Section 3: Convolution, Correlation and Time-Series Flashcards
1
Q
convolution describes
A
the effect of one signal on another
2
Q
correlation describes
A
the similarity of two signals
3
Q
convolution and correlation are particularly important in
A
time series analysis
but also useful in spectroscopy and image analysis
4
Q
A
5
Q
convolution =
A
correlation with g reversed wrt u
6
Q
auto-correlation function of real function f is
A
cross correlation with itself
7
Q
convolution: observed signal often the
A
convolution of the source signal and something else
8
Q
convolution: want to recover the
A
source signal
so need DECONVOLUTION
9
Q
cross-correlation - find the lag/shift needed to get two observations to match via the
A
lag/shift that gives CCF maximum
10
Q
auto-correlation - find a periodic signal or repeating pattern in an observation via the
A
lag distance between odd (or even) peaks in the |ACF|^2
11
Q
Convolution: PSF - the image of a point source produced by telescope and detector is never
A
a point
12
Q
convolution PSF: the image intensity distribution is the
A
convolution of this point source and the point spread function
13
Q
PSF sometimes called the
A
instrumental profile
14
Q
PSF resulting from diffraction telescope aperture is
A
the Airy pattern
15
Q
convolution with the PSF: consider the 1D problem
A
1 point source convolved with PSF = observed sources
16
Q
a single source at position x1 with intensity I(x1) is spread out across
A
position by the PSF
17
Q
1D case: at a position x2, the observed intensity of the single source is
A
o(x2)=I(x1)PSF(x2-x1)
18
Q
convolution with the PSF - consider multiple point sources
A
point sources convolved with PSF = observed sources
19
Q
convolution with PSF multiple sources - source distribution contributes to the intensity as
A
o(x2)=I * PSF
the convolution of the distribution I with the PSF
20
Q
it is possible to correct for effects of the PSF by
A
deconvolving, using the convolution theorem
21
Q
FT of the function h(t)
A
H(s) = F(h(t)) = โซh(t)e^-2piist dt
22
Q
the inverse fourier transform F^-1 of H(s) would recover
A
h(t)
F^-1 (H(s)) = โซ H(s) e^2piist ds = h(t)
23
Q
convolution theorem
A
the FT of the convolution of two functions is equal to the multiplication of their FTs
o=I*PSF
F(o)=F(I) x F(PSF)
24
Q
if the PSF is known we can find I from o via
A
I = F^-1 [F(o)/F(PSF)]
25
if data series has large no of elements, finding the convolution can be slow so can instead
use the convolution theorem
26
as the correlation can also be written in terms of a convolution, we can use correlation theorem to
quickly compute via FT instead of full equation
27
the PSF can be composed of the
instrument and the atmospheric PSF
28
Overall PSF can be measured by
observing a bright point source (eg star or quasar) near the target
object (eg galaxy)
29
For most UV-IR instruments, the image PSF caused by the instrument is broader than
the diffraction pattern
30
Even if you can make the different contributions to the PSF as small as possible you can not get better
than
the instrument's diffraction pattern
ie diffraction limited with FWHM_PSF
31
instrument PSF can arise from
-aperture diffraction
-scattering from roughness/dirt on mirrors or other surfaces
-errors in reflecting surface shape
-optical mis-alignments
32
image of stars are spread out across the detector by
the PSF
33
if we just want to do photometry (counting photons) from a source then can adopt a simpler methods than
PSF deconvolution
"aperture photometry"
34
aperture photometry process
-define 'apertures' (boxes) of different sizes
-sum counts in each aperture
-counts tend to 'true' counts as aperture size increases
35
downsides of aperture photometry
-background noise also increases with larger apertures
-higher chance of getting counts from other sources with larger aperture
36
In spectroscopy, lines recorded have a profile of intensity versus
wavelength that is a combination of
the true line profile and the instrumental lines profile
37
the instrumental profile is sometimes called the
spectral PSF
38
ideally the ILP will be
symmetric and narrower than the true line profile
this is often not the case
39
convolution of an emission spectrum
3 spectral lines convolved with ILP = observed spectrum
40
what makes convolution of an absorption line spectrum different
absorption features wider and shallower in observed vs true spectrum
41
Radiation at ๐๐ is spread
out, so contribution to
intensity at ๐๐ in observed
spectrum is
reduced
42
Adjacent radiation at ๐๐
also spread out, and
contributes to
observed
intensity at ๐๐
43
Observed intensity at ๐๐
found by
summing all contributions
44
All parts of a galaxy along a line-of sight contribute to
its observed spectrum
45
different parts have different LOS velocities which effectively
broadens/smears a spectral line
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overall spectrum can be described as a convolution of a
typical stellar spectrum and the LOS velocity distribution
47
the LOS velocity distribution is
fraction of stars contributing to spectrum with radial velocities between V_LOS and V_LOS +dV_LOS
48
It is convenient to express the observed spectrum in terms of
spectral velocity u
which is defined by u=c lnฮป
49
light observed at spectral velocity u was emitted at
spectral velocity u-v_los
50
Suppose that all stars have intrinsically identical spectra ๐บ(๐).
๐บ(๐) measures
the relative intensity of radiation at spectral velocity u
51
intensity from a star with V_LOS is
S(u-V_LOS)
52
the observed composite spectrum is the convolution of
the stellar spectral velocity with the LOS velocity distribution
53
If the spectral velocity is dominated by a single ๐๐ณ๐ถ๐บ, we can use
cross-correlation to find it
we subtract the mean from the galaxy and stellar spectra then use an arbitrary V_LOSi
54
If ๐๐ณ๐ถ๐บ๐ does not align the two signals, ๐ฎ(๐)๐บ(๐ โ ๐๐ณ๐ถ๐บ) will be
small +ve/-ve numbers at a given u
CCF small
55
If ๐๐ณ๐ถ๐บ๐ does align the two signals,
๐ฎ(๐)๐บ(๐ โ ๐๐ณ๐ถ๐บ) will be
large +ve numbers at a given u
CCF is large
56
we can estimate ๐๐ณ๐ถ๐บ by calculating
the CCF for many trial values of ๐๐ณ๐ถ๐บ and ๐บ(๐ โ ๐๐ณ๐ถ๐บ), and finding its maximum value
57
Periods present in a signal can be found using
auto-correlation
(cross-correlation with the signal itself)
58
the auto-correlation of a time series measures how
well a signal matches a time-shifted version of itself
(not having to fit a model, just matching the data to itself)
59
cross correlation between two time series f and g is
CCF (๐) = integral of f(t) g(t-๐) dt
t=time
๐=variable time lag/shift
60
examples of time-varying signals
cepheid light curve
eclipsing binary
sunspot number
61
To find a period
๐ป present in
data
shift the data by some lag ๐ and cross-correlate with itself โ i.e. auto-correlation
62
When the lag is close to the
period, f(t)f(t-๐) is mostly
positive
63
when the lag is out of phase with the period, f(t)f(t-๐) is
mostly negative
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When the lag is a multiple of the
period get a
max ACF
๐=nT for integer n
๐-0,T,2T,3T...
65
When the lag is half a period out, get a
min ACF
๐=(2n+1)T/2
๐=T/2, 3T/2, 5T/2...
66
in |ACF|^2 you will get a peak when
๐=nT/2
๐=0,T/2, T, 3T/2, 2T, 5T/2, 3T....
67
Even with far โnoisierโ data (i.e. larger non-periodic signal added) the auto-correlation approach can still
recover the periodic signal
68
a periodic function can be expressed as
a sum of sines and cosines
ie a fourier series
69
the fourier transform of this periodic function decomposes it into
those sines and consines
70
we can identify the frequencies/periods via
the power spectrum or power spectral density PSD(v)
71
several routines available to calculate FT or the PSD can also be calculated directly via
PSD(v)^2 =C(v)^2 + S(v)^2
72
As the sines and cosines in
the Fourier series describing
the data add linearly, each
separate period that is
present will contribute
its own peak to the PSD
73
if the signal has the characteristics of white noise (a signal that is entirely random) then the power spectrum is
flat ie PSD=const
this means that there are no periods present in the signal (one part of the signal is entirely uncorrelated with any other)
74
can get "colours" of noise eg
red noise from Brownian motion with PSD prop to v^-2
75
* Suppose that a Cepheid light curve is being observed over several days
* The intrinsic modulation (๐ป โผ few hours) might be overlaid with:
A high frequency signal from the electronics
A low frequency signal due to temperature changes from day to
day affecting the CCD
76
these periodic nosie components can removed by
filtering
77
by plotting the PSD, we can decide which periods correspond to
the unwanted noise
and try get rid of them
78
Having established which frequencies correspond to unwanted noise, these can be
removed from the FT
essentially the FT is truncated at high and low frequencies
79
after removing noise from the FT, doing the inverse FT will then generate
a much cleaner time series, where the interesting effects are enhanced
80
the time series are repeating but very non-sinusoidal and/or the data might not be regularly sampled
due to eg
only observing every so often due to weather/scheduling etc
81
example of a case where fourier methods may not be efficient at identifying periods in data
extra-solar planet transits across the face of the parent star or star spot on rotating star
typically curves are built up by observations over several orbital periods
82
phase folding procedure
1. Guess a trial period, ๐ป๐
2. Divide the time-series ๐ฐ๐ up into bins based on this period
3. Fold/stack all the bins together based on this trial ๐ป๐
4. Average the values inside these stacked bins
5. Plot each average as a function of bin number
83
Time series analysis methods based on Fourier series work best if
data span many periods and the period does not change much
84
time series methods based on fourier series are not good at
searching for an aperiodic signal such as a short-lived burst or a quasi-periodic signal
85
our signal is a linear combination of the
basis functions
86
The problem with Fourier series is that they are strictly periodic, and
Fourier analysis can only decompose a signal into
a set of periodic basis functions
eg sines and cosines
87
Wavelets is a similar approach to Fourier analysis but uses different
basis functions which are finite and/or non-periodic
such basis functions are called mother wavelets
88
mother wavelets
basis functions which are finite and/or non-periodic
89
many different mother wavelets depending on
data type
90
morlet wavelet
plane wave convolved with a gaussian
effectively a gaussian windowed FT so good for changing sinusoids
91
Wavelet analysis assumes that the signal is a superposition of
short time structures, all with the same basic shape
92
Each contributing wavelet has its
amplitude, width (๐) and position (๐)
in time changed so that the wavelets sum equals the signal
93
So for a range of ๐, ๐ values, ๐พ(๐, ๐) is calculated and plotted as
2D wavelet power spectrogram
|๐พ ๐, ๐|^2
94
Colour scale gives regions of higher wavelet power, indicating what
frequencies are occurring when, and how they are changing
time on x-axis, frequency or period on y-axis
95
The region shaded white at the edge of the wavelet power
spectrogram indicates
a,b values at which the wavelet extends outside the time range of the data
96
for small a values, when does wavelet extend outside time range
for b values near time range edges
97
for larger a values, when does wavelet extend outside the time range
occurs for wider range of b values
98
the region where the boundary effects are important resulting in unreliable wavelet power is called the
cone of influence
99
if you add noise to a wavelet spectrogram
get same general pattern but also get extra peaks/blurriness/strings (structure disappearing)
amplitude also weakens
100
in general, when is it best to use wavelets
if multiple of A, f and T are changing
other methods will show something but will not get the full picture
101
in general, when is it best to use phase folding
when you have irregularly sampled data
102
