covariance matrix in signal processing

cross-covariance

In the case of complex random variables, the covariance is defined slightly differently compared to real random variables. For complex random variables ( Z_1 ) and ( Z_2 ), the covariance is defined as:

where ( E[Z] ) is the expected value of ( Z ), and the asterisk ((^*)) denotes the complex conjugate.

Given ( Y_1 = G_1 + N_1 ) and ( Y_2 = G_2 + N_2 ), where ( G_1 ) and ( G_2 ) are complex constants, and ( N_1 ) and ( N_2 ) are complex Gaussian random variables with zero mean and ( \sigma ) variance, we can calculate the covariance as follows:

1. The expected values ( E[Y_1] ) and ( E[Y_2] ) are ( G_1 ) and ( G_2 ) respectively, since the expected value of ( N_1 ) and ( N_2 ) is zero.

2. The covariance between ( Y_1 ) and ( Y_2 ) becomes:

   \[\text{Cov}(Y_1, Y_2) = E[(Y_1 - G_1)(Y_2 - G_2)^*] = E[(G_1 + N_1 - G_1)((G_2 + N_2 - G_2)^*)] = E[N_1 N_2^*] \]

3. Since ( N_1 ) and ( N_2 ) are independent complex Gaussian random variables with zero mean, the expected value of their product ( E[N_1 N_2^*] ) is zero.

Thus, the covariance between ( Y_1 ) and ( Y_2 ) in this complex case is also zero.

auto-covariance

The expression ( E[YY^] ) represents the expected value of the magnitude squared of the complex random variable ( Y ). Given ( Y = G + N ), where ( G ) is a complex constant and ( N ) is a complex Gaussian random variable with zero mean and ( \sigma^2 ) variance, we can calculate ( E[YY^] ) as follows:

1. Expand ( YY^* ):

   \[YY^* = (G + N)(G + N)^* = (G + N)(G^* + N^*) \]

   where ( G^* ) and ( N^* ) are the complex conjugates of ( G ) and ( N ), respectively.

2. Expand the product:

   \[YY^* = GG^* + GN^* + NG^* + NN^* \]

3. Calculate the expected value ( E[YY^*] ):

   \[E[YY^*] = E[GG^* + GN^* + NG^* + NN^*] \]

   Since ( G ) is a constant, ( E[GG^] = GG^ ). The terms ( E[GN^] ) and ( E[NG^] ) are zero because ( N ) has zero mean. The term ( E[NN^*] ) is the variance of ( N ), which is ( \sigma^2 ).

4. Combine the results:

   \[E[YY^*] = GG^* + \sigma^2 \]

So, ( E[YY^*] ) is the sum of the magnitude squared of the complex constant ( G ) and the variance ( \sigma^2 ) of the complex Gaussian random variable ( N ).

codes

%% covariance

rng(100)

N = 100000;
N1 = (randn(1, N) + randn(1, N)*1i);
Y1 = (1 + 2*1i) + N1;

E1 = mean(Y1);

N2 = (randn(1, N) + randn(1, N)*1i);
Y2 = (1 + 2*1i) + N2;
E2 = mean(Y2);

cov1 = Y1 * Y2' / N;

cov2 = (1 + 2*1i) * (1 + 2*1i)';

%% auto-covariance

N = 100000;
N1 = (randn(1, N) + randn(1, N)*1i);
Y1 = (5 + 6*1i) + N1;

autocov = Y1 * Y1' / N;

autocov2 = (5 + 6*1i) * (5 + 6*1i)' + 2;