Document Sample

Review: Complex Numbers Review: Complex Numbers Basics Complex Numbers Review: Complex Numbers A complex number is one of the form CS 450: Introduction to Digital Signal and Image Processing a + bi where √ Bryan Morse i= −1 BYU Computer Science a: real part b: imaginary part Review: Complex Numbers Review: Complex Numbers Basics Basics Complex Arithmetic The Complex Plane Complex numbers can be thought of as points in the When you add two complex numbers, the real parts and complex plane: imaginary parts add independently: (a + bi) + (c + di) = (a + c) + (b + d)i Imaginary When you multiply two complex numbers, you i cross-multiply them like you would polynomials: (a + bi) ∗ (c + di) = ac + a(di) + (bi)c + (bi)(di) -1 Real = ac + (ad + bc)i + (bd)(i 2 ) 1 = ac + (ad + bc)i − bd = (ac − bd) + (ad + bc)i -i Review: Complex Numbers Review: Complex Numbers Magnitude and Phase Magnitude and Phase Magnitude and Phase Magnitude and Phase in the Complex Plane The length is called the magnitude: Imaginary |a + bi| = a2 + b2 z |z| The angle from the real-number axis is called the phase: i b φ(a + bi) = tan−1 a -1 ϕ(z) Real 1 When you multiply two complex numbers, their magnitudes multiply: |xy| = |x||y| -i and their phases add: φ (xy) = φ (x) + φ (y ) Review: Complex Numbers Review: Complex Numbers Complex Conjugates Complex Conjugates Complex Conjugates Complex Conjugates in the Complex Plane Imaginary Complex number z: z z = a + bi i Its complex conjugate: z ∗ = a − bi -1 Real 1 The complex conjugate z∗ has the same real part but opposite imaginary part, and the same magnitude but opposite phase. -i z* Review: Complex Numbers Review: Complex Numbers Complex Conjugates Complex Conjugates Complex Conjugates Linear Algebra with Complex Numbers The inner product of two complex-valued vectors involves Adding z + z ∗ , cancels the imaginary parts to leave a real multiplying each component of one of the vector not by the number: other but by the complex conjugate of the other: (a + bi) + (a − bi) = 2a u·v = u[k] v [k]∗ Multiplying z ∗ z ∗ gives the real number equal to |z|2 : k (a + bi)(a − bi) = a2 − (bi)2 The length of a complex-valued vector is thus a real = a2 + b 2 number: u 2 =u·u = u[k ] u[k]∗ k Review: Complex Numbers Review: Complex Numbers Euler Notation Euler Notation Magnitudes and Phases - revisited Euler’s Formula Euler’s formula uses exponential notation to encode Remember that under complex multiplication complex numbers—uses i in the exponent to differentiate magnitudes multiply from real numbers phases add Euler’s formula (deﬁnition): We can do the same thing using exponents: eiθ = cos θ + i sin θ (a1 eb1 )(a2 eb2 ) = a1 a2 e(b1 +b2 ) eiθ is the vector with magnitude 1.0 and phase θ Let’s encode complex numbers using exponential notation Any complex number z can be written as to make it easier to work with magnitude and phase z = |z| eiφ(z) Review: Complex Numbers Review: Complex Numbers Euler Notation Euler Notation Euler’s Formula: Graphical Interpretation Euler’s Formula: Graphical Interpretation eiθ z = |z|eiφ(z) Imaginary Imaginary z |z| i i eiθ -1 θ Real -1 ϕ(z) Real 1 1 -i -i Review: Complex Numbers Review: Complex Numbers Euler Notation Euler Notation Euler’s Formula: Application Powers of Complex Numbers What is (2 + 2i)(−3 + 3i)? Suppose that we take a complex number Suppose that we already have these numbers in magnitude-phase notation: z = |z| eiφ(z) √ √ |2 + 2i| = 2 2 |−3 + 3i| = 3 2 and raise it to to some power n: φ (2 + 2i) = π √ 4 φ (−3 + 3i) = 3π √ 4 n iπ/4 −3 + 3i = 3 2 ei3π/4 2 + 2i = 2 2 e zn = |z| eiφ(z) = |z|n einφ(z) √ √ (2 + 2i)(−3 + 3i) = 2 2 eiπ/4 3 2 ei3π/4 z n has magnitude |z|n and phase n [φ (z)]. iπ = 12 e = −12 Review: Complex Numbers Review: Complex Numbers Powers of Complex Numbers Powers of Complex Numbers Powers of Complex Numbers: Example Powers of Complex Numbers: Example n What is i n for various n? What is eiπ/4 for various n? Imaginary Imaginary i= eiπ/2 0 eiπ/4 i 0 = ei0 = 1 i 1 i 1 = eiπ/2 = i eiπ/4 i 2 i 2 = ei2π/2 = −1 eiπ/4 i 3 = ei3π/2 = −i -1 Real eiπ/4 3 -1 1 Real i 4 = ei4π/2 = 1 4 1 . . eiπ/4 . -i -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) eiθ = cos θ + i sin θ z = |z| eiφ(z) Complex conjugate: z ∗ = a − bi = |z| e−iφ(z) Raising a complex number to a power: n z n has magnitude |z| and phase n [φ (z)]

DOCUMENT INFO

Shared By:

Categories:

Tags:
Complex Numbers, polar form, polar coordinates, imaginary part, Unit 7, DeMoivre's Theorem, Practice Problems, practice exam, Complex Plane, how to

Stats:

views: | 6 |

posted: | 7/13/2011 |

language: | English |

pages: | 4 |

OTHER DOCS BY ert634

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.