Aristotle University of Thessaloniki Department of Geodesy and

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Signals and Spectral Methods in Geoinformatics Lecture 3: Fourier Transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform and inverse Fourier transform direct inverse A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform and inverse Fourier transform direct from the number domain to the frequency domain inverse A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform and inverse Fourier transform direct from the frequency domain to the number domain inverse A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in the interval [ 0, Τ ] Fourier transform in the interval (- , + ) inverse direct A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform Change of notation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform Change of notation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform Change of notation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying From Fourier series to Fourier transform inverse A. Dermanis direct Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ 3 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier series in a continuously increasing interval Τ ∞ The Fourier series expansion of a function in a continuously larger interval Τ, provides coefficients for continuously denser frequencies ωk. As the length of the interval Τ tends to infinity, the frequencies ωk tend to cover more and more from the set of the real values frequencies ( ) For an infinite interval Τ, i. e. for ( t ) the total real set of frequencies ( ) is required and from the Fourier series expansion we pass to the inverse Fourier transform discrete frequencies ωk wit step Δω = 2π / Τ A. Dermanis continuous frequencies - all possible values ( ) Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function complex form real form A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a complex function direct inverse Notation Usual (mathematically incorect) notation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function Complex function: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function Complex function: Real function: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function Complex function: Real function: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function Complex function: Real function: cosine transform A. Dermanis sine transform Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function Complex function: Real function: cosine transform A. Dermanis sine transform Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform of a real function even function odd function A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying |F (ω )| Fourier transform in polar form A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform in polar form amplitude spectrum phase spectrum polar form: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Fourier transform in polar form even A. Dermanis odd amplitude spectrum even function phase spectrum odd function Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Linearity A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Linearity Symmetry A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Linearity Symmetry Time translation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Linearity Symmetry Time translation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Linearity Symmetry Time translation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Phase translation A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Phase translation Modulation theorem A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem signal carrier frequency A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem signal carrier frequency modulated signal A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem (amplitude) spectrum of signal A. Dermanis (amplitude) spectrum of modulated signal Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem (amplitude) spectrum of signal A. Dermanis (amplitude) spectrum of modulated signal Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem (amplitude) spectrum of signal A. Dermanis (amplitude) spectrum of modulated signal Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Modulation theorem (amplitude) spectrum of signal A. Dermanis (amplitude) spectrum of modulated signal Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Change of time scale: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Change of time scale: Differentiation theorem with respect to time: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Change of time scale: Differentiation theorem with respect to time: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform Change of time scale: Differentiation theorem with respect to time: Differentiation theorem with respect to frequency: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) area =1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) area =1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) area =1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) area =1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) = average value of φ in the interval A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) = average value of φ in the interval A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) 1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) 1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The Dirac delta function δ(t) 1 Dirac delta function Heaviside step function 1 A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform involving the Dirac delta function δ(t) A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Properties of the Fourier transform involving the Dirac delta function δ(t) A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution definition: notation: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution definition: notation: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution definition: notation: property: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution Mathematical mapping: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution Mathematical mapping: The value g(t) of the function g for a particular t follows by multiplying each value f(s) of the function f with a factor (weight) h(t-s) which depends on the “distance” t-s between the particular t and the varying s (-∞<s<+∞). A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution Mathematical mapping: The value g(t) of the function g for a particular t follows by multiplying each value f(s) of the function f with a factor (weight) h(t-s) which depends on the “distance” t-s between the particular t and the varying s (-∞<s<+∞). Thus every value g(t) of the function g is a “weighted mean” of the function f(s) with weights h(t-s) determined by the function h(t). A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution area A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution area A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example area A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution - Example A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution properties A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying Convolution properties A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Convolution is replaced by a simple multiplication in the frequency domain ! A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying The convolution theorem Proof: A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying CONVOLUTION THEOREM for frequencies A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying CONVOLUTION THEOREM for frequencies A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying PARSEVAL THEOREM A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying PARSEVAL THEOREM Similar relation for Fourier series A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying PARSEVAL THEOREM Similar relation for Fourier series Comparison with A. Dermanis Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying END A. Dermanis Signals and Spectral Methods in Geoinformatics