Multidimension spectral estimation is a generalization of spectral estimation, normally formulated for one-dimensional signals, to multidimensional signals or multivariate data, such as wave vectors.
Motivation Multidimensional spectral estimation has gained popularity because of its application in fields like medicine, aerospace, sonar, radar, bio informatics and geophysics. In the recent past, a number of methods have been suggested to design models with finite parameters to estimate the power spectrum of multidimensional signals. In this article, we will be looking into the basics of methods used to estimate the power spectrum of multidimensional signals.
Applications There are many applications of spectral estimation of multi-D signals such as classification of signals as low pass, high pass, pass band and stop band. It is also used in compression and coding of audio and video signals, beam forming and direction finding in radars, Seismic data estimation and processing, array of sensors and antennas and vibrational analysis. In the field of radio astronomy, it is used to synchronize the outputs of an array of telescopes.
Basic Concepts In a single dimensional case, a signal is characterized by an amplitude and a time scale. The basic concepts involved in spectral estimation include autocorrelation, multi-D Fourier transform, mean square error and entropy. When it comes to multidimensional signals, there are two main approaches: use a bank of filters or estimate the parameters of the random process in order to estimate the power spectrum.
Methods
Classical Estimation Theory
It is a technique to estimate the power spectrum of a single dimensional or a multidimensional signal as it cannot be calculated accurately. Given are samples of a wide sense stationary random process and its second order statistics (measurements).The estimates are obtained by applying a multidimensional Fourier transform of the autocorrelation function of the random signal. The estimation begins by calculating a periodogram which is obtained by squaring the magnitude of the multidimensional Fourier transform of the measurements ri(n). 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:
Bartlett suggested a method that averages the spectral estimates to calculate the power spectrum. The measurements are divided into equally spaced segments in time and an average is taken. This gives a better estimate. Based on the wavenumber and index of the receiver/output we can partition the segments. This increases the spectral estimates and decreases the variances between consecutive segments. Welch suggested that we should divide the measurements using data window functions, calculate a periodogram, average them to get a spectral estimate and calculate the power spectrum using Fast Fourier Transform (FFT). This increases the computational speed. Smoothing window will help us smoothen the estimate by multiplying the periodogram with a smoothening spectrum. Wider the main lobe of the smoothening spectrum, smoother it becomes at the cost of frequency resolution.
P ( K x , w ) = ∫ − ∞ ∞ ∫ − ∞ ∞ φ s s ( x , t ) e − j ( w t − k ′ x ) d x d t {\displaystyle P\left(K_{x},w\right)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\varphi _{ss}\left(x,t\right)\,e^{-j\left(wt-k'x\right)}\,dx\,dt}
φ s s ( x , t ) = s [ ( ξ , τ ) s ∗ ( ξ − x , τ − t ) ] {\displaystyle \varphi _{ss}\left(x,t\right)=s\left[\left(\xi ,\tau \right)s^{*}\left(\xi -x,\tau -t\right)\right]}
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