Example2
The second example uses a quadratic trend, a chirp signal, and a third mode with sharp transition between two constant frequencies
$f_{\mathrm{Sig} 2}(t)=6 t^{2}+\cos \left(10 \pi t+10 \pi t^{2}\right)+\left{\begin{array}{ll} \cos (60 \pi t) & t \leq 0.5 \ \cos (80 \pi t-10 \pi) & t>0.5 \end{array}\right. $
For $t \in [0,1]$the chirp’s instantaneous frequency varies linearly between $10\pi$ an $30\pi$.Consequently, the theoretical center frequency of the mode is located at $20\pi$ . The piecewise-constant bi-harmonic has spectral peaks expected at $60\pi$ and $80\pi$
Again, with VMD the estimated center frequencies con- verge to the expected frequencies precisely. Here, we chose to decompose into four modes, thus assigning each half of the piecewise-constant frequency signal to a separate mode. The spectral partitioning can be nicely appreciated in the spectral plot of the different modes. Here, EMD does a better job recov- ering the quadratic trend, and, correspondingly, the first oscilla- tion. However, EMD is unable to separate the two pieces of the piecewise constant frequency signal. .
julia> using VMD,Random,Plots
julia> T = 1000;
julia> t = (1:T)/T;
julia> sample_frequency = 1000;
julia> # center frequencies of components
f_1 = 10;
julia> f_2 = 60;
julia> f_3 = 80;
julia> # modes
v_1 = @. 6t^2;
julia> v_2 = @. cos(10π*t+10π*t^2)
1000-element Array{Float64,1}:
0.9995055730743954
0.9980188300587944
0.995535316310215
0.9920515766278236
0.9875651726383378
0.9820746992776882
0.9755798003302669
0.9680811829874794
0.9595806313877643
0.9500810191007717
⋮
0.730343516655704
0.7910975728967998
0.8449333912108219
0.8913628126476093
0.9299614080033781
0.9603725301679537
0.9823107900032507
0.995564920611647
1.0
julia> v_3 = [t1>=0.5 ? cos(60π*t1) : cos(80π*t1-10π) for t1 in t]
1000-element Array{Float64,1}:
0.9685831611286312
0.8763066800438627
0.7289686274214112
0.5358267949789944
0.3090169943749463
0.06279051952931364
-0.18738131458572646
-0.42577929156507305
-0.6374239897486917
-0.8090169943749481
⋮
0.06279051952930043
0.248689887164824
0.42577929156505256
0.5877852522924629
0.72896862742139
0.8443279255020033
0.9297764858882469
0.9822872507286828
1.0
julia> # composite signal, including noise
f = v_1 + v_2 + v_3 + 0.1*randn(length(v_1));
julia> # some sample parameters for VMD
alpha = 2000; # moderate bandwidth constraint
julia> tau = 0; # noise-tolerance (no strict fidelity enforcement)
julia> K = 4; # 3 modes
julia> DC = false; # no DC part imposed
julia> init = 0; # initialize omegas uniformly
julia> tol = 1e-7;
julia> v = vmd(f ; alpha = alpha,tau = tau,K = K,DC = false,init = init,tol = tol,sample_frequency = sample_frequency)
--iteration times 106 -- error 9.007464294213465e-8
julia> # the first mode frequency
print("1st mode frequency $(n_mode(v,1))")
1st mode frequency 0.16481852051024123
julia> p0 = VMD.plot(v,k=0)
Plot{Plots.GRBackend() n=2}
julia> savefig(p0,"2_0.png")
/home/runner/.julia/packages/GR/BwGt2/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application
GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
julia> p1 = VMD.plot(v,k=1)
Plot{Plots.GRBackend() n=2}
julia> savefig(p1,"2_1.png")
/home/runner/.julia/packages/GR/BwGt2/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application
GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
julia> p2 = VMD.plot(v,k=2)
Plot{Plots.GRBackend() n=2}
julia> savefig(p2,"2_2.png")
/home/runner/.julia/packages/GR/BwGt2/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application
GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
julia> p3 = VMD.plot(v,k=3)
Plot{Plots.GRBackend() n=2}
julia> savefig(p3,"2_3.png")
/home/runner/.julia/packages/GR/BwGt2/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application
GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
julia> p4 = VMD.plot(v,k=4)
Plot{Plots.GRBackend() n=2}
julia> savefig(p4,"2_4.png")
/home/runner/.julia/packages/GR/BwGt2/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application
GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WSplot the original signal and spectrum
plot the 1st decomposed signal and spectrum
plot the 2st signal and spectrum
plot the 3st decomposed signal and spectrum
plot the 4st decomposed signal and spectrum
