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2

OK, so your signal is described in one period, $T_0$, as $$x_{T_0}(t) = \left\{\begin{array}{ll} 0, & -T_0/2 < t \leq 0 \\ 1, & 0 < t \leq T_0/2 \end{array}\right.$$ Let's keep it that way (without any other shortcuts like $\prod$ or $\mathrm{rect(\cdot)}$ etc). One way to find the Fourier coefficients of the even and the odd part of a signal ...

1

Assuming that you want the Fourier Transform of this signal, you can easily obtain it by using a very useful property and a well known FT pair in continuous time. The pair is $$x(t) = e^{-at}u(t), \: \: a>0 \longleftrightarrow X(f) = \frac{1}{a+j2\pi f}$$ and the so-called duality property says that if $$x(t) \longleftrightarrow X(f)$$ is an FT pair, ...

3

If I understand correctly, you want to verify the energy calculation in the frequency domain by computing the energy as $$E_x=\int_{-\infty}^{\infty}|X(f)|^2df\tag{1}$$ with $$X(f)=\mathcal{F}\big\{x(t)\big\}=\frac{A}{A+i2\pi f}\tag{2}$$ From $(2)$ we get $$|X(f)|^2=\frac{A^2}{A^2+(2\pi f)^2}=\frac{1}{1+\left(\frac{2\pi f}{A}\right)^2}\tag{3}$$ With $(... 1 If you are interested one more step into transform theory, I suggest you to go through Exponential Fourier series where the basis functions are complex exponentials, unlike sine and cosine as in Trigonometric Fourier series. I will make an attempt to find the Fourier spectrum of$x(t) = A \cos (2\pi f_0 t)$using Exponential FS and its properties(let me ... 3 Sketching the signal always helps in such cases. Let's consider a cosine of period$T_0$, that is, of fundamental frequency equal to$f_0=1/T_0$. That is the signal on the left. As you can see on the right,$|A\cos(2\pi f_0 t)|$does not share the same period with$A\cos(2\pi f_0 t)$. The period of the signal you are looking for is$T_0^{\prime}=T_0/2$and ... 0 It may help you to look at each term separately if you are not used to the math as in: $$x[n] = 4\sum_{m=-\infty}^{\infty}\delta[n-4m] + 8\sum_{m=-\infty}^{\infty}\delta[n-1-4m]$$ Knowing that$\delta[x]\$ is only 1 when x=0, and 0 everywhere else, ask yourself what does n need to be to make those terms go to zero given that m can be any integer? Sketch ...

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