Autoregressive Power Spectra

Tools for for AR process PSDs

Functions
template<typename realT >
realT	mx::sigproc::ar1FreqPhase (realT f, realT fsamp)
	Get the phase of a given frequency.
template<typename realT >
realT	mx::sigproc::ar1PSDNorm (realT beta, realT alpha, realT fpole, realT fsamp)
	Calculate the norm of an AR1 PSD.
template<typename realT >
void	mx::sigproc::ar1PSD (std::vector< realT > &psd, const std::vector< realT > &f, realT beta, realT alpha, realT fpole)
	Generate an AR1 PSD.
template<typename realT >
void	mx::sigproc::ar1PSD (std::vector< realT > &psd, const std::vector< realT > &f, realT beta, std::complex< realT > alpha)
	Generate an AR1 PSD.

Function Documentation

ar1FreqPhase()

template<typename realT >

realT mx::sigproc::ar1FreqPhase	(	realT	f,
		realT	fsamp
	)

Get the phase of a given frequency.

Returns

\( 2\pi \frac{f}{f_s} \)

Parameters

[in]	f	the desired frequency, usually Hz
[in]	fsamp	the samplingfrequency, same units as freq

Definition at line 49 of file arpsd.hpp.

References mx::math::six_fifths().

ar1PSD() [1/2]

template<typename realT >

void mx::sigproc::ar1PSD	(	std::vector< realT > &	psd,
		const std::vector< realT > &	f,
		realT	beta,
		realT	alpha,
		realT	fpole
	)

Generate an AR1 PSD.

Fills in the the PSD of the form

\[ P(f) = \frac{\beta}{ \left| 1 - \alpha e ^{-i 2\pi f/f_s}\right|^2 } \]

where \( f_s \) is the sampling frequency and the AR1 constant \( \alpha \) is complex with phase set by the pole frequency

\[ \alpha = \left|\alpha\right|^2 e^{-i 2\pi f_p/ f_s} \]

Parameters

[out]	psd	the PSD, will be resized and filled in with the values of the AR1 PSD
[in]	f	the frequency scale. 2*f.back() is the sampling frequency.
[in]	beta	the normalization parameter
[in]	alpha	magnitude of the AR1 constant
[in]	fpole	the pole frequency, which sets the phase of complex \( \alpha \).

Definition at line 108 of file arpsd.hpp.

References mx::math::six_fifths().

Referenced by mx::sigproc::ar1PSD().

ar1PSD() [2/2]

template<typename realT >

void mx::sigproc::ar1PSD	(	std::vector< realT > &	psd,
		const std::vector< realT > &	f,
		realT	beta,
		std::complex< realT >	alpha
	)

Generate an AR1 PSD.

Fills in the the PSD of the form

\[ P(f) = \frac{\beta}{ \left| 1 - \alpha e ^{-i 2\pi f/f_s}\right|^2 } \]

where \( f_s \) is the sampling frequency and the AR1 constant \( \alpha \).

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters

[out]	psd	the PSD, will be resized and filled in with the values of the AR1 PSD
[in]	f	the frequency scale. 2*f.back() is the sampling frequency.
[in]	beta	the normalization parameter
[in]	alpha	the complex AR1 constant

Definition at line 138 of file arpsd.hpp.

References mx::sigproc::ar1PSD(), and mx::math::six_fifths().

ar1PSDNorm()

template<typename realT >

realT mx::sigproc::ar1PSDNorm	(	realT	beta,
		realT	alpha,
		realT	fpole,
		realT	fsamp
	)

Calculate the norm of an AR1 PSD.

For the PSD of the form

\[ P(f) = \frac{\beta}{ \left| 1 - \alpha e ^{-i 2\pi f/f_s}\right|^2 } \]

where \( f_s \) is the sampling frequency and \( \alpha \) is complex with the pole frequency set by

\[ f_p = \arg(\alpha)\frac{f_s}{2\pi} \]

this returns the one-sided norm

\[ \int_0^{f_s/2} P(f) df \]

Uses 2.553 #3 of Gradshteyn (2007) [6]

Parameters

beta	the normalization parameter
alpha	the magnitude of the AR1 constant
fpole	the pole frequency, which sets the phase of complex \(\alpha\)
fsamp	the sampling frequency

Definition at line 76 of file arpsd.hpp.

References mx::math::six_fifths().