# A#06 - Assignment#6 ECSE-2410 Signals Systems Spring 2006...

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Assignment #6 ECSE-2410 Signals & Systems - Spring 2006 Due Tue 09/19/06 1(10). (a) Write the complex expression, ( 29 3 1 ) 1 ( j j j z + + = in polar form , i.e., ) arg( z j e z z = . Express arg ( z ) as p π radians, where p is a fraction. (b) Write the complex number, ( 29 + + = - 2 4 1 1 π π j j e e j j z , in rectangular form, i.e., { } { } z m j z e z + = . 1(12). Given j j z + + = 1 2 , calculate the following: (a) + z z (b) { } z e 2 (c) z z (d) 2 z 3(8). One period of a periodic signal, ) ( 1 t x , is given as < < - < < + - < < - < - = 5 4 , 1 4 2 , 4 2 1 , 2 1 0 , 1 ) ( ~ 1 t t t t t t t x . Sketch ) ( 1 t x for several cycles and find its average value. 4(35). For the periodic signal shown, k a t x ) ( 1 ... ... t 2 1 - 0 2 1 2 (a)(10) Find the exponential Fourier series coefficients, k a . (b)(10)Use equation 3.31 in text to express x ( t ) as a Fourier series in the trigonometric form = + = 1 0 0 ) cos( 2 ) ( k k t k A a t x ϖ (c)(10)Evaluate the first five harmonics and write the series in (b) in expanded form, i.e., ) 5 cos( 2 ... ) 2 cos( 2 ) cos( 2 ) ( 0 5 0 2 0 1 0 t A t A t A a t x ϖ ϖ ϖ + + + + = Note. Some harmonic terms may be zero.

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