00B_BME343_Background

00B_BME343_Background - Mathematical Preliminaries(Chapter...

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Background B-1 Mathematical Preliminaries (Chapter B) • Complex numbers • Sinusoids • Sketching signals (also MATLAB) • Cramer’s rule • Partial fraction expansion • Vectors and matrices • Miscellaneous (on your own) • MATLAB (laboratories will be 70% MATLAB)
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Background B-2 Complex numbers Algebra (al-jabra) introduced from al-Khwarizmi’s book into Italy through Gerard and Fibbonaci Some Italian mathematicians developed an algebraic solution to the cubic equation. For some equations with real solutions, the form of the equations contained square roots of negative numbers. Mathematicians initially used to obtain correct real numbers. Rene’ Descarte introduced the word “imaginary” because in his mind the numbers did not represent anything “real” By the 17 th century, a geometry of complex numbers was worked out by Wallis and was used by Newton and others. Complex numbers were subsequently developed by Euler, Gauss, Cauchy and others John Wallis (1616-1703) One of the first to provide a geometric interpretation for complex 1. Leonhard Euler (1707-1783) Famous Swiss physicist and mathematician
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Background B-3 • Complex numbers are an extension of ordinary numbers • Complex plane (Cartesian and polar coordinates) (Algebra of) Complex numbers Re Im z aj b z a z b =+ = = cos sin ar br θ = = (cos sin ) za j j = + Real Imaginary z r a -b b θ z*
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Background B-4 • The Euler formula • Proof: Maclaurin series to expand the terms Algebra of Complex numbers cos sin j ej θ =+ () 23456 23 45 6 246 35 1 ... 2! 3! 4! 5! 6! 1 ... cos 1 ... sin ... 3! 5! j jjjjj jj j θθθθθ θθ θθθ = + ++++++ = =+ − − + + − + =− + − + =− +
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Background B-5 Algebra of Complex numbers cos sin ar br θ = = 22 1 tan ra b b a =+ ⎛⎞ = ⎜⎟ ⎝⎠ Real Imaginary z r a b θ z r z = = j z z ze = 111 1 j jz j ee zr e r z −∠ == = j z a j e =
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Background B-6 We define z*, the conjugate of z, as To find the conjugate of any number, replace (j) by (–j) in that number A few formulas:
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00B_BME343_Background - Mathematical Preliminaries(Chapter...

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