Complex Demodulation Phase and Amplitude Plot

Complex Demodulation Phase Plot

In the frequency analysis of time series models, a common model is a sinusoidal wave:

where, ∝ is the amplitude, phi is the phase shift and omega is the dominant frequency. The goal of the complex demodulation plot is to improve the frequency estimate.

 The complex demodulation plots formed by two components:

• Vertical axis: Phase
• Horizontal axis: Time

Importance of a Good Initial Estimate for the Frequency:

The non-linear fitting for the sinusoidal waves

The above equation is sensitive to good initial values. The initial value of frequency omega can be obtained by the spectral plot. The complex demodulation phase plot is used to assess whether this estimate is adequate. If the estimate is not adequate then whether it should be increased and decreased.

Complex Demodulation Amplitude Plot:

In the frequency analysis of time series models, a common model is a sinusoidal wave:

where, ∝ is the amplitude, phi is the phase shift and omega is the dominant frequency.

If the slope of the complex demodulation amplitude plot is not zero, then the above equation is finally replaced by the model.

where, a_i is some type of linear model fit with standard least squares. The most common case is linear fit, that is the model becomes as follows:

The complex demodulation amplitude plot is formed by

• Vertical axis: Amplitude
• Horizontal axis: Time

 The complex amplitude demodulation plot answers the following questions:

• Does amplitude change over time?
• Is there any start-up effect that led to the amplitude being unstable at the start?
• Is there any outliers in amplitude?

Implementation:
Code: Python code for complex demodulation analysis 

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