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recursive_osstft.m
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recursive_osstft.m
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function [tfr, stfr, nfreqs, below, over, q_hat_map, if_map] = recursive_osstft(x, k, L, mi, mf, M, n0, q_method, if_method, q_threshold)
% [tfr, stfr, nfreqs, below, over, q_hat_map] = recursive_vsstft(x, k, L, mi, mf, M, n0, q_method, if_method)
%
% Compute the recursive oblique synchrosqueezed STFT
%
% INPUT:
% x : input signal
% k : order
% L : window length
% mi,mf : initial and final frequency bin in range [0,M]
% M : number of frequency bins
% n0 : time-shift used to allow reconstruction with a causal window (should be greater than 0, default: 1)
% q_method : method to use for FM estimation (1: CRE1, 2(default):CRE2t, 3: CRE2w 4: CRE2r)
% if_method : method to use for IF estimation (1: default Biased estimator, 2: unbiased complex estimator)
% q_threshold: threshold applied on window values (default: 10^(-4))
%
%
% OUTPUT:
% tfr: recursive STFT
% stfr: recursive vertical synchrosqueezed STFT
% nfreqs: vector of normalized frequencies (mi:mf)/M
% below, over: stored out of bounds reassigned energy
% q_hat_map: continuous time estimated CRE (\hat q)
% if_map: continuous time estimated IF (\hat\omega)
%
%
% Author: D.Fourer (dominique@fourer.fr)
% Date: 27-06-2016
% Ref: [D. Fourer, F. Auger, K.Czarnecki and S. Meignen, Chirp rate and instantaneous frequency estimation. Proc. IEEE ICASSP 2017]
if ~exist('q_method', 'var')
q_method = 2;
end
if ~exist('if_method', 'var')
if_method = 1;
end
if ~exist('q_threshold', 'var')
%% CRE threshold
q_threshold = 1e-4; %eps
end
order3 = false;
N = length(x);
Nfbins = mf-mi+1;
nfreqs = (mi:mf)/M; %% real normalized frequency
ma = mi:mf; %% absolute frequency index
q_hat_map= zeros(Nfbins, N);
if_map = zeros(Nfbins, N);
tfr = zeros(Nfbins, N);
stfr = zeros(Nfbins, N);
tfr_k = zeros(Nfbins, N);
tfr_kp1 = zeros(Nfbins, N);
tfr_kp2 = zeros(Nfbins, N);
if q_method == 4
order3 = true;
end
if order3
tfr_kp3 = zeros(Nfbins, N);
end
if k >=2, tfr_km1 = zeros(Nfbins, N); end
if k >=3, tfr_km2 = zeros(Nfbins, N); end
below = zeros(1,N);
over = zeros(1,N);
for m = 1:Nfbins,
lambda = (mi+m-1)/M;
pTs = -1.0/L + 1i*2*pi*lambda;
alpha = exp(pTs);
[a,b] = Gk2(k, L, alpha); tfr_k(m,:) = filter(b,a,x);
[a,b] = Gk2(k+1, L, alpha); tfr_kp1(m,:) = filter(b,a,x);
[a,b] = Gk2(k+2, L, alpha); tfr_kp2(m,:) = filter(b,a,x);
if order3
[a,b] = Gk2(k+3, L, alpha); tfr_kp3(m,:) = filter(b,a,x);
end
if k >=2, [a,b] = Gk2(k-1, L, alpha); tfr_km1(m,:) = filter(b,a,x); end
if k >=3, [a,b] = Gk2(k-2, L, alpha); tfr_km2(m,:) = filter(b,a,x); end
tfr(m,:) = tfr_k(m,:);
for n = 1:N
if (abs(tfr_k(m,n)) > eps)
Tsm1_yTg = k*L*tfr_kp1(m,n);
Tsm2_yT2g = k*(k+1)*L^2*tfr_kp2(m,n);
if k >= 2
Ts_yDg = tfr_km1(m,n)/L + pTs*tfr_k(m,n);
elseif k == 1
Ts_yDg = pTs*tfr_k(m,n);
end
if k >= 3
Ts2_yD2g = tfr_km2(m,n)/L^2 + 2*pTs*tfr_km1(m,n)/L + pTs^2*tfr_k(m,n);
elseif k == 2
Ts2_yD2g = 2*pTs/L * tfr_km1(m,n) + pTs^2*tfr_k(m,n);
elseif k == 1
Ts2_yD2g = pTs^2*tfr_k(m,n);
end
yDTg = k*tfr_k(m,n) + k*L*pTs*tfr_kp1(m,n);
yTDg = yDTg - tfr_k(m,n); %yTDg = (k-1)*tfr_k(m,n) + k*L*pTs*tfr_kp1(m,n);
if order3
Tsm3_yT3g = L^3 * k * (k+1) * (k+2) * tfr_kp3(m,n); %L * (k+2)*Tsm2_yT2g;
%Tsm3_yT3g = L * (k+2)*Tsm2_yT2g;
Tsm1_yT2Dg = L * k * (k-1) * tfr_kp1(m,n) + pTs * L^2 * k * (k+1) * tfr_kp2(m,n);
Tsm1_yDT2g = 2 * Tsm1_yTg + Tsm1_yT2Dg;
end
%% used by LM reassignment algorithm
%Tsm1_dpsi_domega = real(Tsm1_yTg /tfr_k(m,n));
%Ts_dpsi_dt = imag(Ts_yDg /tfr_k(m,n));
%d2psi_dtdomega = real(yDTg /tfr_k(m,n) - (Ts_yDg * Tsm1_yTg)/ tfr_k(m,n)^2);
%Ts2_d2psi_dt2 = imag(Ts2_yD2g /tfr_k(m,n) - (Ts_yDg / tfr_k(m,n))^2);
%Tsm2_d2psi_domega2 = -imag(Tsm2_yT2g/tfr_k(m,n) - (Tsm1_yTg / tfr_k(m,n))^2);
ntilde = n - Tsm1_yTg/tfr_k(m,n);
nhat = round(real(ntilde));
%nhat = n - round(real(Tsm1_yTg/tfr_k(m,n)));
%nhat = n - real(Tsm1_yTg/tfr_k(m,n));
mrtilde = M*(Ts_yDg/tfr_k(m,n))/(2.0*pi);
mrhat = round(imag(mrtilde)) - mi+1;
%mrhat = round(M*imag(Ts_yDg/tfr_k(m,n))/(2.0*pi)) - mi;
%mrhat = M*imag(Ts_yDg/tfr_k(m,n))/(2.0*pi) - mi;
if q_method == 1 % CRE1
q_hat_denum = imag(Tsm1_yTg * conj(tfr_k(m,n)));
elseif q_method == 2 % CRE2_t
q_hat_denum = (Tsm1_yTg * Ts_yDg - (yDTg-tfr_k(m,n)) * tfr_k(m,n) );
elseif q_method == 3 %CRE2_w
q_hat_denum = Tsm1_yTg^2 - Tsm2_yT2g * tfr_k(m,n);
elseif q_method == 4 % CRE_robust
% b2 = 1j * (Tsm3_yT3g * tfr_k(m,n) - Tsm2_yT2g * Tsm1_yTg) / tfr_k(m,n)^2;
% a1 = (Tsm1_yTg * Ts_yDg - yTDg * tfr_k(m,n)) / tfr_k(m,n)^2;
%
% b1 = (Tsm1_yDT2g * tfr_k(m,n) - Tsm2_yT2g * Ts_yDg) / tfr_k(m,n)^2;
% %b1 = (Tsm1_yT2Dg * tfr_k(m,n) - Tsm2_yT2g * Ts_yDg) / tfr_k(m,n)^2;
% a2 = 1j * (Tsm1_yTg^2 - Tsm2_yT2g * tfr_k(m,n)) / tfr_k(m,n)^2;
%
% q_hat_denum = b2 * a1 - b1 * a2;
Ar = [Tsm2_yT2g -Tsm1_yTg tfr_k(m,n);...
Tsm1_yDT2g -yTDg Ts_yDg;...
Tsm3_yT3g -Tsm2_yT2g Tsm1_yTg];
%Tsm1_yDT2g Tsm1_yT2Dg
q_hat_denum = det(Ar);
end
if abs(q_hat_denum) > q_threshold %% check if signal is FM
if q_method == 1 %CRE1
alpha_hat = real(Ts_yDg * conj(tfr_k(m,n))) / q_hat_denum;
q_hat = 1j * alpha_hat;
%q_hat = (mrtilde - 1j*mi)/ntilde;
elseif q_method == 2 %CRE2_t
%q_hat = imag((Ts2_yD2g * tfr_k(m,n) - Ts_yDg^2) / q_hat_denum); %modified
q_hat = (Ts2_yD2g * tfr_k(m,n) - (Ts_yDg)^2) / q_hat_denum; %modified
alpha_hat = imag(q_hat);
elseif q_method == 3 %CRE2_w
q_hat = (yTDg * tfr_k(m,n) + tfr_k(m,n).^2 - Tsm1_yTg * Ts_yDg) / q_hat_denum;
alpha_hat = imag(q_hat);
elseif q_method == 4 %CRE2_robust
%y1 = (Ts2_yD2g * tfr_k(m,n) - Ts_yDg^2) / tfr_k(m,n)^2;
%y2 = 1j * ((yTDg * tfr_k(m,n)) - Ts_yDg * Tsm1_yTg) / tfr_k(m,n)^2;
u = [Ts_yDg Ts2_yD2g yTDg+tfr_k(m,n)].';
Ar_inv = pinv(Ar);
rx = Ar_inv(1,:) * u;
q_hat = Ar_inv(2,:) * u - 2 * rx * (n-ntilde);
if abs(q_hat) > 1/q_threshold
q_hat = 0;
end
%q_hat = (b2*y1 - b1 * y2) / q_hat_denum;
alpha_hat = imag(q_hat);
end
else
q_hat = 0;
alpha_hat = 0;
end
q_hat_map(m,n) = q_hat;
%% IF 1, biased when Ax is not constant
if if_method == 1
%% normalized frequency
if_map(m,n) = (imag(mrtilde) + M / (2*pi) * alpha_hat * real(n-ntilde)-mi)/M;
mrhat_q = round(imag(mrtilde) + M / (2*pi) * alpha_hat * real(n-ntilde))-mi+1;
%%mrhat_q = round(mrhat + M / (2*pi) * q_hat * (n-nhat))
%% IF 2, unbiased
elseif if_method == 2
%mrhat_q = imag(mrtilde + round(M / (2*pi) * q_hat * (n-ntilde)))-mi;
if_map(m,n) = (imag(mrtilde + M / (2*pi) * q_hat * (n-ntilde))-mi)/M;
mrhat_q = round(imag(mrtilde + M / (2*pi) * q_hat * (n-ntilde)))-mi+1;
%mrhat_q = mrhat + round(imag(M / (2*pi) * q_hat) *real(n-ntilde)); %-mi;
elseif if_method == 3
%% IF 3, (classical IF (reassignment operator))
mrhat_q = mrhat;
if_map(m,n) = (imag(mrtilde)-mi)/M;
end
%denom = (mu + d2psi_dtdomega) * (1 + mu - d2psi_dtdomega) + Ts2_d2psi_dt2 * Tsm2_d2psi_domega2;
%nhat = n - round((Tsm1_dpsi_domega * (1 + mu - d2psi_dtdomega) - (Tsm2_d2psi_domega2) * (2 * pi * lambda- Ts_dpsi_dt))/denom );
%Ts_omega_tilde = ((mu^2-d2psi_dtdomega^2) + Ts2_d2psi_dt2 * Tsm2_d2psi_domega2)*2*pi*lambda/denom;
%Ts_omega_tilde = Ts_omega_tilde - (Tsm1_dpsi_domega * Ts2_d2psi_dt2 - (mu + d2psi_dtdomega)*Ts_dpsi_dt)/denom;
%mrhat = round(M*Ts_omega_tilde/(2.0*pi))-mi+1; % eq. 25
% %% correct m bounds
% if (mi+mrhat_q>M), mrhat_q=mrhat_q-M; end;
% if (mrhat_q<0), mrhat_q=0; end;
% if (mrhat_q>Nfbins), mrhat_q=Nfbins+1; end;
%
%% correct m bounds
%if (mi+mrhat>M), mrhat=mrhat-M; end;
if (mrhat<0), mrhat=1; end;
if (mrhat>Nfbins), mrhat=Nfbins; end;
if (nhat<0), nhat=0; end;
if (nhat>N), nhat=N+1; end;
if (mrhat==0)
below(n) = below(n) + tfr(m,n);
elseif (mrhat==Nfbins+1)
over(n) = over(n) + tfr(m,n);
elseif (nhat==0 || nhat > N)
% skipped
else
%stfr(mrhat,nhat) = stfr(mrhat,nhat) + tfr(m,n) * exp(1i*pi * (mrhat + ma(m))/M * (nhat-n)) * exp(-2*pi*1i*ma(m)/M * n0);
stfr(mrhat,nhat) = stfr(mrhat,nhat) + tfr(m,n) * exp(1i*pi * ( 2*(mrhat) - alpha_hat * (nhat-n))*(nhat-n)/M) * exp(-2*pi*1i*ma(m)/M * (n0));
% M / (2*pi)
%stfr(mrhat,nhat) = stfr(mrhat,nhat) + tfr(m,n);
%stfr(mrhat_q, n) = stfr(mrhat_q, n) + tfr(m,n) * exp(-2*pi*1i*ma(m)/M * n0);
end;
end;
end;
end;
end