complex.slides.printing.2

complex.slides.printing.2 - Review Complex Numbers Review...

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Unformatted text preview: Review: Complex Numbers Review: Complex Numbers CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Review: Complex Numbers Basics Complex Numbers A complex number is one of the form a + bi where i= a: real part b: imaginary part √ −1 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 Review: Complex Numbers Basics The Complex Plane Complex numbers can be thought of as points in the complex plane: Imaginary i -1 Real 1 -i Review: Complex Numbers Magnitude and Phase Magnitude and Phase The length is called the magnitude: |a + bi | = a2 + 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 ) Review: Complex Numbers Magnitude and Phase Magnitude and Phase in the Complex Plane Imaginary z i |z| ϕ(z) -1 Real 1 -i Review: Complex Numbers Complex Conjugates Complex Conjugates Complex number z : z = a + bi Its complex conjugate: z ∗ = a − bi The complex conjugate z ∗ has the same real part but opposite imaginary part, and the same magnitude but opposite phase. Review: Complex Numbers Complex Conjugates Complex Conjugates in the Complex Plane Imaginary z i -1 Real 1 -i z* Review: Complex Numbers Complex Conjugates Complex Conjugates Adding z + z ∗ , cancels the imaginary parts to leave a real number: (a + bi ) + (a − bi ) = 2a Multiplying z ∗ z ∗ gives the real number equal to |z |2 : (a + bi )(a − bi ) = a2 − (bi )2 = a2 + b 2 Review: Complex Numbers Complex Conjugates Linear Algebra with Complex Numbers The inner product of two complex-valued vectors involves multiplying each component of one of the vector not by the other but by the complex conjugate of the other: u [k ] v [k ]∗ u·v = k The length of a complex-valued vector is thus a real number: u 2 =u·u = u [k ] u [k ]∗ k Review: Complex Numbers Euler Notation Magnitudes and Phases - revisited Remember that under complex multiplication magnitudes multiply phases add We can do the same thing using exponents: (a1 eb1 )(a2 eb2 ) = a1 a2 e(b1 +b2 ) Let’s encode complex numbers using exponential notation to make it easier to work with magnitude and phase Review: Complex Numbers Euler Notation Euler’s Formula Euler’s formula uses exponential notation to encode complex numbers—uses i in the exponent to differentiate from real numbers Euler’s formula (definition): ei θ = cos θ + i sin θ ei θ is the vector with magnitude 1.0 and phase θ Any complex number z can be written as z = |z | ei φ(z ) Review: Complex Numbers Euler Notation Euler’s Formula: Graphical Interpretation ei θ Imaginary i eiθ θ -1 Real 1 -i Review: Complex Numbers Euler Notation Euler’s Formula: Graphical Interpretation z = |z |ei φ(z ) Imaginary z i |z| ϕ(z) -1 Real 1 -i Review: Complex Numbers Euler Notation Euler’s Formula: Application What is (2 + 2i )(−3 + 3i )? Suppose that we already have these numbers in magnitude-phase notation: √ √ |−3 + 3i | = 3 2 |2 + 2i | = 2 2 π φ (2 + 2√ = π i) 4 φ (−3 + 3√ = 34 i) 2 + 2i = 2 2 ei π/4 −3 + 3i = 3 2 ei 3π/4 (2 + 2i )(−3 + 3i ) √ 2 2 ei π/4 = = 12 ei π = −12 Review: Complex Numbers Euler Notation Powers of Complex Numbers Suppose that we take a complex number z = |z | ei φ(z ) and raise it to to some power n: zn = |z | ei φ(z ) n = |z |n einφ(z ) z n has magnitude |z |n and phase n [φ (z )]. √ 3 2 ei 3π/4 Review: Complex Numbers Powers of Complex Numbers Powers of Complex Numbers: Example What is i n for various n? Imaginary i = ei π/2 i 0 = ei 0 = 1 i 1 = ei π/2 = i i 2 = ei 2π/2 = −1 i 3 = ei 3π/2 = −i i 4 = ei 4π/2 = 1 . . . i -1 Real 1 -i Review: Complex Numbers Powers of Complex Numbers Powers of Complex Numbers: Example What is ei π/4 n for various n? Imaginary ei π/4 0 ei π/4 1 ei π/4 2 ei π/4 3 i -1 Real 4 ei π/4 1 -i Review: Complex Numbers Summary Summary: Complex Numbers Can represent in (real,imaginary) Cartesian form Can represent in (magnitude,phase) polar form Magnitude = distance from 0 (same idea as absolute value) Phase = angle with the real axis Euler’s theorem: exponential notation for (magnitude,phase) ei θ = cos θ + i sin θ z = |z | ei φ(z ) Complex conjugate: z ∗ = a − bi = |z | e−i φ(z ) Raising a complex number to a power: n z n has magnitude |z | and phase n [φ (z )] ...
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