Homework1_SOL_Summer.docx - B Olson ECE 3101 Homework SOLUTIONS cos mω t sin nω t 0 and 0(m and n integers 1 Verify that the following basis functions

Homework1_SOL_Summer.docx - B Olson ECE 3101 Homework...

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B. Olson ECE 3101 Homework - SOLUTIONS 1) ) Verify that the following basis functions are orthonormal: cos ( 0 t ) and sin ( 0 t ) (m and n integers) by computing the inner products given below. You may choose t 0 to simplify your calculations. 2 T to to + T sin ( 0 t ) cos ( 0 t ) dt = 0 2 T to to + T sin ( 0 t ) sin ( 0 t ) dt = 1 2 T to to + T sin ( 0 t ) sin ( 0 t ) dt = 0 ( m n ) 2 T T / 2 T / 2 sin ( 0 t ) cos ( 0 t ) dt = 0 odd × even = odd 2 T T / 2 T / 2 sin ( 0 t ) sin ( 0 t ) dt = 4 T 0 T / 2 sin 2 ( 0 t ) dt = sin 2 A = 1 2 1 2 cos2 A = 4 T 1 2 0 T / 2 dt 4 T 1 2 0 T / 2 cos ( 2 0 t ) dt = 2 T ( T 2 ) 4 T 1 2 1 2 0 sin ( 0 t ) | t = 0 T / 2 = 1 [ 0 0 ] = 1 2 T to to + T sin ( 0 t ) sin ( 0 t ) dt = 2 T T / 2 T / 2 sin ( 0 t ) sin ( 0 t ) dt = 4 T 0 T / 2 sin ( 0 t ) sin ( 0 t ) dt = sin A sin B = 1 2 ( cos ( A B ) cos ( A + B ) ) = 4 T 1 2 0 T / 2 cos ( ( n m ) ω 0 t ) dt 4 T 1 2 0 T / 2 cos ( ( n + m ) ω 0 t ) dt = 2 T ( 1 ( n m ) ω 0 ) sin ( ( n m ) ω 0 t ) | t = 0 T / 2 2 T ( 1 ( n + m ) ω 0 ) sin ( ( n + m ) ω 0 t ) | t = 0 T / 2 = 2 T ( 1 ( n m ) ω 0 ) [ sin ( ( n m ) 2 π T T 2 ) sin ( 0 ) ] 2 T ( 1 ( n + m ) ω 0 ) [ sin ( ( n + m ) 2 π T T 2 ) sin ( 0 ) ] = 2 T ( 1 ( n m ) ω 0 ) [ sin ( π ( n m ) ) 0 ] 2 T ( 1 ( n + m ) ω 0 ) [ sin ( π ( n + m ) ) 0 ] = 2 T ( 1 ( n m ) ω 0 ) [ 0 0 ] 2 T ( 1 ( n + m ) ω 0 ) [ 0 0 ] = 0 ( n and m are integers )
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2) Indicate if the following functions exhibit even or odd symmetry or neither: a) odd b) even c) even x even = even d) even + even = even e) odd x odd = even f) odd x odd = even g) even x odd = odd h) even + odd (neither even or odd) 3) By inspection, indicate which integrals would evaluate to be zero: a) Non-zero b) Zero c) Zero if to= 0, otherwise non-zero d) Non-zero e) Zero f) Zero 4) Determine the Fourier Coefficients a) By inspection: a v = 5, a 3 = ½, b 7 = -6 b) By inspection: a v = 7, a 5 = 4, b 8 = 8 c) cos 2 (5 w 0 t) = =1/2 + ½ cos(10 w 0 t) : a v = 1/2, a 10 = ½ d) cos(5 w 0 t + /6) = cos(5 w 0 t )cos( /6) - sin(5 w 0 t )sin( /6) cos( /6) = 3 / 2 and sin( /6) = 1 / 2 cos(5 w 0 t + /6) =( 2 3 / ¿¿ cos(5 w 0 t ) - 1 ¿ 2 ¿ ¿ sin(5 w 0 t ) , a 5 =( 2 3 / ¿¿ , b 5 = 1/2 5) a) The function is even, thus bn coefficients would be zero b) The function is odd, thus a n coefficients would zero including av
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6) For the periodic functions shown below, indicate the Fourier coefficients that would be zero. 7) t f(t) T/2 -T/2 t f(t) T/2 -T/2 t f(t) -T/2 +T/2 The function is odd and has quarter wave symmetry: av = 0 (average value is zero), ak = 0, bk = 0 (k – even) The function can be viewed as the sum of an odd function with quarter wave symmetry plus a DC offset av ≠ 0 (average value is NOT zero), ak = 0, bk = 0 (k – even) The function has half wave symmetry. av = 0 (average value is zero), ak = 0 (k – even), bk = 0 (k – even)
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7) For the following function determine Fourier Coefficients: a v , and a k and b k for the first three non-zero terms. Write the corresponding Fourier Series describing the function. Note the k = 1 term is tricky; consider using L’hopital’s rule when (k-1) = 0.
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