If we have a transmitter antenna and M receiver antennas then it is well known that by using maximum ratio combining an M-fold improvement in the SNR can be obtained. But what if the receiver antennas are correlated and the covariance matrix R is known. Can the SNR improvement factor (which I guess will be between 1 and M) then be calculated as a simple function of R ? Any references to where where this problem has been tackled will be greatly appreciated. Thank you.
Correlated antenna gain
Started by ●October 3, 2003
Reply by ●October 4, 20032003-10-04
Hi, if h_1,.., h_M are the complex channel gains from the transmitter to ecah of the receive antennas, then, irrespective of the correlation between the antenna elements at the receiver, the instantaneous SNR at the output of the combiner is given by gamma = (E_s/N_0)(|h_1|^2 + |h_2|^2 +...+|h_M|^2), where E_s/N_0 is the SNR per symbol per diversity branch. We can take average of gamma without worrying about the correlation between the antenna elements. With the assumption of the marginal pdf of |h_i|^2 is exponentially distributed with mean unity, we obtain an average value of gamma is ME_s/N_0, which is same as the case with uncorrelated fades. However, average SNR does not give a true picture of what's happening (i.e., the average probability of error). So, one should look at the average BER. For more details, pls look at Schwartz-Bennet-Stein's classic book "Communication Systems and Techniques". In principle, one can obtain the average SNR by first finding the marginals from the joint pdf (of |h_1|^2,..|h_M|^2) and taking the expected value of each one. hope this helps Ramesh taqai@hotmail.com (taqai) wrote in message news:<9b5caca3.0310031713.19879fa6@posting.google.com>...> If we have a transmitter antenna and M receiver antennas > then it is well known that by using maximum ratio combining > an M-fold improvement in the SNR can be obtained. > > But what if the receiver antennas are correlated and the > covariance matrix R is known. Can the SNR improvement factor (which > I guess will be between 1 and M) then be calculated as a simple > function of R ? Any references to where where this problem has > been tackled will be greatly appreciated. > > Thank you.
Reply by ●October 4, 20032003-10-04
taqai@hotmail.com (taqai) wrote in message news:<9b5caca3.0310031713.19879fa6@posting.google.com>...> If we have a transmitter antenna and M receiver antennas > then it is well known that by using maximum ratio combining > an M-fold improvement in the SNR can be obtained. > > But what if the receiver antennas are correlated and the > covariance matrix R is known. Can the SNR improvement factor (which > I guess will be between 1 and M) then be calculated as a simple > function of R ? Any references to where where this problem has > been tackled will be greatly appreciated.What do you mean by "correlated antennas"? Anyway, there is a trick you can use to get rid of non-white noise in your measurements. Assume your measured data x(n) is modeled as x(n)=s(n)+w(n) [1] where s is the "useful" signal and w is non-white noise that is uncorrelated with the signal s. In this case the data autocovariance matrix Rxx becomes Rxx=Rss+Rww [2] where Rss is the signal autocovariance and Rww is the noise autocovariance. Ideally, Rww should be like Rww= sigma^2*I [3] where sigma^2 is the noise power and "I" is the identity matrix, but here the problem is that Rww is non-diagonal. Now, for stationary noise, Rww is Hermitian: Rww=Rww^H [4] (superscript H denotes the "complex conjugate and transposed"). Since it is Hermitian, Rww can be decomposed as Rww=U*U^H [5] Insert [5] in [2]: Rxx=Rss+U*U^H [6] Define the "prewhitened" data covariance matrix Rzz, and manipulate to replace the non-white noise term with a white one: Rzz=U^(-1)*Rxx*U^(-H) [8.a] =U^(-1)(Rss+U*U^H)*U^(-H) [8.b] =U^(-1)*Rss*U^(-H)+I [8.c] Here U^(-H) means (U^H)^(-1). The "easiest" expression would be [8.c], and it is based on the (known) noise covariance matrix Rww. It doesn't look like it, but SNR is preserved even though sigma^2 doesn't appear. The transform matrix U scales the eigenvectors (and eigen values) of Rss to preserve SNR. To find out details, the norm of U should be investigated. I wouldn't be surprised if the trace of Rww turns out to be very useful when investigating the noise power. CAVEAT -- I've written this off memory, so there are no guarantees with respect to mathematical details. You should be able to see the main line of arguments, though. Rune
Reply by ●October 4, 20032003-10-04
Thanks to both of you for your replies. I was actually referring to correlated antanna gains by talking of "correlated antennas", i.e. we can assume that the noise remains independent. (With full correlation for example we would have h_1=....=h_M.) Isn't it true that in that case the average antenna gain would be lower as the correlation between antennas starts to grow ?
Reply by ●October 5, 20032003-10-05
Hi Taqai, I agree with your intuition. But the reality is quite different. This is as follows: Take the case of dual diversity. Let G1 and G2 be the average SNRs on the individual diversity branches and rho be the complex correlation between the antenna elements. For simplicity assuming Rayleigh fading diversity channel. Then from Eqn. (10-10-27) of Schwartz-Bennet-Stein's book, we can write the correlation matrix as |G1, rho*sqrt(G1*G2)| |conjugate(rho)*sqrt(G1*G2) G2 | Using the above we can obtain the average of gamma_MRC = gamma_1+gamma_2 as G1+G2, which is independent of rho, the complex correlation. Let us take the number of antennas more than 2. In this case, it is a non-trivial matter to show that the average SNR is exactly given by the trace of the correlation matrix of the receive antenna elements. The trace is also the sum of eigen-values of R. In summary, the average SNR does not depend on rho as the marginal pdf of SNR on each diversity branch is not a function of rho. Mathematically speaking, the main diagonal of R does not contain any entry which is a function of rho. For further reading, I encourage you to go thru Section 10-10 of the above mentioned reference. sincerely Ramesh taqai@hotmail.com (taqai) wrote in message news:<9b5caca3.0310041736.15f98831@posting.google.com>...> Thanks to both of you for your replies. > > I was actually referring to correlated antanna gains by > talking of "correlated antennas", i.e. we can assume that the > noise remains independent. > (With full correlation for example we would have h_1=....=h_M.) > > Isn't it true that in that case the average antenna gain would be > lower as the correlation between antennas starts to grow ?
Reply by ●October 10, 20032003-10-10
"Ramesh" <ecerams@rediffmail.com> wrote in message news:b31e6a26.0310032156.3cd8d9a4@posting.google.com...> However, average SNR does not give a true picture of what's happening > (i.e., the average probability of error). So, one should look at the > average BER.Ramesh, it is very good point. In fact, the correlation between antennas in some system, e.g. beamforming system, help to improve SNR gain, i.e. N times for N elements case. However, such correlation decreases the diversity gain, so it makes larger of the average BER. One simple way to find the diversity order of the correlated antenna system is to investigate the rank of the correlation matrix or the number of non-zero eigen values of that. The rank is, thus, the diversity order. To be in detail, we can evaluate the exact diversity effect by manupulating the eigenvalue theirselves with more complex formulation.> hope this helps > Ramesh--- Regards, James K. (txdiversity@hotmail.com) - Private opinions: These are not the opinions from my affiliation.
Reply by ●October 14, 20032003-10-14
taqai wrote:> If we have a transmitter antenna and M receiver antennas > then it is well known that by using maximum ratio combining > an M-fold improvement in the SNR can be obtained.10 Log 10(M) in dB with independent noise at each sensor and equal shading. With shading, AG= ( \sum_{i} (a_{i})^{2} ) / \sum_{i} a_{i} The assumption of independent noise is usually not justified. The only two casses I know of is thermal noise of the front end electronics, and flow noise in hydroacoustic arrays.> > But what if the receiver antennas are correlated and the > covariance matrix R is known. Can the SNR improvement factor (which > I guess will be between 1 and M) then be calculated as a simple > function of R ? Any references to where where this problem has > been tackled will be greatly appreciated. > > Thank you.I think you want to calculate "Array Gain". AG = SNR_out/ SNR_in If so, you need two covariance Matricies, one for signal and the other for noise. Look in "Sonar Signal Processing", by R.O. Nielson.
Reply by ●October 14, 20032003-10-14
Ramesh wrote:> Hi, > > if h_1,.., h_M are the complex channel gains from the transmitter to > ecah of the receive antennas, then, irrespective of the correlation > between the antenna elements at the receiver, the instantaneous SNR at > the output of the combiner is given by > gamma = (E_s/N_0)(|h_1|^2 + |h_2|^2 +...+|h_M|^2), > where E_s/N_0 is the SNR per symbol per diversity branch.Correlation matters. If the noise is correlated, it will not on average cancel out.> We can take average of gamma without worrying about the correlation > between the antenna elements. With the assumption of the marginal pdf > of |h_i|^2 is exponentially distributed with mean unity, we obtain an > average value of gamma is ME_s/N_0, which is same as the case with > uncorrelated fades. > > However, average SNR does not give a true picture of what's happening > (i.e., the average probability of error). So, one should look at the > average BER. For more details, pls look at Schwartz-Bennet-Stein's > classic book "Communication Systems and Techniques". In principle, one > can obtain the average SNR by first finding the marginals from the > joint pdf (of |h_1|^2,..|h_M|^2) > and taking the expected value of each one. > > hope this helps > Ramesh > > > taqai@hotmail.com (taqai) wrote in message news:<9b5caca3.0310031713.19879fa6@posting.google.com>... > >>If we have a transmitter antenna and M receiver antennas >>then it is well known that by using maximum ratio combining >>an M-fold improvement in the SNR can be obtained. >> >>But what if the receiver antennas are correlated and the >>covariance matrix R is known. Can the SNR improvement factor (which >>I guess will be between 1 and M) then be calculated as a simple >>function of R ? Any references to where where this problem has >>been tackled will be greatly appreciated. >> >>Thank you.
Reply by ●October 14, 20032003-10-14
Rune Allnor wrote:> taqai@hotmail.com (taqai) wrote in message news:<9b5caca3.0310031713.19879fa6@posting.google.com>... > >>If we have a transmitter antenna and M receiver antennas >>then it is well known that by using maximum ratio combining >>an M-fold improvement in the SNR can be obtained. >> >>But what if the receiver antennas are correlated and the >>covariance matrix R is known. Can the SNR improvement factor (which >>I guess will be between 1 and M) then be calculated as a simple >>function of R ? Any references to where where this problem has >>been tackled will be greatly appreciated. > > > What do you mean by "correlated antennas"? > > Anyway, there is a trick you can use to get rid of non-white noise > in your measurements. Assume your measured data x(n) is modeled as > > x(n)=s(n)+w(n) [1] > > where s is the "useful" signal and w is non-white noise that is > uncorrelated with the signal s. > > In this case the data autocovariance matrix Rxx becomes > > Rxx=Rss+Rww [2] > > where Rss is the signal autocovariance and Rww is the noise autocovariance. > Ideally, Rww should be like > > Rww= sigma^2*I [3] > > where sigma^2 is the noise power and "I" is the identity matrix, but > here the problem is that Rww is non-diagonal. > > Now, for stationary noise, Rww is Hermitian: > > Rww=Rww^H [4] > > (superscript H denotes the "complex conjugate and transposed"). > Since it is Hermitian, Rww can be decomposed as > > Rww=U*U^H [5] > > Insert [5] in [2]: > > Rxx=Rss+U*U^H [6] > > Define the "prewhitened" data covariance matrix Rzz, and manipulate > to replace the non-white noise term with a white one: > > Rzz=U^(-1)*Rxx*U^(-H) [8.a] > =U^(-1)(Rss+U*U^H)*U^(-H) [8.b] > =U^(-1)*Rss*U^(-H)+I [8.c] > > Here U^(-H) means (U^H)^(-1). > > The "easiest" expression would be [8.c], and it is based on the (known) > noise covariance matrix Rww. It doesn't look like it, but SNR is preserved > even though sigma^2 doesn't appear. The transform matrix U scales the > eigenvectors (and eigen values) of Rss to preserve SNR. To find out > details, the norm of U should be investigated. I wouldn't be surprised if > the trace of Rww turns out to be very useful when investigating the > noise power. > > CAVEAT -- I've written this off memory, so there are no guarantees with > respect to mathematical details. You should be able to see the main > line of arguments, though. > > RuneIf I may add, The whitening matrix U^{-1} isn't unitary and it acts on signal as well as noise as Rune shows. The signal is squashed by the whitening. I once had a fellow of the IEEE try to sell me snake-oil by claiming that he could arbitrarily increase array gain by packing a bunch of sensors closer together and using ABF to whiten the noise.
Reply by ●October 14, 20032003-10-14
Stan Pawlukiewicz wrote: ...> > I once had a fellow of the IEEE try to sell me snake-oil by claiming > that he could arbitrarily increase array gain by packing a bunch of > sensors closer together and using ABF to whiten the noise. > >As Stephan Sprenger once wrote, "Papier ist duldig." The snake oil probably did work on paper. Look at the fantastic directionality of a binomial antenna array. Pity you can't build one that works! Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
