This section addresses the methods and mathematical techniques behind signal recognition and signal analysis. It considers the time domain and frequency domain analysis of signals. This section also discusses various transforms and their usefulness in the analysis of multi-dimensional waves.
3D sampling Sampling The first step in any signal processing approach is analog to digital conversion. The geophysical signals in the analog domain has to be converted to digital domain for further processing. Most of the filters are available in 1D as well as 2D.
Analog to digital conversion As the name suggests, the gravitational and electromagnetic waves in the analog domain are detected, sampled and stored for further analysis. The signals can be sampled in both time and frequency domains. The signal component is measured at both intervals of time and space. Ex, time-domain sampling refers to measuring a signal component at several instances of time. Similarly, spatial-sampling refers to measuring the signal at different locations in space. Traditional sampling of 1D time varying signals is performed by measuring the amplitude of the signal under consideration in discrete intervals of time. Similarly sampling of space-time signals (signals which are functions of 4 variables – 3D space and time), is performed by measuring the amplitude of the signals at different time instances and different locations in the space. For example, the Earth's gravitational data is measured with the help of
gravitational wave sensor or gradiometer by placing it in different locations at different instances of time.
Spectrum analysis Multi-dimensional Fourier transform The Fourier expansion of a time domain signal is the representation of the signal as a sum of its frequency components, specifically sum of sines and cosines.
Joseph Fourier came up with the Fourier representation to estimate the heat distribution of a body. The same approach can be followed to analyse the multi-dimensional signals such as gravitational waves and electromagnetic waves. The 4D Fourier representation of such signals is given by :S (K, \omega) = \iint s(x,t) e^{-j (\omega t - k' x)} \, dx\, dt •
ω represents temporal frequency and
k represents spatial frequency. •
s(
x,
t) is a 4-dimensional space-time signal which can be imagined as travelling plane waves. For such plane waves, the plane of propagation is perpendicular to the direction of propagation of the considered wave.
Wavelet transform The motivation for development of the Wavelet transform was the Short-time Fourier transform. The signal to be analysed, say
f(
t) is multiplied with a window function
w(
t) at a particular time instant. Analysing the Fourier coefficients of this signal gives us information about the frequency components of the signal at a particular time instant. The STFT is mathematically written as: :\{x(t)\}(\tau,\omega) \equiv X(\tau, \omega) = \int_{-\infty}^{\infty} x(t) w(t-\tau) e^{-j \omega t} \, dt The Wavelet transform is defined as :X(a,b) = \frac{1}{\sqrt{a}} \int\limits_\ \Psi( \frac{t-b}{a}) x(t) dt A variety of window functions can be used for analysis. Wavelet functions are used for both time and frequency localisation. For example, one of the windows used in calculating the Fourier coefficients is the Gaussian window which is optimally concentrated in time and frequency. This optimal nature can be explained by considering the time scaling and time shifting parameters
a and
b respectively. By choosing the appropriate values of
a and
b, we can determine the frequencies and the time associated with that signal. By representing any signal as the linear combination of the wavelet functions, we can localize the signals in both time and frequency domain. Hence wavelet transforms are important in geophysical applications where spatial and temporal frequency localisation is important.
Time frequency localisation using wavelets Geophysical signals are continuously varying functions of space and time. The wavelet transform techniques offer a way to decompose the signals as a linear combination of shifted and scaled version of basis functions. The amount of "shift" and "scale" can be modified to localize the signal in time and frequency.
Beamforming Simply put, space-time signal filtering problem can be thought as localizing the speed and direction of a particular signal. The design of filters for space-time signals follows a similar approach as that of 1D signals. The filters for 1-D signals are designed in such a way that if the requirement of the filter is to extract frequency components in a particular non-zero range of frequencies, a
bandpass filter with appropriate passband and stop band frequencies in determined. Similarly, in the case of multi-dimensional systems, the wavenumber-frequency response of filters is designed in such a way that it is unity in the designed region of (
k,
ω) a.k.a. wavenumber – frequency and zero elsewhere. :P\left (K_x,w\right)=\int_{-\infty}^\infty \int_{-\infty}^\infty \varphi_{ss}\left(x,t\right)\ e^{-j\left(w t-k' x\right)}\,dx\,dt :\varphi_{ss}\left(x,t\right)=s\left[\left(\xi,\tau\right)s*\left(\xi-x,\tau-t\right)\right] The spectral estimates can be obtained by finding the square of the magnitude of the Fourier transform also called as Periodogram. The spectral estimates obtained from the periodogram have a large variance in amplitude for consecutive periodogram samples or in wavenumber. This problem is resolved using techniques that constitute the classical estimation theory. They are as follows: 1.Bartlett suggested a method that averages the spectral estimates to calculate the power spectrum. Average of spectral estimates over a time interval gives a better estimate. :P_B\left(w\right) = \frac{1}{\mathrm{det}\,N}\sum_{l}|\sum_{n}\ x\left(n+MI\right)\ e^{-j\left(w' n\right)}|^2 Bartlett's case :P_W\left(w\right) = \frac{1}{\mathrm{det}\,N}\sum_{l}|\sum_{n}\ g\left(n\right)\ x\left(n+MI\right)\ e^{-j\left(w' n\right)}|^2 Welch's case 4. The periodogram under consideration can be modified by multiplying it with a window function. Smoothing window will help us smoothen the estimate. Wider the main lobe of the smoothing spectrum, smoother it becomes at the cost of frequency resolution. :P_M\left(w\right) = \frac{1}{detN}|\sum_{n}\ g\left(n\right)\ x\left(n\right)\ e^{-j\left(w' n\right)}|^2 Modified periodogram For further details on spectral estimation, please refer
Spectral Analysis of Multi-dimensional signals == Applications==