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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 ...


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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 ...


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The best way to accomplish your desired result is to estimate the parameters of the interfering tone(s) and remove it(them) from the signal in the time domain. To get you started on how to go about this, I recommend you read these two articles of mine: Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT Phase and Amplitude Calculation for a Pure ...


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Much depends on what you want to deduce from your data. In general, if your sample rate is not uniform, your measurements should be accurately time stamped. The nearer to an average of 1/10 a second your sample intervals are the better. A typical hueristic used in Engineering is the 1/10 rule, so if your samples are within 1/100 of a second of 1/10 ...


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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 $h(X,Y)$.


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did i make some mistake ? Yes. The cutoff frequency is specified relative to the Nyquist Frequency and not the sample rate From the documentation Wn must be 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate. Also: you design a 10th order butterworth and filtfilt() doubles that again to 20. That will give you significant domain "...


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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 ...


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The signal in question is a rectified sine-wave. This signal has a non-zero average, i.e. a DC component. It also has AC components at f, 3f, 5f, etc. where f is the frequency of the non-rectified signal.


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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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Synthetic Method: If $\{\hat X(t)\}$ and $\{\hat Y(t)\}$ are zero-mean uncorrelated low-pass WSS processes with identical autocorrelation function $R(\tau)$ and identical power spectral density $S(f)$ enjoying the property that $S(f) = 0$ for $|f|>B$, then $$\hat{N}(t) = \hat X(t)\cos(2\pi f_ct) - \hat Y(t)\sin(2\pi f)ct$$ is a band-pass process whose ...


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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 ...


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Assume $Y = g(X)$ be the function of RV $X$, then by using the following $$E\{ g(X) \} = \int g(x) f_X(x) dx $$ variance of $Y$ can be computed without the computation of pdf $f_Y(y)$ as: $$ \begin{align} \text{Var(Y)} &= E\{ (Y-\mu_Y)^2 \} = E\{ Y^2 \} - (\mu_Y)^2 \\ & = E\{ g^2(X) \} - E\{ g(X) \}^2 \\ & = \int g^2(x) f_x(x) dx - \left(\...


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Actually AC and DC are terms that come from descriptions of types of electrical supply currents. There is an interesting history between Tesla and Edison on this, you can look it up. AC stands for alternating current, in signal processing terms it is ideally a pure sinusoid. DC stands for direct current, ideally rock steady. The terms have been co-opted ...


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When you have a continuous-time periodic signal, you should use CTFS (continuous-time Fourier series) to determine whether the signal is DC, AC or a mixture (AC + DC). Given the periodic signal $x(t) \neq 0$, perform a CTFS analysis; $x(t) \longleftrightarrow a_k$ : if $a_0 \neq 0$, but all remaning $a_k = 0$ for $k = \pm1,\pm2,...$ then it's a pure DC ...


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I believe it is interesting to look at such operator properties from an algebraic point of view. Properties are for instance, for a generic operator $\bigcirc$: commutativity: $a \bigcirc b = b \bigcirc a $ associativity: $a \bigcirc( b \bigcirc c) = (a \bigcirc b) \bigcirc c $ alternativity: $a \bigcirc( a \bigcirc b) = (a \bigcirc a) \bigcirc b $ ...


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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 ...


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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 ...


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"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 ...


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I did not read carefully through the whole question as I don't have time now, but have you tried some form of robust peak detection? See e.g., https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.find_peaks.html Then you can set parameters such as the minimum (and maximum) distance, the prominence, the minimal height. If you look for minima ...


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