Confidence intervals of amplitude and phase for a noisy sine wave

[demute]

Homework Statement

I have two series of data consisting of samples of a noisy sine wave and need to determine the amplitude and phase and the confidence intervals. I determined the amplitude and phase but don't know how to calculate the confidence intervals, help!

The Attempt at a Solution

By doing an fft, I got the frequency.
By least squares fitting the data to this frequency, a*sin(k*t)+b*cos(k*t), I got estimates of a and b
The amplitude can now be determined by sqrt(a^2+b^2) and the phase by atan(b/a) (if I remember it correct), so I suppose I can get the confidence intervals of the phase and amplitude if I know the confidence intervals of a and b. How do I get these?

The mean and variance of the noise (=difference between the sample data and the sine wave) was also calculated and for the first data series the noise was more or less evenly distributed in [-a,a] for some a. For the other series it's more or less bell shaped in [-b,b]...

 

[Nino MMP]

In order to calculate the confidence intervals of the amplitude and phase, you need to first estimate the variance of the noise. This can be done by taking the mean of the squared differences between the sample data and the sine wave. Once you have the variance, you can calculate the standard error of the estimates of a and b using standard error propagation techniques. Then, the confidence intervals can be calculated using the standard errors and the desired confidence level.