Hellinger distance

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Library Technician Anton from Strathroy, has many passions that include r/c helicopters, property developers in condo new launch singapore and coin collecting. Finds the beauty in planing a trip to spots around the globe, recently only returning from Old Town of Corfu. In statistical signal processing, the goal of spectral density estimation is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. The purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

SDE should be distinguished from the field of frequency estimation, which assumes a limited (usually small) number of generating frequencies plus noise and seeks to find their frequencies. SDE makes no assumption on the number of components and seeks to estimate the whole generating spectrum.

Techniques

Techniques for spectrum estimation can generally be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an auto-regressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.

Following is a partial list of spectral density estimation techniques:

• Welch's method
• Bartlett's method
• Autoregressive moving average estimation, based on fitting to an ARMA model
• Multitaper
• Maximum entropy spectral estimation
• Least-squares spectral analysis, based on least-squares fitting to known frequencies
• Non-uniform discrete Fourier transform

Finite number of tones

Template:Cleanup merge Template:Expert-subject Frequency estimation is the process of estimating the complex frequency components of a signal in the presence of noise given assumptions about the number of the components. This contrasts with the general methods above which does not make prior assumptions about the components.

The most common methods involve identifying the noise subspace to extract these components. The most popular methods of noise subspace based frequency estimation are Pisarenko's Method, MUSIC, the eigenvector solution, and the minimum norm solution.

For example, consider a signal, $x(n)$, consisting of a sum of $p$ complex exponentials in the presence of white noise, $w(n)$. This may be represented as

$x(n)=\sum _{i=1}^{p}A_{i}e^{jn\omega _{i}}+w(n)$.

Thus, the power spectrum of $x(n)$ consists of $p$ impulses in addition to the power due to noise.

The noise subspace methods of frequency estimation are based on eigen decomposition of the autocorrelation matrix into a signal subspace and a noise subspace. After these subspaces are identified, a frequency estimation function is used to find the component frequencies from the noise subspace.

Pisarenko's Method

${\hat {P}}_{PHD}(e^{j\omega })={\frac {1}{|\mathbf {e} ^{H}\mathbf {v} _{min}|^{2}}}$

MUSIC

${\hat {P}}_{MU}(e^{j\omega })={\frac {1}{\sum _{i=p+1}^{M}|\mathbf {e} ^{H}\mathbf {v} _{i}|^{2}}}$,

Eigenvector Method

${\hat {P}}_{EV}(e^{j\omega })={\frac {1}{\sum _{i=p+1}^{M}{\frac {1}{\lambda _{i}}}|\mathbf {e} ^{H}\mathbf {v} _{i}|^{2}}}$

Minimum Norm

${\hat {P}}_{MN}(e^{j\omega })={\frac {1}{|\mathbf {e} ^{H}\mathbf {a} |^{2}}};\mathbf {a} =\lambda \mathbf {P} _{n}\mathbf {u} _{1}$

Single tone

If one only wants to estimate the single loudest frequency, one can use a pitch detection algorithm.

If one wants to know all the (possibly complex) frequency components of a received signal (including transmitted signal and noise), one uses a discrete Fourier transform or some other Fourier-related transform.

References

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Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• P Stoica and R Moses, Spectral Analysis of Signals. Prentice Hall, NJ, 2005 (Chinese Edition, 2007). AVAILABLE FOR DOWNLOAD.