Answer Happy

3. (Fourier Transform - 6 points) The Fourier and inverse Fourier transform can be used to transform information between time and frequency domain representations. Figure 1 below illustrates a few examples of this transformation for a few common waveforms, including the corresponding Fourier coefficients. In Fourier analysis a time domain signal can be decomposed into a series of its constituent sinusoidal functions as described by: f(t) = ao + En=1 ancos(nw, t) + bn sin(nwot) where where wo is the angular frequency, i.e. Wo = 21/T where T is the period of f(t). Using the information in Figure 1, report the first four terms (i.e. for n=0 to n=3) of the Fourier decomposition for the time-domain signal presented in Figure 2; i.e., present the series of sine and cosine functions that would reconstruct the original time-domain signal.

Time Domain Frequency Domain a. Pulse 20 = Ad T dk/T 4, = 2A sin (nad) be = 0 (d - 0.27 in this example) Of 2f3f4f5f6f b. Square = 0 Inna À DOUD OLI 0 (all even harmonies are zero) of 2f3f4f5f6f c. Triangle ag = 0 47 - 44 À VVVV (191) bi-O (all even harmonies are zero) t=0 O f 2f3f4f5f6f Figure 1 Time and frequency domain representations of some common waveforms along with the Fourier coefficients based on the functional result of the Fourier transform. "All even harmonics are zero" means that a, = 0 when n is even. -4 -2 Figure 2 Time domain triangle wave where the y-axis is measured in units of Volts and the x-axis is measured in units of microseconds