I have not seen var_n = 1/snr. At first glance it cannot be correct, because snr is a dimensionless variable, but var_n has units of power squared if n is noise power. So if var_n=1/snr, then var_n must be normalized in some way that remains to be specified.

What is more,

1/snr = (mean noise power) / (mean signal power) = N / S

         =  noise power normalized by signal power = N'

so if var_n=1/snr is true, then it means that not only that noise is normalized in some way, but it happens to follow a statistical distribution whose mean is equal to its variance. This would only be true for certain kinds of noise, and is certainly not generally true of noise in practice.

The use of var_n = 1 also suggests that some form of normalization of noise is being applied to make it dimensionless. var_n = 1 is an assumption commonly made in the absence of other information about the noise statistics, or under the assumption that it is Gaussian. It is usually used for illustration and education purposes, for learning the principles of signal detection in noise for instance, but it very often does not apply to actual cases of signal noise (in sonar or radar for instance). It is a good simplification for teaching purposes, because is suppresses distracting attention to units and noise statistics.

I hope that helps.

Ronald

Dear Ronald Kessel,

Thanks for your feedback, my concern is: Can I use the variance of the synthetic noise that is added to the transmitted signal and use it for threshold calculation? In such scenario can we pretend that such energy detector with this known noise power is IDEAL?

Best,

The average SNR is defined gamma = S/var_n, where S is the average signal power and N is the noise variance. If you consider that S=1 then var_n = (S/var_n)^-1. Equivalently, if you consider that N = 1 then S = S/var_n. 

Now let consider a classical energy detector, which uses the following test statistics

T_k = (1/N) sum_{i=1}^{N}|y_k|^2. 

If you consider that s is a zero mean complex Gaussian distributed random variable, and assume that N is sufficiently large, the test statistics is well-approximated by the Gaussian distribution, and the false alarm probability can be expressed as

P_{fa} = Q( (gamma_th-var_n^2)/(var_n^2 sqrt(N)) )     (1)

where Q(x) is the Q-function and gamma_th is the threshold to decide whether the channel that is being sensed is empty or not.

Furthermore, the detection probability is given by

P_{d} = Q( (gamma_th-var_n^2 (1+gamma)) /  (var_n^2 (1+gamma)/sqrt(N)))

For a target false alarm probability, the threshold can be solved from (1).

So the answer is that you can use the var_n for threshold calculation. 

Dear Alexandros-Apostolos A. Boulogeorgos,

Many thanks for sufficient feedback, I can see that you are confirming the use of var_n in the probability of detection / false alarm equations. I would like ask you only one more question:

Can I use that known var_n and apply the noise uncertainty which is defined in the following Matlab to measure the degradation in the detector performance? or that is not possible because WE KNOW THE NOISE PRIORY.

Noise uncertainty:

S1=rand(1,1); % Generating a random variable corresponding to noise variance

rho_dB=0; % The uncertainty factor in dB

rho_linear=10^(rho_dB/10); % convert to linear scale

S_Uncertainty = ((rho_linear-(1/rho_linear))*S1)+1/rho_linear;

var_new=var_n*S_Uncertainty;

Best,

Dear Mahdi Badrawi,

In practical application, to estimate the noise variance, when it is decided that there is no signal in the channel of interest, you use the samples of T_{k} to estimate the noise variance. In the next detection, you will use the previous estimation of the noise variance. 

From theoretical point of view, I think that you will be able to use a normal distribution to model the uncertainty of the noise variance, as you propose.

Best Regards,

Alexandros 

Dear Alexandros,

Appreciate your feedback,

Best regards,

Mahdi