Question

In a particular trace analysis via FTIR, a set of sixteen interferograms was collected. The signal-to-noise ratio (S/N) associated with a particular spectral peak was approximately 4:1. How many interferograms would have to be collected and averaged if the goal is to obtain an S/N = 20:1?

Short answer

The number of interferograms would have to be collected and averaged if the goal is to obtain an S/N = 20:1 is 400.

Step-by-step solution

Step 1. Given information

In a particular trace analysis via FTIR, a set of sixteen interferograms was collected. The signal-to-noise ratio (S/N) associated with a particular spectral peak was approximately 4:1. The number of interferograms that have to be collected and averaged if the goal is to obtain an$S/N=20:1$.

Step 2. Explanation

A set of sixteen interferometers is collected, the signal to noise ratio is 4: 1.

The expression for the signal to noise ratio after n repetitions is given by:

${\left(\frac{S}{N}\right)}_{\mathrm{n}}=k(n{)}^{1/2}\dots \dots (\mathrm{I})$

Substitute n=16, we get

${\left(\frac{S}{N}\right)}_{16}=k(16{)}^{1/2}\dots \dots .(II)$

For n we will have

${\left(\frac{S}{N}\right)}_{\mathrm{n}}=k(n{)}^{1/2}\dots \dots (\text{III})$

Divide the Equation (III) by Equation (II), we get

$\frac{{\left(\frac{s}{N}\right)}_{\mathrm{n}}}{{\left(\frac{s}{N}\right)}_{16}}=\frac{k(n{)}^{1/2}}{k(16{)}^{1/2}}\dots \dots ..\left(\mathrm{IV}\right)$

Substitute ${\left(\frac{S}{N}\right)}_{\mathrm{n}}=20$

and ${\left(\frac{S}{N}\right)}_{16}=4$

we get

$\frac{20}{4}=\frac{k(n{)}^{1/2}}{k(16{)}^{1/2}}$

Solve for n,

we get n=400.