Fourier Transform Applications and Vector Orthogonality

interest of the Fourier transform
• Electronics
• Signal Theory
• Telecommunications
• Optics
• Acoustics
• Radar
• Image processing
convolution product of discrete FFT (Fast FourierTransform) => time ? N log N
-Needed a FFTper each of the two images on which you must do the convolution product. And antitransformadaa the acabarde make the product. In total, we need 3 Fast Fourier transform.
• Image Space => time NN ?
space filtering of specific frequencies freqüè preferences eliminacióde
• Periodic measurement of elements
• We will say that two vectors, orthogonal vectors are orthogonal if their inner product is zero: (u, v) = 0
• The domestic product, in this case is defined as
?Uv = 0
domestic product or domestic product dot product of two vectors in a vector space is a operaciódonada by: V x V -> K (where V is the vector space and K is the body on which estàdefinit, which must comply properties: [ax + by, z] = a [x, z] + b [y, z] .. [x, y] = [y, x] Hermitian .. [x, x] ? 0 positive definite (where x, y and z are arbitrary vectors and b are scalars)

When a set of orthogonal basis vectors (a) .. U1u2 that satisfies (u, UJ) = 0 and for any j ? I, j = 1 ,..., n
states that the elements of this set are mutually orthogonal and form an orthogonal basis in space R n. Any vector space wd'aquest can be expressed as a linear combination of the basis w = a1u1 + a2u2 +...+ Anuns
• The concept of orthogonal functions orthogonality can be extended to sets of functions. We will say that members of a set of functions S = (f1 (t), f2 (t) ... Fn (t )...}
form an orthogonal set on the interval to <t <b if ? Fn (t) fm (t) dt = 0 if n ? M
In particular orthogonal basis functions under the transformations Fourierens particularly interested in a set of mutually orthogonal functions. S (... I3wot-e ^ e ^ q) ...-i2wot meet t / To eînwot ? ... Dt = 0 if n ? M 1 if n = m
sè estuaries of functions Fourierper periodi Sea dikesf (t) T or a funcióperiòdica period. Then we can express as an infinite sum of complex exponentials of the form: (perquès'ha seen that this is a basic feature set) The terms F (n) No sound coefficients Fourier?