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Instatistical signal processing, the goal ofspectral density estimation (SDE) or simplyspectral estimation is toestimate thespectral density (also known as thepower spectral density) of a signal from a sequence of time samples of the signal.^[1] Intuitively speaking, the spectral density characterizes thefrequency content of the signal. One purpose of estimating the spectral density is to detect anyperiodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Some SDE techniques assume that a signal is composed of a limited (usually small) number of generating frequencies plus noise and seek to find the location and intensity of the generated frequencies. Others make no assumption on the number of components and seek to estimate the whole generating spectrum.

This articlemay need to be cleaned up. It has been merged fromFrequency domain.

Spectrum analysis, also referred to asfrequency domain analysis or spectral density estimation, is the technical process of decomposing a complex signal into simpler parts. As described above, many physical processes are best described as a sum of many individual frequency components. Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities) versus frequency (orphase) can be calledspectrum analysis.

Spectrum analysis can be performed on the entire signal. Alternatively, a signal can be broken into short segments (sometimes calledframes), and spectrum analysis may be applied to these individual segments.Periodic functions (such as${\displaystyle \sin (t)}$) are particularly well-suited for this sub-division. General mathematical techniques for analyzing non-periodic functions fall into the category ofFourier analysis.

TheFourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form. This means that the original function can be completely reconstructed (synthesized) by aninverse Fourier transform. For perfect reconstruction, the spectrum analyzer must preserve both theamplitude andphase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as acomplex number, or as magnitude (amplitude) and phase inpolar coordinates (i.e., as aphasor). A common technique in signal processing is to consider the squared amplitude, orpower; in this case the resulting plot is referred to as apower spectrum.

Because of reversibility, the Fourier transform is called arepresentation of the function, in terms of frequency instead of time; thus, it is afrequency domain representation. Linear operations that could be performed in the time domain have counterparts that can often be performed more easily in the frequency domain. Frequency analysis also simplifies the understanding and interpretation of the effects of various time-domain operations, both linear and non-linear. For instance, onlynon-linear ortime-variant operations can create new frequencies in the frequency spectrum.

In practice, nearly all software and electronic devices that generate frequency spectra utilize adiscrete Fourier transform (DFT), which operates onsamples of the signal, and which provides a mathematical approximation to the full integral solution. The DFT is almost invariably implemented by an efficient algorithm calledfast Fourier transform (FFT). The array of squared-magnitude components of a DFT is a type of power spectrum calledperiodogram, which is widely used for examining the frequency characteristics of noise-free functions such asfilter impulse responses andwindow functions. But the periodogram does not provide processing-gain when applied to noiselike signals or even sinusoids at low signal-to-noise ratios^[why?]. In other words, the variance of its spectral estimate at a given frequency does not decrease as the number of samples used in the computation increases. This can be mitigated by averaging over time (Welch's method^[2])  or over frequency (smoothing). Welch's method is widely used for spectral density estimation (SDE). However, periodogram-based techniques introduce small biases that are unacceptable in some applications. So other alternatives are presented in the next section.

Many other techniques for spectral estimation have been developed to mitigate the disadvantages of the basic periodogram. These techniques can generally be divided intonon-parametric,parametric, and more recentlysemi-parametric (also called sparse) methods.^[3] The non-parametric approaches explicitly estimate thecovariance or the spectrum of the process without assuming that the process has any particular structure. Some of the most common estimators in use for basic applications (e.g.Welch's method) are non-parametric estimators closely related to the periodogram. By contrast, the parametric approaches assume that the underlyingstationary stochastic process has a certain structure that can be described using a small number of parameters (for example, using anauto-regressive or moving-average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. When using the semi-parametric methods, the underlying process is modeled using a non-parametric framework, with the additional assumption that the number of non-zero components of the model is small (i.e., the model is sparse). Similar approaches may also be used for missing data recovery^[4] as well assignal reconstruction.

Following is a partial list of spectral density estimation techniques:

• Non-parametric methods for which the signal samples can beunevenly spaced in time (records can be incomplete)
  • Least-squares spectral analysis, based onleast squares fitting to known frequencies
  • Lomb–Scargle periodogram, an approximation of theLeast-squares spectral analysis
  • Non-uniform discrete Fourier transform
• Non-parametric methods for which the signal samples must be evenly spaced in time (records must be complete):
  • Periodogram, themodulus squared of the discrete Fourier transform
  • Bartlett's method is the average of the periodograms taken of multiple segments of the signal to reduce variance of the spectral density estimate
  • Welch's method a windowed version of Bartlett's method that uses overlapping segments
  • Multitaper is a periodogram-based method that uses multiple tapers, or windows, to form independent estimates of the spectral density to reduce variance of the spectral density estimate
  • Singular spectrum analysis is a nonparametric method that uses asingular value decomposition of thecovariance matrix to estimate the spectral density
  • Short-time Fourier transform
  • Critical filter is a nonparametric method based oninformation field theory that can deal with noise, incomplete data, and instrumental response functions
• Parametric techniques (an incomplete list):
  • Autoregressive model (AR) estimation, which assumes that thenth sample is correlated with the previousp samples.
  • Moving-average model (MA) estimation, which assumes that thenth sample is correlated with noise terms in the previousp samples.
  • Autoregressive moving-average (ARMA) estimation, which generalizes the AR and MA models.
  • MUltiple SIgnal Classification (MUSIC) is a popularsuperresolution method.
  • Estimation of signal parameters via rotational invariance techniques (ESPRIT) is another superresolution method.
  • Maximum entropy spectral estimation is anall-poles method useful for SDE when singular spectral features, such as sharp peaks, are expected.
• Semi-parametric techniques (an incomplete list):
  • SParse Iterative Covariance-based Estimation (SPICE) estimation,^[3] and the more generalized${\displaystyle (r,q)}$-SPICE.^[5]
  • Iterative Adaptive Approach (IAA) estimation.^[6]
  • Lasso, similar toleast-squares spectral analysis but with a sparsity enforcing penalty.^[7]

In parametric spectral estimation, one assumes that the signal is modeled by astationary process which has a spectral density function (SDF)${\displaystyle S(f;a_1,\dots ,a_p)}$ that is a function of the frequency${\displaystyle f}$ and${\displaystyle p}$ parameters${\displaystyle a_1,\dots ,a_p}$.^[8] The estimation problem then becomes one of estimating these parameters.

The most common form of parametric SDF estimate uses as a model anautoregressive model${\displaystyle \text{AR}(p)}$ of order${\displaystyle p}$.^[8]: 392 A signal sequence${\displaystyle \{Y_t\}}$ obeying a zero mean${\displaystyle \text{AR}(p)}$ process satisfies the equation

$${\displaystyle Y_t={\varphi }_1{Y}_{t-1}+{\varphi }_2{Y}_{t-2}+\cdots +{\varphi }_p{Y}_{t-p}+{\epsilon }_t,}$$

where the${\displaystyle {\varphi }_1,\dots ,{\varphi }_p}$ are fixed coefficients and${\displaystyle {\epsilon }_t}$ is a white noise process with zero mean andinnovation variance${\displaystyle {\sigma }_p^2}$. The SDF for this process is

with${\displaystyle \mathrm{\Delta }t}$ the sampling time interval and${\displaystyle f_N}$ theNyquist frequency.

There are a number of approaches to estimating the parameters${\displaystyle {\varphi }_1,\dots ,{\varphi }_p,{\sigma }_p^2}$ of the${\displaystyle \text{AR}(p)}$ process and thus the spectral density:^{[8]: 452-453}

• TheYule–Walker estimators are found by recursively solving the Yule–Walker equations for an${\displaystyle \text{AR}(p)}$ process
• TheBurg estimators are found by treating the Yule–Walker equations as a form of ordinary least squares problem. The Burg estimators are generally considered superior to the Yule–Walker estimators.^[8]: 452 Burg associated these withmaximum entropy spectral estimation.^[9]
• Theforward-backward least-squares estimators treat the${\displaystyle \text{AR}(p)}$ process as a regression problem and solves that problem using forward-backward method. They are competitive with the Burg estimators.
• Themaximum likelihood estimators estimate the parameters using amaximum likelihood approach. This involves a nonlinear optimization and is more complex than the first three.

Alternative parametric methods include fitting to amoving-average model (MA) and to a fullautoregressive moving-average model (ARMA).

Frequency estimation is the process ofestimating thefrequency, amplitude, and phase-shift of asignal in the presence ofnoise given assumptions about the number of the components.^[10] This contrasts with the general methods above, which do not make prior assumptions about the components.

If one only wants to estimate the frequency of the single loudestpure-tone signal, one can use apitch detection algorithm.

If the dominant frequency changes over time, then the problem becomes the estimation of theinstantaneous frequency as defined in thetime–frequency representation. Methods for instantaneous frequency estimation include those based on theWigner–Ville distribution and higher orderambiguity functions.^[11]

If one wants to knowall the (possibly complex) frequency components of a received signal (including transmitted signal and noise), one uses a multiple-tone approach.

A typical model for a signal${\displaystyle x(n)}$ consists of a sum of${\displaystyle p}$ complex exponentials in the presence ofwhite noise,${\displaystyle w(n)}$

${\displaystyle x(n)=\sum _{k=1}^p{A}_k{e}^{in{\omega }_k}+w(n)}$.

The power spectral density of${\displaystyle x(n)}$ is composed of${\displaystyle p}$impulse functions in addition to the spectral density function due to noise.

The most common methods for frequency estimation involve identifying the noisesubspace to extract these components. These methods are based oneigendecomposition of theautocorrelation 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. The most popular methods of noise subspace based frequency estimation arePisarenko's method, themultiple signal classification (MUSIC) method, the eigenvector method, and the minimum norm method.

Pisarenko's method

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

MUSIC

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

Eigenvector method

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

Minimum norm method

${\displaystyle {\hat{P}}_{\text{MN}}\left(e^{j\omega }\right)=\frac{1}{{\left|{\mathbf{e}}^H\mathbf{a}\right|}^2};\mathbf{a}=\lambda {\mathbf{P}}_n{\mathbf{u}}_1}$

Suppose${\displaystyle x_n}$, from${\displaystyle n=0}$ to${\displaystyle N-1}$ is a time series (discrete time) with zero mean. Suppose that it is a sum of a finite number of periodic components (all frequencies are positive):

where

The variance of${\displaystyle x_n}$ is, for a zero-mean function as above, given by

$${\displaystyle \frac{1}{N}\sum _{n=0}^{N-1}{x}_n^2.}$$

If these data were samples taken from an electrical signal, this would be its average power (power is energy per unit time, so it is analogous to variance if energy is analogous to the amplitude squared).

Now, for simplicity, suppose the signal extends infinitely in time, so we pass to the limit as${\displaystyle N\to \mathrm{\infty }.}$ If the average power is bounded, which is almost always the case in reality, then the following limit exists and is the variance of the data.

$${\displaystyle \underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{n=0}^{N-1}{x}_n^2.}$$

Again, for simplicity, we will pass to continuous time, and assume that the signal extends infinitely in time in both directions. Then these two formulas become

$${\displaystyle x(t)=\sum _k{A}_k\sin \left(2\pi {\nu }_{k}t+{\varphi }_k\right)}$$

and

$${\displaystyle \underset{T\to \mathrm{\infty }}{lim}\frac{1}{2T}{\int }_{-T}^{T}x(t{)}^{2}dt.}$$

The root mean square of${\displaystyle \sin }$ is${\displaystyle 1/\sqrt{2}}$, so the variance of${\displaystyle A_k\sin (2\pi {\nu }_{k}t+{\varphi }_k)}$ is${\displaystyle \frac{1}{2}{A}_k^2.}$ Hence, the contribution to the average power of${\displaystyle x(t)}$ coming from the component with frequency${\displaystyle {\nu }_k}$ is${\displaystyle \frac{1}{2}{A}_k^2.}$ All these contributions add up to the average power of${\displaystyle x(t).}$

Then the power as a function of frequency is${\displaystyle \frac{1}{2}{A}_k^2,}$ and its statisticalcumulative distribution function${\displaystyle S(\nu )}$ will be

${\displaystyle S}$ is astep function, monotonically non-decreasing. Its jumps occur at the frequencies of theperiodic components of${\displaystyle x}$, and the value of each jump is the power or variance of that component.

The variance is the covariance of the data with itself. If we now consider the same data but with a lag of${\displaystyle \tau }$, we can take thecovariance of${\displaystyle x(t)}$ with${\displaystyle x(t+\tau )}$, and define this to be theautocorrelation function${\displaystyle c}$ of the signal (or data)${\displaystyle x}$:

$${\displaystyle c(\tau )=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{2T}{\int }_{-T}^{T}x(t)x(t+\tau )dt.}$$

If it exists, it is an even function of${\displaystyle \tau .}$ If the average power is bounded, then${\displaystyle c}$ exists everywhere, is finite, and is bounded by${\displaystyle c(0)}$, which is the average power or variance of the data.

It can be shown that${\displaystyle c}$ can be decomposed into periodic components with the same periods as${\displaystyle x}$:

$${\displaystyle c(\tau )=\frac{1}{2}\sum _k{A}_k^2\cos (2\pi {\nu }_k\tau ).}$$

This is in fact the spectral decomposition of${\displaystyle c}$ over the different frequencies, and is related to the distribution of power of${\displaystyle x}$ over the frequencies: the amplitude of a frequency component of${\displaystyle c}$ is its contribution to the average power of the signal.

The power spectrum of this example is not continuous, and therefore does not have a derivative, and therefore this signal does not have a power spectral density function. In general, the power spectrum will usually be the sum of two parts: a line spectrum such as in this example, which is not continuous and does not have a density function, and a residue, which is absolutely continuous and does have a density function.

• Porat, B. (1994).Digital Processing of Random Signals: Theory & Methods. Prentice Hall.ISBN 978-0-13-063751-2.
• Priestley, M.B. (1991).Spectral Analysis and Time Series. Academic Press.ISBN 978-0-12-564922-3.
• Stoica, P.; Moses, R. (2005).Spectral Analysis of Signals. Prentice Hall.ISBN 978-0-13-113956-5.
• Thomson, D. J. (1982). "Spectrum estimation and harmonic analysis".Proceedings of the IEEE.70 (9):1055–1096.Bibcode:1982IEEEP..70.1055T.CiteSeerX 10.1.1.471.1278.doi:10.1109/PROC.1982.12433.S2CID 290772.

• Statistical signal processing
• Signal estimation
• Frequency-domain analysis
• Spectrum (physical sciences)

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