Randomly Modulated Periodic Signals

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Randomly Modulated Periodic Signals

• Melvin J. Hinich

Applied Research Laboratories University of Texas
at Austin
hinich_at_austin.utexas.edu
http//web.austin.utexas.edu/hinich/
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Rotating Cylinder Data
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Cat Brain EEG Seizure
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Daily Sunspots
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Hourly Alberta Electricity Demand
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Hourly Alberta Electricity Prices
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One Week of Yen/US Returns
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Canadian /US Rates of Return
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Bassoon Note
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Flute Note
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Definition of a RMP
A signal is called a randomly modulated
periodicity with period TNd if it is of the
form
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Random Modulations
are jointly dependent random processes that
satisfy two conditions
Periodic block stationarity
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Finite Dependence
Condition needed to ensure that averaging over
frames yields asymptotically gaussian estimates
are independently distributed if
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Fourier Series for Components
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Signal Plus Noise
The modulation is part of the signal
It is not measurement noise
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Artificial Data Examples
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Five Standard Deviations
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Three Standard Deviations
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Two Standard Deviations
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One Standard Deviation
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No Correlation in the Modulation
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Block Data into Frames
The data block is divided into M frames of
length T
T is chosen by the user to be the period of the
periodic component
The t-th observation in the mth frame is
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Signal Coherence Function
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Frame Rate Synchronization
The frame length T is chosen by the user to be
the hypothetical period of the randomly modulated
periodic signal.
If T is not an integer multiple of the true
period then coherence is lost.
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Signal Coherence Spectrum
The signal-to-noise ratio is
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Estimating Signal Coherence
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Statistical Measure of Modulation SNR
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Chi Squared Statistic
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Coherent Part of the Mean Frame
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Spectrum of the Variance
Fourier Series Expansion
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Sunspot Spectra
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Power Signal Coherence Spectra - Demand
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Power Signal Coherence Spectra - Prices
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Yen/ Spectrum Coherencies
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Exchange Rates Data
25 exchange rate series sampled half-hourly for
the whole of 1996 with weekends removed. The
weekend period is from 2300 GMT on Friday when
North American financial centres close until
2300 GMT on Sunday when Australasian markets
open. The incorporation of such prices would lead
to spurious zero returns and would potentially
render trading strategies which recommended a buy
or sell at this time to be nonsensical. Removal
of these weekend observations leaves 12,575
observations for subsequent analysis.
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Summary Statistics
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Coherence Results I
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Coherence Results II
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Coherent Part of Yen/ Mean Frame
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Descriptive Statistics of Canadian/US Daily Data

• Sample size 7569 20.7 Years
• Mean 0.557E-04 Std Dev 0.270E-02
• Skew 0.933E-01 Kurtosis 3.62
• Max value 0.191E-01 Min value
  -0.184E-01
• Frames 40 Frame Length 186.8
• Frequencies in band 93
• Maximum coherence probability 0.999999
• Coherence threshold 0.35

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Canadian/US Daily Data Spectra
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Waterfall Spectrograms of RMPNoise SNR - 38
dB
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Sigcoh Probability Waterfall of RMPNoise SNR -
38 dB
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Waterfall Spectrograms of RMPNoise SNR - 44
dB
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Sigcoh Probability Waterfall of RMPNoise SNR -
44 dB
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Winglet Vibration Data
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Winglet Spectrum Coherencies
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Prewhitened Winglet Spectra
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Tuned Prewhitened Winglet Spectra
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Noise Like Signal