Enumerations
enum	CorrelationScale { CorrelationScale::None, CorrelationScale::Biased, CorrelationScale::Unbiased }
	The ScaleOpt enum defines the normalization option of the correlation function. More...

enum	SpectralScale { SpectralScale::Linear, SpectralScale::Logarithmic }
	The SpectralScale enum represent the scale used to represent the power spectral density. More...

Functions
template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>
void	cepstrum (InputIt first, InputIt last, OutputIt d_first)
	Computes the cepstrum of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>
void	conv (InputIt first1, InputIt last1, InputIt first2, OutputIt d_first)
	Computes the convolution between the range [first1, last1) and the range [first2, last2), and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>
void	xcorr (InputIt first, InputIt last, OutputIt d_first, CorrelationScale scale=CorrelationScale::None)
	Computes the autocorrelation of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>
void	xcorr (InputIt first1, InputIt last1, InputIt first2, OutputIt d_first, CorrelationScale scale=CorrelationScale::None)
	Computes the correlation between the range [first1, last1) and the [first2, last2), and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	dct (InputIt first, InputIt last, OutputIt d_first)
	Computes the Discrete-Cosine-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	idct (InputIt first, InputIt last, OutputIt d_first)
	Computes the Inverse-Discrete-Cosine-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename Integer >
constexpr Integer	make_fft_size (Integer real_size) noexcept
	Computes the expected DFT size for a real-to-complex DFT transform. More...

template<typename Integer >
constexpr Integer	make_ifft_size (Integer complex_size) noexcept
	Computes the expected IDFT size for a complex-to-real IDFT transform. More...

template<typename InputIt , typename OutputIt >
void	cdft (InputIt first, InputIt last, OutputIt d_first)
	Computes the complex-to-complex Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	cidft (InputIt first, InputIt last, OutputIt d_first)
	Computes the complex-to-complex Inverse-Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	dft (InputIt first, InputIt last, OutputIt d_first)
	Computes the real-to-complex Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	idft (InputIt first, InputIt last, OutputIt d_first)
	Computes the complex-to-real Inverse-Discrete-Fourier-Transform of the complex range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt >
void	hartley (InputIt first, InputIt last, OutputIt d_first)
	Computes the Discrete-Hartley-Transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt , typename Allocator = std::allocator<std::complex<meta::value_type_t<InputIt>>>>
void	hilbert (InputIt first, InputIt last, OutputIt d_first)
	Computes the Discrete-Time analytic signal using Hilbert transform of the range [first, last) and stores the result in another range, beginning at d_first. More...

template<typename InputIt , typename OutputIt , typename Allocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>
void	periodogram (InputIt first, InputIt last, OutputIt d_first, SpectralScale scale)
	Computes the periodogram of the range [first, last) and stores the result in another range, beginning at d_first. More...

Enumeration Type Documentation

◆ CorrelationScale

enum edsp::spectral::CorrelationScale

strong

The ScaleOpt enum defines the normalization option of the correlation function.

Enumerator
None	Raw, unscaled cross-correlation.
Biased	Biased estimate of the cross-correlation.
Unbiased	Unbiased estimate of the cross-correlation.

◆ SpectralScale

enum edsp::spectral::SpectralScale

strong

The SpectralScale enum represent the scale used to represent the power spectral density.

Enumerator
Linear	Linear scale
Logarithmic	Logarithmic scale

Function Documentation

◆ cdft()

template<typename InputIt , typename OutputIt >

void edsp::spectral::cdft	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the complex-to-complex Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

The DFT transforms a sequence of \( N \) complex numbers \( x_n \) into another sequence of complex numbers \( X_k \):

\[ {\displaystyle {\begin{aligned}X_{k}&=\sum _{n=0}^{N-1}x_{n}\cdot e^{-{\frac {2\pi i}{N}}kn}\\&=\sum _{n=0}^{N-1}x_{n}\cdot [\cos(2\pi kn/N)-i\cdot \sin(2\pi kn/N)],\end{aligned}}} \]

Template Parameters

T	Underlying type to perform the arithmetic operations (float, double or long double).

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: complex_idft

◆ cepstrum()

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>

void edsp::spectral::cepstrum	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the cepstrum of the range [first, last) and stores the result in another range, beginning at d_first.

The Cepstrum is the result of taking the inverse Fourier transform (IDFT) of the logarithm of the estimated spectrum of a signal:

\[ {\displaystyle C_K =\left|{\mathcal {F}}^{-1}\left\{\log \left(\left|{\mathcal {F}}\{f(t)\}\right|^{2}\right)\right\}\right|^{2}} \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

◆ cidft()

template<typename InputIt , typename OutputIt >

void edsp::spectral::cidft	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the complex-to-complex Inverse-Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

The IDFT transforms a sequence of \( N \) complex numbers \( X_k \) into another sequence of complex numbers \( x_n \):

\[ {\displaystyle x_{n}={\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}e^{{\frac {2\pi i}{N}}kn}\quad \quad n=0,\dots ,N-1.} \]

Template Parameters

T	Underlying type to perform the arithmetic operations (float, double or long double).

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: complex_dft

◆ conv()

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>

void edsp::spectral::conv	(	InputIt	first1,
		InputIt	last1,
		InputIt	first2,
		OutputIt	d_first
	)

inline

Computes the convolution between the range [first1, last1) and the range [first2, last2), and stores the result in another range, beginning at d_first.

The result of conv can be interpreted as an estimate of how the shape of one range is modified by the other. It is defined as:

\[ {\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m].} \]

Parameters

first1	Input iterator defining the beginning of the first input range.
last1	Input iterator defining the ending of the first input range.
first2	Input iterator defining the beginning of the second input range.
d_first	Output iterator defining the beginning of the destination range.

◆ dct()

template<typename InputIt , typename OutputIt >

void edsp::spectral::dct	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the Discrete-Cosine-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

The DCT-II is computed as follows:

\[ {\displaystyle X_{k}=\sum _{n=0}^{N-1}{x_{n}\cos \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)k\right]}} \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

◆ dft()

template<typename InputIt , typename OutputIt >

void edsp::spectral::dft	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

Computes the real-to-complex Discrete-Fourier-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

When the input data are purely real number, the DFT output satisfies the “Hermitian” redundancy: \( F(i) \) is the conjugate of \( F(N - i) \), where \( N \) is the size of the DFT. To benefit from this property (speed and space advantages), the input and output arrays are of different sizes and types: the input is \( N \) real numbers, while the output is \( \frac{N}{2} + 1 \) complex numbers (the non-redundant outputs) where the division is rounded down.

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: make_fft_size

◆ hartley()

template<typename InputIt , typename OutputIt >

void edsp::spectral::hartley	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the Discrete-Hartley-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

The DHT transforms a sequence of \( N \) real numbers \( x_n \) into another sequence of real numbers \( H_k \):

\[ {\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\operatorname {cas} \left({\frac {2\pi }{N}}nk\right)=\sum _{n=0}^{N-1}x_{n} \left[\cos \left({\frac {2\pi }{N}}nk\right)+\sin \left({\frac {2\pi }{N}}nk\right)\right]\quad \quad k=0,\dots ,N-1} \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: complex_idft

◆ hilbert()

template<typename InputIt , typename OutputIt , typename Allocator = std::allocator<std::complex<meta::value_type_t<InputIt>>>>

void edsp::spectral::hilbert	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the Discrete-Time analytic signal using Hilbert transform of the range [first, last) and stores the result in another range, beginning at d_first.

The Discrete-time analytic is the complex helical sequence obtained from a real data sequence. The analytic signal \( A_k = A_r + iA_i \) has a real part, \( A_r \), which is the original data, and an imaginary part, \( A_i \), which contains the Hilbert transform. The imaginary part is a version of the original real sequence with a 90° phase shift.

The Hilbert transform is computed as follows:

\[ {\displaystyle {\widehat {s}}(t)={\mathcal {H}}\{s\}(t)=(h*s)(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {s(\tau )}{t-\tau }}\,d\tau .\,} \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: complex_idft

◆ idct()

template<typename InputIt , typename OutputIt >

void edsp::spectral::idct	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

inline

Computes the Inverse-Discrete-Cosine-Transform of the range [first, last) and stores the result in another range, beginning at d_first.

The IDCT-II is computed as follows:

\[ {\displaystyle X_{k}={\frac {1}{2}}x_{0}+\sum _{n=1}^{N-1}{x_{n}\cos \left[{\frac {\pi }{N}}n\left(k+{\frac {1}{2}}\right)\right]}} \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

◆ idft()

template<typename InputIt , typename OutputIt >

void edsp::spectral::idft	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first
	)

Computes the complex-to-real Inverse-Discrete-Fourier-Transform of the complex range [first, last) and stores the result in another range, beginning at d_first.

When the input data are purely real number, the DFT output satisfies the “Hermitian” redundancy: \( F(i) \) is the conjugate of \( F(N - i) \), where \( N \) is the size of the DFT. To benefit from this property (speed and space advantages), the input and output arrays are of different sizes and types: the input is \( N \) real numbers, while the output is \( \frac{N}{2} + 1 \) complex numbers (the non-redundant outputs) where the division is rounded down.

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.

See also: make_ifft_size

◆ make_fft_size()

template<typename Integer >

constexpr Integer edsp::spectral::make_fft_size ( Integer real_size )

noexcept

Computes the expected DFT size for a real-to-complex DFT transform.

Returns: Size of the DFT

◆ make_ifft_size()

template<typename Integer >

constexpr Integer edsp::spectral::make_ifft_size ( Integer complex_size )

noexcept

Computes the expected IDFT size for a complex-to-real IDFT transform.

Returns: Size of the IDFT

◆ periodogram()

template<typename InputIt , typename OutputIt , typename Allocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>

void edsp::spectral::periodogram	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first,
		SpectralScale	scale
	)

inline

Computes the periodogram of the range [first, last) and stores the result in another range, beginning at d_first.

The periodogram is an estimate of the spectral density of a signal.

\[ {\displaystyle S\left({\tfrac {k}{NT}}\right)=\left|\sum _{n}x_{N}[n]\cdot e^{-i2\pi {\frac {kn}{N}}}\right|^{2}} \]

where T, is the inverse of the sample rate \( f_s \).

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.
scale	Scale to be used in the output

◆ xcorr() [1/2]

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>

void edsp::spectral::xcorr	(	InputIt	first,
		InputIt	last,
		OutputIt	d_first,
		CorrelationScale	scale = `CorrelationScale::None`
	)

inline

Computes the autocorrelation of the range [first, last) and stores the result in another range, beginning at d_first.

The result of xcorr can be interpreted as an estimate of the correlation between two random sequences or as the deterministic correlation between two deterministic signals.

\[ R_{xx}(k) = \sum_{n=-\infty}^{\infty} x(n)x(n-k) \]

Parameters

first	Input iterator defining the beginning of the input range.
last	Input iterator defining the ending of the input range.
d_first	Output iterator defining the beginning of the destination range.
scale	Scale factor to use.

◆ xcorr() [2/2]

template<typename InputIt , typename OutputIt , typename RAllocator = std::allocator<meta::value_type_t<InputIt>>, typename CAllocator = std::allocator<std::complex<meta::value_type_t<OutputIt>>>>

void edsp::spectral::xcorr	(	InputIt	first1,
		InputIt	last1,
		InputIt	first2,
		OutputIt	d_first,
		CorrelationScale	scale = `CorrelationScale::None`
	)

inline

Computes the correlation between the range [first1, last1) and the [first2, last2), and stores the result in another range, beginning at d_first.

The result of xcorr can be interpreted as an estimate of the correlation between two random sequences or as the deterministic correlation between two deterministic signals.

\[ R_{x_1 x_2}(k) = \sum_{n=-\infty}^{\infty} x_1(n)x_2(n-k) \]

Parameters

first1	Input iterator defining the beginning of the first input range.
last1	Input iterator defining the ending of the first input range.
first2	Input iterator defining the beginning of the second input range.
d_first	Output iterator defining the beginning of the destination range.
scale	Scale factor to use.

Enumerations

Functions

Enumeration Type Documentation

◆ CorrelationScale

◆ SpectralScale

Function Documentation

◆ cdft()

◆ cepstrum()

◆ cidft()

◆ conv()

◆ dct()

◆ dft()

◆ hartley()

◆ hilbert()

◆ idct()

◆ idft()

◆ make_fft_size()

◆ make_ifft_size()

◆ periodogram()

◆ xcorr() [1/2]

◆ xcorr() [2/2]