Confidence intervals of amplitude and phase for a noisy sine wave

In summary, to calculate the confidence intervals for the amplitude and phase of two series of data consisting of noisy sine waves, you can use an fft to determine the frequency and then use least squares fitting to estimate the values of a and b. From there, the amplitude can be found using the formula sqrt(a^2+b^2) and the phase can be found using the formula atan(b/a). To get the confidence intervals, you will need to estimate the variance of the noise by taking the mean of the squared differences between the sample data and the sine wave. Then, use standard error propagation techniques to calculate the standard errors of the estimates of a and b and use those to determine the confidence intervals at the desired confidence level.
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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]...
 
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  • #2
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.
 

Related to Confidence intervals of amplitude and phase for a noisy sine wave

1. What is a confidence interval?

A confidence interval is a range of values that is likely to contain the true value of a population parameter, such as the amplitude or phase of a sine wave. It is calculated from a sample of data and provides a measure of the uncertainty associated with estimating the true value.

2. How are confidence intervals calculated for amplitude and phase of a sine wave?

Confidence intervals for amplitude and phase are typically calculated using statistical methods, such as the t-distribution or the normal distribution. These methods take into account the sample size, variability of the data, and the desired level of confidence (usually 95%). The resulting interval represents the range of values within which the true amplitude or phase is likely to fall.

3. How do noisy data affect confidence intervals for amplitude and phase?

Noisy data can increase the uncertainty associated with estimating the amplitude and phase of a sine wave. This is because noise can obscure the true underlying signal, making it more difficult to accurately estimate the parameters. As a result, the confidence intervals for amplitude and phase may be wider when the data is noisy compared to when the data is clean.

4. What is the significance of confidence intervals for amplitude and phase in a noisy sine wave?

Confidence intervals for amplitude and phase provide a measure of the uncertainty associated with estimating the true values in a noisy sine wave. They also allow us to make statistical inferences about the population parameters based on a sample of data. For example, we can use the confidence intervals to determine if the amplitude and phase are significantly different from a certain value, or if there is a significant difference between two sets of data.

5. How can confidence intervals for amplitude and phase be used in practical applications?

Confidence intervals for amplitude and phase can be used in a variety of practical applications, such as signal processing, data analysis, and scientific research. They can help to assess the accuracy of measurements and make predictions about future data points. In addition, they can aid in decision making by providing a measure of uncertainty and allowing for comparisons between different sets of data.

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