2-Complex Numbers_2

2-Complex Numbers_2 - Complex Numbers in Communications...

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Unformatted text preview: Complex Numbers in Communications Engineering 1 − = = i j Euler’s Formula: θ θ θ sin cos j e j ± = ± Proof: Substitute complex arguments into a real-valued Taylor Series expansion (why?): .... ! 7 ! 6 ! 5 ! 4 ! 3 ! 2 1 .... ! 7 ! 5 ! 3 ) sin( .... ! 6 ! 4 ! 2 1 ) cos( 7 6 5 4 3 2 7 5 3 6 4 2 + − ± + − ± = + ± ± = ± + − + − = ± θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ j j j j e j j j j j j m m m m Useful Related Expressions: 2 cos θ θ θ j j e e − + = , j e e j j 2 sin θ θ θ − − = Principal Nth root of a complex number, ( ) θ θ sin cos j r z + = : ( ) [ ] ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + = N k j N k r j r z N N N π θ π θ θ θ 2 sin 2 cos sin cos / 1 / 1 / 1 for k = 0,1,2,…N-1 Polar Circle Z-Plane Real Imaginary r Nth Roots of z=re j θ θ Polar Circle Z-Plane Real Imaginary r Nth Roots of z=re j θ θ Principal Nth root of a complex number ( N = 16 for this example) Magnitude and Phase of a Complex Phasor (number)...
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2-Complex Numbers_2 - Complex Numbers in Communications...

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