Intro to Spectral Analysis and
Matlab
Q: How Could
you quantify
how much lower
the tone of a
race car is after
it passes you
compared to as
it is coming
towards you?
How would you
set the
experiment up?
Running the Experiment .
Data is often recorded in
the time domain. The
stored dataset is called a
timeseries. It is a set of
time and amplitude pairs.
Frequency Domain
(Do a Fourier Transform on Timeseries)
We have converted to the
Frequency Domain. This
dataset is called a Spectra.
It is a set of frequency and
Amplitude pairs.
Time
Domain
What’s the Frequency?
What’s the Period?
What will this look like in the Frequency Domain?
180
180
160
160
140
140
Amplitude
Amplitude
120
120
100
100
80
80
60
60
40
20
0
00
0.5
0.5
11
1.5
1.5
22
2.5
2.5
Period(s)
(s)
Period
33
3.5
3.5
44
4.5
4.5
55
What’s the new (red) period?
How Does its amplitude Compare to the 1 s
signal?
180
160
140
Amplitude
120
100
80
60
40
20
0
0
0.5
1
1.5
2
2.5
Period (s)
3
3.5
4
4.5
5
Power Spectral Densities
Secondary
Microseism (~8 s)
Primary Microseism
(~ 16 s)
QSPA PSD PDF
The Mysterious Case of HOWD
Sampling Frequency
• Digital signals aren’t continuous
– Sampled at discrete times
• How often to sample?
– Big effect on data volume
How many samples/second
are needed?
Are red points enough?
Aliasing
FFT will give wrong frequency
Nyquist frequency
1/2 sampling frequency
Nyquist frequency
• Can only accurately measure frequencies
<1/2 of the sampling frequency
– For example, if sampling frequency is 200
Hz, the highest theoretically measurable
frequency is 100 Hz
• How to deal with higher frequencies?
– Filter before taking spectra
Summary
• Infinite sine wave is spike in frequency
domain
• Can create arbitrary seismogram by adding
up enough sine waves of differing
amplitude, frequency and phase
• Both time and frequency domains are
complete representations
– Can transform back and forth – FFT and iFFT
• Must be careful about aliasing
– Always sample at least 2X highest frequency
of interest
To create arbitrary seismogram
• Becomes integral in the limit
• Fourier Transform
– Computer: Fast Fourier Transform - FFT
Exercise plots
Sine_wave column 2
Sine_wave column 2
Sine_wave column 2 and 3
Sine_wave column 2 and 3 sum
Spectra, column 2
Spectra, columns 2, 3
Spectra, column 2, 3, 2 and 3 sum
Multi_sine, individual columns
Multi_sine, individual columns
Multi_sine spectra
Spike in time
Spike in time, frequency
Rock, sed, bog time series
Rock spectra
Rock (black), Sed (red), bog (blue)
Spectral ratio sed/rock
Basin Thickness
• Sediment site
• 110 m/s /2.5 Hz = 44 m wavelength
• Basin thickness = 11 m
• Peat Bog
• 80 m/s /1 Hz = 80 m
• Basin thickness = 20 m
Station LKWY, Utah
raw
Filtered
2-19 Hz
Filtered
twice
Station LKWY, Utah
raw
Filtered
2-19 Hz
Filtered
twice
Zoomed in once
Zoomed in once
Zoomed in again
Triggered earthquakes