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How to derive the stationary Kalman filter predictor?

In its chapter on Kalman filters, my DSP book states, seemingly out of the blue, that the stationary Kalman filter for a system $$\begin{cases} x(t+1) &= Ax(t) + w(t) \\ y(t) &= Cx(t) + v(t) \end{cases}$$ has the predictor $$\hat{x}(t+1|t) =…
Andreas
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How Does a Convolution of an Image Can Be Expressed as a Matrix Multiplication (Matrix Form)?

I know this question may not be very relevant to programming, but if I don't understand the theory behind image processing I'll never be able to implement something in practice. If I got it right Gaussian filters are convolved with an image for…
Matteo
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Automatic Cropping of Arbitrary Shapes

I have an arbitrary shape defined by a binary mask (gray = shape, black = background). I would like to find a largest possible rectangle containing only gray pixels (such rectangle is pictured in yellow): The shape is always "one piece" but it is…
Libor
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Genetic or evolutionary algorithms for filter design?

For what types of filter design problems might genetic or evolutionary algorithms be useful? What kinds of genetic or evolutionary algorithms are used for DSP problems? Edit: I expanded the question to include the larger set of evolutionary…
hotpaw2
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Relationship between DCT and PCA

I have a basic implementation knowledge of the 2D 8x8 DCT used in image & video compression. Whilst reading about Principle Component Analysis, I can see a lot of similarity, albeit PCA is clearly more generic. When I've read about DCT previously it…
trican
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7 answers

Why does a longer observation time improve DFT resolution, but repeating a signal does not?

As was proven here: https://math.stackexchange.com/questions/228614/why-doesnt-repeating-a-signal-give-rise-to-a-finer-resolution-of-dft-fft repeating a certain sequence does not improve DFT frequency resolution. However, it is known that…
Ariane
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How to Deduce a Linear System's Impulse Response from a Set of Input and Output Signals?

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the systems are $y_1[n]$, $y_2[n]$ and $y_3[n]$ as…
Belbesy
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Fourier transform is an isomorphism...but we don’t get when each frequency appears?

Statistician here who wants to get some DSP knowledge for time series analysis. I’ve known for years that if we hit a function with a Fourier transform, we have an inverse Fourier transform that will recover the original function. However, doesn’t…
Dave
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Scale and Rotation invariant Template Matching

I'm looking for a method for scale and rotation invariant Template matching. I already tried some, but they didn't work so good for my examples or took for ever to execute . SIFT and SURF Feature detection failed totally. I also tried to implement a…
Arndt Bieberstein
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Deconvolution - Richardson Lucy vs. Wiener Filter

I am studying some deconvolution techniques, In order to remove motion blur, like: Richardson-Lucy Wiener Are there any pros / cons of using one versus another? For example which are the pros / cons of Richardson-Lucy technique?
dynamic
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Is it numerically more stable to implement filtering as multiplication or convolution?

I'm writing a program to filter a 20,000-sample signal with a fifth-order Butterworth filter offline. I've got access to an FFT implementation. There seems to be two alternatives for implementing the filtering: convolving the signal with the…
Andreas
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Choices of convention and notation for the Fourier transform?

The definitions of the Fourier transform and inverse Fourier transform I learned in college were $$ F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t}\ dt $$ $$ f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t}…
rtollert
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DSP Concepts Visually Explained

Similar to this question: Visually stunning math concepts which are easy to explain, what are some great visualizations of basic DSP concepts such as FFTs, filters, etc?
progressiveCavemen
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What are the various definitions for $\rm SNR$, and their associated methods for measuring it?

The definition of $\rm SNR$ seems to be somewhat of a tower of babel in industry. What definitions of $\rm SNR$ are there (feel free to site application), and how exactly can it be measured for that applications? My specific questions on $\rm SNR$…
Spacey
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how do I calculate SNR of noisy signal?

I am having problems in understanding how to do it practically I have a wav file that contains pure speech and another ave file that just contains the background noise (can be various things, such as white noise, crowd noise, a recording of blowing…
user13267
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