0
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0answers
42 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
1
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0answers
12 views

number of possible component in sinusoidal model

suppose that we have following model $y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$ my question is not related how to determine number of ...
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0answers
25 views

is following model stationary?

I am interested if following model is stationary,model is represented by following formula $$ x(n) = \sum_{p=1}^{P} a_p \cos(2\pi f_pn + \phi_p) + \epsilon(n) $$ $n$ is changing from $1$ to $N$, I ...
0
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0answers
7 views

How can I calculate distribution of minima of sections of a continuous path (from a stochastic process)?

I have a long slab whose width is defined by a stochastic process, whose complete statistics I am aware of, say. I now cut it into smaller sections of uniform length, and calculate the minimum width ...
0
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0answers
38 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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1answer
261 views

Power spectrum for discrete signals.

If $x(t)$ is a real (aperiodic) power signal, i.e. \begin{equation} 0<\lim_{T\rightarrow\infty} \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2 dt<\infty \end{equation} $x_T (t)$ is a truncated version of ...
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0answers
122 views

Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?

Definitions Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists $$r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} ...
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1answer
271 views

how can I get minimum error probability for this decision problem?

I have the decision problem for 4 hypotheses as follows: $$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$ where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$ In ...
0
votes
1answer
125 views

ROC analysis: detection probability when false alarm is maximized.

I am wondering if detection probability always goes to 1 as false alarm probability goes to 1. Let's assume binary hypothesis problem: $\mathcal{H}_0: x(t) =n(t)$ $\mathcal{H}_1: x(t) = s(t) + ...
1
vote
1answer
111 views

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: ...
2
votes
0answers
81 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
2
votes
1answer
673 views

Hilbert transform of white noise

What is the Hilbert transform of a white noise $\xi(t)$? By the Hilbert transform I mean: http://mathworld.wolfram.com/HilbertTransform.html Thank you.
1
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0answers
638 views

Partial differentiation of vector to find Jacobian (extended Kalman filter)

I am working through some coursework on self-tuning control and part of one of the questions requires the use of the extended Kalman filter for joint parameter and state estimation. For completeness, ...
1
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0answers
129 views

What is the difference between various kalman filters?

What is the difference between additive and multiplicative kalman filters, as well as some other kinds? I'm also looking for reference texts and articles that describe the algorithms, so ...
1
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3answers
658 views

Sensor fusioning in Kalman filter

I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled? Say for instance that you have an accelerometer and a gyro and want to present the "horizon level", like ...