Use the Fourier integral to determine all the Fourier series coefficients of the "sine...

Use the Fourier integral to determine all the Fourier series coefficients of the "sine-cubed" signal. In other words, evaluate the integral

for all k.

Hints: Find the period first, so that the integration interval is known. In addition, you might find it easier to convert the sin3(•) function to exponential form (via the inverse Euler formula for sin(•)) before doing the Fourier integral on each of four different terms. If you then invoke the orthogonality property on each integral, you should get exactly the same answer as (3.29).

Fig. 3: Spectrum of the "sine-cubed" signal derived from its Fourier series coefficients. Only the range from –5 to +5Hz is shown. The complex amplitude of each spectrum line is equal to the Fourier series coefficient ak for that frequency, kf0.