EEE 4304 Lecture Notes - Lecture 5: Synthetic Aperture Radar, Fast Fourier Transform, Partial Differential Equation
24 May 2018
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Department
Course
Professor
1
APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES
(David Sandwell, Copyright, 2004)
(Reference – The Fourier Transform and its Application, second edition, R.N. Bracewell,
McGraw-Hill Book Co., New York, 1978.)
Fourier analysis is a fundamental tool used in all areas of science and engineering. The fast
fourier transform (FFT) algorithm is remarkably efficient for solving large problems. Nearly
every computing platform has a library of highly-optimized FFT routines. In the field of Earth
science, fourier analysis is used in the following areas:
Solving linear partial differential equations (PDE’s):
Gravity/magnetics Laplace ∇2Φ = 0
Elasticity (flexure) Biharmonic ∇4Φ = 0
Heat Conduction Diffusion ∇2Φ - δ Φ/ δt = 0
Wave Propagation Wave ∇2Φ - δ2Φ/ δt2= 0
Designing and using antennas:
Seismic arrays and streamers
Multibeam echo sounder and side scan sonar
Interferometers – VLBI – GPS
Synthetic Aperture Radar (SAR) and Interferometric SAR (InSAR)
Image Processing and filters:
Transformation, representation, and encoding
Smoothing and sharpening
Restoration, blur removal, and Wiener filter
Data Processing and Analysis:
High-pass, low-pass, and band-pass filters
Cross correlation – transfer functions – coherence
Signal and noise estimation – encoding time series
In this remote sensing course we will use fourier analysis to understand and evaluate apertures
(antennas and telescopes) as well as to filter images.
Fourier analysis deals with complex numbers so perhaps it is time to dust off your book on
advanced calculus. Here is a very brief review of the things you’ll need. A complex number
€
z=x+iy
is composed of real and imaginary numbers. Remember
€
i=−1
. Functions can have
real and imaginary components as well. For example a general, complex-valued function of a
real variable is
€
f(x)=u(x)+iw(x)
. The most important complex-valued function for this
discussion is the complex exponential function
€
ei
θ
=cos
θ
+isin
θ
. After a little algebra it is easy
to show that
€
cos
θ
=ei
θ
+e−i
θ
2
and
€
sin
θ
=ei
θ
−e−i
θ
2i
. This is the level of math you’ll need to
understand fourier analysis.
2
Definitions of fourier transforms in 1-D and 2-D
The 1-dimensional fourier transform is defined as:
where x is distance and k is wavenumber where k = 1
/λ
and
λ
is wavelength. These equations
are more commonly written in terms of time t and frequency ν where ν = 1/T and T is the period.
The 2-dimensional fourier transform is defined as:
where x = (x, y) is the position vector, k = (kx, ky) is the wavenumber vector, and
(k . x) = kx x + ky y.
The next two page show some examples of fourier transform pairs. These figures were taken
from Bracewell [1978].
F(k) =f(x)e−i2
π
kx
−∞
∞
∫dx F(k) =ℑf(x)
[ ]
- forward transform
f(x) =F(k)ei2
π
kx
−∞
∞
∫dk f(x) =ℑ−1F(k)
[ ]
- inverse transform
F(k)=f(x
−∞
∞
∫
−∞
∞
∫)e−i2
π
(k⋅x)d2x F(k)=ℑ2f(x)
[ ]
f(x)=F(k
−∞
∞
∫
−∞
∞
∫)ei2
π
(k⋅x)d2k f(x)=ℑ2
−1F(k)
[ ]
3
Document Summary
Applications and review of fourier transform/series (david sandwell, copyright, 2004) (reference the fourier transform and its application, second edition, r. n. Fourier analysis is a fundamental tool used in all areas of science and engineering. The fast fourier transform (fft) algorithm is remarkably efficient for solving large problems. Nearly every computing platform has a library of highly-optimized fft routines. In the field of earth science, fourier analysis is used in the following areas: Synthetic aperture radar (sar) and interferometric sar (insar) Signal and noise estimation encoding time series. In this remote sensing course we will use fourier analysis to understand and evaluate apertures (antennas and telescopes) as well as to filter images. Fourier analysis deals with complex numbers so perhaps it is time to dust off your book on advanced calculus. Here is a very brief review of the things you"ll need. A complex number z = x + iy is composed of real and imaginary numbers.
