# S/N Formula with probability

Hello I am looking for the S/N formula that incorporates the probability of catching the signal, and setting an arbitrary treshold level above which we detect the signal.

I can't figure out the formula but I have this nomograph where it is represented:

For example for a 98% probability detection with a false alarm rate of 10-3 we get a S/N or 12 dB.

I would like to know the formula being used there.

• Probably a gaussian distribution effect. – Andy aka Aug 22 '17 at 18:02

The OP is looking for an S/N formula, as per the first paragraph of their statement, and also has a specific question about a nomograph figure they though might be relevant. These two things are addressed separately.

Any SNR (aka S/N) formula will necessarily depend on the specific circumstances, e.g., how is the measurement noise distributed, is it homoscedastic or not, what are the characteristics of the signal, and so on. Since none of that is given, I will do one of the simplest possible examples and leave it to the OP to consider their actual scenario.

First, assume the measurement noise is additive, white Gaussian noise with zero mean. Also assume the noise is homoscedastic, i.e., its population standard deviation, $$\\sigma \$$, is constant. In the absence of noise, assume the system response, denoted by $$\\mu \$$, is constant. For illustrative purposes, let $$\\mu = 5\$$ and $$\\sigma = 1\$$.

Let p be the probability of false positives, i.e., false alarms, and q be the probability of false negatives. Then $$\1-q \$$ is the detection probability. The figure below depicts the situation where $$\p = 0.001 \$$ and $$\1-q = 0.98 \$$, i.e., $$\q = 0.02 \$$ .

The decision level (aka decision threshold, among many other names) is the vertical blue line at $$\\mu + z_p\sigma \$$, where $$\z_p \$$ is the critical z value for p. It is easily computed via $$\z_p = -normsinv(p) \$$ in Excel or via lookup table. For $$\p = 0.001 \$$, $$\z_p = 3.090 \$$, so the decision level is at 8.090. The lower (black) Gaussian is the operative distribution when only noise is present and a random sample has only 0.1% probability of being above the decision level, i.e., of being a false alarm.

For $$\q = 0.02 \$$, $$\z_q = -normsinv(q) \$$, so $$\z_q = 2.054 \$$. Therefore, $$\\mu + (z_p + z_q)\sigma \$$ is where the center of the upper (red) Gaussian must be located in order for random samples from it to have 98% probability of being above the decision level, i.e., of being detected. The center of the upper Gaussian distribution is the true value of the noiseless signal and it equals 10.144 in the present example. Therefore, amplitude SNR = $$\(z_p + z_q)\sigma /\sigma = (z_p + z_q) \$$ and SNR = $$\(z_p + z_q)^2 = 26.46 \$$, i.e., 14.2 dB.

Second, the OP’s figure pertains to the performance of a quadrature receiver with incoherent detection and has nothing to do with shifted Gaussian distributions in the above figure. Whalen 1, as his Fig. 7.5 on page 205, gives the exactly equivalent figure, except with axes swapped and his figure has curves for both coherent (signal completely known) and incoherent (signal known except for phase) detection:

Whalen gives the equations for probability of false alarm, $$\P_{fa}\$$, and probability of detection, $$\P_D\$$. These are based on the Rayleigh and Rician distributions.

The Rayleigh distribution is given by $$p_q(q) = \frac{q}{\sigma_T^2} e^{-q^2 / 2\sigma_T^2}$$

where $$\T\$$ = sampling period, $$\N_0\$$ = unilateral power spectral density of white Gaussian noise, $$\\sigma_T^2 =N_0 T/4\$$, $$\E = A^2 T/2\$$, and $$\SNR = E/N_0\$$.

Then Whalen’s equation 7-24 is $$P_{fa} = \int\limits_{\eta }^{\infty} p_q(q) \mathrm dq = e^{-\beta^2 / 2}$$

where $$\\beta = \eta / \sigma_T\$$ is a normalized detection threshold. With $$\z \equiv q/\sigma_T\$$ and $$\\alpha^2 = 2E/N_0\$$, Whalen’s equation 7-27 is

$$P_D = \int\limits_{\beta }^{\infty} z e^{-(z^2 + \alpha^2)/2} I_0(\alpha z) \mathrm dz$$

where $$\I_0(x)\$$ is the modified Bessel function of zero order. As Whalen says, this is the Marcum Q function. Since there is no closed form for the Marcum Q function, this explains the use of a nomograph back in 1971.

Reference:

1. A.D. Whalen, Detection of Signals in Noise, Academic Press, New York, ©1971, pp. 202-205.

Draw two Gaussian distributions, symmetric about a central "zero volts" line.

Move the two GD mean-values to be 2 sigma above and below "zero volts".

The separation is 2+2 or 4 sigma, which is 4 RMS, which is 12dB SNR.

Now call one of the GD the Signal GD; some of the time that GD will produce a voltage smaller than "zero volts", meaning an event was not detected. At the 2-sigma level, the area of the tails is 0.045; the area of that tail below "zero volts" is 0.022, and 1 - 0.022 is 0.978 (close to 98%).

About that FalseAlarm of 0.001? I dunno.

• If you want to maximize the chance of detecting a present signal, subject to a maximum probability of a false alarm on a not-present signal, you just lower your detector threshold until you reach your maximum false-alarm rate. – The Photon Aug 23 '17 at 4:42