ComplexReview_handouts

# ComplexReview_handouts - Complex Numbers Review Complex...

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Complex Numbers Review Dr. Deepa Kundur Complex Numbers Review 1 / 12 Complex Numbers Review Complex Exponential Function Continuous-time : x ( t ) = A e j ( ω t + φ ) Recall, Euler’s relation: e j θ = cos θ + j sin θ = A (cos( ω t + φ ) + j sin( ω t + φ )) = A cos( ω t + φ ) + j A sin( ω t + φ ) complex function Discrete-time : x [ n ] = A e j ( Ω n + φ ) = A (cos(Ω n + φ ) + j sin(Ω n + φ )) = A cos(Ω n + φ ) + j A sin(Ω n + φ ) complex function The complex exponential function has similar properties to sinusoids – e.g., periodicity Complex Numbers Review 2 / 12 Complex Numbers Review Complex Numbers and the Quadratic Equation I complex numbers are natural solutions to: aw 2 + bw + c = 0 where b 2 - 4 ac < 0 w = - b ± b 2 - 4 ac 2 a I we deﬁne j , - 1 Complex Numbers Review 3 / 12 Complex Numbers Review Complex Numbers and Coordinate Systems

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ComplexReview_handouts - Complex Numbers Review Complex...

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