Questions tagged [approximation]

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relation between Nyquist sampling theory and regression

Regression is mainly about estimating a function when only a finite number of samples of the function are available. Theories usually care about asymptotic performance as the number of samples tends ...
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92 views

“Improved” MZT/IIM type One pole LPF

Some time ago, while playing with all kind of approximation of common math functions, I came up to this idea to calculate a1 coefficient for a 1st order LPF filter: \begin{array}{l}a1=-(1+x+\frac12x^...
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36 views

Some difficult-to-approximate signals to test an signal extracting algorithm

I am searching for some strange signals in order to test an algorithm that I am developing. I've found an article by Donoho and Johnstone where they propose to test the algorithms using Blocks, Bumps, ...
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1answer
35 views

Amount of exactly approximated samples with Padé approximation

I've already had some good discussion with Fat32 in this question yesterday. Today I'm confused again. The author of Statistical digital signal processing and modeling M.h. Hayes and Fat32 both ...
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1answer
48 views

Self study question about Padé Approximation and transfer functions

Given a signal $x[n] = [1, 2, 3, 4, 5]$. How many transfer functions can be found with Padé approximation which have a causal impulse response and start with those five samples. How many of them are ...
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1answer
110 views

Ratio of expected values of squares of errors in power quantities, in dB

An amplitude quantity $a$ is estimated by two estimators $\hat a_1$ and $\hat a_2.$ The error of estimator $\hat a_2$ is compared to that of $\hat a_1$. One possible comparison is the ratio of ...
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1answer
119 views

What's wrong with my implementation of the Matching Pursuit algorithm?

I have recently attempted to implement a simple version of the Matching Pursuit algorithm, following the description detailed on Wikipedia. Although the algorithm itself seems dead simple, my ...
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2answers
203 views

z-Transform Methods: Definition vs. Integration Rule

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
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1answer
81 views

Techniques to reject noisy neural network input

Suppose an artificial neural network is used to approximate a sine wave (shown in red in the graph below), given the linear input variable $x$ (scaled such that the ANN input is $x_{\rm nn}\in[-1;1]$)....
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1answer
81 views

Dimensional reduction from DWT with threshold

I have been trying to find out how can the discrete wavelet transform (DWT) be possible to reduce dimension of data. Then I saw the question which is seemingly related to my work: Feature extraction/...
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1answer
57 views

How to determine which Fouriers Series terms to use to approximate a signal?

I have a signal (a time-series of air temperature values) that I can approximate quite well with a Fourier series. However, the number of terms in the series grows rapidly, to the point that 30 - 40 ...
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4answers
2k views

Finding polynomial approximations of a sine wave

I want to approximate the sine wave given by $\sin\left(\pi x\right)$ by applying a polynomial waveshaper to a simple triangle wave, generated by the function $$T\left(x\right)=1-4\left|\tfrac{1}{2}-...
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2answers
534 views

Pade Approximation of dead time

For the time delay $e^{-sT}$ I shall find the Pade Approximation for $M = 0$ and $N = 1$. $f(s) = \sum_i^{\infty} a_is^i \approx \dfrac{\sum_{n=0}^{N} b_is^i}{\sum_{m=0}^{M} c_is^i}$ $e^{-sT} = \...
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2answers
526 views

Kaiser window approximation

Are there any "good" low-computational-cost approximations for parameterized Kaiser window generation suitable for small systems/languages that do not include Bessel functions in their standard ...
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0answers
193 views

What approximations are involved in STFT of a convolution?

I have two equations in time domain: $$x_k(t)=\sum_{i=1}^N s_i(t)\star h_{ki}(t)+n^a_k(t)$$ and $$x_k(t)=\sum_{i=1}^N s_i(t)\star h_{ki}(t)\star n^c_k(t)$$ which explain the additive and convolutive ...
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1answer
500 views

How to approximate the sample rate?

I have a simple program that captures audio from an audio device. I have configured a nominal sample rate of 48000Hz and a buffer size of 1 millisecond. The audio sub system should execute my capture ...
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1answer
81 views

Biorthogonal Wavelet Expansion

Consider a multiresolution analysis (MRA) $\{V_j\}$ in $L^2(\mathbb R)$ generated by the scaling function $\phi(x)$ and such that for each $j\in\mathbb Z$, $V_j\subset V_{j-1}$. For a given function $...
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4answers
1k views

Relation between curve fitting and filtering

Before I ask the question: I am from mechanical background and would appreciate a detailed answer. Generally, for automobiles, I am trying to understand the following: Given, say $1\;\mathrm{sec}$ ...
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1answer
932 views

Adaptive Piecewise Constant Approximation (APCA) with wavelets/DWT

I am trying to approximate a vector or a time series, in order to have as little changes as possible. To do so, I pretend to apply the Adaptive piecewise constant approximation (APCA) algorithm. ...
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1answer
655 views

Determinant of Hessian approximation (SURF)

I have a question regarding formula in SURF article by Bay et al. Theory Given a point $p=(x,y)$ in an image $I$, the Hessian matrix $\mathcal{H}$ in $x$ at scale $\sigma$ is defined as follows $$ \...
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1answer
318 views

Wavelet transform: How to compute the initial coefficients when only samples are available?

In standard MRA we have that the space of functions at scale J can be expressed as $$V_j = V_0\oplus \left(\bigoplus_{j=0}^{J} W_j\right)$$ where $V_0$ is spanned by the orthonormal system of the ...
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1answer
832 views

Approximation of Hilbert Transform Using Very Short Hilbert Transform FIR / IIR Filter

Hilbert transform is a quite sensitive topic here, since Gabor's paper, Theory of Communication, J. Inst. Electr. Engineering, London, 1946. Perhaps even more important than the Fourier transform. ...
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2answers
103 views

How to choose basis functions that contribute most efficiently per term to an approximation of an image f(x,y)?

GOAL: I would like to approximate some positive, scalar function, $f(x,y) > 0$, on a 2D field of finite size i.e. $x=[a,b],y=[c,d]$ OBSTACLE: I am familiar with the set of basis functions used ...
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1answer
146 views

Looking for a better algorithm for finding the slope of an input signal

I've got an input signal that while sloping up, approximately bounces off an imaginary line. I'm looking for the equation to that imaginary line. I have a method, but is very brute force and it ...
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1answer
158 views

Wavelet decomposition

For the following code, X=[2 2 3 100 4 0 98 100 90 2 3 67 98 0 6 6 89 9 21 78] [C,L]=wavedec(X,N,'db1'); where N is the ...
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0answers
58 views

Unexpected peak after up & down sampling

I am working on signals to check how approximation can change the results. I am testing on example that can change frequency from 22kHz to 20 KHz. I prepare this script in MatLab ...
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0answers
114 views

Second-order sections from complex coefficients

I have an impulse response ir that I need to approximate with an iir because of realtime requirements. For stability I want to break down the 10th order IIR generated with Steiglitz-Mcbride to biquads ...
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1answer
76 views

How to determine best fit from two?

I'm studying some variances ($\sigma^2$), that in my case, they must depend by velocity squared $v^2$. So, in my experimental proofs, I have plotted a graph of $\sigma^2/v^2$, and I was hoping to ...
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0answers
196 views

MATLAB implementation Spline Fitting

Let $s(t)$ be a signal that can be approximated by a uniform spline function of order $K$ (say $K=2$): $$s(t)\approx\sum_{n\in \mathbb{Z}}c_n\beta_+^{(K)}(t-n) $$ Suppose that we know the ...
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1answer
98 views

Calculating Signal Approximation Error (Theoretical signal - Real hardware signal)?

I've got a FIR filter, which I developed in IC. Now, when I got my results, I ended up with two signals: Theoretical signal calculated through MATLAB as a response to my input. Real response given by ...
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1answer
232 views

Standard deviation of Gaussian corresponding to perfect low pass filter [1,1]

For a time discrete signal, the rect-filter (for 1D signal [1,1]) is a perfect low pass filter. I was wondering, what is the best gaussian approximation of this filter? For a current problem, I ...
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1answer
888 views

How to use Expectation maximization to estimate poisson noise matlab [closed]

I delved into the state of the art of algorithms for Poisson noise estimation in order to estimate the variance, I found that the Expectation Maximization algorithm is very used and it is very ...
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2answers
572 views

Mathematical proof of relationship between derivative and convolution

What I'm looking for is the mathematical proof of why we can calculate derivatives as a simple convolution of a mask, and how do we get that mask? I know it has something to do with how the derivative ...
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0answers
81 views

How to fit a function (find a lower bound) to (for) the plots?

I am dealing with a complicated optimization problem with two parameters $N, K$. Setting $K =$ constant and finding the minimum for different $N$'s lead to a set of function values. I can do this for $...
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1answer
102 views

How to combine similar signals to generate another one?

Assume there are 11 signals. These signals are very similar to each other in both time and frequency domains. (And the similarity between them is not random; they are similar because they have somehow ...
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1answer
776 views

Signal approximation using linear combination of functions

How I can approximate the signal $x(t)=0.001\,t^3 \exp(-0.1t)$ in the interval $[0,100]$ using a linear combination of the following functions: $f_1(t)=A_1$ $f_2(t)=A_2\cos(0.05t)$ $f_3(t)=A_3\cos(...
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2answers
10k views

Using continuous verses discrete wavelet transform in digital applications

I am familiar with much of the mathematical background behind wavelets. However when implementing algorithms on a computer with wavelets I am less certain about whether I should be using continuous or ...
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1answer
112 views

Establishing the function of a decaying signal

I have a set of samples from a signal. The signal is erratic in behaviour at the beginning but then it settles down and oscillates around the value of 100 (the frequency of oscillation is not ...
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1answer
593 views

Approximating pixel location in scale space

I'm implementing a scale-space with o octaves and s scale levels in each octave. Each octave is half the size the previous. I have keypoints found in each octave, I need to approximate the real ...
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2answers
2k views

Estimate Taylor series coefficients from samples of a function

Say I have measurements of a function $y = y(x)$, sampled at $x_i$ with some noise, that could be approximated by a Taylor series expansion. Is there an accepted way of estimating the coefficients for ...
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5answers
14k views

What Is the Best First Order IIR (AR Filter) Approximation to a Moving Average Filter (FIR Filter)?

Assume the following first order IIR Filter: $$ y[n] = \alpha x[n] + (1 - \alpha) y[n - 1] $$ How can I choose the parameter $ \alpha $ s.t. the IIR approximates as good as possible the FIR which is ...
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1answer
564 views

What approximation techniques exist for the square super-root function?

I need to implement an approximation to the inverse of $x^x$, i.e. the square super-root (ssrt) function. For example, $\mathrm{ssrt}(2) \approx 1.56$ means that $1.56^{1.56} \approx 2$. I'm not as ...