11

One approach would be to cast the problem as least-squares smoothing. The idea is to locally fit a polynomial with a moving window, then evaluate the derivative of the polynomial. This answer about Savitzky-Golay filtering has some theoretical background on how it works for nonuniform sampling. In this case, code is probably more illuminating as to the ...


11

I will focus on the reason of the factor $1/2$ and leave aside the estimation things. The exact understanding should be : if a scalar Gaussian random variable (rv) is circular symmetric, its real and imaginary parts must be uncorrelated (= independent if they are assumed jointly Gaussian) and identically distributed with zero mean. Thus, your Matlab code is ...


10

I've never seen the word "Formula" with "AMDF". My understanding of the definition of AMDF is $$ Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \Big| x[n+n_0] - x[n+n_0+k] \Big| $$ $n_0$ is the neighborhood of interest in $x[n]$. Note that you are summing up only non-negative terms. So $Q_x[k,n_0] \ge 0$. We call "$k$" the "lag". clearly if ...


9

Read the original paper: Schmidt, R. O. "Multiple Emitter Location and Signal Parameter Estimation." IEEE Transactions on Antennas and Propagation. Vol. AP-34, March, 1986, pp. 276–280 You may also want to look up "Pisarenko's Method", "Prony's Method" and read about related problems such as ESPRIT (Roy, R.; Kailath, T. (1989). "Esprit - Estimation Of ...


8

Are you sure you shouldn't be using numpy.correlate instead of numpy.convolve? To estimate delay, you want to cross-correlate your signals, not convolve them. You'll possibly end up with a much larger delay by convolving. Trying something simple: x = [1, 0, 0, 0, 0 ]; y = [0, 0, 0, 0, 1 ]; conv = numpy.convolve(x,y); conv array([0, 0, 0, 0, 1, 0, 0, 0, ...


7

Maximum likelihood (ML) estimator Here will be derived a maximum-likelihood estimator of the power of the clean signal, but it doesn't seem to be improving things in terms of root mean square error, for any SNR, compared to spectral power subtraction. Introduction Let's introduce the normalized clean amplitude $a$ and normalized noisy magnitude $m$ ...


6

The measurement that is important to you will depend on the application. If you are looking for an over all measurement of all signal power to noise power then you define signal to be the power in all signal bands and the noise all of the powers in the noise bands. However, if you are trying to use an sub-band adaptive filter to correct for some sort of ...


6

My intuition is that it would be difficult to derive the right decision threshold you expect to find: $$\tau = \frac{1}{2}\left(\mu_0 + \mu_1\right) - \frac{\sigma^2}{\lVert\mu_0 - \mu_1\rVert^2} \log \frac{\pi}{1 - \pi}\left(\mu_0 - \mu_1\right)$$ From the global statistics you are considering (sample mean: $\pi \mu_0 + (1 - \pi) \mu_1$ ; standard ...


6

Given measurements $$\begin{align} Z_1 &= x + N_1\\ Z_2 &= x + N_2 \end{align}$$ where $N_1$ and $N_2$ are independent zero-mean Gaussian random variables with variances $\sigma_1^2$ and $\sigma_2^2$ respectively, it can be shown that the minimum-mean-square-error (MMSE) estimate of $x$ in terms of $Z_1$ and $Z_2$ is a linear function of $Z_1$ and $...


6

Well, in continuous time, a sinusoid with a bias can be seen as the output of the linear system \begin{align*} \begin{bmatrix}\dot x_1\\\dot x_2\\\dot x_3\end{bmatrix} &= \begin{bmatrix}0 & 1 & 0\\-\omega^2&0&0\\0 &0 &0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\\ y &= \begin{bmatrix}1&0&1\end{bmatrix} \...


6

Slope from all samples obtained To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without going through all the samples again. On the page you cite is the equation for calculation of the slope $m_n$ that together with $b_n$ minimizes the sum of ...


6

Common Approaches for Commercial Denoisers Commercial denoisers are different than what you'd see on most papers. While on papers the results are mostly using objective metrics (PSNR / SSIM) and are evaluated vs. Additive White Gaussian Noise (AWGN) with high level of noise real world images are mostly with moderate level of noise with Mixed Poisson ...


5

Complex A and B are a bit messy. What about using the signal model $$ f(t) = \sum_{k=1}^{2} A_k\, e^{j(w_k t + \phi_k)} $$ where $w$ is the frequency and $\phi$ is phase. Since sinusoids are orthogonal, probably the easiest thing to do is to Fourier transform the signal and pick out the two largest peaks in the positive part of the spectrum. These will ...


5

I'm unsure if this is the answer you are looking for, but why not save and share $|H_i|^2$ in addition to the least squares estimates? If you have $w_1^*=\frac 1 {|H_1|^2} H_1^t d_1$, $w_2^*=\frac 1 {|H_2|^2} H_2^t d_2$, and also know $|H_1|^2$ and $|H_2|^2$, you get the total least squares estimate with: $w^*=\frac 1 {|H_1|^2+|H_2|^2} (H_1^t d_1 + H_2^t ...


5

Well, because in a lot of real world problems this If I have a deterministic signal with a fixed number of samples, shouldn't I be able to directly determine its spectral information? is just not the case. Very often, measured signals are more of a random process. A simple and common case would be to have the desired signal and some additive noise, very ...


5

The first algorithm that springs to mind is the Goertzel Algorithm. That algorithm usually assumes that the frequency of interest is an integer multiple of the fundamental frequency. However, this paper applies the (generalized) algorithm to the case you are interested in. Another problem is that the signal model is incorrect. It uses 2*%pi*(1:siglen)*(Fc/...


5

You want a method that removes noise while preserving edges. This cannot be achieved well by linear filtering, as you noticed yourself. I know of two approaches that might work well for your problem. The first is median filtering, where samples inside a window are replaced by their median. The following plot shows the result of median filtering with a window ...


5

To answer your final question: Why is it that the estimation community ignores the use of a discrete lowpass filter for estimation? As far as I can tell it's the best approach to estimate the signal above. That's because you're feeding them the wrong signal model. TL;DR: Use the right tool for the job! Gory Details Any time you start in with the ...


5

So this is just the start of an answer. I'll have to keep updating it as I go. The first attempt is to say that the quantities you are interested in are the location of the center of the four LEDs, and the roll, pitch, and yaw (rotation angles) of the LEDs. That means your Kalman FIlter state will be: $$ \mathbf{x}_k = \left[x_k\ y_k\ \alpha_k\ \beta_k\ \...


5

*STOP! If you only want a hint and not the complete solution please see Stanley P.'s or Peter K.'s answers. * Since you do not specify if there is model for the temperature evolving over time $n$, I will derive an estimator which is a combination of $Y_1$ and $Y_2$ for each fixed $n$. Let $\alpha \in (0,1)$ and suppose the estimate of $X$ can be written as $...


4

There is a problem in checking whether the homography is OK. The algorithm for checking correct homographies may interest someone, so I will write it down here: 1) Create a quadrilateral $ABDC$ with vertex coordinates (in homogenous coordinates): $$\begin{eqnarray} A:& (-w/2,-h/2, 1.0) \\ B:& (w/2,-h/2, 1.0) \\ C:& (-w/2,h/2, 1.0) \\ D: &(...


4

Is this crazy? Good? Done before and has a name? I wouldn't say that it is "Good" because clipping is not noise that can be modeled as an additive component with some distribution. fit a curve over the top 3 or 5 highest bins, and the position of the peak is the signal median The position of that peak would be the modal mean, not the median The ...


4

Generally, the frequency estimation problem decouples from the amplitude + phase estimation problem. As I said in the comments, you could just do: $$ \hat{A} = \left|\sum_{n=0}^{N-1} x_n e^{-j 2\pi\hat{f} n} \right| $$ where $\hat{f}$ is your frequency estimate. Regarding your issues: because the number of points isn't high enough to hit the right ...


4

I assume that you have a wide-sense stationary discrete-time input noise signal $x(n)$, a linear time-invariant filter with impulse response $h(n)$, and an output noise process $y(n)$. If we assume that $x(n)$ is zero mean (which is not necessary, but makes it easier) and if we model $x(n)$ as perfectly band-limited, we get for its variance $$\sigma_x^2=\...


4

If you are willing to use multiple neighboring FFT bins, not just 2, then windowed Sinc interpolation between the complex bin results can produce a very accurate estimate, depending on the width of the window. Windowed Sinc interpolation is commonly found in high quality audio upsamplers, so papers on that subject will have suitable interpolation formulas ...


4

If you use Flanagan [1] it is computed from the phase difference of successive phase spectra Δϕ (Instantaneous Frequency) and if you reconstruct the magnitude using a correct factor (Instantaneous Magnitude) [2] use a normalized sinc function: $$ \frac{\sin( \pi x ) }{ (\pi x)}$$ And at the end use Parabolic interpolation around the peak magnitude you can ...


4

The usual way to estimate the amplitude of a particular frequency is to use the Goertzel algorithm. There is a good write-up by Rick Lyons here. Even though Rick's writeup is about single tone detection, it can be applied when multiple tones are present, too.


4

I doubt that there is such a rule because your assumption- the negative of computational work more than offsets the benefits of lower error probabilities- assumes that computations are expensive and errors are not. The "costs" of computational work and errors depends entirely on the application.


4

Here's the way I think about a discrete Wiener Filter Consider a sequence of observations $\mathbf{y} \in \Re^n $ Form a matrix from the input $\mathbf{x} \in \Re^{n+r-1}$ by shifting columns one sample each: $$ X= \begin{bmatrix} x_1 & x_2 & ... & x_r \\ x_2 & x_3 & & x_{r+1} \\ x_3 & x_4 & & x_{r+2} \\ ... & &...


4

The Hermitian symmetry is used to obtain a real-valued time-domain signal. It is a special case of OFDM called discrete multitone (DMT). It exploits a property of the discrete Fourier transform (DFT), namely that the DFT of a real-valued signal has Hermitian symmetry. The motivation is usually the channel: if the signal shall be transmitted over a low-pass ...


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