The goal of *spectral estimation* is
to describe the distribution (over frequency) of the power contained
in a signal, based on a finite set of data. Estimation of power spectra
is useful in a variety of applications, including the detection of
signals buried in wideband noise.

The *power spectral density* (PSD) of a stationary random
process *x*(*n*) is
mathematically related to the autocorrelation sequence by the discrete-time Fourier transform. In terms of normalized
frequency, this is given by

$${P}_{xx}(\omega )=\frac{1}{2\pi}{\displaystyle \sum _{m=-\infty}^{\infty}{R}_{xx}(m){e}^{-j\omega m}}.$$

This can be written as a function of physical frequency *f* (e.g., in hertz) by using the relation *ω* = 2*πf* / *f _{s}*, where

$${P}_{xx}(f)=\frac{1}{{f}_{s}}{\displaystyle \sum _{m=-\infty}^{\infty}{R}_{xx}(m){e}^{-j2\pi mf/{f}_{s}}}.$$

The correlation sequence can be derived from the PSD by use of the inverse discrete-time Fourier transform:

$${R}_{xx}(m)={\displaystyle \underset{-\pi}{\overset{\pi}{\int}}{P}_{xx}(\omega )\text{\hspace{0.17em}}{e}^{j\omega m}d\omega}={\displaystyle \underset{-{f}_{s}/2}{\overset{{f}_{s}/2}{\int}}{P}_{xx}(f)\text{\hspace{0.17em}}{e}^{j2\pi mf/{f}_{s}}\text{\hspace{0.05em}}df}.$$

The average power of the sequence *x*(*n*) over
the entire Nyquist interval is represented by

$${R}_{xx}(0)={\displaystyle \underset{-\pi}{\overset{\pi}{\int}}{P}_{xx}(\omega )d\omega ={\displaystyle \underset{-{f}_{s}/2}{\overset{{f}_{s}/2}{\int}}{P}_{xx}(f)df}}.$$

The average power of a signal over a particular frequency band [*ω*_{1}, *ω*_{2}], 0 ≤ *ω*_{1} ≤ *ω*_{2} ≤ *π*,
can be found by integrating the PSD over that band:

$${\overline{P}}_{[{\omega}_{1},{\omega}_{2}]}={\displaystyle {\int}_{{\omega}_{1}}^{{\omega}_{2}}{P}_{xx}}(\omega )\text{\hspace{0.17em}}d\omega ={\displaystyle {\int}_{-{\omega}_{2}}^{-{\omega}_{1}}{P}_{xx}}(\omega )\text{\hspace{0.17em}}d\omega .$$

You can see from the above expression that *P _{xx}*(

The units of the PSD are power
(e.g., watts) per unit of frequency. In the case of *P _{xx}*(

For real–valued signals, the PSD is symmetric about DC,
and thus *P _{xx}*(

The one-sided PSD is given by

$${P}_{\text{one-sided}}(\omega )=\{\begin{array}{ll}0,\hfill & -\pi \le \omega <0,\hfill \\ 2{P}_{xx}(\omega ),\hfill & 0\le \omega \le \pi .\hfill \end{array}$$

The average power of a signal over the frequency band, [*ω*_{1},*ω*_{2}] with 0 ≤ *ω*_{1} ≤ *ω*_{2} ≤ *π*,
can be computed using the one-sided PSD as

$${\overline{P}}_{[{\omega}_{1},{\omega}_{2}]}={\displaystyle {\int}_{{\omega}_{1}}^{{\omega}_{2}}{P}_{\text{one-sided}}}(\omega )d\omega .$$

The various methods of spectrum estimation available in the toolbox are categorized as follows:

*Nonparametric methods* are
those in which the PSD is estimated directly from the signal itself.
The simplest such method is the *periodogram*.
Other nonparametric techniques such as *Welch's method* [8], the *multitaper
method *(*MTM*) reduce the variance of
the periodogram.

*Parametric methods* are those
in which the PSD is estimated from a signal that is assumed to be
output of a linear system driven by white noise. Examples are the *Yule-Walker
autoregressive *(*AR*)* method* and
the *Burg method*. These methods
estimate the PSD by first estimating the parameters (coefficients)
of the linear system that hypothetically generates the signal. They
tend to produce better results than classical nonparametric methods
when the data length of the available signal is relatively short.
Parametric methods also produce smoother estimates of the PSD than
nonparametric methods, but are subject to error from model misspecification.

*Subspace methods*, also known
as *high-resolution methods* or *super-resolution
methods*, generate frequency component estimates for a signal
based on an eigenanalysis or eigendecomposition of the autocorrelation
matrix. Examples are the *multiple
signal classification *(*MUSIC*)* method* or
the *eigenvector *(*EV*)* method*.
These methods are best suited for line spectra — that is, spectra
of sinusoidal signals — and are effective in the detection
of sinusoids buried in noise, especially when the signal to noise
ratios are low. The subspace methods do not yield true PSD estimates:
they do not preserve process power between the time and frequency
domains, and the autocorrelation sequence cannot be recovered by taking
the inverse Fourier transform of the frequency estimate.

All three categories of methods are listed in the table below
with the corresponding toolbox function names. More information about
each function is on the corresponding function reference page. See Parametric Modeling for
details about `lpc`

and other parametric estimation
functions.

**Spectral Estimation Methods/Functions**

Method | Description | Functions |
---|---|---|

Periodogram | Power spectral density estimate | |

Welch | Averaged periodograms of overlapped, windowed signal sections | |

Multitaper | Spectral estimate from combination of multiple orthogonal windows (or "tapers") | |

Yule-Walker AR | Autoregressive (AR) spectral estimate of a time-series from its estimated autocorrelation function | |

Burg | Autoregressive (AR) spectral estimation of a time-series by minimization of linear prediction errors | |

Autoregressive (AR) spectral estimation of a time-series by minimization of the forward prediction errors | ||

Modified Covariance | Autoregressive (AR) spectral estimation of a time-series by minimization of the forward and backward prediction errors | |

MUSIC | Multiple signal classification | |

Eigenvector | Pseudospectrum estimate |

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