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Think it this way; assuming $x[-1] = 0$, then recusively compute the output $x[k]$ for $k \ge 0$ such as
$$\begin{align}
x[0] &= v[0] \\
x[1] &= a x[0] + v[1] \\
x[2] &= a x[1] + v[2] = a^2 x[0] + a v[1] + v[2] = a^2 v[0] + a v[1] + v[2] \\
x[2] &= \left( a^2 v[0] + a v[1] \right) + v[2] \\
... &= ... \\
x[k] &= \left( a^k v[0] + a^{...
3
First of all, it's not correct to say "poles should (always) be inside the unit circle for an LTI system to be stable" ; unless it's implied that system is also causal. Otherwise, if the system is noncausal, then its poles should be outside of unit circle for the system being stable.
For IIR systems that are described by LCCDEs causality must be externally ...
3
In general LTI System is invertible if it has neither zeros nor poles in the Fourier Domain (Its spectrum).
The way to prove it is to calculate the Fourier Transform of its Impulse Response.
The intuition is simple, if it has no zeros in the frequency domain one could calculate its inverse (Element wise inverse) in the frequency domain.
Few remarks for the ...
2
The mass-spring-damper combination is an LTI system described by the following continuous-time linear differential equation
$$ \ddot{x} = - \frac{k}{m} x - \frac{b}{m} \dot{x} + \frac{1}{m} u $$
where $u$ is the deterministic input force (N), $k$ is the spring constant (N/m) and $b$ is the damping coefficient (N.s/m).
Assuming two states as the position $...
2
D ear N atalie, as you also know that these are just letters of English (org Greek) alphabet which by themselves have no meaning or explanations other than what you have arbitrarily imposed on them and the following is by convention what's being imposed on them in the mathematics, physics and the standard DSP literature as accepted unless otherwise stated ...
2
You have to be clear what you mean by "invertible". Commonly, you want the inverse system to be causal and stable, and that puts certain restrictions on the original system.
In the case of systems with rational transfer functions, you just have to look at the zeros of the transfer function, because they become the poles of the inverse system. If all zeros ...
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It happens that if $X$ and $Y$ are independent then so will their functions $g(X)$ and $h(Y)$ be; but not $g(X,Y)$ and $h(X,Y)$.
2
If $x(t)$ is a finite-energy signal with Fourier transform $X(f)$, then $x(t)\cos(2\pi f_c t)$ is also a finite-energy signal with Fourier transform $\left.\left.\frac 12 \right[X(f-f_c) + X(f+f_c)\right]$. This is just the modulation theorem of Fourier transform theory.
The energy spectral density of $x(t)$ is $S_x(f) = |X(f)|^2$ while the energy spectral ...
1
I understand the sampling process is completely handled by the sensor itself, i.e. it delivers digital data. Then you just collect the desired number of samples from the sensor and calculate its FFT. The frequency range depends only on the sampling rate ($f_{\text{max}}=f_s/2)$, the frequency resolution depends only on the number of samples $N$, in your case ...
1
The canonical definition of independence of two random variables $X$ and $Y$ is
$X$ and $Y$ are called independent random variables if
for every choice of Borel sets $B_1, B_2$, the events $\{X \in B_1\}$ and $\{Y \in B_2\}$ are independent events, that is, $$P\{X \in B_1, Y \in B_2\} = P\{X \in B_1\}P\{Y \in B_2\}
\tag{1}$$
If you don't know what ...
1
The notation for discrete-time signals, $x[n]$, was first noticed by me in the original Oppenheim and Schaffer (1975), even though i didn't see that book until 1981.
So before, discrete-time signals in DSP used the same notation in mathematics used for sequences (infinite or not), with the subscript, $x_n$. In both notations, the argument or subscript play ...
1
The sequence $x[k]$ only depends on the current input $v[k]$ and on past inputs $v[k-l]$, $l>0$. Consequently, $x[k-1]$ only depends on past inputs $v[k-l]$, $l>0$. And since $E\{v[k]v[k-l]\}=0$ for $l>0$, you also have $E\{v[k]x[k-1]\}=0$.
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there's an ISO for that.
https://www.iso.org/obp/ui/#iso:std:iso:18431:-4:ed-1:v1:en
You must buy. No, I need not.
Anyway, as FAT32 says, they are just symbols. Which symbols stand for what is just convention, and as you have pointed out different "Authoritays" have slightly different conventions.
What is important is how the math works and that is ...
1
The term LTI system is a bit broad, so perhaps restricting ourselves to single input-single output systems makes sense. Let's just look at $s$ (Laplace) for now. The $Z$ transform follows in a straightforward way.
Also, if the system is not stable, a practical inverse filter will not recover the input. Convolution commutes so following a stable filter with ...
1
"Invariant" here can be misleading, see for instance: What is the difference between “equivariant to translation” and “invariant to translation”.
For the system perspective (which could be called shift-equivariant, see above), a shift-invariant (sometimes called translation-invariant or time-invariant) system is a system (with action denoted by $S$) does ...
1
Vanilla implementation of each method for image of size m x n and kernel of size k x l will yield:
Spatial Domain Convolution - O(mnkl) as for each pixel in the image we do kl multiplications (Additions are discarded).
Frequency Domain Convolution - O(mn log(mn) + mn) as the complexity of the FFT is mn log(mn) and we add the multiplication (You could add ...
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