what are the dominant frequencies? fourier transforms decompose a data sequence into a set of...
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What are the dominant frequencies?
Fourier transforms decompose a data sequence into a set of discrete spectral estimates – separate the variance of a time series as a function of frequency.
A common use of Fourier transforms is to find the frequency components of a signal buried in a time domain signal.
1
0
2)(1 N
t
NtnietfN
FFT
FAST FOURIER TRANSFORM (FFT)
In practice, if the time series f(t) is not a power of 2, it should be padded with zeros
What is the statistical significance of the peaks?
Each spectral estimate has a confidence limit defined by a chi-squared distribution 2
Spectral Analysis Approach
1. Remove mean and trend of time series
2. Pad series with zeroes to a power of 2
3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series
4. Compute the Fourier transform of the series, multiplied times the window
5. Rescale Fourier transform by multiplying times 8/3 for the Hanning Window
6. Compute band-averages or block-segmented averages
7. Incorporate confidence intervals to spectral estimates
Sea level at Mayport, FL
July 1, 2007 (day “0” in the abscissa) to September 1, 2007
mm
Raw data and Low-pass filtered data
High-pass filtered data
1. Remove mean and trend of time series (N = 1512)
2. Pad series with zeroes to a power of 2 (N = 2048)
Cycles per day
m2/c
pd
m2/c
pd
Spectrum of raw data
Spectrum of high-pass filtered data
Day from July 1, 2007
Val
ue
of
the
Win
do
w
Hanning Window
Hamming Window
102121 NnNnw ...,)/cos(
102460540 NnNnw ...),/cos(..
3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series
Day from July 1, 2007
Val
ue
of
the
Win
do
w
Hanning WindowHamming WindowKaiser-Bessel, α = 2Kaiser-Bessel, α = 3
2
00
212
0
0
2
21
20
k
k
kx
xI
Nn
NnII
w
!
)(
/...,)()(
3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series
mm
Raw series x Hanning Window(one to one)
Raw series x Hamming Window(one to one)
Day from July 1, 2007
To reduce side-lobe effects
4. Compute the Fourier transform of the series, multiplied times the window
mm
High-pass series x Hanning Window(one to one)
High pass series x Hamming Window(one to one)
Day from July 1, 2007
To reduce side-lobe effects
4. Compute the Fourier transform of the series, multiplied times the window
High pass series x Kaiser-Bessel Windowα=3 (one to one)
m
Day from July 1, 20074. Compute the Fourier transform of the series, multiplied times the window
Cycles per day
m2/c
pd
m2/c
pd
Original from Raw Data
with Hanning window
with Hamming window
Windows reduce noise produced by side-lobe effects
Noise reduction is effected at different frequencies
Cycles per day
m2/c
pd
m2/c
pd
with Hanning window
with Hamming and Kaiser-Bessel (α=3) windows
5. Rescale Fourier transform by multiplying:
times 8/3 for the Hanning Window
times 2.5164 for the Hamming Window
times ~8/3 for the Kaiser-Bessel (Depending on alpha)
6. Compute band-averages or block-segmented averages
7. Incorporate confidence intervals to spectral estimates
Upper limit:
2
,21
2,2
Lower limit:
1-alpha is the confidence (or probability)nu are the degrees of freedomgamma is the ordinate reference value
0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005 7.88 6.63 5.02 3.84 2.71 1.32 0.45 0.10 0.02 0.00 0.00 0.00 0.00 10.60 9.21 7.38 5.99 4.61 2.77 1.39 0.58 0.21 0.10 0.05 0.02 0.01 12.84 11.34 9.35 7.81 6.25 4.11 2.37 1.21 0.58 0.35 0.22 0.11 0.07 14.86 13.28 11.14 9.49 7.78 5.39 3.36 1.92 1.06 0.71 0.48 0.30 0.21 16.75 15.09 12.83 11.07 9.24 6.63 4.35 2.67 1.61 1.15 0.83 0.55 0.41 18.55 16.81 14.45 12.59 10.64 7.84 5.35 3.45 2.20 1.64 1.24 0.87 0.68 20.28 18.48 16.01 14.07 12.02 9.04 6.35 4.25 2.83 2.17 1.69 1.24 0.99 21.95 20.09 17.53 15.51 13.36 10.22 7.34 5.07 3.49 2.73 2.18 1.65 1.34 23.59 21.67 19.02 16.92 14.68 11.39 8.34 5.90 4.17 3.33 2.70 2.09 1.73 25.19 23.21 20.48 18.31 15.99 12.55 9.34 6.74 4.87 3.94 3.25 2.56 2.16 26.76 24.72 21.92 19.68 17.28 13.70 10.34 7.58 5.58 4.57 3.82 3.05 2.60 28.30 26.22 23.34 21.03 18.55 14.85 11.34 8.44 6.30 5.23 4.40 3.57 3.07 29.82 27.69 24.74 22.36 19.81 15.98 12.34 9.30 7.04 5.89 5.01 4.11 3.57 31.32 29.14 26.12 23.68 21.06 17.12 13.34 10.17 7.79 6.57 5.63 4.66 4.07 32.80 30.58 27.49 25.00 22.31 18.25 14.34 11.04 8.55 7.26 6.26 5.23 4.60 34.27 32.00 28.85 26.30 23.54 19.37 15.34 11.91 9.31 7.96 6.91 5.81 5.14 35.72 33.41 30.19 27.59 24.77 20.49 16.34 12.79 10.09 8.67 7.56 6.41 5.70 37.16 34.81 31.53 28.87 25.99 21.60 17.34 13.68 10.86 9.39 8.23 7.01 6.26 38.58 36.19 32.85 30.14 27.20 22.72 18.34 14.56 11.65 10.12 8.91 7.63 6.84 40.00 37.57 34.17 31.41 28.41 23.83 19.34 15.45 12.44 10.85 9.59 8.26 7.43 41.40 38.93 35.48 32.67 29.62 24.93 20.34 16.34 13.24 11.59 10.28 8.90 8.03 42.80 40.29 36.78 33.92 30.81 26.04 21.34 17.24 14.04 12.34 10.98 9.54 8.64 44.18 41.64 38.08 35.17 32.01 27.14 22.34 18.14 14.85 13.09 11.69 10.20 9.26 45.56 42.98 39.36 36.42 33.20 28.24 23.34 19.04 15.66 13.85 12.40 10.86 9.89 46.93 44.31 40.65 37.65 34.38 29.34 24.34 19.94 16.47 14.61 13.12 11.52 10.52 48.29 45.64 41.92 38.89 35.56 30.43 25.34 20.84 17.29 15.38 13.84 12.20 11.16 49.64 46.96 43.19 40.11 36.74 31.53 26.34 21.75 18.11 16.15 14.57 12.88 11.81 50.99 48.28 44.46 41.34 37.92 32.62 27.34 22.66 18.94 16.93 15.31 13.56 12.46 52.34 49.59 45.72 42.56 39.09 33.71 28.34 23.57 19.77 17.71 16.05 14.26 13.12 53.67 50.89 46.98 43.77 40.26 34.80 29.34 24.48 20.60 18.49 16.79 14.95 13.79 55.00 52.19 48.23 44.99 41.42 35.89 30.34 25.39 21.43 19.28 17.54 15.66 14.46 56.33 53.49 49.48 46.19 42.58 36.97 31.34 26.30 22.27 20.07 18.29 16.36 15.13 57.65 54.78 50.73 47.40 43.75 38.06 32.34 27.22 23.11 20.87 19.05 17.07 15.82 58.96 56.06 51.97 48.60 44.90 39.14 33.34 28.14 23.95 21.66 19.81 17.79 16.50 60.27 57.34 53.20 49.80 46.06 40.22 34.34 29.05 24.80 22.47 20.57 18.51 17.19
Probability1234567891011121314151617181920212223242526272829303132333435
Deg
rees
of
fre
edo
m
Includes low frequency
N=1512
Excludes low frequency
N=1512
N=1512
N=1512
Least Squares Fit to Main Harmonics
The observed flow u’ may be represented as the sum of M harmonics:
u’ = u0 + ΣjM
=1 Aj sin (j t + j)
For M = 1 harmonic (e.g. a diurnal or semidiurnal constituent):
u’ = u0 + A1 sin (1t + 1)
With the trigonometric identity: sin (A + B) = cosBsinA + cosAsinB u’ = u0 + a1 sin (1t ) + b1 cos (1t )
taking:a1 = A1 cos 1
b1 = A1 sin 1
The squared errors between the observed current u and the harmonic representation may be expressed as 2 :
2 = ΣN [u - u’ ]2 = u 2 - 2uu’ + u’ 2
Then:
2 = ΣN {u 2 - 2uu0 - 2ua1 sin (1t ) - 2ub1 cos (1t ) + u02 + 2u0a1 sin (1t ) +
2u0b1 cos (1t ) + 2a1 b1 sin (1t ) cos (1t ) + a12 sin2 (1t ) +
b12 cos2 (1t ) }
Using u’ = u0 + a1 sin (1t ) + b1 cos (1t )
Then, to find the minimum distance between observed and theoretical values we need to minimize
2 with respect to u0 a1 and b1, i.e., δ 2/ δu0 , δ 2/ δa1 , δ 2/ δb1 :
δ2/ δu0 = ΣN { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0
δ2/ δa1 = ΣN { -2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0
δ2/ δb1 = ΣN {-2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0
ΣN { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0
ΣN {-2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0
ΣN { -2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0
Rearranging:
ΣN { u = u0 + a1 sin (1t ) + b1 cos (1t ) }
ΣN { u sin (1t ) = u0 sin (1t ) + b1 sin (1t ) cos (1t ) + a1 sin2(1t ) }
ΣN { u cos (1t ) = u0 cos (1t ) + a1 sin (1t ) cos (1t ) + b1 cos2(1t ) }
And in matrix form:
ΣN u cos (1t ) ΣN cos (1t ) ΣN sin (1t ) cos (1t ) ΣN cos2(1t ) b1
ΣN u N ΣN sin (1t ) Σ N cos (1t ) u0
ΣN u sin (1t ) = ΣN sin (1t ) ΣN sin2(1t ) ΣN sin (1t ) cos (1t ) a1
B = A X X = A-1 B
Finally...
The residual or mean is u0
The phase of constituent 1 is: 1 = atan ( b1 / a1 )
The amplitude of constituent 1 is: A1 = ( b12 + a1
2 )½
Pay attention to the arc tangent function used. For example, in IDL you should use atan (b1,a1) and in MATLAB, you should use atan2
For M = 2 harmonics (e.g. diurnal and semidiurnal constituents):
u’ = u0 + A1 sin (1t + 1) + A2 sin (2t + 2)
ΣN cos (1t ) ΣN sin (1t ) cos (1t ) ΣN cos2(1t ) ΣN cos (1t ) sin (2t ) ΣN cos (1t ) cos (2t )
N ΣN sin (1t ) Σ N cos (1t ) ΣN sin (2t ) Σ N cos (2t )
ΣN sin (1t ) ΣN sin2(1t ) ΣN sin (1t ) cos (1t ) ΣN sin (1t ) sin (2t ) ΣN sin (1t ) cos (2t )
Matrix A is then:
ΣN sin (2t ) ΣN sin (1t ) sin (2t ) ΣN cos (1t ) sin (2t ) ΣN sin2(2t ) ΣN sin (2t ) cos (2t )
ΣN cos (2t ) ΣN sin (1t ) cos (2t ) ΣN cos (1t ) cos (2t ) ΣN sin (2t ) cos (2t ) ΣN cos2 (2t )
Remember that: X = A-1 B
and B =ΣN u cos (1t )
ΣN u sin (2t )
ΣN u cos (2t )
ΣN u
ΣN u sin (1t )
u0
a1
b1
a2
b2
X =
Goodness of Fit:
Σ [< uobs > - upred] 2
-------------------------------------
Σ [<uobs > - uobs] 2
Root mean square error:
[1/N Σ (uobs - upred) 2] ½
Fit with M2 only
Fit with M2, K1
Fit with M2, S2, K1
Fit with M2, S2, K1,M4, M6
Tidal Ellipse Parameters
21
)sin(221 22
ppaaaac uvvuvuQ
ua, va, up, vp are the amplitudes and phases of the east-west and north-south components of velocity
amplitude of the clockwise rotary component
21
)sin(221 22
ppaaaacc uvvuvuQ amplitude of the counter-clockwise rotary component
papa
papac vvuu
vvuu
sincos
cossintan 1 phase of the clockwise rotary component
papa
papacc vvuu
vvuu
sincos
cossintan 1 phase of the counter-clockwise rotary component
The characteristics of the tidal ellipses are: Major axis = M = Qcc + Qc
minor axis = m = Qac - Qc
ellipticity = m / MPhase = -0.5 (thetacc - thetac)Orientation = 0.5 (thetacc + thetac)
Ellipse Coordinates:tmtMy
tmtMx
sincoscossin
sinsincoscos
M2
S2
K1
Fit with M2 only (dotted) and M2 + M4 (continuous)
Fit with M2 only (dotted) and M2 + M6 (continuous)
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