Léonard Seydoux and Alexandre Fournier seydoux,fournier@ipgp.fr September 2026 at the institut de physique du globe de Paris.
Let's first think about signal processing, then about digital signal processing
A system transforms an input signal into an output signal:a seismometer turns ground motion into counts
Linear: the response to a sum of inputs is the sum of their responses,with and
Time-invariant: the system behaves the same at any time,so a delayed input gives the same output, delayed
A system that satisfies both properties is called LTI
Seismic waves in the Earth, sensors, or digital filters are LTI to a good approximation
An LTI system is fully described by its response to a single impulse
Hit the system with a very short pulse and record the output:this is the impulse response . For the Earth, it is the Green's function
Hitting later gives the same response, later
Two hits give the sum of two responses
Any input can be seen as a series of small hits every ,with weights , each producing its own response
When the hits get infinitely close (), the sums become integrals
The output of an LTI system is the convolution of the input with
Dry piano
Hall response
Piano in the hall
Every instant of the piano triggers a copy of the hall response
Piano, hall and microphone become source, medium and receiver: Stein & Wysession (2003), fig. 1.1-1
Deconvolution: remove to get the ground motion, to study the source, or to image the Earth
A sinusoid comes out as the same sinusoid,with a gain and a phase shift
Any signal is also a sum of sinusoids. The Fourier transform tells how much of each frequency it contains, and can be inverted
What does a convolution look like in the frequency domain? With the change of variable , the double integral splits in two
Where is the frequency response: gain and phase shift
Convolution in time is a product in frequency
In frequency, removing the instrument response is a division
Unstable where : keep the band where it is large
With ObsPy: trace.remove_response(inventory)
trace.remove_response(inventory)
Piano record 24 bits 44.1 kHz 16.7 M levels file size 1.6 MB
Piano record 8 bits 8 kHz 256 levels file size 95 kB
16 times smaller, but audio quality is low. Resolution and sampling rate both matter
Sampling keeps the values of every ; it multiplies the signal by a Dirac comb Ш
Ш
with and the sampling rate .
The Fourier transform of a comb is a comb, and by the convolution theorem, the product in time becomes a convolution in frequency
Sampling copies the spectrum at every multiple of
If , the copies overlap:high frequencies disguise as low ones, this is aliasing
Sampling theorem: a signal can be reconstructed if ,that is, if its highest frequency is below the Nyquist frequency
In practice, low-pass filter before sampling: seismic digitizers do it,and this filter is part of the instrument response
On a computer, we only have the samples .Their Fourier transform is the discrete-time Fourier transform (DTFT)
with the frequency in radians per sample, so that
A computer handles samples, so we use the discrete Fourier transform
numpy.fft