Considering a model of the form:
$y[n] = x[n] + noise[n]$ and
$y[n] = u[n] + noise[n]$
where the term noise
represents Additive White Gaussian Noise of variance $\sigma^2_{noise}$.
I want to understand if there is any difference in the value of the terms: $\sigma^2_x/\sigma^2_{noise}$ and $\sigma^2_u/\sigma^2_{noise}$ by varying the noise level.
$x$ represents a signal generated from Gaussian distribution whose variance is $\sigma^2_x$. In Matlab, I have used the command rand()
to generate $x$.
$u$ represents another signal taking values zero or 1 having variance $\sigma^2_u$. In Matlab, I have used rand()>0.5
to generate thie binary valued data.
Based on my understanding the numerator in the ratio is the autocorrelation of the data mathematically expressed as $E[\mathbf{x}[n]\mathbf{x}[n]^T]$ and $E[\mathbf{u}[n]\mathbf{u}[n]^T]$. Why are these terms giving same result?
In most papers, the variance of the data irrespective of whether it takes numeric or symbols is considered to be 1.
Question1) Is my approach of implementing the ratio correct? I cannot understand what the value of $\sigma^2_x$ and $\sigma^2_u$ should be in practice. Implementing this way would give the same value for the ratio for both the different kinds of data $x$ and $u$. Would the auto correlation for numeric values x
and binary data u
be the same value?
Question2) If the ratio is the same then what is the need for using binary signal?
N = 128;
var_x =1;
var_u = 1;
signal_x = rand(1,N);
signal_u = rand(1,N)>=0.5; % this creates 0/1 data
noise = 0;
index = 1;
for noise = 0:5:30
noise_var = 10^(-noise/10);
ratio_x(index)= 10*log10(var_x/noise_var); %in dB
ratio_u(index)= 10*log10(var_u/noise_var);
index = index+1;
end
i
with 0, and noise does not seem to change $\endgroup$ratio_x(0)
won't work $\endgroup$ratio_x
andratio_u
are zero $\endgroup$i = 0:5:30
some inner indices yield zero values $\endgroup$