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# pmtm

Multitaper power spectral density estimate

## Syntax

``pxx = pmtm(x)``
``pxx = pmtm(x,nw)``
``pxx = pmtm(x,nw,nfft)``
``[pxx,w] = pmtm(___)``
``[pxx,f] = pmtm(___,fs)``
``[pxx,w] = pmtm(x,nw,w)``
``[pxx,f] = pmtm(x,nw,f,fs)``
``[___] = pmtm(___,method)``
``[___] = pmtm(x,e,v)``
``[___] = pmtm(x,dpss_params)``
``[___] = pmtm(___,'DropLastTaper',dropflag)``
``[___] = pmtm(___,freqrange)``
``[___,pxxc] = pmtm(___,'ConfidenceLevel',probability)``
``pmtm(___)``

## Description

example

````pxx = pmtm(x)` returns Thomson's multitaper power spectral density (PSD) estimate, `pxx`, of the input signal, `x`. When `x` is a vector, it is treated as a single channel. When `x` is a matrix, the PSD is computed independently for each column and stored in the corresponding column of `pxx`. The tapers are the discrete prolate spheroidal (DPSS), or Slepian, sequences. The time-halfbandwidth, `nw`, product is 4. By default, `pmtm` uses the first 2`nw` – 1 DPSS sequences. If `x` is real-valued, `pxx` is a one-sided PSD estimate. If `x` is complex-valued, `pxx` is a two-sided PSD estimate. The number of points, `nfft`, in the discrete Fourier transform (DFT) is the maximum of 256 or the next power of two greater than the signal length.```

example

````pxx = pmtm(x,nw)` use the time-halfbandwidth product, `nw`, to obtain the multitaper PSD estimate. The time-halfbandwidth product controls the frequency resolution of the multitaper estimate. `pmtm` uses 2`nw` – 1 Slepian tapers in the PSD estimate.```

example

````pxx = pmtm(x,nw,nfft)` uses `nfft` points in the DFT. If `nfft` is greater than the signal length, `x` is zero-padded to length `nfft`. If `nfft` is less than the signal length, the signal is wrapped modulo `nfft` .```
````[pxx,w] = pmtm(___)` returns the normalized frequency vector, `w`. If `pxx` is a one-sided PSD estimate, `w` spans the interval [0,π] if `nfft` is even and [0,π) if `nfft` is odd. If `pxx` is a two-sided PSD estimate, `w` spans the interval [0,2π).```

example

````[pxx,f] = pmtm(___,fs)` returns a frequency vector, `f`, in cycles per unit time. The sampling frequency, `fs`, is the number of samples per unit time. If the unit of time is seconds, then `f` is in cycles/sec (Hz). For real–valued signals, `f` spans the interval [0,`fs`/2] when `nfft` is even and [0,`fs`/2) when `nfft` is odd. For complex-valued signals, `f` spans the interval [0,`fs`). `fs` must be the fourth input to `pmtm`. To input a sample rate and still use the default values of the preceding optional arguments, specify these arguments as empty, `[]`.```
````[pxx,w] = pmtm(x,nw,w)` returns the two-sided multitaper PSD estimates at the normalized frequencies specified in the vector, `w`. `w` must contain at least two elements.```
````[pxx,f] = pmtm(x,nw,f,fs)` returns the two-sided multitaper PSD estimates at the frequencies specified in the vector, `f`. `f` must contain at least two elements. The frequencies in `f` are in cycles per unit time. The sampling frequency, `fs`, is the number of samples per unit time. If the unit of time is seconds, then `f` is in cycles/second (Hz).```

example

````[___] = pmtm(___,method)` combines the individual tapered PSD estimates using the method, `method`. `method` can be one of: `'adapt'` (default), `'eigen'`, or `'unity'`.```

example

````[___] = pmtm(x,e,v)` uses the tapers in the N-by-K matrix `e` with concentrations `v` in the frequency band [-w,w]. N is the length of the input signal, `x`. Use `dpss` to obtain the Slepian tapers and corresponding concentrations.```
````[___] = pmtm(x,dpss_params)` uses the cell array, `dpss_params`, to pass input arguments to `dpss` except the number of elements in the sequences. The number of elements in the sequences is the first input argument to `dpss` and is not included in `dpss_params`. An example of this usage is `pxx = pmtm(randn(1000,1),{2.5,3})`.```

example

````[___] = pmtm(___,'DropLastTaper',dropflag)` specifies whether `pmtm` drops the last taper in the computation of the multitaper PSD estimate. `dropflag` is a logical. The default value of `dropflag` is `true` and the last taper is not used in the PSD estimate.```

example

````[___] = pmtm(___,freqrange)` returns the multitaper PSD estimate over the frequency range specified by `freqrange`. Valid options for `freqrange` are `'onesided'`, `'twosided'`, and `'centered'`.```

example

````[___,pxxc] = pmtm(___,'ConfidenceLevel',probability)` returns the `probability` × 100% confidence intervals for the PSD estimate in `pxxc`. ```

example

````pmtm(___)` with no output arguments plots the multitaper PSD estimate in the current figure window. ```

## Examples

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Obtain the multitaper PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive N(0,1) white noise.

Create a sine wave with an angular frequency of rad/sample with additive N(0,1) white noise. The signal is 320 samples in length. Obtain the multitaper PSD estimate using the default time-halfbandwidth product of 4 and DFT length. The default number of DFT points is 512. Because the signal is real-valued, the PSD estimate is one-sided and there are 512/2+1 points in the PSD estimate.

```n = 0:319; x = cos(pi/4*n)+randn(size(n)); pxx = pmtm(x);```

Plot the multitaper PSD estimate.

`pmtm(x)`

Obtain the multitaper PSD estimate with a specified time-halfbandwidth product.

Create a sine wave with an angular frequency of rad/sample with additive N(0,1) white noise. The signal is 320 samples in length. Obtain the multitaper PSD estimate with a time-halfbandwidth product of 2.5. The resolution bandwidth is rad/sample. The default number of DFT points is 512. Because the signal is real-valued, the PSD estimate is one-sided and there are 512/2+1 points in the PSD estimate.

```n = 0:319; x = cos(pi/4*n)+randn(size(n)); pmtm(x,2.5)```

Obtain the multitaper PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive N(0,1) white noise. Use a DFT length equal to the signal length.

Create a sine wave with an angular frequency of rad/sample with additive N(0,1) white noise. The signal is 320 samples in length. Obtain the multitaper PSD estimate with a time-halfbandwidth product of 3 and a DFT length equal to the signal length. Because the signal is real-valued, the one-sided PSD estimate is returned by default with a length equal to 320/2+1.

```n = 0:319; x = cos(pi/4*n)+randn(size(n)); pmtm(x,3,length(x))```

Obtain the multitaper PSD estimate of a signal sampled at 1 kHz. The signal is a 100 Hz sine wave in additive N(0,1) white noise. The signal duration is 2 s. Use a time-halfbandwidth product of 3 and DFT length equal to the signal length.

```fs = 1000; t = 0:1/fs:2-1/fs; x = cos(2*pi*100*t)+randn(size(t)); [pxx,f] = pmtm(x,3,length(x),fs);```

Plot the multitaper PSD estimate.

`pmtm(x,3,length(x),fs)`

Obtain a multitaper PSD estimate where the individual tapered direct spectral estimates are given equal weight in the average.

Obtain the multitaper PSD estimate of a signal sampled at 1 kHz. The signal is a 100 Hz sine wave in additive N(0,1) white noise. The signal duration is 2 s. Use a time-halfbandwidth product of 3 and a DFT length equal to the signal length. Use the `'unity'` option to give equal weight in the average to each of the individual tapered direct spectral estimates.

```fs = 1000; t = 0:1/fs:2-1/fs; x = cos(2*pi*100*t)+randn(size(t)); [pxx,f] = pmtm(x,3,length(x),fs,'unity');```

Plot the multitaper PSD estimate.

`pmtm(x,3,length(x),fs,'unity')`

This example examines the frequency-domain concentrations of the DPSS sequences. The example produces a multitaper PSD estimate of an input signal by precomputing the Slepian sequences and selecting only those with more than 99% of their energy concentrated in the resolution bandwidth.

The signal is a 100 Hz sine wave in additive N(0,1) white noise. The signal duration is 2 s.

```fs = 1000; t = 0:1/fs:2-1/fs; x = cos(2*pi*100*t)+randn(size(t));```

Set the time-halfbandwidth product to 3.5. For the signal length of 2000 samples and a sampling interval of 0.001 seconds, this results in a resolution bandwidth of [-1.75,1.75] Hz. Calculate the first 10 Slepian sequences and examine their frequency concentrations in the specified resolution bandwidth.

```[e,v] = dpss(length(x),3.5,10); stem(1:length(v),v,'filled') ylim([0 1.2]) title('Proportion of Energy in [-w,w] of k-th Slepian Sequence')```

Determine the number of Slepian sequences with energy concentrations greater than 99%. Using the selected DPSS sequences, obtain the multitaper PSD estimate. Set `'DropLastTaper'` to `false` to use all the selected tapers.

```hold on plot(1:length(v),0.99*ones(length(v),1))```

`idx = find(v>0.99,1,'last')`
```idx = 5 ```
`[pxx,f] = pmtm(x,e(:,1:idx),v(1:idx),length(x),fs,'DropLastTaper',false);`

Plot the multitaper PSD estimate.

```figure pmtm(x,e(:,1:idx),v(1:idx),length(x),fs,'DropLastTaper',false)```

Obtain the multitaper PSD estimate of a 100 Hz sine wave in additive N(0,1) noise. The data are sampled at 1 kHz. Use the `'centered'` option to obtain the DC-centered PSD.

```fs = 1000; t = 0:1/fs:2-1/fs; x = cos(2*pi*100*t)+randn(size(t)); [pxx,f] = pmtm(x,3.5,length(x),fs,'centered');```

Plot the DC-centered PSD estimate.

`pmtm(x,3.5,length(x),fs,'centered')`

The following example illustrates the use of confidence bounds with the multitaper PSD estimate. While not a necessary condition for statistical significance, frequencies in the multitaper PSD estimate where the lower confidence bound exceeds the upper confidence bound for surrounding PSD estimates clearly indicate significant oscillations in the time series.

Create a signal consisting of the superposition of 100-Hz and 150-Hz sine waves in additive white N(0,1) noise. The amplitude of the two sine waves is 1. The sampling frequency is 1 kHz. The signal is 2 s in duration.

```fs = 1000; t = 0:1/fs:2-1/fs; x = cos(2*pi*100*t)+cos(2*pi*150*t)+randn(size(t));```

Obtain the multitaper PSD estimate with 95%-confidence bounds. Plot the PSD estimate along with the confidence interval and zoom in on the frequency region of interest near 100 and 150 Hz.

```[pxx,f,pxxc] = pmtm(x,3.5,length(x),fs,'ConfidenceLevel',0.95); plot(f,10*log10(pxx)) hold on plot(f,10*log10(pxxc),'r-.') xlim([85 175]) xlabel('Hz') ylabel('dB') title('Multitaper PSD Estimate with 95%-Confidence Bounds')```

The lower confidence bound in the immediate vicinity of 100 and 150 Hz is significantly above the upper confidence bound outside the vicinity of 100 and 150 Hz.

Generate 1024 samples of a multichannel signal consisting of three sinusoids in additive white Gaussian noise. The sinusoids' frequencies are , , and rad/sample. Estimate the PSD of the signal using Thomson's multitaper method and plot it.

```N = 1024; n = 0:N-1; w = pi./[2;3;4]; x = cos(w*n)' + randn(length(n),3); pmtm(x)```

## Input Arguments

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Input signal, specified as a row or column vector, or as a matrix. If `x` is a matrix, then its columns are treated as independent channels.

Example: `cos(pi/4*(0:159))+randn(1,160)` is a single-channel row-vector signal.

Example: `cos(pi./[4;2]*(0:159))'+randn(160,2)` is a two-channel signal.

Data Types: `single` | `double`
Complex Number Support: Yes

Time-halfbandwidth product, specified as a positive scalar. In multitaper spectral estimation, the user specifies the resolution bandwidth of the multitaper estimate [–W,W] where W = k/NΔt for some small k > 1. Equivalently, W is some small multiple of the frequency resolution of the DFT. The time-halfbandwidth product is the product of the resolution halfbandwidth and the number of samples in the input signal, N. The number of Slepian tapers whose Fourier transforms are well-concentrated in [–W,W] (eigenvalues close to unity) is 2NW – 1.

Number of DFT points, specified as a positive integer. For a real-valued input signal, `x`, the PSD estimate, `pxx` has length (`nfft`/2 + 1) if `nfft` is even, and (`nfft` + 1)/2 if `nfft` is odd. For a complex-valued input signal,`x`, the PSD estimate always has length `nfft`. If `nfft` is specified as empty, the default `nfft` is used.

Data Types: `single` | `double`

Sample rate, specified as a positive scalar. The sample rate is the number of samples per unit time. If the unit of time is seconds, then the sample rate has units of Hz.

Normalized frequencies, specified as a row or column vector with at least two elements. Normalized frequencies are in rad/sample.

Example: `w = [pi/4 pi/2]`

Data Types: `double`

Frequencies, specified as a row or column vector with at least two elements. The frequencies are in cycles per unit time. The unit time is specified by the sample rate, `fs`. If `fs` has units of samples/second, then `f` has units of Hz.

Example: `fs = 1000; f = [100 200]`

Data Types: `double`

Weights on individual tapered PSD estimates, specified as one of `'adapt'`, `'eigen'`, or `'unity'`. The default is Thomson's adaptive frequency-dependent weights, `'adapt'`. The calculation of these weights is detailed on pp. 368–370 in [1]. The `'eigen'` method weights each tapered PSD estimate by the eigenvalue (frequency concentration) of the corresponding Slepian taper. The `'unity'` method weights each tapered PSD estimate equally.

DPSS (Slepian) sequences, specified as a N-by-K matrix where N is the length of the input signal, `x`. The matrix `e` is the output of `dpss`.

Eigenvalues for DPSS (Slepian) sequences, specified as a column vector. The eigenvalues for the DPSS sequences indicate the proportion of the sequence energy concentrated in the resolution bandwidth, [-W, W]. The eigenvalues range lie in the interval (0,1) and generally the first 2NW-1 eigenvalues are close to 1 and then decrease toward 0.

Input arguments for `dpss`, specified as a cell array. The first input argument to `dpss` is the length of the DPSS sequences and is omitted from `dpss_params`. The length of the DPSS sequences is obtained from the length of the input signal, `x`.

Example: `{3.5,5}`

Flag indicating whether to drop or keep the last DPSS sequence, specified as a logical. The default is `true` and `pmtm` drops the last taper. In a multitaper estimate, the first 2NW –  DPSS sequences have eigenvalues close to unity. If you use less than 2NW – 1 sequences, it is likely that all the tapers have eigenvalues close to 1 and you can specify `dropflag` as `false` to keep the last taper.

Frequency range for the PSD estimate, specified as a one of `'onesided'`, `'twosided'`, or `'centered'`. The default is `'onesided'` for real-valued signals and `'twosided'` for complex-valued signals. The frequency ranges corresponding to each option are

• `'onesided'` — returns the one-sided PSD estimate of a real-valued input signal, `x`. If `nfft` is even, `pxx` has length `nfft`/2 + 1 and is computed over the interval [0,π] rad/sample. If `nfft` is odd, the length of `pxx` is (`nfft` + 1)/2 and the interval is [0,π) rad/sample. When `fs` is optionally specified, the corresponding intervals are [0,`fs`/2] cycles/unit time and [0,`fs`/2) cycles/unit time for even and odd length `nfft` respectively.

• `'twosided'` — returns the two-sided PSD estimate for either the real-valued or complex-valued input, `x`. In this case, `pxx` has length `nfft` and is computed over the interval [0,2π) rad/sample. When `fs` is optionally specified, the interval is [0,`fs`) cycles/unit time.

• `'centered'` — returns the centered two-sided PSD estimate for either the real-valued or complex-valued input, `x`. In this case, `pxx` has length `nfft` and is computed over the interval (–π,π] rad/sample for even length `nfft` and (–π,π) rad/sample for odd length `nfft`. When `fs` is optionally specified, the corresponding intervals are (–`fs`/2, `fs`/2] cycles/unit time and (–`fs`/2, `fs`/2) cycles/unit time for even and odd length `nfft` respectively.

Data Types: `char`

Coverage probability for the true PSD, specified as a scalar in the range (0,1). The output, `pxxc`, contains the lower and upper bounds of the `probability` × 100% interval estimate for the true PSD.

## Output Arguments

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PSD estimate, returned as a real-valued, nonnegative column vector or matrix. Each column of `pxx` is the PSD estimate of the corresponding column of `x`. The units of the PSD estimate are in squared magnitude units of the time series data per unit frequency. For example, if the input data is in volts, the PSD estimate is in units of squared volts per unit frequency. For a time series in volts, if you assume a resistance of 1 Ω and specify the sample rate in hertz, the PSD estimate is in watts per hertz.

Data Types: `single` | `double`

Normalized frequencies, returned as a real-valued column vector. If `pxx` is a one-sided PSD estimate, `w` spans the interval [0,π] if `nfft` is even and [0,π) if `nfft` is odd. If `pxx` is a two-sided PSD estimate, `w` spans the interval [0,2π). For a DC-centered PSD estimate, `w` spans the interval (–π,π] for even `nfft` and (–π,π) for odd `nfft`.

Data Types: `double`

Cyclical frequencies, returned as a real-valued column vector. For a one-sided PSD estimate, `f` spans the interval [0,`fs`/2] when `nfft` is even and [0,`fs`/2) when `nfft` is odd. For a two-sided PSD estimate, `f` spans the interval [0,`fs`). For a DC-centered PSD estimate, `f` spans the interval (–`fs`/2, `fs`/2] cycles/unit time for even length `nfft` and (–`fs`/2, `fs`/2) cycles/unit time for odd length `nfft`.

Data Types: `double`

Confidence bounds, returned as a matrix with real-valued elements. The row size of the matrix is equal to the length of the PSD estimate, `pxx`. `pxxc` has twice as many columns as `pxx`. Odd-numbered columns contain the lower bounds of the confidence intervals, and even-numbered columns contain the upper bounds. Thus, `pxxc(m,2*n-1)` is the lower confidence bound and `pxxc(m,2*n)` is the upper confidence bound corresponding to the estimate `pxx(m,n)`. The coverage probability of the confidence intervals is determined by the value of the `probability` input.

Data Types: `single` | `double`

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### Discrete Prolate Spheroidal (Slepian) Sequences

The derivation of the Slepian sequences proceeds from the discrete-time — continuous frequency concentration problem. For all 2 sequences index-limited to 0,1,...,N – 1, the problem seeks the sequence having the maximal concentration of its energy in a frequency band [–W,W] with |W| < 1/2Δt.

This amounts to finding the eigenvalues and corresponding eigenvectors of an N-by-N self-adjoint positive semi-definite operator. Therefore, the eigenvalues are real and nonnegative and eigenvectors corresponding to distinct eigenvalues are mutually orthogonal. In this particular problem, the eigenvalues are bounded by 1 and the eigenvalue is the measure of the sequence's energy concentration in the frequency interval [–W,W].

The eigenvalue problem is given by

`$\sum _{n=0}^{N-1}\frac{\mathrm{sin}\left(2\pi W\left(n-m\right)\right)}{\pi \left(n-m\right)}{g}_{n}={\lambda }_{k}\left(N,W\right){g}_{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }m=0,1,2,\dots ,N-1$`

The 0th-order DPSS sequence, g0 is the eigenvector corresponding to the largest eigenvalue. The 1-st order DPSS sequence, g1 is the eigenvector corresponding to the next largest eigenvalue and is orthogonal to the 0-th order sequence. The 2nd-order DPSS sequence, g2, is the eigenvector corresponding to the third largest eigenvalue and is orthogonal to the 0-th order and 1-st order DPSS sequences. Because the operator is N-by-N, there are N eigenvectors. However, it can be shown that for a given sequence length N and a specified bandwidth [-W,W], there are approximately 2NW – 1 DPSS sequences with eigenvalues very close to unity.

### Multitaper Spectral Estimation

The periodogram is not a consistent estimator of the true power spectral density of a wide-sense stationary process. To produce a consistent estimate of the PSD, the multitaper method averages modified periodograms obtained using a family of mutually orthogonal tapers (windows). In addition to mutual orthogonality, the tapers also have optimal time-frequency concentration properties. Both the orthogonality and time-frequency concentration of the tapers is critical to the success of the multitaper technique. See Discrete Prolate Spheroidal (Slepian) Sequences for a brief description of the Slepian sequences used in Thomson's multitaper method.

The multitaper method uses K modified periodograms with each one obtained using a different Slepian sequence as the window. Let

`${S}_{k}\left(f\right)=\Delta t|\sum _{n=0}^{N-1}{g}_{k,n}{x}_{n}{e}^{-i2\pi fn\Delta t}{|}^{2}$`

denote the modified periodogram obtained with the k-th Slepian sequence, gk,n.

In the simplest form, the multitaper method simply averages the K modified periodograms to produce the multitaper PSD estimate.

`${S}^{\left(\text{MT}\right)}\left(f\right)=\frac{1}{K}\sum _{k=0}^{K-1}{S}_{k}\left(f\right)$`

Note the difference between the multitaper PSD estimate and Welch's method. Both methods reduce the variability in the periodogram by averaging over approximately uncorrelated estimates of the PSD. However, the two approaches differ in how they produce these uncorrelated PSD estimates. The multitaper method uses the entire signal in each modified periodogram. The orthogonality of the Slepian tapers decorrelates the different modified periodograms. Welch's overlapped segment averaging approach uses segments of the signal in each modified periodogram and the segmenting decorrelates the different modified periodograms.

The preceding equation corresponds to the `'unity'` option in `pmtm`. However, as explained in Discrete Prolate Spheroidal (Slepian) Sequences, the Slepian sequences do not possess equal energy concentration in the frequency band of interest. The higher the order of the Slepian sequence, the less concentrated the sequence energy is in the band [-W,W] with the concentration given by the eigenvalue. Consequently, it can be beneficial to use the eigenvalues to weight the K modified periodograms prior to averaging. This corresponds to the `'eigen'` option in `pmtm`.

Using the sequence eigenvalues to produce a weighted average of modified periodograms accounts for the frequency concentration properties of the Slepian sequences. However, it does not account for the interaction between the power spectral density of the random process and the frequency concentration of the Slepian sequences. Specifically, frequency regions where the random process has little power are less reliably estimated in the modified periodograms using higher order Slepian sequences. This argues for an frequency-dependent adaptive process, which accounts not only for the frequency concentration of the Slepian sequence, but also for the power distribution in the time series. This adaptive weighting corresponds to the `'adapt'` option in `pmtm` and is the default for computing the multitaper estimate.

## References

[1] Percival, D. B., and A. T. Walden, Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge, UK: Cambridge University Press, 1993.

[2] Thomson, D. J., "Spectrum estimation and harmonic analysis." Proceedings of the IEEE®. Vol. 70, 1982, pp. 1055–1096.