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Homework1

# Homework1 - < e stands for “real” part of a complex...

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ELCT 321 Digital Signal Processing Homework Assignment # 1 (Reviews of Complex Number) by Dr. Yong-June Shin Assigned: August 23, 2011 Due: August 29, 2011 1. (20 Points) Express each of following complex numbers in Cartesian form and plot them in the complex plane, indicating the real and imag- inary parts of each number. (a) - 3 j · (1+ j ) 2 1 - j (b) 1 2 · e - j 27 π/ 4 (c) 3 e j 1500 π + 3 e j 1701 π (d) 6 · e jπ/ 3 1+ j 2. (10 Points) Simplify the following complex-valued expressions: ={ j · e jπ/ 6 } (Note: The symbol = stands for “imaginary” part of complex number. For example, = (1 - j ) = - 1) 3. (20 Points) Solve the following equation for a complex variable z : z 4 + 1 = j (Be sure to find all possible answers, and express your answers in polar form and plot all the solutions on the complex plane with magnitude and angle) 4. (10 Points) Solve the following equation for θ : < e { (1 + j ) e } = - 1 (Note: The symbol

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Unformatted text preview: < e stands for “real” part of a complex number) 1 5. (10 Points) Determine real and imaginary parts of following complex number: (1 + j )-2011 (Hint: Use following formula and polar form representation of the com-plex number, i.e., 1 √ 2 (1 + j ) = e jπ/ 4 ) (cos θ + j sin θ ) n = cos nθ + j sin nθ 6. (30 Points) By use of Euler’s formula show that: (a) cos θ = 1-θ 2 2! + θ 4 4!-θ 6 6! +- - - - - - - - - - - - -∞ sin θ = θ-θ 3 3! + θ 5 3!-θ 7 7! +- - - - - - - - - - - - -∞ (b) 1 . Z e aθ sin bθ · dθ = e aθ a 2 + b 2 · ( a sin bθ-b cos bθ ) 2 . Z e aθ cos bθ · dθ = e aθ a 2 + b 2 · ( a cos bθ + b sin bθ ) (c) cos( A + B ) = cos A · cos B-sin A · sin B (Hint: e jθ = cos θ + j sin θ ) 2...
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