Question

Find the complex exponential Fourier series coefficients for:
a. x(t)=3cos(4w0t)
b. x(t)=sin^2(t)
2. Find the DTFT of:
x(n)={1 0<=n<=3
0 otherwise}

Answer 1

a. Find the complex and exponential form:

$$x(t)=3\text{cos}(4w_0t)\iff0+j\frac{3}{\sqrt{2}}\iff\frac{3}{\sqrt{2}}e^{j90^\circ}.$$

Fourier series coefficients from -π to π:

$$a_0=\frac{\int^\pi_{-\pi}\text{cos}(4\omega_0t)dx}{\pi},\\ a_n=\frac{\int^\pi_{-\pi}\text{cos}(4\omega_0t)\text{cos}(tk)dx}{\pi},\\ b_n=\frac{\int^\pi_{-\pi}\text{sin}(tk)\text{cos}(4\omega_0t)dx}{\pi},\\ k=1,2,3,...$$

b. Find the complex and exponential form:

$$x(t)=\text{sin}^2(t)=\frac{1}{2}-\frac{\text{cos(2t)}}{2}\iff\frac{1}{2}-\frac{j}{2\sqrt2}\iff\\\iff\frac{1}{2}-\frac{e^{j90^\circ}}{2\sqrt2}.$$

Fourier series coefficients from -π to π:

$$a_0=\frac{\int^\pi_{-\pi}\text{sin}^2(t)dx}{\pi},\\ a_n=\frac{\int^\pi_{-\pi}\text{sin}^2(t)\text{cos}(tk)dx}{\pi},\\ b_n=\frac{\int^\pi_{-\pi}\text{sin}^2(t)\text{sin}(tk)dx}{\pi},\\ k=1,2,3,...$$

2. Find the DTFT of

$$x(n)=\begin{Bmatrix} 1,& \text{if }1\leq n\leq3 \\ 0&\text{otherwise} \end{Bmatrix}$$

$$X(\Omega)=\sum^3_1e^{-j\Omega n}=e^{-3j\Omega}(e^{j\Omega}+e^{2j\Omega}+1).$$