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# How to find annual amplitude and phase of time series

Asked by Chris Martin on 19 Jul 2013

How to find annual amplitude and phase of time series

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Answer by Wayne King on 19 Jul 2013

There are a number of ways you can try and estimate it. Do you have a priori knowledge of the frequency? If so you can use linear regression to find the amplitude and phase.

For example, assume I know the frequency is 10 Hz for data sampled at 100 Hz.

```    Fs = 100;
t = 0:1/Fs:2-1/Fs;
x = 1.5*cos(2*pi*10*t-pi/4)+randn(size(t));
% create the design matrix
X = ones(length(x),3);
X(:,2) = sin(2*pi*10*t);
X(:,3) = cos(2*pi*10*t);
x = x(:);
beta = X\x;```

The amplitude estimate is:

`   amp_est = norm([beta(2) beta(3)],2)`

The phase angle estimate is:

`   phase_est = atan2(-beta(2),beta(3))`

If you don't know the frequency a priori, then look at the following example:

https://www.mathworks.com/help/signal/ug/frequency-domain-linear-regression.html