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# MMSE standard form ???

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#### bee

##### Full Member level 2
do anyone has got the proof or have any idea of getting this proof that MMSE equation i.e. min E [|| WY-X ||^2] can be standerized to ( (H'H+σ²)^-1 )H'

where ' ----> hermettian transpose
^2 ---> rasie to the power 2
^-1 ----> raise to the power -1
σ ------> noise variance
E [] -----> mean

tthe system considered above is Y = HX+N , where X is transmiites signal, Y is recieved, H is the channel matrix and N is white gaussian noise. W is the MMSE filter coefficent.

I got this principle expression for MMSE receiver is

G=H*(H*H + No/Po I )^-1 *x

Where H -- channel matrix

I --- identity matrix

Iam not understand about the above expression That is what is No/Po and why multiplying with I , how can we achevied MMSE in the above expression etc.... Can you please help me ...

From MMSE sense, you can induce the following relationship.
E[|WY - X|^2] ==> E[|WY - X|Y'] = 0 ==> WE[YY'] - E[XY'] = 0 ==> W = E[XY']/E[YY']

Here, E[XY'] is the cross-correlation, while E[YY'] is the auto-correlation.

So, E[XY'] = XX'H + XN' and E[YY'] = XX'HH' + XHN' + X'H'N + NN'
Since X and N are independent and uncorrelated, XN' = X'N = 0, and NN' = sigma^2

Therefore, W = XX'H'/(XX'HH' + sigma^2).

If you send 2-PAM, then XX' = 1. Now you get W = H'/(HH' + sigma^2)

G=H*(H*H + No/Po I )^-1 *x

Can you help me to understand the above expression...

Help pls...

From the orthogonal principle, E{(WY-X)Y^H} = 0, Y=HX+n
we could get WE(YY^H)=E(XY^H), assuming E(XN^H)=0, i.e., signal and noise are independent, and mean of the noise is zero

E(YY^H)=HE(XX^H)H^H+E(nn^H), Assumped that E(XX^H)=PI, and E(NN^H)=NI, i.e., signal is an independent random vector and noise is also an independent random vector, W=H^H(HH^H+N/PI)^(-1). where P/I reprsents the signal-to-noise ratio. Here, ^H=^(*')

It is clear that the MMSE is one of the Equalization technique, and the equation is as follow:
Y=X.[H*/|H|.^2+(Es/N0)^-1]
Where Y is the received signal
X is the transmitted signal
H is the Faded signal (channel Impulse resp[onse)
Es is a symbol energy for QPSK
N0 is the noise.
I hope this will help
Sh

I have the same problem as you, but stumble across this book:
Multiantenna wireless communication systems by Sergio Barbarossa (Artech House)