Complex Demodulation (From Bloomfield: “Fourier Analysis of Time series: An Introduction Wiley , 1976 Chapter 6)

Many signals can be characterized by a sinusoid that has a slowly varying amplitude and phase , i.e.

(1)

Here Rt is a slowly varying amplitude and is a slowly varying phase.

The aim of complex demodulation is to extract approximations of Rt and t. It may be regarded as a local version of harmonic analysis.—in that it aims to define the amplitude and phase of an oscillation in the neighborhood of t rather than the entire series.

Draw some examples and give physical (when possible) insights.

Two closely spaced Harmonics (i.e. tides)

AM radio

Sunspot data

First lest consider the complex analog of (1)

The extraction of Rt and t. is trivial is  is known, i.e.

From which the amplitude and phase can be find with the MATLAB command amp and phase, i.e.

We say that yt is obtained from xt by complex demodulation

Now the real form of (1) can be written as:

So it is the first term that we want because it is trivial to remove Rt and t. from it (as shown above), but this requires removing the second term with frequency  before proceeding.

Any ideas how you would remove this second term?

In general the data being analyzed does not consist solely of a perturbed sinusoid. The data may contain a low frequency term, such as the sea-level data we’ve looked at where we consider the tidal period motion the High Frequency motion and the meteorologically forced variability the low frequency component. The figure below shows the hourly data (from 1961) from the Battery record in blue and the low pass signal in red. The highpassed signal is plotted in the lower panel. Rt would then be the envelope on the high passed record and the would be the slowying varying phase. If we demodulated this record at the M2 tidal frequency (12.4206 hours) the slowly varying phase would captures the fact that successive high-tides occur in less than 12.42 hours as tidal amplitude is increasing while successive high tides occur at intervals greater than 12.42 hours when tidal amplitude is decreasing.

For this well call the low frequency component zt and this is a slowly time-varying quantity. Based on the filter that I used, with a cut-off frequency (32 hours) the time variability of this signal is ~ 32 hours. Thus the original record can be written as:

and

So by the complex demodulation the low frequency component zt has been shifted to a higher frequency and the signal of interest (Rt, ) has been shifted to near zero frequency. So they can be separated by a simple filtering process.

So the process of demodulation simply involves

1)Shifting the frequency of interest to zero by multiplying the original record by

2)Applying a linear filter to the above results.

So what should be the cut-off frequency of the filter we choose in step 2?

Some knowledge of the frequency content of the above record is helpful in determining the filter design.

For example in the case of the tidal motion—we are interested in characterizing the time variability of the amplitude and phase of the semidiurnal signal, This is a product of the beating of a set of semidiurnal tidal constituents (M2, S2, N2 etc). What I want to remove from demodulated record is not only the low frequency component zt but also the diurnal signal- which will now reside in the semi-diurnal band of the of the demodulated signal.

HERE WE PUT THE FREQUENCY REPSPONSE OF THE LANCOZS FILTER.

Draw figure on board of the demodulated record in frequency space—and thus we want a filter tight enough to remove this variability. Particularly note the near-by diurnal peak that will bleed into our estimate of Rt if we are not careful about our filter design. However, we want to remain the variability of Rt over the spring/neap cycle (14 days).

Show some examples.

1)Show one that passes some of the diurnal signal

2)Improve that so that it nicely captures the spring/neap variability

3)Go even more so that the S/N variability is removed.

Show result of Phase from 2)

Another use of demodulation is that it can also be used to define a precise frequency of the record. For example in the ocean (at atmosphere) there is significant variability near the local inertial frequency. This is a function of latitude. At the poles it’s 12 hours, at our latitude its 18.8 hours and increases to infinity at the equator. The actual frequency of the inertial motion is shifted by a number of processes beyond the scope of this class—but for

So suppose the signal had a frequency , i,e,


So if we demodulate at frequency 

The phase ()will be

So the slope of with time will be , i.e.

Show some examples with synthetic data:

1