EE200_Weber_9-16

EE200_Weber_9-16 - EE 200 Fourier Series Coefficients The...

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13 EE 200 Fourier Series Coefficients The coefficients for the continuous time Fourier series can be calculated from the expression for the periodic function. Each coefficient is the result of multiplying x by a complex exponential and then averaging the product over one period. a k = 1 T 0 x ( t ) e " j (2 # / T 0 ) kt 0 T 0 $ dt a 0 = 1 T 0 x ( t ) 0 T 0 $ dt
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14 EE 200 Fourier Series Coefficients We can get some insight as to why this works by calculating the coefficients of x(t) where x(t) is written in the form of its Fourier series expansion a m = 1 T 0 x ( t ) e " j (2 # / T 0 ) mt dt 0 T 0 $ = 1 T 0 a k e j (2 / T 0 ) kt k = "% % ( ) * + , 0 T 0 $ e " j (2 / T 0 ) mt dt = 1 T 0 a k k = "% % e j (2 / T 0 ) kt e " j (2 / T 0 ) mt dt 0 T 0 $ = 1 T 0 a k k = "% % e j (2 / T 0 )( k " m ) t dt 0 T 0 $
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15 EE 200 Fourier Series Coefficients In the expression below, the k index on the summation varies over all values. For one term in the summation k=m , and that term evaluates to a m For all other values of k , the k-m
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This note was uploaded on 10/13/2009 for the course EE 200 at USC.

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EE200_Weber_9-16 - EE 200 Fourier Series Coefficients The...

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