9
votes
Does windowing affect Parseval's theorem?
Parseval's theorem will hold, but take into account that your signal in the time domain will no longer be $x[n]$. Namely, if you have that
$$\sum_{n=0}^{N-1} \Big| x[n] \Big|^2 = \frac{1}{N} \sum_{...
- 5,040
8
votes
Accepted
why total energy of a finite duration continuous signal becomes infinite after sampling
If you multiply a continuous-time finite energy signal $f(t)$ with an impulse train you get
$$\tilde{f}(t)=\sum_{n=-\infty}^{\infty}f(nT)\delta(t-nT)\tag{1}$$
where $T$ is the sampling interval and $\...
- 92.5k
7
votes
Accepted
Is the Dirac delta (impulse) signal a power signal or an energy signal?
[Added a reference on Schwartz's impossibility theorem for products of distribution]
The continuous Dirac delta $\delta$ is not considered a true function or signal, but a distribution. From its ...
- 32.3k
7
votes
Accepted
Is there an equivalent of Parseval's theorem for wavelets?
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
the ...
- 32.3k
7
votes
Why is energy zero at zero bandwidth?
The curves represent energy (or power) densities; thus, energy is what you get when you integrate them over a range. For example, you could get the power in the range 1000 Hz to 2000 Hz by integrating ...
- 32.5k
6
votes
Accepted
Continuous vs discrete signal energy
I think you are correct. People are being fast and loose with the expression in your Eq. (2), but it captures the behaviour of the energy of the signal up to a constant $T_s$ factor (the sampling ...
- 691
5
votes
Accepted
Random signals as power signals
Note that the condition
$$\int_{-\infty}^{\infty}|f(t)|^2dt<\infty\tag{1}$$
(i.e., that the signal $f(t)$ has finite energy) is very restrictive when we try to model signals, even though ...
- 92.5k
5
votes
Accepted
How to compute the energy of two signals?
You simply have to apply the definition of energy. Assuming all signals involved are real, the energy of $g_3(t)$ is given by
\begin{align}
E &= \int_{-\infty}^\infty g_3^2(t) \, dt \\
&= \...
- 15.4k
5
votes
Does windowing affect Parseval's theorem?
While the Fourier transform, discrete or continuous, can be regarded as unitary transform i.e a naturally norm preserving change between orthonormal bases in a normed complex vector space, the ...
- 4,603
5
votes
Accepted
Average Energy of modified QAM Constellation
The procedure is always the same. You need to compute the expectation $E\{|A_k|^2\}$, where $A_k$ are the complex symbols of the constellation:
$$E\{|A_k|^2\}=\sum_kP_k|A_k|^2\tag{1}$$
$P_k$ is the ...
- 92.5k
5
votes
Accepted
Is there a relationship between the energies of the inputs to a convolution and the energy of its output?
Probably not an equality directly, but upper bounds. Let us look at the continuous case first, which is easier to derive. There is a Young's convolution inequality: with proper integrability ...
- 32.3k
5
votes
Is there instantaneous energy for signals? Why is $\big|x(t)\big|^2$ instantaneous power?
Consider
$$ \begin{align}
E_x(t) &= \int\limits_{-\infty}^{t} p_x(u) \, \mathrm{d}u \\
\\
&= \int\limits_{-\infty}^{t} \big|x(u) \big|^2 \, \mathrm{d}u \\
\end{align} $$
For a capacitor ...
- 21.5k
5
votes
Accepted
Can the total energy of a signal diverge while its average power converges to zero?
That's indeed possible, at least in theory. Just come up with a signal which when squared and integrated diverges as $t\to\infty$, but slower than linearly. E.g.,
$$x(t)=\frac{1}{\sqrt[4]{|t|}}$$
$$\...
- 92.5k
5
votes
A necessary condition for the signal energy be finite by Lathi
I would say that for all practical purposes Lathi is right. He would have been completely correct if he had added "if the limit exists". But for an engineering text that would be overly ...
- 92.5k
4
votes
Accepted
Does windowing affect Parseval's theorem?
Windowing your data, $x[n]$, with window $w[n]$ is equivalent to windowing the square of your data (or square-magnitude), $\Big|x[n]\Big|^2$, with the square of the window $w^2[n]$. So think of this ...
- 21.5k
4
votes
Does windowing affect Parseval's theorem?
[EDIT: 20180307, expanded some details] Globaly no, windowing does NOT affect Parseval's theorem (because theorems are only affected, more precisely not applicable, when their hypotheses are not met), ...
- 32.3k
4
votes
Accepted
Is $\arctan(t)$ an energy or power signal?
The typical inverse tangent function maps the input range of $ t \in (-\infty,\infty)$ into an output range of $(-\pi/2, \pi/2)$ as in the figure below:
Based on this, its values are bounded for all $...
- 28.4k
4
votes
Accepted
Calculate the energy of x[n]
It's always helpful to write down the first few elements and check if you can see the pattern. You can quickly see that $\sin(\frac{\pi}{2}n)$ is simply ...
- 48.2k
3
votes
Accepted
Even and odd signal energy property
Your result is correct but note that for complex signals, the even and odd parts are defined by
$$x_e(t)=\frac12\left[x(t)+x^*(-t)\right]\tag{1}$$
and
$$x_o(t)=\frac12\left[x(t)-x^*(-t)\right]\tag{2}$...
- 92.5k
3
votes
Accepted
Total Variation of a Signal - Is It Proportional to Signal Energy?
No it is not.
Total Variation is like the amount of changes in the signal.
Though changes require energy it doesn't mean they are proportional.
For instance, imagine that during a Window we see a ...
- 20.5k
3
votes
Is the Dirac delta (impulse) signal a power signal or an energy signal?
$\delta(x)$ doesn't really exist at all for any particular $x$. Like Laurent Duval said, Dirac is not an $\mathbb{R}\to\mathbb{R}$ function, rather the whole mapping
$$\backslash f \mapsto f(a) \equiv ...
- 1,269
3
votes
Is the Dirac delta (impulse) signal a power signal or an energy signal?
You're right that the square of a Dirac delta impulse is undefined, so energy and power cannot be defined in the usual way for signals containing Dirac impulses.
However, in analogy with discrete-...
- 92.5k
3
votes
Definition of minimum-phase system
one thing about a non-minimum phase system (with a rational transfer function), is that it can be thought of as the series concatenation (or cascade) of a minimum-phase system, having identical ...
- 21.5k
3
votes
Random signals as power signals
I think simple.
We want to model a random physical phenomenon for analysis purpose. One way is to model it by a stochastic process $X(t)$, i.e. a time series of random variables $\left\lbrace X(t_k) =...
- 6,725
3
votes
Probability of False Alarm versus SNR
Hi: In statistics we call that the probability of type I error ( rejecting when true ) and type II error ( accepting when false ). The way it's done there is that, once you make an assumption about ...
- 1,117
3
votes
Accepted
Fourier transform exercise
The signal, whose total energy you want to calculate, is periodic therefore it will have infinite energy...
To see that note, the following Fourier transform pair:
$$
x(t) = \cos(2\pi f_0 t) \...
- 28.4k
3
votes
Energy of compressed signals
Without information on $Φ$, you can obtain almost anything, since $\lambda Φ$ could be a valid CS matrix as well. Generally, one imposes structure contraints, such as unit energy for their rows or ...
- 32.3k
3
votes
Accepted
Inconsistency between the units of power spectral density and the definition that people often give
The OP is correct in their dimensional analysis
$|X(f)|^2$ is NOT the power spectral density, despite what other authors might claim. Other authors probably call this the power spectral density ...
- 312
3
votes
Accepted
How are the constellation point coordinates determined in digital modulation
Constellation diagrams exist in what is called signal space which is an abstraction used to describe finite-energy signals. The coordinate axes, even if they are marked $x$ and $y$ as in Marcus Muller'...
- 20.9k
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