Fundamental Frequency and Period Review

Fundamental Frequency and Period Review - ECE 3337 HOMEWORK...

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Unformatted text preview: ECE 3337 HOMEWORK 1 1) ZTF problem 1.9 (b) & (d) What are the fundamental periods of the signals given below? (Assume units of seconds for the t-variable) (b) x(t) = cos 60 t The fundamental period of x(t) is the smallest number T for which the following equation holds: x(t+T) = x(t) For this particular case: [ ] ( 29 T 60 t 60 cos ) T t ( 60 cos ) T t ( x + = + = + which, according to the trigonometric identity B sin A sin B cos A cos ) B A cos( = , is equal to: ) T 60 sin( ) t 60 sin( ) T 60 cos( ) t 60 cos( ) T t ( x - = + In order x(t+T) = x(t) , it must be truth that: 1 T 60 cos = and T 60 sin = which are truth if s 30 1 s 60 2 T 2 T 60 = = = (d) x(t) = sin 50 t + cos 60 t The fundamental period of t 50 sin x 1 = is s 25 1 T 1 = The fundamental period of t 60 cos x 2 = is s 30 1 T 2 = 2 1 x x + is periodic because 1 2 2 1 n n 5 6 25 30 30 1 25 1 T T = = = = (Eq. 1) is a rational number. If we call T o the fundamental period of 2 1 x x + , then: o 1 1 T n 1 T = (Eq. 2) and o 2 2 T n 1 T = (Eq. 3) Either Eq. 2 or Eq. 3 can be solved to obtain T o , with n 1 and n 2 defined by Eq. 1. From Eq. 2 we obtain: s 2 . s 5 1 s 25 5 T T 5 1 25 1 o o = = = = The same result must be obtained from Eq. 3. 2) ZTF problem 1.10 (a), (c) & (e) Given the two complex numbers A=3+j3 and B=10exp(j /3). (a) Put A into polar form and B into Cartesian form. What is the magnitude of A? The argument of A ? The real part of B ? The imaginary part of B ? Re(A)=3 Im(A)=3 [ ] [ ] 2 3 18 3 3 ) A Im( ) A Re( A 2 2 2 2 = = + = + = [ ] 4 1 tan ) A Re( ) A Im( tan ) A arg( 1 1 = = =-- 10 B = 3 ) B arg( = [ ] ( 29 5 3 cos 10 ) B arg( cos B ) B Re( = = = [ ] ( 29 3 5 3 sin 10 ) B arg( cos B ) B Im( = = = (b) Compute their difference. Show as a vector in the complex plane. ( 29 ( 29 6603 . 5 j 2 3 5 3 j 2 3 5 j 5 3 j 3 B A-- =- +- = +- + =--2-1 1 2 3 4 5-6-4-2 2 4 6 8 10 A B A-B (d) Compute their quotient A/B in two ways: by dividing with both numbers expressed in Cartesian form and by dividing with both numbers expressed in polar form. Show that both answers are equivalent. CARTESIAN FORM ( 29 ( 29 ( 29 j0.1098- 0.4098 100 3 15 15 100 3 15 15 3 5 5 3 15 15 j 3 15 15 3 5 j 5 3 5 j 5 3 5 j 5 3 j 3 B A 2 2 =- + + = = +- + + =-- + + = POLAR FORM 12 j 3 4 j 3 j 4 j e 2 3 . e 10 2 3 e 10 e 2 3 B A - - = = = which can be expressed in cartesian form as: 4098 . 12 cos 2 3 . B A Re = - = 1098 . 12 sin 2 3 . B A Im- = - = 3) ZTF problem 1.11 (a), (c) & (e) Find the periods and the fundamental frequencies of the following signals: (a) ( 29 6 t 10 cos 2 ) t ( x a + = ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 T 10 sin 6 t 10 2sin T 10 cos 6 t 10 2cos T 10 6 t 10 cos 2 6 T t 10 cos 2 ) T t ( x a + - + = = + +...
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Fundamental Frequency and Period Review - ECE 3337 HOMEWORK...

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