New answers tagged fourier-series
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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