1

Randomly Modulated Periodic Randomly Modulated Periodic SignalsSignals

Melvin J. HinichMelvin J. Hinich

hinich@austin.utexas.eduhinich@austin.utexas.edu

Applied Research Applied Research LaboratoriesLaboratoriesUniversity of Texas at AustinUniversity of Texas at Austin

http://web.austin.utexas.edu/hinich/

2

Rotating Cylinder DataRotating Cylinder DataFluid Nonlinearity

-2000

-1500

-1000

-500

0

500

1000

1500

Time

Am

pli

tud

e

3

Cat Brain EEG SeizureCat Brain EEG Seizure

-4000

-2000

0

2000

4000

6000

8000

Time

mV

4

Daily SunspotsDaily Sunspots

Daily Sunspot Measurements

0

50

100

150

200

250

300

1850

1852

1855

1857

1860

1862

1865

1867

1870

1872

1874

1877

1879

1882

1884

1887

1889

1892

1894

1897

1899

1901

1904

1906

1909

1911

1914

1916

1919

1921

1923

1926

1928

1931

1933

1936

Days

Su

nsp

ots

5

Hourly Alberta Electricity DemandHourly Alberta Electricity DemandElectricity Demand ( 5/4/1996 - 6/15/1996)

0

1000

2000

3000

4000

5000

6000

7000

1 169 337 505 673 841

Time

kW H

ou

rs

6

Hourly Alberta Electricity PricesHourly Alberta Electricity PricesA Section of the Alberta Electric Spot Prices Time Series

-200

0

200

400

600

800

1000

1200

1 1681 3361 5041 6721 8401 10081 11761 13441 15121 16801 18481 20161 21841 23521 25201 26881 28561 30241 31921

Time

Pri

ce

7

One Week of Yen/US $ ReturnsOne Week of Yen/US $ Returns

Second Week of Yen/$ Log Returns

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 20 39 58 77 96 115 134 153 172 191 210 229

Time

log

x(t

) -

log

x(t

-1)

8

Canadian $/US $ Rates of ReturnCanadian $/US $ Rates of Return

Canadian$/US$ 2/11/2002 - 2/12/2003

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1/11/2002 2/15/2002 3/22/2002 4/26/2002 5/31/2002 7/5/2002 8/9/2002 9/13/2002 10/18/2002 11/22/2002 12/27/2002 1/31/2003

Days

Rat

es o

f R

etu

rn

9

Bassoon NoteBassoon Note

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

pli

tud

e

10

Flute NoteFlute Note

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time

Am

pli

tud

e

11

Definition of a RMPDefinition of a RMP

A signal is called a randomly modulated periodicity with period T=N if it is of the form

1 110

1 2 2

( ) cos 2 ( )

( ) sin 2

Kk k n k n

nk k k n k n

s u t f tx t N

s u t f t

1 2( ) ( ) 0k n k nEu t Eu t

for each 1, , where 2T

k K K

L

k

kf

T

nt n

12

Random ModulationsRandom Modulations

are jointly dependent random processes that satisfy two conditions:

The vector of the modulationsK

Periodic block stationarity

1 1( ), , ( ) = ( ), , ( ) m mf t t f t T t T u u u u

1 for all 0 mt t T

1 2( ), : 1, ,n k n k nt u t u t k K u

13

Finite DependenceFinite Dependence

' '1 1( ), , ( ) & ( ), , ( ) m rt t t tu u u u

'1 for some & and allmt D t D

' '1 1 & m nt t t t

Condition needed to ensure that averaging over frames yields asymptotically gaussian estimates

are independently distributed if

14

Fourier Series for ComponentsFourier Series for Components

Thus ( ) ( ) ( ) en n nx t s t u t wher

10 1 2

1

( ) cos 2 sin 2K

n k k n k k nk

s t s N s f t s f t

11 2

1

( ) cos 2 sin 2K

n k k n k k nk

u t N u f t u f t

15

Signal Plus NoiseSignal Plus Noise

( ) ( )ns t is the mean of x t

int nu t has a periodic jo distribution

The modulation is part of the signalThe modulation is part of the signal

It is not measurement noiseIt is not measurement noise

16

Artificial Data ExamplesArtificial Data Examples

110

1 2

1 cos 2 ( )

1 sin 2

Kk n k n

nk k n k n

u t f tx t s N

u t f t

1 1 1k n k n nu t u t T e t

2 2 2k n k n nu t u t T e t

17

Five Standard DeviationsFive Standard Deviations 10 Harmonics Modulation = 5 = 0.9 Frame=100

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

18

Three Standard DeviationsThree Standard Deviations10 Harmonics Modulation = 3 = 0.9 Frame=100

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Am

plit

ud

e

19

Two Standard DeviationsTwo Standard Deviations10 Harmonics Modulation = 2 = 0.9 Frame =100

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Am

pliu

tde

20

One Standard DeviationOne Standard DeviationRandomly Modulated Pulses

10 Harmonics Modulation = 1 = 0.9 Frame =100

-1

-0.5

0

0.5

1

1.5

Time

Am

plit

ud

e

21

No Correlation in the ModulationNo Correlation in the Modulation

Four Randomly Modulated Pulses Frame = 100 =5

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Am

pli

tud

e

22

Block Data into FramesBlock Data into Frames

The data block is divided into M frames of length T

The t-th observation in the mth frame is

( 1) n 0, , 1x m T n N L

T is chosen by the user to be the period of the periodic component

23

Signal Coherence FunctionSignal Coherence Function1

0

( ) (( 1) )exp( 2 )N

m n k nn

X k x m T t i f t

( )m k mX k s U k 1

0

( ) ( )exp( 2 )N

m m n k nn

U k u t i f t

1 2

1( )

2k k ks s is

22 ( )u mk E U k

24

Frame Rate SynchronizationFrame 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.

25

Signal Coherence SpectrumSignal Coherence Spectrum

The signal-to-noise ratio is The signal-to-noise ratio is

2 2( ) ( )x k uk s k

2

21x

x

x

kk

k

2

2 2( )

( )k

x

k u

sk

s k

26

Estimating Signal CoherenceEstimating Signal Coherence

1

0

ˆ ( ) ( )exp( 2 )ˆN

n k nn

X k x t i f t

2

2 2

ˆ ( )( )ˆ

ˆ ( ) ( )ˆx

u

X kk

X k k

22 1

1

ˆ( ) ( )ˆM

u mm

k M X k X k

: 0, , 1 ˆ

nx t n N is the mean frame

averaged over the M frames

L

27

StatisticalStatistical MeasureMeasure of Modulation SNRof Modulation SNR

2

2

2

ˆ

ˆ

ˆ

x

x

X kMZ k

N k

Mk

N

22 2ˆ( ) ( ) ( )ˆ ˆx uk X k k

28

Chi Squared StatisticChi Squared Statistic

2ˆ x

MZ k k

N

2

k x

Mk

N

22

mod

&

buted.k

If the ulation is stationary the distribution of

each Z k is approximately

they are independently distri

29

Coherent Part of the Mean FrameCoherent Part of the Mean Frame

1

0

1 ˆ ( ) ( )exp( 2 )ˆK

coherent kk

x t c k X k i f tN

1 if

0 xc k k threshold

c k otherwise

30

Spectrum of the VarianceSpectrum of the Variance

01 1

2 cos 2 2 sin 2K K

n k k n k k nk k

s t a a f t b f t

Fourier Series ExpansionFourier Series Expansion

- s t a stationary random process

- k k kVar a Var b S f Spectrum

2

1

K

s kk

S f

31

Sunspot Spectra

Power Spectrum of Detrended Daily Sunspot Data Prewhitened by an AR(10) Fit

10

15

20

25

30

35

40

3966 152.5 77.76 52.18 39.27 31.48 26.26 22.53 19.73 17.55 15.8 14.37 13.18 12.17 11.3 10.55 9.89 9.31 8.794 8.332 7.916 7.54 7.198

Period

dB

32

Signal Coherence Spectrum of Detrended & Prewhitened Daily Sunspot Data

0.75

0.77

0.79

0.81

0.83

0.85

0.87

0.89

0.91

0.93

3966

.00

39.27

19.73

13.18 9.

897.

926.

605.

664.

954.

403.

963.

603.

303.

052.

832.

642.

482.

332.

202.

09

Period

Sig

nal

Co

her

ence

33

3966

.00

360.5

5

188.8

6

127.9

4

96.73

77.76

65.02

55.86

48.96

43.58

39.27

35.73

32.78

30.27

28.13

26.26

24.63

23.19

21.91

20.76

S1

Period in Days

Time Segments

Coherence Probabilitygram of Four Segments of the Sunspots Data

0.96-0.97 0.97-0.98 0.98-0.99 0.99-1

34

Power & Signal Coherence Spectra - DemandPower & Signal Coherence Spectra - DemandPower & Signal Coherence Spectra of the Residuals from an AR(12) Fit of the Alberta

Electricity Hourly Spot Demand

0

0.25

0.5

0.75

1

168.

0

42.0

24.0

16.8

12.9

10.5 8.8

7.6

6.7

6.0

5.4

4.9

4.5

4.2

3.9

3.7

3.4

3.2

3.1

2.9

2.8

2.6

2.5

2.4

2.3

2.2

2.1

2.0

Period in Hours

Signal Coherence

0

10

20

30

40

50

60

dBCoherence Log Spectrum

35

Power & Signal Coherence Spectra - PricesPower & Signal Coherence Spectra - PricesPower & Signal Coherence Spectra of the Residuals from an AR(12) Fit of the

Alberta Electricity Hourly Spot Prices

10

20

30

40

50

60

168.0 24.0 12.9 8.8 6.7 5.4 4.5 3.9 3.4 3.1 2.8 2.5 2.3 2.1

Period in Hours

dB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Signal CoherenceSpectrum Coherence

36

Yen/$ Spectrum & CoherenciesYen/$ Spectrum & CoherenciesYen/$ Spectrum & Peaks of the Coherence Probabilities

5 dB in Spectrum = 1.78 * Amplitude

-3

-2

-1

0

1

2

3

240.00 21.82 11.43 7.74 5.85 4.71 3.93 3.38 2.96 2.64 2.38 2.16

Period

dB

0.9975

0.998

0.9985

0.999

Coherence Probability

37

Exchange Rates DataExchange Rates Data

25 exchange rate series sampled half-hourly for the whole of 1996 with weekends removed. The weekend period is from 23:00 GMT on Friday when North American financial centres close until 23:00 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.

38

Summary StatisticsSummary Statistics

DEM_ JPY

GBP_ DEM

GBP_ USD

USD_ ITL

USD_ JPY

Mean 3.4E-4 9.7E-4 5.6E-4 -2.1E-4 6.5E-4

Var 6.5E-3 4.6E-3 4.8E-4 9.0E-3 6.2E-3

Skew -0.049 -0.004 -0.167 -0.011 -0.019

Kurt 5.642 83.51 13.01 15.719 9.723

Min -0.707 -1.966 -1.137 -0.924 -0.770

Max 0.659 1.992 1.203 0.966 0.758

Lag 1 -0.198 -0.306 -0.205 -0.315 -0.150

39

Coherence Results ICoherence Results I

Period 30 min

Spectrum (dB)

Coherence p-value

DEM_JPY

60 hours -2.164 0.314 0.003 2 hours 2 minutes -0.165 0.307 0.004

GBP_DEM

6 hours -1.357 0.312 0.003 2 hours 8 minutes -0.775 0.329 0.002 1 hour 38 minutes -1.534 0.317 0.003

GBP_USD

12 hours -3.688 0.302 0.005 8 hours -2.882 0.295 0.007

USD_CHF

15 hours -1.320 0.336 0.001 8 hours -2.110 0.345 0.001

3 hours 32 minutes -0.803 0.299 0.006 2 hours 0.096 0.283 0.010

40

Coherence Results IICoherence Results IIPanel E: USD_DEM

15 hours 0.390 0.352 0.001

8 hours -0.365 0.331 0.002

Panel F: USD_ITL

12 hours -4.166 0.308 0.004

5 hours 43 min -3.135 0.329 0.002

1 hour 49 min 0.139 0.353 0.001

Panel G: USD_JPY

60 hours -1.669 0.341 0.001

15 hours -0.956 0.362 0.000

8 hours -1.399 0.349 0.001

2 hours 37 min 0.046 0.298 0.006

1 hour 5 min 0.907 0.297 0.006

41

Coherent Part of Yen/$ Mean FrameCoherent Part of Yen/$ Mean FrameYen/$ Coherent Part of the Mean Frame

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 20 39 58 77 96 115 134 153 172 191 210 229

Time

42

Descriptive Statistics of Canadian$/US$ Daily DataDescriptive Statistics of Canadian$/US$ Daily Data

• Sample size = 7569 20.720.7 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 = 4040 Frame Length = 186.8186.8

• Frequencies in band = 93

• Maximum coherence probability = 0.999999

• Coherence threshold = 0.35

43

Canadian$/US$ Daily Data SpectraCanadian$/US$ Daily Data Spectra

-2

0

2

186.80 31.13 16.98 11.68 8.90 7.18 6.03 5.19 4.56 4.06 3.66 3.34 3.06 2.83 2.63 2.46 2.31 2.17 2.05

Period in Days

dB

0.99

0.995

1

Coherence ProbabilitySpectrum - US$/Canadian$ Coherence Spectrum

44

Waterfall SpectrogramsWaterfall Spectrograms of RMP+Noise of RMP+Noise SNR = - 38 dBSNR = - 38 dB

3.0 3.5 4.0 4.4 4.9 5.4 5.9 6.4 6.8 7.3 7.8 8.3 8.8 9.2 9.7 10.2 10.7 11.2 11.6 12.1 12.6 13.1S1

S16

S31

S46

S61

Frequency Hz

Time Slices

RMP Waterfall Plot of Normalized Spectrograms - u = 20 & e = 10 SNR = - 38 dB

0.95 < Peak Values < 1.0

0.99-1

0.98-0.99

0.97-0.98

0.96-0.97

0.95-0.96

0.94-0.95

45

Sigcoh Probability Waterfall of RMP+Noise Sigcoh Probability Waterfall of RMP+Noise SNR = - 38 dBSNR = - 38 dB

1.0

5.0

9.0

13.0

17.0

21.0

25.0

29.0

S1

S16

S31

S46

S61

S76

Frequency Hz

Time Slices

Waterfall Plot of Signal Coherence Probabilities - u = 20 & e = 10 - 38 dB

0.95 < Peaks < 1.0

0.99-1

0.98-0.99

0.97-0.98

0.96-0.97

0.95-0.96

0.94-0.95

46

Waterfall SpectrogramsWaterfall Spectrograms of RMP+Noise of RMP+Noise SNR = - 44 dBSNR = - 44 dB

3.0 3.5 4.0 4.4 4.9 5.4 5.9 6.4 6.8 7.3 7.8 8.3 8.8 9.2 9.7 10.2 10.7 11.2 11.6 12.1 12.6 13.11

16

31

46

61

Frequency Hz

Figure 3

Time Slices

RMP Waterfall Plot of Normalized Spectrograms - u = 20 & e = 1 SNR = - 44 dB

0.95 < Peak Values < 1.0

0.99-1

0.98-0.99

0.97-0.98

0.96-0.97

0.95-0.96

0.94-0.95

47

Sigcoh Probability Waterfall of RMP+Noise Sigcoh Probability Waterfall of RMP+Noise SNR = - 44 dBSNR = - 44 dB

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

1

16

31

46

61

Frequency Hz

Figure 4

Time Slices

Waterfall Plot of Signal Coherence Probabilities - u = 20 & e = 20 SNR = - 44 dB

0.95 < Peaks < 1.0

0.99-1

0.98-0.99

0.97-0.98

0.96-0.97

0.95-0.96

0.94-0.95

48

Winglet Vibration DataWinglet Vibration Data

Block of the Winglet Data

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Am

pli

tud

e

49

Winglet Spectrum & CoherenciesWinglet Spectrum & Coherencies

Winglet Spectrum & Peaks of the Coherence Probabilities 20 dB in Spectrum = 10 * Amplitude

-14

-9

-4

1

6

11

0.0010.0220.043 0.0640.0850.1060.127 0.1480.1690.1900.2110.232 0.2530.2740.295

Frequency in kHz

dB

0.995

0.996

0.997

0.998

0.999

1

Coherence Probability Spectrum Coherence Probability

50

Prewhitened Winglet SpectraPrewhitened Winglet SpectraAR Prewhitened Winglet Data

-24

-22

-20

-18

-16

-14

-12

-10

0.070 0.082 0.094 0.106 0.118 0.130 0.142 0.154 0.166 0.178 0.190 0.202 0.214 0.226 0.238 0.250 0.262 0.274 0.286

Frequency in kHz

dB

0.995

0.996

0.997

0.998

0.999

1

Coherence ProbabilitySpectrum Coherence Probability

51

Tuned Prewhitened Winglet SpectraTuned Prewhitened Winglet SpectraAR Prewhitened Winglet Spectra Tuned to 1.018 Hz

-24

-22

-20

-18

-16

-14

-12

-10

0.070 0.088 0.105 0.122 0.140 0.157 0.174 0.191 0.209 0.226 0.243 0.261 0.278 0.295

Frequency in kHz

dB

0.995

0.996

0.997

0.998

0.999

1

Coherence Probability

Spectrum Coherence Probability

52

Noise Like SignalNoise Like Signal

50 Harmonics Modulation = 40 = 0.9 Frame = 200

-6

-4

-2

0

2

4

6

Time

Am

pli

tud

e