4 votes
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Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

I will ask you something that will give you intuition. How would you calculate the Gradient of an image? Image is a discretization of reality, so how would you estimate the gradient of the "Reality" ...
Royi's user avatar
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4 votes
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Causal Noise Free 1st Order Derivative in Discrete Domain

If there was no noise in the input, then the causal filter: $$y[k] = \frac{\frac{3}{2}x[k] - 2x[k-1] + \frac{1}{2}x[k-2]}{T_s}$$ would give the asymptotically optimal 3-sample linear approximation ...
Olli Niemitalo's user avatar
4 votes

Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

Hi: I've been wondering about the same exact thing myself and the light bulb finally turned on a few days ago when I went back to Kalman's 1960 paper. ( I've read it many times but not recently ). ...
mark leeds's user avatar
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4 votes
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Multiplying the imaginary part of DFT with a linear ramp to get a derivative

I've changed my mind during the answer. Instead of using frequency domain processing to get the derivative signal, I will provide a more direct and clean approach to get the derivative of a signal by ...
Fat32's user avatar
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3 votes

Multiplying the imaginary part of DFT with a linear ramp to get a derivative

From your comments, it look as if you want to manually create the ramp mask, and apply it in the frequency-domain so as to achieve the result of differentiation in time-domain. Considering the ideal ...
Fat32's user avatar
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3 votes
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Understanding the resulting image matrix when differentiating image

Your interpretation is correct: directional derivation operators highlight variation in a given direction. Here, you use the $2$-point discrete derivative in the $x$-direction (along image rows). It ...
Laurent Duval's user avatar
3 votes

Backward finite difference differentiation filter frequency response

In discrete-time signal processing terminology a backward (one sample) difference filter is $$y[n] = x[n] - x[n-1]$$ which is a causal, LTI system with an FIR impulse response of $$h[n] = \delta[n] -\...
Fat32's user avatar
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2 votes

Causal Noise Free 1st Order Derivative in Discrete Domain

Well, Since basically the Derivative Operation is Linear Filter you can chose your own optimal trade off between Noise Sensitivity and Bandwidth. If you look at Finite Differences Coefficients page ...
Royi's user avatar
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2 votes
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The Gradient Operator of a Vectorized Image in Matrix Form

It is pretty simple to create those Matrices. The real issue with them is their size which is enormous for real world images. For small kernels they are sparse which saves the day. Indeed for the ...
Royi's user avatar
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2 votes

Causal Noise Free 1st Order Derivative in Discrete Domain

You can use differentiator filter that acts as a differentiator in the band of interest, and as an attenuator at all other frequencies, effectively removing high frequency noise. Take a look at the ...
Andrei Keino's user avatar
2 votes
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Frequency response of numerical differentiation by polynomial interpolation / finite difference

Polynomials are not integrable, hence their Fourier interpretation is complicated. However, locally, there are a few works. Under least-squares, polynomial interpolation's linearity works well with ...
Laurent Duval's user avatar
2 votes

Numerical differentiation on circular data

I'm not certain how I should compute the angular velocities from the Euler angles because the data contains discontinuities at the boundaries ±180 degrees. Welcome to the wonderful world of trying to ...
TimWescott's user avatar
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2 votes
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Compute the Second Order Derivative of Digital Image with Finite Differences

Pay attention that convolution mean flipping the kernel both on the x and y axis. Hence the first element is a multiplication of ...
Royi's user avatar
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2 votes
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FM demodulation using complex differentiation

Note that ideally you would approximate the derivative at time instance $t=nT_s$ by the following central difference quotient: $$\frac{dx(nT_s)}{dt}\approx\frac{x[(n+1)T_s]-x[(n-1)T_s]}{2T_s}\tag{1}$$ ...
Matt L.'s user avatar
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1 vote
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How to calculate a 3D Fourier Transform?

Do not expect to see anything interesting in the Fourier transform spectra, whether 3D or 1D spectra, of forced oscillations with resonant frequency. Any natural oscillations that may (and should) ...
V.V.T's user avatar
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1 vote
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Undo finite difference with arbitrary timesteps

Does such a procedure exist? Of course. All systems that you are describing are just FIR filters, i.e. $$y[n] = \sum_{k=0}^{N-1} h[k] \cdot x[n-k] $$ and their Z-transform is $$H(z) = \sum_{k=0}^{N-1}...
Hilmar's user avatar
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1 vote

Particular Solution to Difference Equation

The solution you came up with is the correct homogeneous solution (i.e., when $x[n]=0$). Thing is, this is a non-homogeneous difference equation, and its solutions are of the form $$y[n]=y_h[n]+y_p[n]=...
cjferes's user avatar
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1 vote

How to solve this Laplace integral for an averaging function in an iterative numerical (finite difference) model?

The variable of the weighted mean value zm would convey a more expressive power if denoted as zmean. The RHS integral in formula (14) of your reference is a "g(x)-weighted average of f(x) on the ...
V.V.T's user avatar
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1 vote
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Calculate the Derivative of Gradient Field of an Image

Consider the expansion of the term below $$\|\nabla x -p\|_2^2 = (\nabla x -p)^T(\nabla x -p)$$ $$\|\nabla x -p\|_2^2 = (x^T\nabla^T -p^T)(\nabla x -p)$$ $$\|\nabla x -p\|_2^2 = (x^T\nabla^T\nabla x -...
Dsp guy sam's user avatar
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1 vote

Time invariance and linearity of recursive system

The system is both linear and time-invariant. Theory says linearity means that for an input $$ax_1[n] + bx_2[n]$$ your output would be $$ay_1[n] + by_2[n]$$ where $y_1[n], y_2[n]$ are the outputs of ...
GKH's user avatar
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1 vote

Linear Difference Equation and Method of Least Squares

The notation $ \hat{y} \left( k \mid k - 1 \right) $ usually means this is an estimated value of $ y \left( k \right) $ given all the available data up to time index $ k - 1 $. So generally speaking, ...
Royi's user avatar
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1 vote

Frequency response of numerical differentiation by polynomial interpolation / finite difference

For finite differences, as suggested by Andy, you can use MATLAB/Octave's freqz, but with some tricks to get the phase frequency response right. I'm using Wikipedia'...
Olli Niemitalo's user avatar
1 vote

Using Parks-Mcclellan (or others) for FIR differentiator?

There is no fundamental difference between the design of standard frequency selective filters (such as low pass, high pass, etc.) and differentiators. The only difference is that for a (linear phase) ...
Matt L.'s user avatar
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1 vote

Given a Horizontal Derivative Kernel What Would Be the Output Image?

The solution provided is correct. Here is why. To compute the derivative at index(3,5) d(3,5) = value_at(4,5) - value_at(2,5) d(3,5) = 150 - 0 d(3,5) = 150 ...
harshkn's user avatar
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