Periodic matrix conversions

convert(PM1,A::PM2) -> B::PM1

Convert the discrete-time periodic matrix A of type PM2 to the discrete-time periodic matrix B of type PM1, where PM1 and PM2 are of types PeriodicMatrix, SwitchingPeriodicMatrix, PeriodicArray or SwitchingPeriodicArray.

convert(PM1,A::PM2) -> B::PM1

Convert the continuous-time periodic matrix A of type PM2 to the continuous-time periodic matrix B of type PM1, where PM1 and PM2 are of types PeriodicFunctionMatrix, HarmonicArray, PeriodicTimeSeriesMatrix, PeriodicSwitchingMatrix, PeriodicSymbolicMatrix or PeriodicFunctionMatrix.

 ts2hr(A::PeriodicTimeSeriesMatrix; atol = 0, rtol, n, squeeze = true) -> Ahr::HarmonicArray

Compute the harmonic (Fourier) approximation of a periodic matrix specified by a time series matrix. The periodic matrix A(t) is specified as a continuous-time periodic time series matrix A, with m matrices contained in the vector of matrices A.values, where A.values[k] is the value of A(t) at time moment (k-1)T/m, with T = A.period being the period. The resulting harmonic approximation Ahr(t) of A(t) has the form

                 p
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T)+As_i*sin(i*2*π*t/T) ) 
                i=1

where A_0 is the constant term (the mean value), Ac_i and As_i are the coefficient matrices of the i-th cosinus and sinus terms, respectively. The order of the approximation p is determined using the maximum order specified by n (default: n = (m-1)/2) and the absolute and relative tolerances atol and rtol, as follows: p is the minimum between n, (m-1)/2 and the maximum index k such that Ac_k and/or As_k are nonzero. The tolerance used to assess nonzero elements is tol = max(atol, rtol*maxnorm), where maxnorm is the maximum norm of the matrices contained in A.values. The default values of tolerances are atol = 0 and rtol = 10*p*ϵ, where ϵ is the working machine precision.

The resulting harmonic approximation Ahr(t) is returned in the harmonic array object Ahr (see HarmonicArray).

 pfm2hr(A::PeriodicFunctionMatrix; nsample, NyquistFreq) -> Ahr::HarmonicArray

Convert a periodic function matrix into a harmonic array representation. If A(t) is a periodic function matrix of period T, then the harmonic array representation Ahr is determined by applying the fast Fourier transform to the sampled arrays A(iΔ), i = 0, ..., k, where Δ = T/k is the sampling period and k is the number of samples specified by the keyword argument nsample = k (default: k = 128). If the Nyquist frequency f is specified via the keyword argument NyquistFreq = f, then k is chosen k = 2*f*T to avoid signal aliasing.

 ts2pfm(At::PeriodicTimeSeriesMatrix; method = "linear") -> A::PeriodicFunctionMatrix

Compute the periodic function matrix corresponding to an interpolated periodic time series matrix. For the given periodic time series matrix At, a periodic function matrix A(t) is defined as the mapping A(t) = t -> etpf(t), where etpf(t) is a periodic interpolation/extrapolation object, as provided in the Interpolations.jl package. The keyword parameter method specifies the interpolation/extrapolation method to be used as follows:

method = "constant" - use periodic B-splines of degree 0 (periodic constant interpolation);

method = "linear" - use periodic B-splines of degree 1 (periodic linear interpolation) (default);

method = "quadratic" - use periodic B-splines of degree 2 (periodic quadratic interpolation);

method = "cubic" - use periodic B-splines of degree 3 (periodic cubic interpolation).

 hr2psm(Ahr::HarmonicArray, nrange) -> A::Matrix{Num}

Convert a range of harmonic components nrange of the harmonic array Ahr to a symbolic matrix A. The default range is nrange = 0:n, where n is the order of the maximum harmonics.

 psm2hr(A::PeriodicSymbolicMatrix; nsample, NyquistFreq) -> Ahr::HarmonicArray

Convert a periodic symbolic matrix into a harmonic array representation. If A(t) is a periodic symbolic matrix of period T, then the harmonic array representation Ahr is determined by applying the fast Fourier transform to the sampled arrays A(iΔ), i = 0, ..., k, where Δ = T/k is the sampling period and k is the number of samples specified by the keyword argument nsample = k (default: k = 128). If the Nyquist frequency f is specified via the keyword argument NyquistFreq = f, then k is chosen k = 2*f*T to avoid signal aliasing.

 pm2pa(At::PeriodicMatrix) -> A::PeriodicArray

Convert a discrete-time periodic matrix object into a discrete-time periodic array object.

The discrete-time periodic matrix object At contains a p-vector At of real matrices At[i], i = 1,..., p, the associated time period T and the number of subperiods k. The resulting discrete-time periodic array object A of period T and number of subperiods k is built from a m×n×p real array A, such that A[:,:,i] contains At[i], for i = 1,..., p. For non-constant dimensions, the resulting m and n are the maximum row and column dimensions, respectively, and the resulting component matrices A[:,:,i] contain At[i], appropriately padded with zero entries.

 ffm2hr(Afun::FourierFunctionMatrix; atol = 0, rtol = √ϵ, squeeze = true) -> Ahr::HarmonicArray

Compute the harmonic (Fourier) representation of a Fourier periodic matrix object.

The Fourier function matrix object Afun of period T is built using the Fourier series representation of a periodic matrix Afun(t) of subperiod T′ = T/k, where each entry of Afun(t) has the form

         p
  a_0 +  ∑ ( ac_i*cos(i*t*2*π/T′)+as_i*sin(i*2*π*t/T′) ) ,
        i=1

where k ≥ 1 is the number of subperiods (default: k = 1).

The harmonic array object Ahr of period T is built using the harmonic representation of a periodic matrix Ahr(t) of subperiod T′′ = T/k′ in the form

                 p′
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T′′)+As_i*sin(i*2*π*t/T′′) ) ,
                i=1

where p′ is the maximum index j, such that Ac_j and/or As_j are nonzero. The tolerance used to assess nonzero elements is tol = max(atol, rtol*maxnorm), where maxnorm is the maximum absolute value of the coefficients ac_i and as_i in Afun(t). The default values of tolerances are atol = 0 and rtol = √ϵ, where ϵ is the working machine precision. The resulting harmonic approximation Ahr(t) is returned in the harmonic array object Ahr (see HarmonicArray).

 ffm2psm(Af::FourierFunctionMatrix, nrange atol = 0, rtol = 10*n*ϵ,) -> A::Matrix{Num}

Convert a range of harmonic components nrange of the Fourier function matrix Af to a symbolic matrix A. The default range is nrange = 0:n, where n is the order of the maximum harmonics. The tolerance used to assess nonzero coefficients is tol = max(atol, rtol*maxnorm), where maxnorm is the maximum absolute value of the coefficients ac_i and as_i in Af(t). The default values of tolerances are atol = 0 and rtol = 10*n*ϵ, where ϵ is the working machine precision.

 hr2bt(Ahr::HarmonicArray, N; P, nperiod]) -> Abt::Matrix

Build the block Toeplitz matrix of a harmonic (Fourier) array representation of a periodic matrix.

The harmonic representation object Ahr of period T of a periodic matrix Ahr(t) of subperiod T′ = T/k is in the form

                 p
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T′)+As_i*sin(i*2*π*t/T′) ) ,
                i=1

where k ≥ 1 is the number of subperiods. Ahr(t) can be equivalently expressed in the Fourier series representation form

            p
 Ahr(t) =   ∑ A_i*exp(im*i*ωh*t) ,
           i=-p

where ωh = 2π/T′, A_i = (Ac_i-im*As_i)/2 and A_{-i} = (Ac_i+im*As_i)/2. N is the number of selected harmonic components (or Fourier modes) used for approximation. The keyword parameter P is the number of full periods to be considered (default: P = 1) and nperiod is the number of subperiods to be considered, such that 1 ≤ nperiod ≤ k (default: nperiod = k).

For a given number N ≥ p, if the number of period is P = 1 and the number of subperiods is nperiod = 1, then the banded block Toeplitz matrix Abt with (2N+1)×(2N+1) blocks is constructed

       ( A_0  A_{-1} …  A_{-p}        0    )
       ( A_1   A_0             ⋱           )
       (  ⋮         ⋱            ⋱         )
 Abt = ( A_p             ⋱          A_{-p} )
       (        ⋱           ⋱         ⋮    )
       (  0        A_p      …         A_0  )

If N < p, then a truncated full block Toeplitz matrix is built using the first N harmonic components.

Generally, for given P ≥ 1 and nperiod ≥ 1, the block Toeplitz matrix Abt is constructed with (2N*np+1)×(2N*np+1) blocks, with np = P*nperiod, such that each A_i is preceeded in its column by np-1 zero blocks, each A_{-i} is preceeded in its row by np-1 zero blocks and all diagonal blocks are equal toA_0.

 hr2btupd(Ahr::HarmonicArray, N; P, nperiod, shift]) -> Ab::Matrix

Build the updated block Toeplitz matrix of a harmonic (Fourier) array representation of a periodic matrix.

If BT is the block Toeplitz matrix of the harmonic array representation of the n × n periodic matrix Ahr of subperiod T′ (see HarmonicArray) as constructed with hr2bt, then the updated matrix Ab = BT-NT is constructed, with NT a block-diagonal matrix with n × n diagonal blocks. The k-th diagonal block of NT is the diagonal matrix im*(μ + k*ωh)*I, where μ is a shift specified via the keyword parameter shift = μ (default: μ = 0) and ωh is the base frequency computed as ωh = 2π*nperiod/(P*T′). The value of shift must satisfy 0 ≤ μ ≤ ωh/2.