H}_{\mathrm{qq}}|}^2}{{\left|H_{\mathrm{qq}}\right|}^2+{\mathrm{\σ}}_n^2}}{N_s-\underset{q=0}{\overset{N_s-1}{\mathrm{\Σ}}}\frac{{{|H}_{\mathrm{qq}}|}^2}{{{|H}_{\mathrm{qq}}|}^2+{\mathrm{\σ}}_n^2}}---\left(17\right)$

In a GOFDM system, be used for multiplexing OFDM symbol lengths Nm=16, it is multiplexing that 8 OFDM symbols carry out, k=8, forming length is the GOFDM frame of Ns=128, adopts the MMSE equilibrium.The SNR that the MCS mode of each frame is calculated by formula (17) according to threshold value _MMSEDecision.

Simulation result

Simulation parameter is as shown in table 1:

System	OFDM (FFT=128) GOFDM (little FFT=16, multiplexing #=8) CP-Single Carrier (CP length=16)
		Carrier frequency	2.4G
Sample frequency	5MHz
		Channel model	ITU-PB
Translational speed	3km/h
		Channel equalization	MMSE (channel condition information is known)
AMC	MSC based on frame selects
		Coding	Convolution code (CC)
The emulation amount	Imitative 10,000 frames of each point

Modulation coding mode is as shown in table 2:

MCS	Information bit length	Code length	Delete the length after surplus	Symbol lengths
					QPSK R＝1/2	128	256	256	128
QPSK R＝3/4	192	384	256	128

16QAM R＝1/2	256	512	512	128
					16QAM R＝3/4	384	768	512	128
64QAM R＝2/3	512	1024	768	128
					64QAM R＝3/4	576	1152	768	128

The signal-noise ratio threshold of modulation coding mode correspondence is as shown in table 3:

Threshold (dB)	＜5.91	5.91	8.85	12.62	16.98	18.97
							MCS	QPSK R＝1/2	QPSK R＝3/4	16QAM R＝1/2	16QAM R＝3/4	64QAM R＝2/3	64QAM R＝3/4

The thresholding of signal to noise ratio snr is set formula based under Gaussian channel, the simulation performance of the throughput of system of each modulation coding mode, as shown in Figure 3.The intersection point of adjacent two curves is exactly a threshold value.Employing is based on the CP-SC of the AMC of frame, and the throughput of system performance of GOFDM and OFDM is seen Fig. 4, and the selection of MCS is the received signal to noise ratio that calculates according to formula (17).The throughput that can significantly see the GOFDM system is between CP-SC and OFDM.

Claims (8)

1, a kind of signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding is characterized in that, will receive data and be expressed as and contain U ^HFunction, U is a unitary matrice,

And $U^{H}U={\mathrm{UU}}^H=I_{N_s\×N_s},$ $W_{N_m}=\frac{1}{\sqrt{N_m}}{\left[\begin{array}{cccc}1& W_{N_m}^{1.1}& ...& W_{N_m}^{1.N_m}\\ 1& W_{N_m}^{2.1}& ...& W_{N_m}^{2.N_m}\\ ...& ...& ...& ...\\ 1& W_{N_m}^{N_m.1}& ...& W_{N_m}^{N_m.N_m}\end{array}\right]}_{N_m\×N_m}$ , and $W_{N_m}^H{W}_{N_m}=W_{N_m}{W}_{N_m}^H=I_{N_m\×N_m}$ Wherein, W _NmBeing the FFT unitary matrice, also can be quadrature Walsh-Hadamad transformation matrix, ^HRepresenting matrix grip transposition altogether.I _{Nm * Nm}It for diagonal element 1 diagonal matrix.

With the above-mentioned U that contains ^HThe reception data be used to calculate signal to noise ratio, the signal to noise ratio of acquisition is a vector corresponding to the received signal to noise ratio of each constellation symbol.

2, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 1, it is characterized in that, comprise step: long for the data sequence of Ns is modulated onto on the constellation point at transmitting terminal, going here and there and changing length overall is that the transmission sequence d of Ns is divided into the short sequence d that the K group length is Nm ^k, NS=Nm * k, O≤k≤K-1 is the IFFT that Nm is ordered respectively with described short sequence, then obtains s ^k, can be expressed as: $S^k=W_{N_m}^H{d}^k.$

3, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 2 is characterized in that, comprises step: with sequence s ^kCascade, also string converts sequence s to again, and s is expressed as: s=U ^HD.

4, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 3 is characterized in that, comprises step: send after sequence s adds Cyclic Prefix.

5, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 4 is characterized in that, comprises step: at receiving terminal, at first will obtain sequence r behind the data removal Cyclic Prefix that receive, r is expressed as:

$r=W_{N_s}^H{\mathrm{HW}}_{N_s}{U}^{H}d+n,$ Wherein,

Be N _s* N _sDiagonal matrix, diagonal element is the frequency response of channel, all the other are zero, n is a white Gaussian noise.

6, the signal-to-noise ration estimation method that is used for white adaptation modulating-coding according to claim 5 is characterized in that, comprises step: the sequence behind the above-mentioned removal Cyclic Prefix is passed through N _sPoint FFT can transform to frequency domain, obtains data sequence R:

$R={\mathrm{HW}}_{N_s}{U}^{H}d+W_{N_s}n.$

7, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 6 is characterized in that, comprises step:

Above-mentioned data sequence R is carried out the zF equilibrium, be about to data sequence R and multiply by frequency channels matrix (H ^HH) ^-1H ^H, obtain $\hat{R}=W_{N_s}{U}^{H}d{+H}^{-1}{W}_{N_s}n;$

Carrying out the IFFT conversion that NS orders again obtains: $\hat{r}=U^{H}d+W_{N_s}^H{H}^{-1}{W}_{N_s}n;$

Will

Divide into groups, do the FFT conversion that Nm is ordered, obtain $\hat{d}=d+{\mathrm{UW}}_{N_s}^H{H}^{-1}{W}_{N_s}n;$

Obtaining received signal to noise ratio at last is: ${\mathrm{SNR}}_{\mathrm{ZF}}=\frac{{\mathrm{\σ}}_s^2.I}{{\mathrm{UW}}_{N_s}^H{\mathrm{\σ}}_n^2{\left(H^{H}H\right)}^{-1}{W}_{N_s}{U}^H}$ Wherein, σ _n ²Be noise variance, σ _s ²

Be signal energy, be assumed to 1, I _{Ns * Ns}It for diagonal element 1 diagonal matrix.

8, the signal-to-noise ration estimation method that is used for Adaptive Modulation and Coding according to claim 6 is characterized in that, comprises step:

Above-mentioned data sequence R is carried out the MMSE equilibrium, be about to sequence R and multiply by (H ^HH+I σ _n ²)-1H ^H, obtain

$\hat{R}={(H^{H}H+{\mathrm{I\σ}}_n^2)}^{-1}{H}^H{\mathrm{HW}}_{N_s}{U}^{H}d+{(H^{H}H+{\mathrm{I\σ}}_n^2)}^{-1}{H}^H{W_N}_{s}n;$

Carrying out the IFFT conversion that Ns orders again obtains

, be expressed as:

$\hat{r}=W_{N_s}^H{(H^{H}H+{\mathrm{I\σ}}_n^2)}^{-1}{H}^H{\mathrm{HW}}_{N_s}{U}^{H}d+W_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{W_N}_{s}n;$

Will

Divide into groups, do the FFT conversion that Nm is ordered, obtain

, can be expressed as:

Wherein, first is the signal of expectation, and middle one is remaining intersymbol interference ISI, and last is an additive Gaussian noise; Remaining intersymbol interference and additive noise sum are designated as $J:J=I-{\mathrm{UW}}_{N_s}^H{(H^{H}H+{\mathrm{I\σ}}_n^2)}^{-1}{H}^H{\mathrm{HW}}_{N_s}{U}^H;$

At last, received signal to noise ratio can be expressed as: ${\mathrm{SNR}}_{\mathrm{MMSE}}=\frac{{\mathrm{UW}}_{N_s}^H{(H^{H}H+{\mathrm{I\σ}}_n^2)}^{-1}{H}^H{\mathrm{HW}}_{N_s}{U}^H}{I-{\mathrm{UW}}_{N_s}^H{(H^{H}H+I{\mathrm{\σ}}_n^2)}^{-1}{H}^H{\mathrm{HW}}_{N_s}{U}^H}.$

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