BC:8.2 The periodic waveform, g(t), shown below is a...

BC:8.2 The periodic waveform, g(t), shown below is a halfwave rectified sine where only the positive parts of the sine are nonzero, and the signal is zero whenever the sine would be negative. Note the time units on the graph are milliseconds. Use the definition of the Fourier series to determine a formula for the (possibly complex) coefficients, a_k, for the exponential Fourier Series representation of the function. g(t) = sum_{k=-infty}^{infty} a_k expleft(jfrac{2pi}{T_0}kt ight) Waveform for Problem BC:8.2 (sin(250pi t)) g(t) [figure showing half-wave rectified sine with positive half-cycles on intervals 0–4 ms, 8–12 ms, etc.; time ticks at 0, 4, 8, 12 ms] Determine a formula for the coefficients, a_k, using the definition integral. (Hint: it is recommended to treat a_0 as a special case, and it is useful to expand the sine using the Euler identity for other values of a_k. If any a_k is 0/0 use L'Hôpital's rule to evaluate for that value of k.) Evaluate the coefficients a_0, a_1, a_{-1}, a_2 and a_{-2}. If the values are complex, express them in polar form with the angle in degrees.

Transcript

00:01 So here the solution for this problem is as from the given equation which is j of t is equal to sigma k equals to infinity which is negative infinity to positive infinity of alpha of k multiplied with xp of j multiplied with 2 pi divided by t not multiplied with k multiplied with k multiplied with t.

00:27 So this will come as sigma k equals to negative infinity to positive infinity of alpha of k multiplied with e power j multiplied with 2 pi divided by t0 multiplied with k t so we can write it as alpha of k is equals to 1 divided by t not multiplied with integral t not of g of t multiplied with e power negative j 2 pi divided by t not multiplied with kt multiplied with d t so we'll get the equation as alpha of k is equal to 1 divided by 8 multiplied with 10 power negative 3 multiplied with integral 0 to 4 multiplied with 10 power negative 3 of sine multiplied with 250 pi multiplied with t whole multiplied with e power negative j multiplied with 2 pi divided by t not multiplied with k t multiplied with d t so this will come as alpha of k is equals to 1 divided by 8 multiplied with 10 power negative 3 multiplied with integral 0 to 4 multiplied with 10 20 negative 3 of e power d multiplied with 250 pi multiplied with t minus e power negative j multiplied with 250 pi multiplied with t whole multiplied with e power negative j multiplied with 2 pi divided by t0 multiplied with k t whole multiplied with d t so this will be equals to alpha of k is equals to 1 divided by 16 j multiplied with 10 power negative 3 integral 0 to 4 multiplied with 10 power negative 3 multiplied with e power negative j multiplied with 250 pi whole multiplied with t multiplied with e power negative j multiplied with 250 pi multiplied with t whole multiplied with d t plus integral 0 to 4 multiplied with 10 negative 3 of e power negative j multiplied with 250 multiplied with t multiplied with e power negative j multiplied with 250 pi multiplied with k t whole multiplied with d t so by simplifying this further we'll get the equation as alpha of k is equals to 1 divided by 16 j multiplied with 10 power negative 3 whole multiplied with e power j j power j j multiplied with e power j multiplied with 250 pi plus 1 minus k divided by j multiplied with 250 pi integral of 0 to 4 multiplied with 10 power negative 3 whole minus e power negative j multiplied with 250 pi plus 1 plus k divided by minus j multiplied with 250 pi multiplied with 1 plus k integral 0 to 4 multiplied with 10 power negative 3 so by simplifying this we'll get the equation as alpha of k is equals to 1 divided by 16 j multiplied with 10 power negative 3 multiplied with e power j pi multiplied with 1 minus k whole minus 1 divided by j multiplied with 250 pi multiplied with 1 minus k whole minus e multiplied with 1 minus k whole minus 1 divided by minus j multiplied with 250 pi multiplied with 1 plus k so here we got the 5 of k as like this now the 5 of 0 will be equals to 1 divided by t not multiplied with integral 0 to t0 of g of t multiplied with d t so this will be equals to 1 divided by 8 multiplied with 10 0 x 0 to 4 multiplied with 10 power negative 3 of sine of 250 pi multiplied with t whole multiplied with d t so we will get the alpha of 0 0 as 1 divided by pi and we will get the 5 of 1 as 1 divided by 16 multiplied with j multiplied with 10 negative 3 multiplied with e power j pi multiplied with 1 minus 1 whole minus 1 divided by j multiplied with 250 pi multiplied with 1 minus 1 whole minus e power negative j pi multiplied with 1 minus 1 plus 1 whole minus 1 divided by minus j multiplied with 250 pi multiplied with 1 plus 1 so by simplifying this we'll get the alpha 1 as equals to j divided by 4 or negative j divided by 4 which means 0 .25 of 90 degrees or 0 .25 of 90 degrees or 0.

07:38 25 of minus 90 degrees...

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