Window selection of the pseudo WV and the adaptive algorithm

last update: 9/30 (2017)



In [1]:

    
using Distributions
import DSP



In [ ]:

    
include("../juwvid.jl")









    Out[ ]:





juwvid

Window Size dependence of the Pseudo Wigner Ville Distribution

generating the signal with the nonlinear IF



In [3]:

    
nsample=1024
x=collect(linspace(-100,100,nsample));
ifreq=exp.(-(x.*x)/100) + 1;
PyPlot.plot(x,ifreq)
PyPlot.xlabel("t")
PyPlot.ylabel("input instantaneous frequency")









    












    Out[3]:





PyObject Text(24,0.5,'input instantaneous frequency')



In [4]:

    
iw=2*pi*ifreq;
phase=cumsum(iw)/nsample*(x[end]-x[1]);
y=cos.(phase);
PyPlot.plot(x,y)









    












    Out[4]:





1-element Array{PyCall.PyObject,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f847e61da20>



In [5]:

    
z=DSP.Util.hilbert(y);
tfrp4=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",4);
indf4=extif.maxif(abs.(tfrp4));
fn4=juwutils.index_to_frequency(indf4, NaN, x[2]-x[1],nsample);

tfrp8=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",8);
indf8=extif.maxif(abs.(tfrp8));
fn8=juwutils.index_to_frequency(indf8, NaN, x[2]-x[1],nsample);

tfrp16=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",16);
indf16=extif.maxif(abs.(tfrp16));
fn16=juwutils.index_to_frequency(indf16, NaN, x[2]-x[1],nsample);

tfrp32=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",32);
indf32=extif.maxif(abs.(tfrp32));
fn32=juwutils.index_to_frequency(indf32, NaN, x[2]-x[1],nsample);









    



Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.



In [6]:

    
fig=PyPlot.figure()
ax = fig[:add_subplot](2,2,1)
PyPlot.plot(x,fn4,color="red")
PyPlot.plot(x,fn8,color="yellow")
PyPlot.plot(x,fn16,color="green")
PyPlot.plot(x,fn32,color="blue")
PyPlot.plot(x,ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(0.9,2.1)
PyPlot.title("instantaneous frequency")
ax = fig[:add_subplot](2,2,2)
PyPlot.plot(x,ifreq-fn4,color="red")
PyPlot.plot(x,ifreq-fn8,color="yellow")
PyPlot.plot(x,ifreq-fn16,color="green")
PyPlot.plot(x,ifreq-fn32,color="blue")
PyPlot.plot(x,ifreq-ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(-0.3,0.1)
PyPlot.title("Difference")
ax = fig[:add_subplot](2,2,3)
PyPlot.plot(x,fn4,color="red")
PyPlot.plot(x,fn8,color="yellow")
PyPlot.plot(x,fn16,color="green")
PyPlot.plot(x,fn32,color="blue")
PyPlot.plot(x,ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(0.9,2.1)
PyPlot.xlim(-20.0,20.0)
PyPlot.xlabel("t")
ax = fig[:add_subplot](2,2,4)
PyPlot.plot(x,ifreq-fn4,color="red")
PyPlot.plot(x,ifreq-fn8,color="yellow")
PyPlot.plot(x,ifreq-fn16,color="green")
PyPlot.plot(x,ifreq-fn32,color="blue")
PyPlot.plot(x,ifreq-ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(-0.3,0.1)
PyPlot.xlim(-20.0,20.0)
PyPlot.xlabel("t")









    












    Out[6]:





PyObject Text(0.5,24,'t')



In [7]:

    
# Figure 3
fig=PyPlot.figure()
ax = fig[:add_subplot](2,2,1)
a=juwplot.tfrshow(abs.(tfrp4),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.ylabel("frequency")
PyPlot.title("width = N/4")
ax = fig[:add_subplot](2,2,2)
a=juwplot.tfrshow(abs.(tfrp8),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.title("width = N/8")
ax = fig[:add_subplot](2,2,3)
a=juwplot.tfrshow(abs.(tfrp16),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.xlabel("t")
PyPlot.ylabel("frequency")
PyPlot.title("width = N/16")
ax = fig[:add_subplot](2,2,4)
a=juwplot.tfrshow(abs.(tfrp32),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.xlabel("t")
PyPlot.title("width = N/32")









    












    Out[7]:





PyObject Text(0.5,1,'width = N/32')

ADDING Gaussian noises



In [8]:

    
using Distributions
d = Normal()
ynoise=y+rand(d,nsample)*std(y)*0.5
PyPlot.plot(x,ynoise)









    












    Out[8]:





1-element Array{PyCall.PyObject,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f847b5ed2b0>



In [9]:

    
z=DSP.Util.hilbert(ynoise);
tfrp4=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",4);
indf4=extif.maxif(abs.(tfrp4));
fn4=juwutils.index_to_frequency(indf4, NaN, x[2]-x[1],nsample);

tfrp8=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",8);
indf8=extif.maxif(abs.(tfrp8));
fn8=juwutils.index_to_frequency(indf8, NaN, x[2]-x[1],nsample);

tfrp16=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",16);
indf16=extif.maxif(abs.(tfrp16));
fn16=juwutils.index_to_frequency(indf16, NaN, x[2]-x[1],nsample);

tfrp32=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",32);
indf32=extif.maxif(abs.(tfrp32));
fn32=juwutils.index_to_frequency(indf32, NaN, x[2]-x[1],nsample);









    



Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.
Single pseudo Wigner Ville
Use fft.
Assuming nft = nsample.



In [10]:

    
fig=PyPlot.figure()
ax = fig[:add_subplot](2,2,1)
PyPlot.plot(x,fn4,color="red")
PyPlot.plot(x,fn8,color="yellow")
PyPlot.plot(x,fn16,color="green")
PyPlot.plot(x,fn32,color="blue")
PyPlot.plot(x,ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(0.9,2.1)
PyPlot.title("instantaneous frequency")
ax = fig[:add_subplot](2,2,2)
PyPlot.plot(x,ifreq-fn4,color="red")
PyPlot.plot(x,ifreq-fn8,color="yellow")
PyPlot.plot(x,ifreq-fn16,color="green")
PyPlot.plot(x,ifreq-fn32,color="blue")
PyPlot.plot(x,ifreq-ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(-0.3,0.1)
PyPlot.title("Difference")
ax = fig[:add_subplot](2,2,3)
PyPlot.plot(x,fn4,color="red")
PyPlot.plot(x,fn8,color="yellow")
PyPlot.plot(x,fn16,color="green")
PyPlot.plot(x,fn32,color="blue")
PyPlot.plot(x,ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(0.9,1.1)
PyPlot.xlim(-100.0,-30.0)
PyPlot.xlabel("t")
ax = fig[:add_subplot](2,2,4)
PyPlot.plot(x,ifreq-fn4,color="red")
PyPlot.plot(x,ifreq-fn8,color="yellow")
PyPlot.plot(x,ifreq-fn16,color="green")
PyPlot.plot(x,ifreq-fn32,color="blue")
PyPlot.plot(x,ifreq-ifreq,lw=2,ls="dotted",color="black")
PyPlot.ylim(-0.05,0.05)
PyPlot.xlim(-100.0,-30.0)
PyPlot.xlabel("t")









    












    Out[10]:





PyObject Text(0.5,24,'t')



In [11]:

    
fig=PyPlot.figure()
ax = fig[:add_subplot](2,2,1)
a=juwplot.tfrshow(abs.(tfrp4),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.ylabel("frequency")
PyPlot.title("width = N/4")
ax = fig[:add_subplot](2,2,2)
a=juwplot.tfrshow(abs.(tfrp8),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.title("width = N/8")
ax = fig[:add_subplot](2,2,3)
a=juwplot.tfrshow(abs.(tfrp16),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.xlabel("t")
PyPlot.ylabel("frequency")
PyPlot.title("width = N/16")
ax = fig[:add_subplot](2,2,4)
a=juwplot.tfrshow(abs.(tfrp32),x[2]-x[1],x[1],x[end],NaN,NaN,0.7)
PyPlot.xlabel("t")
PyPlot.title("width = N/32")









    












    Out[11]:





PyObject Text(0.5,1,'width = N/32')

An adaptive algorithm of pseudo Wigner-Ville Distribution

This sample aims to reproduce Figure 5.3 of the book "Time-frequency signal analysis with applications" by Stankovic, Dakovic, and Thayaparan.

Generating sample data with Gaussian noise



In [12]:

    
nsample=1024
x,z,iw,ynorm=sampledata.genstepfm(nsample);
d = Normal()
ynoise=real(z)+rand(d,nsample)/3.16
nsample=length(x);dx=x[2]-x[1]
PyPlot.plot(x,ynoise)









    












    Out[12]:





1-element Array{PyCall.PyObject,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f847d603240>

Input Instantaneous Frequency



In [13]:

    
PyPlot.plot(x,iw/(2*pi),lw=3)
PyPlot.title("no noise")
PyPlot.xlabel("time")
PyPlot.ylabel("Instantaneous frequency")









    












    Out[13]:





PyObject Text(24,0.5,'Instantaneous frequency')

Wigner Ville Distribution



In [14]:

    
### one can see large artifact in the middle of time.
z=DSP.Util.hilbert(ynoise);
tfr=cohenclass.tfrwv(z);
indfwv=extif.maxif(abs.(tfr));

fig=PyPlot.figure()
a=juwplot.tfrshow(abs.(tfr),dx,x[1],x[end],NaN,NaN,1.0)
PyPlot.xlabel("time")
PyPlot.ylabel("frequency")
PyPlot.ylim(0,250)









    



Single Wigner Ville






    












    



Use fft.






    Out[14]:





(0, 250)

The Pseudo Wigner Ville with N/4 window and N/16

you can see a smoothing effect for a narrow width



In [15]:

    
# window size=256 (default window size of tfrpwv=N/4 )
tfrp=cohenclass.tfrpwv(z)
indfpwv256=extif.maxif(abs.(tfrp))
fig=PyPlot.figure()
a=juwplot.tfrshow(abs.(tfrp),dx,x[1],x[end],NaN,NaN,1.0)
PyPlot.xlabel("time")
PyPlot.ylabel("frequency")
PyPlot.ylim(0,250)









    



Single pseudo Wigner Ville






    












    



Use fft.






    Out[15]:





(0, 250)



In [16]:

    
# window size=N/8 
tfrp=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",8)
indfpwv128=extif.maxif(abs.(tfrp))
fig=PyPlot.figure()
a=juwplot.tfrshow(abs.(tfrp),dx,x[1],x[end],NaN,NaN,1.0)
PyPlot.xlabel("time")
PyPlot.ylabel("frequency")
PyPlot.ylim(0,250)









    



Single pseudo Wigner Ville






    












    



Use fft.






    Out[16]:





(0, 250)



In [17]:

    
# window size = N/32
tfrp=cohenclass.tfrpwv(z,NaN,NaN,NaN,NaN,NaN,0,"fft",32)
indfpwv32=extif.maxif(abs.(tfrp))
fig=PyPlot.figure()
a=juwplot.tfrshow(abs.(tfrp),dx,x[1],x[end],NaN,NaN,1.0)
PyPlot.xlabel("time")
PyPlot.ylabel("frequency")
PyPlot.ylim(0,250)









    



Single pseudo Wigner Ville






    












    



Use fft.






    Out[17]:





(0, 250)

In summary, there is a trade-off relation. The larger window size exhibits more artifact and finer resolution. The small window size exhibits less artifact and poor resolution.

Then, the adaptive window solves the trade-off



In [18]:

    
fin=collect(linspace(1,512,1024));
varrat=ones(length(x))./(3.16*3.16);



In [19]:

    
### kappa=2.0, delta kappa=0.39, 
### ndiv_static_crit=4 i.e. accept the criterion if delta frequency < nsample/4
### use Hamming window
include("../pwvaw.jl")
tfra,windows=pwvaw.awpwv(z,varrat,NaN,NaN,fin,NaN,2.0,0.39,4,"Hamming");









    



Single pseudo Wigner Ville with the Adaptive Algorithm






    



WARNING: replacing module pwvaw
WARNING: abs(x::AbstractArray{T}) where T <: Number is deprecated, use abs.(x) instead.
Stacktrace:
 [1] depwarn(::String, ::Symbol) at ./deprecated.jl:70
 [2] abs(::Array{Complex{Float64},1}) at ./deprecated.jl:57
 [3] awpwv(::Array{Complex{Float64},1}, ::Array{Float64,1}, ::Float64, ::Float64, ::Array{Float64,1}, ::Float64, ::Float64, ::Float64, ::Int64, ::String, ::Int64, ::String, ::Bool) at /home/kawahara/juwvid/pwvaw.jl:96
 [4] awpwv(::Array{Complex{Float64},1}, ::Array{Float64,1}, ::Float64, ::Float64, ::Array{Float64,1}, ::Float64, ::Float64, ::Float64, ::Int64, ::String) at /home/kawahara/juwvid/pwvaw.jl:21
 [5] include_string(::String, ::String) at ./loading.jl:522
 [6] execute_request(::





    



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ZMQ.Socket, ::IJulia.Msg) at /home/kawahara/anaconda3/share/julia/site/v0.6/IJulia/src/execute_request.jl:180
 [7] (::Compat.#inner#14{Array{Any,1},IJulia.#execute_request,Tuple{ZMQ.Socket,IJulia.Msg}})() at /home/kawahara/anaconda3/share/julia/site/v0.6/Compat/src/Compat.jl:332
 [8] eventloop(::ZMQ.Socket) at /home/kawahara/anaconda3/share/julia/site/v0.6/IJulia/src/eventloop.jl:8
 [9] (::IJulia.##15#18)() at ./task.jl:335
while loading In[19], in expression starting on line 5






    



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In [20]:

    
fig=PyPlot.figure()
a=juwplot.tfrshow(abs.(tfra),dx,x[1],x[end],fin[1],fin[end],0.7)
PyPlot.xlabel("time")
PyPlot.ylabel("frequency")









    












    Out[20]:





PyObject Text(24,0.5,'frequency')



In [21]:

    
PyPlot.plot(collect(1:length(windows)),map(Float64,windows))
PyPlot.xlabel("time")
PyPlot.xlabel("Window size (pixel)")









    












    Out[21]:





PyObject Text(0.5,24,'Window size (pixel)')



In [22]:

    
indfn=extif.maxif(abs.(tfra));
fx=juwutils.index_to_frequency(indfn, fin, dx,nsample);









    



Assuming nft = nsample.

Comparison between WV, PWV (fixed window=256 and 32), and PWV (adaptive)

red is input
yellow (WV), green (pseudo WV, large window),orange (pseudo WV, small window)
blue (adaptive, pseudo WV)



In [ ]:

    
fwv=juwutils.index_to_frequency(indfwv,NaN, dx,nsample);
fwv32=juwutils.index_to_frequency(indfpwv32,NaN, dx,nsample);
fwv256=juwutils.index_to_frequency(indfpwv256,NaN, dx,nsample);
PyPlot.xlabel("time")
PyPlot.ylabel("instantaneous frequency")
PyPlot.plot(x,fwv,color="C0")
PyPlot.plot(x,fwv32,color="C4")
PyPlot.plot(x,fwv256,color="C2")
PyPlot.plot(x,fx,color="C3",lw=2)
PyPlot.plot(x,abs.(iw/(2*pi)),color="black",lw=3, ls="dotted")
PyPlot.legend(["WV","PWV 32","PWV 256","adaptive","input"])



In [ ]:



In [ ]: