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# HW1 (S 2007) - Solve the following equation for a complex...

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ELCT 321 Digital Signal Processing Homework Assignment # 1 (Reviews of Complex Number) by Dr. Yong-June Shin Assigned: January 23, 2007 Due: January 30, 2007 1. (10 Points) Let us deﬁne “Hyperbolic Cosine” and “Hyperbolic Sine” functions as follows: cosh( x ) = 1 2 ( e x + e - x ) sinh( x ) = 1 2 ( e x - e - x ) Using the Euler’s formula determine values of following expressions: cosh( jπ/ 2) = ? sinh( - jπ/ 4) = ? 2. (20 Points) Evaluate the following by reducing the answer to rectan- gular form: ( - j ) - 1 / 2 (ﬁnd two answers) (Hint: - j = e j 2 πk · e - jπ/ 2 where k is an integer) 3. (10 Points) Simplify the following complex-valued expressions: ={ je jπ/ 6 } (Note: The symbol = stands for “imaginary” part of complex number. For example, = (1 - j ) = - 1) 1

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4. (20 Points)
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Unformatted text preview: Solve the following equation for a complex variable z : z 4 =-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) 5. (20 Points) Evaluate following complex number: (1 + j ) 2007 (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. (20 Points) By use of Euler’s formula show that: Z e ax sin bxdx = 1 a 2 + b 2 · e ax · ( a sin bx-b cos bx ) Z e ax cos bxdx = 1 a 2 + b 2 · e ax · ( a cos bx + b sin bx ) 2...
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HW1 (S 2007) - Solve the following equation for a complex...

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