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    How to distinguish between linear and non-linear circuits?

    IIRC, 2 things for certain can classify circuits as non-linear: 1. Non linear elements such as diodes. 2. Changing frequency of sources or input waveforms. Mathematically, a linear circuit will obey the superposition principle with regards to signals across the output.
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    Semigroup property for convolution

    I submitted both the approaches, one with the incorrect fourier transform and the other with the convolution integral. The convolution integral, reduced using partial fraction decomposition or the residue theorem (neither of which I was able to produce as a solution) reduces it to the form...
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    Semigroup property for convolution

    Scaling property and linearity property. Note a and b are greater than zero. The fourier transform of $$ \mathcal{F} \bigg \{ \frac{1}{1 + {(\frac{x}{a})}^{2} } \bigg \} = \frac{1}{|a|}\cdot \sqrt{ \frac{\pi}{2} } \cdot e^{-|\frac{\omega}{a} | } $$ Hence: $$ \frac{1}{a \pi} \cdot \mathcal{F}...
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    Semigroup property for convolution

    Hi The fourier transform of: $$ \mathcal{F} \bigg\{ \frac{1}{a \pi} \cdot \frac{1}{1 + { \frac{x}{a} }^{2} } \bigg \} =\frac{1}{ a^2 \pi} \cdot \sqrt{ \frac{\pi}{2} } \cdot e^{-| \frac{\omega}{a} |} $$ My idea is that, since, the fourier transform of the convolution results in a mulitplication...
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    Semigroup property for convolution

    The convolution is defined on $$ l^{1}(\mathbb{Z}) $$ and $$ l^{1}(\mathbb{R} ) $$ spaces. Respectively, for continuous and discrete (absolutely integrable and absolutely summable functions): $$ x[n] * y[n] = \displaystyle \sum_{k \in \mathbb{Z} } x[k] \cdot y[n - k] $$ And $$ x(t) * y(t) =...
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    Semigroup property for convolution

    Summary: Show that for this family of functions the following semigroup property with respect to convolution holds. Hi. My task is to prove that for the family of functions defined as: $$ f_{a}(x) = \frac{1}{a \pi} \cdot \frac{1}{1 + \frac{x^{2}}{a^{2}} } $$ The following semigroup property...
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    MATLAB Bilinear spline interpolation MATLAB using MESHGRID and SURF

    My apologies, $$ u_{i , j} $$ Is the matrix, whereas $$ \Omega $$ I sismply the cartesian product as the professor has given above. I must say I am very grateful for the time you took to actually write the code, thank you. I will check the details and post the results in a few hours as I am at...
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    MATLAB Bilinear spline interpolation MATLAB using MESHGRID and SURF

    Hello. So, I must provide a solution for an image processing course I am taking (implemented in MATLAB). The task is as follows: 1. I must provide a MATLAB script that takes in a DISCRETE N x N matrix (Greyscale picture) and does Bilinear spline interpolation on it. This is the spline...
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    How to determine the power of this circuit?

    Hi I thought I might add this quick and easy online circuit simulator, that would help you when practicing questions: https://www.falstad.com/circuit/
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    Engineering Help finding the damping ratio formula for this circuit

    Hi BO. the standard form of second order systems is like this: $$ x(t) = y''(t) + 2 \delta y'(t) + \omega_{r}^{2} y(t) $$ I would do a thevenin transform from the point of view of the branch of the inductor and capacitor in series, you get the following thevenin parameters: $$ v_{th}(t) =...
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    I Why use Epsilon Delta proofs?

    Hi NockWodz I can say that we use epsilon delta proofs to prove that a limit exists because thats literally the definition of a limit. Hence, to prove that some objects exists or is equal to some other well defined object, the way is to prove that it matches the definition of that object. In...
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    I Ambiguous Results for two Fourier transform techniques

    Yes, I apologise for not being more thorough. So: $$ \Re\big( X(j \omega) \big) = X(j \omega)_{Even} \,\,\,\,\, j \cdot \Im\big(X(j\omega) \big) = X(j \omega)_{odd} $$ Here: $$ X(j \omega) = \dfrac{A}{b + j\omega)}= \dfrac{A \cdot(b - j\omega) }{b^{2} + \omega^2} = ´\dfrac{Ab}{b^{2} +...
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    I Fourier series representation

    Hi Check the dirichlet conditions Any function that satisfies these, has a fourier series rep.
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    Fourier Series = Re(Power Series)

    Hi Joe 1. There exist algebraic methods for DC (constant ) and sinusoidal functions (sinusoidal steady state analysis) in circuits, to find voltage and current and power. There does not exist any algebraic method for non sinusoidal, non DC functions, like triangles, square waves, bipolar...
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    I Ambiguous Results for two Fourier transform techniques

    Hi Painter The Fourier Transform of a function is defined as: $$ x(j\omega) = \displaystyle \int_{-\infty}^{\infty} f(t) \cdot e^{-j(\omega t)} \,\,\,\, \text{dt} $$ The trigonometric fourier series for a function is defined as: $$ f(t) = \dfrac{a_{0}}{2} + \displaystyle \sum_{k=1}^{k \to...
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    Convolution in a Continous Linear Time Invariant System

    Hi Saravanan The dirac pulse: $$ \delta(t) = \begin{cases} \infty & \text{ $ t = 0 $} \\ 0 & \text{Otherwise} \end{cases} $$ And its discrete time equivalent (equivalent in the sense that it plays much of the same role as the pulse) the unit impulse sequence: $$ \delta[ n ] = \begin{cases}1...
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    Output as the convolution of the Impulse response and input

    Hi chn I believe the answer can be found by thinking about the definition of convolution, also are you using the correct definition of the unit step function? $$ \text{The unit step function} \,\,\,\,\,\, \epsilon(t) = \begin{cases} 1 & \text{if $ t $ } \gt 0 \\ 0 & \text{Otherwise}...
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    Physical Significance of the Laplace Transform

    $$ \textbf{Correction} $$ The fourier transform of the $$ \epsilon(t) $$ exists, and is: $$ \pi \delta(\omega) + \dfrac{1}{j \omega} $$
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    Physical Significance of the Laplace Transform

    Correct! Actually in our studies we always assumed the damped sinusoid begins at t = 0 So I should have redefined it as: $$ f(t) = \epsilon(t) \cdot e^{-(\sigma t)} \cdot \cos(\omega_{0} t) $$ Where ## \epsilon(t) ##is the unit step function, so the function is zero for all t less than zero...
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    Physical Significance of the Laplace Transform

    It says not secure on google but it seems like a good site created by signal processing lovers like us out of altruism hehe and thank you you are welcome. http://dspcan.homestead.com/files/idxpages.htm http://cnyack.homestead.com/files/idxpages.htm
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    Physical Significance of the Laplace Transform

    Hi. I also agree with the post, because the fourier transform of the dirac impulse and the sine and cosine both exist. If by step impulse it is meant that: $$ \delta(t) $$ Then the fourier transform exists and is a constant 1, which is the same as its laplace transform. $$ \delta(t) \iff X(j...
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    Physical Significance of the Laplace Transform

    Hi No unfortunately not, because the mathematical definition of ## f(t) ## is: $$ f(t) = 1 $$ Whereas an exponentially decaying sinusoidal signal of time constant 1 whose peak magnitude is 0.707 and frequency of oscillations of this decaying signal is 1 Hz is this: $$ g(t) = \dfrac{e^{-t} \cdot...
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    Physical Significance of the Laplace Transform

    Hi. I would like to add another aspect which I believe might contribute a little more to the understanding. $$ \textbf{The role of the damping exponential} \,\,\,\, e^{-(\sigma t) } $$ The Laplace transform extends the set of signals for which the notion of a spectrum is available. It does...
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    LaTeX Excluding circuittikz, how can I create circuit diagrams

    The best I am hoping for is that I can draw it in some software, export it as tikz code or another package, and render it in latex. I do not want to write the code for it, but to be able to create a schematic and import it so its rendered in latex.
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    LaTeX Excluding circuittikz, how can I create circuit diagrams

    Hello. I want to create circuit diagrams, excluding the package tikz and not wanting to write up a schematic in latex, how can I create circuit diagrams? I would think one of you would have been creating and importing complex diagrams into latex. I do not want to take pictures and post them, I...
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    Electrical Engineering: Is this signal stable?

    Hi I think the title of this thread should be is this system stable instead of is this signal stable, because ## \textbf{BIBO stability} ## is a system property and not a signal property. There are a few system properties: 1. BIBO stability. 2. Causality. 3. Linearity. 4. Time invariance. 5...
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    How does a Capacitor protect against voltage spikes?

    Hi One way to understand this, is to see the step response of RC and RL systems. In the case of RC systems it opposes sudden changes in voltages, and in the case of an RL network it opposes sudden changes in current. Check out the step response of an RC network: $$ \textbf{Step response of an RC...
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    Deriving the impedance equation of reactive passive components

    Hi Tim. I am sorry if I caused any confusion, I know it is not easy if you have been out of it for some time. if you simply divide that complex number by the complex number for current, you would get the impedence of the inductor, as we derived the current using the impedence voltage divider...
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    Deriving the impedance equation of reactive passive components

    Hi. I provided a method up top, but: Here is an easy way to do it, using the laplace transform, and its generalisation the fourier transform: $$ \textbf{Deriving the impedence of an inductor using the laplace and fourier transform} $$ $$ i_{L}(t)= \dfrac{1}{L} \cdot \displaystyle \int v(t)...
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    A Hint as to how to solve an integral differential equation

    Hello. The differential equation for a second order system like an RLC circuit (series) driven by a voltage source and an RLC circuit (parallel) driven by a current source is derived, as suggested above, by taking the derivative on both sides: $$ v(t) = i(t) R + L i'(t) + \dfrac{1}{C}...
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