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complex.slides.printing

# complex.slides.printing - Review Complex Numbers Review...

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Review: Complex Numbers Review: Complex Numbers CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science

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Review: Complex Numbers Basics Complex Numbers A complex number is one of the form a + bi where i = - 1 a : real part b : imaginary part
Review: Complex Numbers Basics Complex Arithmetic When you add two complex numbers, the real parts and imaginary parts add independently: ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i When you multiply two complex numbers, you cross-multiply them like you would polynomials: ( a + bi ) * ( c + di ) = ac + a ( di ) + ( bi ) c + ( bi )( di ) = ac + ( ad + bc ) i + ( bd )( i 2 ) = ac + ( ad + bc ) i - bd = ( ac - bd ) + ( ad + bc ) i

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Review: Complex Numbers Basics The Complex Plane Complex numbers can be thought of as points in the complex plane: Imaginary Real i 1 -i -1
Review: Complex Numbers Magnitude and Phase Magnitude and Phase The length is called the magnitude : | a + bi | = a 2 + b 2 The angle from the real-number axis is called the phase : φ ( a + bi ) = tan - 1 b a When you multiply two complex numbers, their magnitudes multiply: | xy | = | x || y | and their phases add: φ ( xy ) = φ ( x ) + φ ( y )

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Review: Complex Numbers Magnitude and Phase Magnitude and Phase in the Complex Plane Imaginary Real i 1 -i -1 z |z| ϕ (z)
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