# Spectral density estimation

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In statistical signal processing, the goal of spectral density estimation (SDE) 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. [1] Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements. When the data consist of multiple variables and one is estimating the relationship between them, estimation is known as regression analysis.

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

## Contents

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.

## Overview

Spectrum analysis, also referred to as frequency 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, or phases), versus frequency can be called spectrum analysis.

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

Spectrum analysis can be performed on the entire signal. Alternatively, a signal can be broken into short segments (sometimes called frames), 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 of Fourier analysis.

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

The Fourier 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 an inverse Fourier transform. For perfect reconstruction, the spectrum analyzer must preserve both the amplitude and phase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude (amplitude) and phase in polar coordinates (i.e., as a phasor). A common technique in signal processing is to consider the squared amplitude, or power; in this case the resulting plot is referred to as a power spectrum.

The Fourier transform (FT) decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest, one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time. In image processing the notion of a time domain is replaced by that of a spatial domain where the intensity of a signal is identified by its spatial position rather than at any point in time.

The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its arbitrary initial value. The graph to the right shows how one cycle constitutes 360° of phase. The graph also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57.3°.

Because of reversibility, the Fourier transform is called a representation of the function, in terms of frequency instead of time; thus, it is a frequency 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, only non-linear or time-variant operations can create new frequencies in the frequency spectrum.

A time-variant system is a system that is not time invariant (TIV). Roughly speaking, its output characteristics depend explicitly upon time. In other words, a system in which certain quantities governing the system's behavior change with time, so that the system will respond differently to the same input at different times.

In practice, nearly all software and electronic devices that generate frequency spectra utilize a discrete Fourier transform (DFT), which operates on samples of the signal, and which provides a mathematical approximation to the full integral solution. The DFT is almost invariably implemented by an efficient algorithm called fast Fourier transform (FFT). The squared-magnitude components of a DFT are a type of power spectrum called periodogram, which is widely used for examining the frequency characteristics of noise-free functions such as filter impulse responses and window functions. But the periodogram does not provide processing-gain when applied to noiselike signals or even sinusoids at low signal-to-noise ratios. 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.

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous, and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of samples.

A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where is the data size. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

## Techniques

Many other techniques for spectral estimation have been developed to mitigate the disadvantages of the basic periodogram. These techniques can generally be divided into non-parametric and parametric methods. The non-parametric approaches explicitly estimate the covariance 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 underlying stationary stochastic process has a certain structure that 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.

Following is a partial list of non-parametric spectral density estimation techniques:

Below is a partial list of parametric techniques:

### Parametric estimation

In parametric spectral estimation, one assumes that the signal is modeled by a stationary process which has a spectral density function (SDF) ${\displaystyle S(f;a_{1},\ldots ,a_{p})}$ that is a function of the frequency ${\displaystyle f}$ and ${\displaystyle p}$ parameters ${\displaystyle a_{1},\ldots ,a_{p}}$. [3] The estimation problem then becomes one of estimating these parameters.

The most common form of parametric SDF estimate uses as a model an autoregressive model ${\displaystyle {\text{AR}}(p)}$ of order ${\displaystyle p}$. [3] :392 A signal sequence ${\displaystyle \{Y_{t}\}}$ obeying a zero mean ${\displaystyle {\text{AR}}(p)}$ process satisfies the equation

${\displaystyle Y_{t}=\phi _{1}Y_{t-1}+\phi _{2}Y_{t-2}+\cdots +\phi _{p}Y_{t-p}+\epsilon _{t},}$

where the ${\displaystyle \phi _{1},\ldots ,\phi _{p}}$ are fixed coefficients and ${\displaystyle \epsilon _{t}}$ is a white noise process with zero mean and innovation variance${\displaystyle \sigma _{p}^{2}}$. The SDF for this process is

${\displaystyle S(f;\phi _{1},\ldots ,\phi _{p},\sigma _{p}^{2})={\frac {\sigma _{p}^{2}\Delta t}{\left|1-\sum _{k=1}^{p}\phi _{k}e^{-2i\pi fk\Delta t}\right|^{2}}}\qquad |f|

with ${\displaystyle \Delta t}$ the sampling time interval and ${\displaystyle f_{N}}$

There are a number of approaches to estimating the parameters ${\displaystyle \phi _{1},\ldots ,\phi _{p},\sigma _{p}^{2}}$ of the ${\displaystyle {\text{AR}}(p)}$ process and thus the spectral density: [3] :452-453

• The Yule-Walker estimators are found by recursively solving the Yule-Walker equations for an ${\displaystyle {\text{AR}}(p)}$ process
• The Burg 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. [3] :452 Burg associated these with maximum entropy spectral estimation. [4]
• The forward-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.
• The maximum likelihood estimators estimate the parameters using a maximum likelihood approach. This involves a nonlinear optimization and is more complex than the first three.

Alternative parametric methods include fitting to a moving average model (MA) and to a full autoregressive moving average model (ARMA).

## Frequency estimation

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. [5] This contrasts with the general methods above, which do not make prior assumptions about the components.

### Finite number of tones

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

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

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 noise subspace to extract these components. These methods 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. The most popular methods of noise subspace based frequency estimation are Pisarenko's method, the multiple 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
${\displaystyle {\hat {P}}_{\text{MN}}\left(e^{j\omega }\right)={\frac {1}{\left|\mathbf {e} ^{H}\mathbf {a} \right|^{2}}};\ \mathbf {a} =\lambda \mathbf {P} _{n}\mathbf {u} _{1}}$

### Single tone

If one only wants to estimate the single loudest frequency, one can use a pitch detection algorithm. If the dominant frequency changes over time, then the problem becomes the estimation of the instantaneous frequency as defined in the time–frequency representation. Methods for instantaneous frequency estimation include those based on the Wigner-Ville distribution and higher order ambiguity functions. [6]

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.

## Example calculation

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):

{\displaystyle {\begin{aligned}x_{n}&=\sum _{k}A_{k}\cdot \sin(2\pi \nu _{k}n+\phi _{k})\\&=\sum _{k}\left(\overbrace {a_{k}} ^{A_{k}\sin(\phi _{k})}\cos(2\pi \nu _{k}n)+\overbrace {b_{k}} ^{A_{k}\cos(\phi _{k})}\sin(2\pi \nu _{k}n)\right)\end{aligned}}}

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 ${\displaystyle N\rightarrow \infty }$. 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.

${\displaystyle \lim _{N\rightarrow \infty }{\frac {1}{N}}\sum _{n=0}^{N-1}x_{n}^{2}.}$

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

${\displaystyle x(t)=\sum _{k}A_{k}\cdot \sin(2\pi \nu _{k}t+\phi _{k})}$

and

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

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

Then the power as a function of frequency is ${\displaystyle A_{k}^{2}/2}$, and its statistical cumulative distribution function ${\displaystyle S(\nu )}$ will be

${\displaystyle S(\nu )=\sum _{k:\nu _{k}<\nu }{\frac {1}{2}}A_{k}^{2}.}$

${\displaystyle S}$ is a step function, monotonically non-decreasing. Its jumps occur at the frequencies of the periodic 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 the covariance of ${\displaystyle x(t)}$ with ${\displaystyle x(t+\tau )}$, and define this to be the autocorrelation function ${\displaystyle c}$ of the signal (or data) ${\displaystyle x}$:

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

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 ${\displaystyle x}$:

${\displaystyle c(\tau )=\sum _{k}{\frac {1}{2}}A_{k}^{2}\cos(2\pi \nu _{k}\tau ).}$

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.

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