Question

3. Derive the following formula for the complex Fourier Series rep- resentation of a full wave rectified sine wave of unit amplitude. Remember that the period of the rectified signal is half of that of the sine wave +00 2π

Answer 1

Full wave rectified sine wave x(t) = sin t: 0 < t < pi time period (T) = pi, f_0 = 1/T = 1/pi rightarrow By using exponential Fourier series x[n] = -1/T integral^T_0 x(t) e^- j 2 pi F_0 n t dt = 1/pi integral^pi_0 sin t e^- j 2 pi 1/pi n t dt = 1/pi integral^pi_0 sin t e^- j 2 nt dt we know that integral e^a t sin b t d t = e^a t/a^^= 2 + b^2 [a sin b t - b cos b t] where a = -j 24, b = 1 x[n] = 1/pi[e^- j 2 n t /(- j 2 n)^2 + 10[- j 2 n sin t - cos t]]^pi_0 = 1/pi [1 - 4 x^2] [e^- j 21 pi (0 + 1) - e^(0 - 1)] = 1/ pi [1 - 4 n^2] [e^- j 2 n pi + 1] \r\n x[n] = 1/pi [1 - 4 n^2] [1 + e^- j 2 n pi] = 1/pi [1 - 4 n^2] [1 + cos 2 pi - j sin 2 n pi]