Question

A measurement with a signal-to-noise ratio of 100 / 1 can be thought of as a signal, S, with 1 % uncertainty, e. That is, the measurement is$\mathbf{S}\mathbf{\pm }\mathbf{e}\mathbf{=}\mathbf{100}\mathbf{\pm }\mathbf{1}$.

(a) Use the rules for propagation of uncertainty to show that, if you add two such signals, the result is total signal$\mathbf{=}\mathbf{200}\mathbf{\pm }\mathbf{\surd }\mathbf{2}$ , giving a signal-to-noise ratio of$\mathbf{200}\mathbf{/}\mathbf{\surd }\mathbf{2}\mathbf{=}\mathbf{141}\mathbf{/}\mathbf{1}$ .

(b) Show that if you add four such measurements, the signal-tonoise ratio increases to 200 / 1.

(c) Show that averaging n measurements increases the signal-to-noise ratio by a factor of$\sqrt{\mathbf{n}}$ compared with the value for one measurement.

Short answer

(a) The principles for uncertainty propagation are followed, and the result is a total signal $\sqrt{2}$

(b) Four measurements had a signal-to-noise ratio of 400: 2=200: 1.

(c)$\sqrt{n}$ times higher than the initial signal/noise ratio.

Step-by-step solution

Definition of signal to noise

• The signal-to-noise ratio (SNR) is calculated by dividing the mean value of a signal (S) by the background standard deviation (N).
• When S/N falls below 2 or 3, the signal becomes impossible to see.

Determine the signal to noise ratio

(a)

When it combine two of these signals, it get:

$100\pm 1+100\pm 1-200\pm \surd 2$

where the uncertainty is determined:

$$\begin{gathered} e=\sqrt{e_1^2+e_2^2} \\ e=\sqrt{1^2+1^2} \\ =\sqrt{2} \end{gathered}$$

The signal-to-noise ratio is calculated as follows:

$\frac{200}{\sqrt{2}}=\frac{141}{1}$

Determine the singal to noise ratio increase by 200/1

(b)

When four of these measurements are added together, it get:

$100\pm 1+100\pm 1+100\pm 1+100\pm 1-400\pm 2$

where there is uncertainty:

$$\begin{gathered} \mathrm{e}=\sqrt{{\mathrm{e}}_1^2+{\mathrm{e}}_2^2+{\mathrm{e}}_3^2+{\mathrm{e}}_4^2} \\ \mathrm{e}=\sqrt{1^2+1^2+1^2+1^2} \\ =2 \end{gathered}$$

The signal-to-noise ratio is calculated as follows:

$\frac{400}{2}=\frac{200}{1}$

Determine the signal-to-noise ratio by a factor of n compared with the value for one measurement.

(c)

The signal to noise ratio of the initial measurement is:

$\frac{100}{1}$

As a result, the average signal is as follows:

role="math" localid="1663653596294" $\frac{\mathrm{n}\times 100}{\mathrm{n}}\times 100$

The average amount of noise is:

$\frac{\sqrt{\mathrm{n}}}{\mathrm{n}}=\frac{1}{\sqrt{\mathrm{n}}}$

The average signal to average noise ratio is:

$\frac{100}{\sqrt{\mathrm{n}}}=100\times \sqrt{\mathrm{n}}$