There is certainly differences (usually of a factor of PI) in the various
definitions used for PSDs, but a simple sign wave has an infinite power
density at the sine wave frequency. Are we agreed on that?
Use of windowing will modify this comment somewhat (so it probably won't
really go to infinity) but the basic fact remains. The units of a PSD are
amp^2/Hz. The MS of a signal between two frequencies should equal the
area under the PSD between those frequencies (with allowance for different
definitions/factors of PI). As I said, for a sign wave the frequency band
can be made arbitrarily small about the sine wave frequency, but the power
between these bands remains constant. Therefore the PSD goes to infinity.
Otherwise it isn't a density.
Joseph Park <[EMAIL PROTECTED]>
Sent by: [EMAIL PROTECTED]
26/10/2007 10:49 AM
To
cc
matplotlib-users@lists.sourceforge.net
Subject
Re: [Matplotlib-users] PSD amplitudes
is the suggestion that the matplotlib algorithm is correct in computing
PSD amplitudes?
btw, increasing nFFT increases the number of points used in the FFT, which
increases the spectral frequency resolution (smaller binwidth) but for a
limited data set
of N points, as is the case in the example, decreases the number of data
averages
thereby decreasing the spectral amplitude resolution (accuracy). keep in
mind that
just changing nFFT without making a corresponding change in overlap will
oversample
the data, thereby skewing the amplitudes.
in any case, the amplitude change is not approaching infinity, even if you
set nFFT to
6000, which is the length of the timeseries, the amplitudes are ~35dB,
adjust variable ymax
to see this.
to review issues of spectral/amplitude resolution, windowing/overlap, etc,
a good
reference is Random Data by Bendat &Piersol:
http://www.amazon.com/Random-Data-Analysis-Measurement-Procedures/dp/0471317330
i remain unconvinced that the PSD amplitudes are reasonable, which only
leaves Matlab
as an alternative... that's a hard pill to swallow... matplotlib is
clearly preferable.
[EMAIL PROTECTED] wrote:
If you lower the resolution (ie increase nFFT) in your program you will
see that the PSD does indeed increase. I think it may be on the way to
infinity.
Joseph Park <[EMAIL PROTECTED]>
Sent by: [EMAIL PROTECTED]
26/10/2007 10:05 AM
To
matplotlib-users@lists.sourceforge.net
cc
Subject
Re: [Matplotlib-users] PSD amplitudes
Shouldn't the PSD for a simple sine wave tend to infinity
the spectral resolution will impact the amplitude, if you
are not dealing with a density. by definition a spectral density
has applied the bandwidth resolution correction. the PSD amplitude
should correspond to the RMS amplitude of the sine wave. in the
example a 1VRMS amplitude sine wave (time domain) should have a
PSD power of 20*log(1V) = 0dB. The windowing function will impact
this ideal number a bit, but certainly not by 25dB.
[EMAIL PROTECTED] wrote:
Are you sure that the answer should be zero? Shouldn't the PSD for a
simple sine wave tend to infinity (depending on the resolution)?
Joseph Park <[EMAIL PROTECTED]>
Sent by: [EMAIL PROTECTED]
26/10/2007 06:50 AM
To
matplotlib-users@lists.sourceforge.net
cc
Subject
[Matplotlib-users] PSD amplitudes
Please try the attached script.
The answer should be ~0 dB for each of the frequencies.
Most likely a simple scaling issue/parameter of which i'm ignorant.
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##----------------------------------------------------------------------------
## Name: psd_scale.py
##
## Purpose: Test Power Spectral Density of 1Vrms data
## Depends on Python SciPy and NumPy
##
## Author: J Park
##
## Created: 10/17/07
##
## Modified:
##----------------------------------------------------------------------------
try:
from numpy import * # www.numpy.org numpy.scipy.org
except ImportError:
print "Failed to import numpy."
try:
import pylab as mp # matplotlib.sourceforge.net
from matplotlib.font_manager import fontManager, FontProperties
except ImportError:
print "Failed to import pylab."
# Default Parameters
nFFT = 1024
overlap = 512
freqSample = 100.
PlotAll = False
WriteOutput = False
##----------------------------------------------------------------------------
## Main module
def main():
deltaF = freqSample/nFFT # Frequency resolution in Hz
deltaT = 1./freqSample # Sample interval
print 'Sample interval %e (s)' % (deltaT)
print 'Frequency resolution %e (Hz)' % (deltaF)
# Setup Plots
# ----------------------------------------------------------------------
mp.figure(1)
mp.title ( "PSD" )
mp.ylabel( "(dB)" )
mp.xlabel( "Frequency (Hz)" )
legendFont = FontProperties(size='small')
ymin = 0
ymax = 30
xmin = 0
xmax = 50
xticks = 5
yticks = 5
if PlotAll:
mp.figure(2)
mp.title ( "Input Timeseries" )
mp.ylabel( "Amplitude" )
mp.xlabel( "time (s)" )
# Create some synthetic data with unity RMS amplitude = 0 dB
# ----------------------------------------------------------------------
t = mp.arange(0., 60., deltaT) # 60 seconds at deltaT interval
A = 1.414
y0 = A * sin( 2. * math.pi * 5 * t )
y1 = A * sin( 2. * math.pi * 10 * t )
y2 = A * sin( 2. * math.pi * 20 * t )
y3 = A * sin( 2. * math.pi * 30 * t )
y4 = A * sin( 2. * math.pi * 40 * t )
y5 = A * sin( 2. * math.pi * 45 * t )
dataList = [ y0, y1, y2, y3, y4, y5 ]
for data in dataList:
inputDataLen = len( data )
numAverages = math.floor( inputDataLen / (overlap) ) - 1
normalizedRandomError = 1./math.sqrt( numAverages )
print "%d points" % ( inputDataLen ),
print "%d averages" % (numAverages),
print "normalized random error %.3f" % ( normalizedRandomError )
mp.figure(1)
(Pxx, freqs) = mp.psd( data,
NFFT = nFFT,
Fs = freqSample,
noverlap = overlap,
lw = 2,
label = '' )
Pxx_dB = 10.*log10(Pxx)
if PlotAll:
mp.figure(2)
mp.plot(t, data, label='' )
# Write Output data
#
----------------------------------------------------------------------
if WriteOutput:
PxxLen = len(Pxx)
OutputFile = "PSD.dat"
fdOutFile = open( OutputFile, 'a' )
fdOutFile.write( "Freq\t\tPower(dB)\n" )
for i in range(PxxLen):
fdOutFile.write( "%.4e\t%.3f\n" % ( freqs[i], Pxx_dB[i] ) )
fdOutFile.close()
print "Wrote ", PxxLen, " points to ", OutputFile
# Show the Plot
# ----------------------------------------------------------------------
mp.figure(1)
mp.axis([xmin, xmax, ymin, ymax])
mp.xticks( arange(xmin, xmax+1, xticks) )
mp.yticks( arange(ymin, ymax , yticks) )
mp.title('')
mp.xlabel('Frequency (Hz)')
mp.ylabel(r'$\tt{dB re V^2/Hz}$')
#mp.legend( loc='upper right', prop=legendFont )
if WriteOutput:
plotFileName = "PSD.png"
mp.savefig( plotFileName )
print "Wrote png image to ", plotFileName
if PlotAll:
mp.figure(2)
#mp.legend( loc='lower left', prop=legendFont )
mp.show()
print "Normal Exit"
## Main module
##----------------------------------------------------------------------------
##----------------------------------------------------------------------------
## Provide for cmd line invocation
if __name__ == "__main__":
main()
-------------------------------------------------------------------------
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email, nor take or omit to take any action in reliance on it. United Group
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