spectral density is by convention a 1Hz
binwidth, not an arbitrary one, units of A^2/Hz.
perhaps if you manually compute the spectral density of a sine wave,
you will easily see
that they don't have infinite power, R is the autocorrelation of the
Asin(wt):
Back to the original question:
Is there evidence that the matplotlib PSD spectral amplitudes are
accurate?
say by comparison with Matlab results, or a synthetic signal as in the
example, or
from considerations of basic DSP as in the references?
[EMAIL PROTECTED] wrote:
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.
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.
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)?
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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