def fft_convolve(fftd_data, fftd_kern):
"""Perform a convolution with the complex floating point vectors
'fftd_data' and 'fftd_kern'.
"""
# Note: The initial FFTs should be done like:
# fftd_kern = presto.rfft(kernel, -1)
# fftd_data = presto.rfft(data, -1)
prod = np.multiply(fftd_data, fftd_kern)
prod.real[0] = fftd_kern.real[0] * fftd_data.real[0]
prod.imag[0] = fftd_kern.imag[0] * fftd_data.imag[0]
return presto.rfft(prod, 1).astype(np.float32)
def make_fftd_kern(downfact, fftlen):
kern = np.zeros(fftlen, dtype=np.float32)
# These offsets produce kernels that give results
# equal to scipy.signal.convolve
if downfact % 2: # Odd number
kern[:downfact/2+1] += 1.0
kern[-(downfact/2):] += 1.0
else: # Even number
kern[:downfact/2+1] += 1.0
if (downfact > 2):
kern[-(downfact/2-1):] += 1.0
# The following normalization preserves the
# RMS=1 characteristic of the data
return presto.rfft(kern / np.sqrt(downfact), -1)
def do_trials(self, Ntrials):
for trial in xrange(Ntrials):
data = self.calc_time_series()
ft = rfft(data)
# pows = spectralpower(ft)/ft[0].real
# print 'Number of counts =', sum(data)
# print 'Max power =', max(pows)
# print 'Max freq =', argmax(pows)
# plotxy(pows)
# closeplot()
[hipow, hifreq, rderivs] = \
maximize_r(ft, self.freq, norm = ft[0].real)
self.meas_pows.append(hipow)
self.calc_props()
def do_trials(self, Ntrials):
for trial in range(Ntrials):
data = self.calc_time_series()
ft = rfft(data)
# pows = spectralpower(ft)/ft[0].real
# print 'Number of counts =', sum(data)
# print 'Max power =', max(pows)
# print 'Max freq =', argmax(pows)
# plotxy(pows)
# closeplot()
[hipow, hifreq, rderivs] = \
maximize_r(ft, self.freq, norm = ft[0].real)
self.meas_pows.append(hipow)
self.calc_props()
def fft_convolve(fftd_data, fftd_kern, lo, hi):
"""
fft_convolve(fftd_data, fftd_kern, lo, hi):
Perform a convolution with the complex floating point vectors
'fftd_data' and 'fftd_kern'. The returned vector will start at
at bin 'lo' (must be an integer), and go up to but not
include bin 'hi' (also an integer).
"""
# Note: The initial FFTs should be done like:
# fftd_kern = rfft(kernel, -1)
# fftd_data = rfft(data, -1)
prod = Num.multiply(fftd_data, fftd_kern)
prod.real[0] = fftd_kern.real[0] * fftd_data.real[0]
prod.imag[0] = fftd_kern.imag[0] * fftd_data.imag[0]
return rfft(prod, 1)[lo:hi].astype(Num.float32)
def fft_convolve(fftd_data, fftd_kern, lo, hi):
"""
fft_convolve(fftd_data, fftd_kern, lo, hi):
Perform a convolution with the complex floating point vectors
'fftd_data' and 'fftd_kern'. The returned vector will start at
at bin 'lo' (must be an integer), and go up to but not
include bin 'hi' (also an integer).
"""
# Note: The initial FFTs should be done like:
# fftd_kern = rfft(kernel, -1)
# fftd_data = rfft(data, -1)
prod = Num.multiply(fftd_data, fftd_kern)
prod.real[0] = fftd_kern.real[0] * fftd_data.real[0]
prod.imag[0] = fftd_kern.imag[0] * fftd_data.imag[0]
return rfft(prod, 1)[lo:hi].astype(Num.float32)
def make_fftd_kerns(downfacts, fftlen):
fftd_kerns = []
for downfact in downfacts:
kern = Num.zeros(fftlen, dtype=Num.float32)
# These offsets produce kernels that give results
# equal to scipy.signal.convolve
if downfact % 2: # Odd number
kern[:downfact / 2 + 1] += 1.0
kern[-(downfact / 2):] += 1.0
else: # Even number
kern[:downfact / 2 + 1] += 1.0
if (downfact > 2):
kern[-(downfact / 2 - 1):] += 1.0
# The following normalization preserves the
# RMS=1 characteristic of the data
fftd_kerns.append(rfft(kern / Num.sqrt(downfact), -1))
return fftd_kerns
def make_fftd_kerns(downfacts, fftlen):
fftd_kerns = []
for downfact in downfacts:
kern = Num.zeros(fftlen, dtype=Num.float32)
# These offsets produce kernels that give results
# equal to scipy.signal.convolve
if downfact % 2: # Odd number
kern[:downfact/2+1] += 1.0
kern[-(downfact/2):] += 1.0
else: # Even number
kern[:downfact/2+1] += 1.0
if (downfact > 2):
kern[-(downfact/2-1):] += 1.0
# The following normalization preserves the
# RMS=1 characteristic of the data
fftd_kerns.append(rfft(kern / Num.sqrt(downfact), -1))
return fftd_kerns
def smoothed(self, factor):
"""Return a smoothed numpy array of the chunk.
Overlap is removed.
Input:
factor: number of bins to smooth by.
"""
if factor == 1:
# No smoothing
smoothdata = self.data.filled()
elif factor > 1:
if self.fftd_data is None:
self.fftd_data = presto.rfft(self.data.filled(), -1)
kernel = fftd_kern_cache[factor]
smoothdata = utils.fft_convolve(self.fftd_data, kernel)
else:
raise ValueError("Smoothing factor must be larger than 0.")
return smoothdata[OVERLAP:-OVERLAP]
def main():
parser = OptionParser(usage)
parser.add_option(
"-x",
"--xwin",
action="store_true",
dest="xwin",
default=False,
help="Don't make a postscript plot, just use an X-window")
parser.add_option("-p",
"--noplot",
action="store_false",
dest="makeplot",
default=True,
help="Look for pulses but do not generate a plot")
parser.add_option(
"-m",
"--maxwidth",
type="float",
dest="maxwidth",
default=0.0,
help="Set the max downsampling in sec (see below for default)")
parser.add_option("-t",
"--threshold",
type="float",
dest="threshold",
default=5.0,
help="Set a different threshold SNR (default=5.0)")
parser.add_option("-s",
"--start",
type="float",
dest="T_start",
default=0.0,
help="Only plot events occuring after this time (s)")
parser.add_option("-e",
"--end",
type="float",
dest="T_end",
default=1e9,
help="Only plot events occuring before this time (s)")
parser.add_option("-g",
"--glob",
type="string",
dest="globexp",
default=None,
help="Process the files from this glob expression")
parser.add_option("-f",
"--fast",
action="store_true",
dest="fast",
default=False,
help="Use a faster method of de-trending (2x speedup)")
(opts, args) = parser.parse_args()
if len(args) == 0:
if opts.globexp == None:
print full_usage
sys.exit(0)
else:
args = []
for globexp in opts.globexp.split():
args += glob.glob(globexp)
useffts = True
dosearch = True
if opts.xwin:
pgplot_device = "/XWIN"
else:
pgplot_device = ""
fftlen = 8192 # Should be a power-of-two for best speed
chunklen = 8000 # Must be at least max_downfact less than fftlen
detrendlen = 1000 # length of a linear piecewise chunk of data for detrending
blocks_per_chunk = chunklen / detrendlen
overlap = (fftlen - chunklen) / 2
worklen = chunklen + 2 * overlap # currently it is fftlen...
max_downfact = 30
default_downfacts = [2, 3, 4, 6, 9, 14, 20, 30, 45, 70, 100, 150]
if args[0].endswith(".singlepulse"):
filenmbase = args[0][:args[0].rfind(".singlepulse")]
dosearch = False
elif args[0].endswith(".dat"):
filenmbase = args[0][:args[0].rfind(".dat")]
else:
filenmbase = args[0]
# Don't do a search, just read results and plot
if not dosearch:
info, DMs, candlist, num_v_DMstr = \
read_singlepulse_files(args, opts.threshold, opts.T_start, opts.T_end)
orig_N, orig_dt = int(info.N), info.dt
obstime = orig_N * orig_dt
else:
DMs = []
candlist = []
num_v_DMstr = {}
# Loop over the input files
for filenm in args:
if filenm.endswith(".dat"):
filenmbase = filenm[:filenm.rfind(".dat")]
else:
filenmbase = filenm
info = infodata.infodata(filenmbase + ".inf")
DMstr = "%.2f" % info.DM
DMs.append(info.DM)
N, dt = int(info.N), info.dt
obstime = N * dt
# Choose the maximum width to search based on time instead
# of bins. This helps prevent increased S/N when the downsampling
# changes as the DM gets larger.
if opts.maxwidth > 0.0:
downfacts = [
x for x in default_downfacts if x * dt <= opts.maxwidth
]
else:
downfacts = [x for x in default_downfacts if x <= max_downfact]
if len(downfacts) == 0:
downfacts = [default_downfacts[0]]
if (filenm == args[0]):
orig_N = N
orig_dt = dt
if useffts:
fftd_kerns = make_fftd_kerns(downfacts, fftlen)
if info.breaks:
offregions = zip([x[1] for x in info.onoff[:-1]],
[x[0] for x in info.onoff[1:]])
outfile = open(filenmbase + '.singlepulse', mode='w')
# Compute the file length in detrendlens
roundN = N / detrendlen * detrendlen
numchunks = roundN / chunklen
# Read in the file
print 'Reading "%s"...' % filenm
timeseries = Num.fromfile(filenm, dtype=Num.float32, count=roundN)
# Split the timeseries into chunks for detrending
numblocks = roundN / detrendlen
timeseries.shape = (numblocks, detrendlen)
stds = Num.zeros(numblocks, dtype=Num.float64)
# de-trend the data one chunk at a time
print ' De-trending the data and computing statistics...'
for ii, chunk in enumerate(timeseries):
if opts.fast: # use median removal instead of detrending (2x speedup)
tmpchunk = chunk.copy()
tmpchunk.sort()
med = tmpchunk[detrendlen / 2]
chunk -= med
tmpchunk -= med
else:
# The detrend calls are the most expensive in the program
timeseries[ii] = scipy.signal.detrend(chunk, type='linear')
tmpchunk = timeseries[ii].copy()
tmpchunk.sort()
# The following gets rid of (hopefully) most of the
# outlying values (i.e. power dropouts and single pulses)
# If you throw out 5% (2.5% at bottom and 2.5% at top)
# of random gaussian deviates, the measured stdev is ~0.871
# of the true stdev. Thus the 1.0/0.871=1.148 correction below.
# The following is roughly .std() since we already removed the median
stds[ii] = Num.sqrt(
(tmpchunk[detrendlen / 40:-detrendlen / 40]**2.0).sum() /
(0.95 * detrendlen))
stds *= 1.148
# sort the standard deviations and separate those with
# very low or very high values
sort_stds = stds.copy()
sort_stds.sort()
# identify the differences with the larges values (this
# will split off the chunks with very low and very high stds
locut = (sort_stds[1:numblocks / 2 + 1] -
sort_stds[:numblocks / 2]).argmax() + 1
hicut = (sort_stds[numblocks / 2 + 1:] -
sort_stds[numblocks / 2:-1]).argmax() + numblocks / 2 - 2
std_stds = scipy.std(sort_stds[locut:hicut])
median_stds = sort_stds[(locut + hicut) / 2]
lo_std = median_stds - 4.0 * std_stds
hi_std = median_stds + 4.0 * std_stds
# Determine a list of "bad" chunks. We will not search these.
bad_blocks = Num.nonzero((stds < lo_std) | (stds > hi_std))[0]
print " pseudo-median block standard deviation = %.2f" % (
median_stds)
print " identified %d bad blocks out of %d (i.e. %.2f%%)" % \
(len(bad_blocks), len(stds),
100.0*float(len(bad_blocks))/float(len(stds)))
stds[bad_blocks] = median_stds
print " Now searching..."
# Now normalize all of the data and reshape it to 1-D
timeseries /= stds[:, Num.newaxis]
timeseries.shape = (roundN, )
# And set the data in the bad blocks to zeros
# Even though we don't search these parts, it is important
# because of the overlaps for the convolutions
for bad_block in bad_blocks:
loind, hiind = bad_block * detrendlen, (bad_block +
1) * detrendlen
timeseries[loind:hiind] = 0.0
# Convert to a set for faster lookups below
bad_blocks = set(bad_blocks)
# Step through the data
dm_candlist = []
for chunknum in range(numchunks):
loind = chunknum * chunklen - overlap
hiind = (chunknum + 1) * chunklen + overlap
# Take care of beginning and end of file overlap issues
if (chunknum == 0): # Beginning of file
chunk = Num.zeros(worklen, dtype=Num.float32)
chunk[overlap:] = timeseries[loind + overlap:hiind]
elif (chunknum == numchunks - 1): # end of the timeseries
chunk = Num.zeros(worklen, dtype=Num.float32)
chunk[:-overlap] = timeseries[loind:hiind - overlap]
else:
chunk = timeseries[loind:hiind]
# Make a set with the current block numbers
lowblock = blocks_per_chunk * chunknum
currentblocks = set(Num.arange(blocks_per_chunk) + lowblock)
localgoodblocks = Num.asarray(
list(currentblocks - bad_blocks)) - lowblock
# Search this chunk if it is not all bad
if len(localgoodblocks):
# This is the good part of the data (end effects removed)
goodchunk = chunk[overlap:-overlap]
# need to pass blocks/chunklen, localgoodblocks
# dm_candlist, dt, opts.threshold to cython routine
# Search non-downsampled data first
# NOTE: these nonzero() calls are some of the most
# expensive calls in the program. Best bet would
# probably be to simply iterate over the goodchunk
# in C and append to the candlist there.
hibins = Num.flatnonzero(goodchunk > opts.threshold)
hivals = goodchunk[hibins]
hibins += chunknum * chunklen
hiblocks = hibins / detrendlen
# Add the candidates (which are sorted by bin)
for bin, val, block in zip(hibins, hivals, hiblocks):
if block not in bad_blocks:
time = bin * dt
dm_candlist.append(
candidate(info.DM, val, time, bin, 1))
# Prepare our data for the convolution
if useffts: fftd_chunk = rfft(chunk, -1)
# Now do the downsampling...
for ii, downfact in enumerate(downfacts):
if useffts:
# Note: FFT convolution is faster for _all_ downfacts, even 2
goodchunk = fft_convolve(fftd_chunk,
fftd_kerns[ii], overlap,
-overlap)
else:
# The normalization of this kernel keeps the post-smoothing RMS = 1
kernel = Num.ones(downfact, dtype=Num.float32) / \
Num.sqrt(downfact)
smoothed_chunk = scipy.signal.convolve(
chunk, kernel, 1)
goodchunk = smoothed_chunk[overlap:-overlap]
#hibins = Num.nonzero(goodchunk>opts.threshold)[0]
hibins = Num.flatnonzero(goodchunk > opts.threshold)
hivals = goodchunk[hibins]
hibins += chunknum * chunklen
hiblocks = hibins / detrendlen
hibins = hibins.tolist()
hivals = hivals.tolist()
# Now walk through the new candidates and remove those
# that are not the highest but are within downfact/2
# bins of a higher signal pulse
hibins, hivals = prune_related1(
hibins, hivals, downfact)
# Insert the new candidates into the candlist, but
# keep it sorted...
for bin, val, block in zip(hibins, hivals, hiblocks):
if block not in bad_blocks:
time = bin * dt
bisect.insort(
dm_candlist,
candidate(info.DM, val, time, bin,
downfact))
# Now walk through the dm_candlist and remove the ones that
# are within the downsample proximity of a higher
# signal-to-noise pulse
dm_candlist = prune_related2(dm_candlist, downfacts)
print " Found %d pulse candidates" % len(dm_candlist)
# Get rid of those near padding regions
if info.breaks: prune_border_cases(dm_candlist, offregions)
# Write the pulses to an ASCII output file
if len(dm_candlist):
#dm_candlist.sort(cmp_sigma)
outfile.write(
"# DM Sigma Time (s) Sample Downfact\n")
for cand in dm_candlist:
outfile.write(str(cand))
outfile.close()
# Add these candidates to the overall candidate list
for cand in dm_candlist:
candlist.append(cand)
num_v_DMstr[DMstr] = len(dm_candlist)
if (opts.makeplot):
# Step through the candidates to make a SNR list
DMs.sort()
snrs = []
for cand in candlist:
snrs.append(cand.sigma)
if snrs:
maxsnr = max(int(max(snrs)), int(opts.threshold)) + 3
else:
maxsnr = int(opts.threshold) + 3
# Generate the SNR histogram
snrs = Num.asarray(snrs)
(num_v_snr, lo_snr, d_snr, num_out_of_range) = \
scipy.stats.histogram(snrs,
int(maxsnr-opts.threshold+1),
[opts.threshold, maxsnr])
snrs = Num.arange(maxsnr-opts.threshold+1, dtype=Num.float64) * d_snr \
+ lo_snr + 0.5*d_snr
num_v_snr = num_v_snr.astype(Num.float32)
num_v_snr[num_v_snr == 0.0] = 0.001
# Generate the DM histogram
num_v_DM = Num.zeros(len(DMs))
for ii, DM in enumerate(DMs):
num_v_DM[ii] = num_v_DMstr["%.2f" % DM]
DMs = Num.asarray(DMs)
# open the plot device
short_filenmbase = filenmbase[:filenmbase.find("_DM")]
if opts.T_end > obstime:
opts.T_end = obstime
if pgplot_device:
ppgplot.pgopen(pgplot_device)
else:
if (opts.T_start > 0.0 or opts.T_end < obstime):
ppgplot.pgopen(short_filenmbase +
'_%.0f-%.0fs_singlepulse.ps/VPS' %
(opts.T_start, opts.T_end))
else:
ppgplot.pgopen(short_filenmbase + '_singlepulse.ps/VPS')
ppgplot.pgpap(7.5, 1.0) # Width in inches, aspect
# plot the SNR histogram
ppgplot.pgsvp(0.06, 0.31, 0.6, 0.87)
ppgplot.pgswin(opts.threshold, maxsnr, Num.log10(0.5),
Num.log10(2 * max(num_v_snr)))
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCLNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "Signal-to-Noise")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Number of Pulses")
ppgplot.pgsch(1.0)
ppgplot.pgbin(snrs, Num.log10(num_v_snr), 1)
# plot the DM histogram
ppgplot.pgsvp(0.39, 0.64, 0.6, 0.87)
# Add [1] to num_v_DM in YMAX below so that YMIN != YMAX when max(num_v_DM)==0
ppgplot.pgswin(
min(DMs) - 0.5,
max(DMs) + 0.5, 0.0, 1.1 * max(num_v_DM + [1]))
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "DM (pc cm\u-3\d)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Number of Pulses")
ppgplot.pgsch(1.0)
ppgplot.pgbin(DMs, num_v_DM, 1)
# plot the SNR vs DM plot
ppgplot.pgsvp(0.72, 0.97, 0.6, 0.87)
ppgplot.pgswin(min(DMs) - 0.5, max(DMs) + 0.5, opts.threshold, maxsnr)
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "DM (pc cm\u-3\d)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Signal-to-Noise")
ppgplot.pgsch(1.0)
cand_ts = Num.zeros(len(candlist), dtype=Num.float32)
cand_SNRs = Num.zeros(len(candlist), dtype=Num.float32)
cand_DMs = Num.zeros(len(candlist), dtype=Num.float32)
for ii, cand in enumerate(candlist):
cand_ts[ii], cand_SNRs[ii], cand_DMs[ii] = \
cand.time, cand.sigma, cand.DM
ppgplot.pgpt(cand_DMs, cand_SNRs, 20)
# plot the DM vs Time plot
ppgplot.pgsvp(0.06, 0.97, 0.08, 0.52)
ppgplot.pgswin(opts.T_start, opts.T_end,
min(DMs) - 0.5,
max(DMs) + 0.5)
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "Time (s)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "DM (pc cm\u-3\d)")
# Circles are symbols 20-26 in increasing order
snr_range = 12.0
cand_symbols = (cand_SNRs - opts.threshold) / snr_range * 6.0 + 20.5
cand_symbols = cand_symbols.astype(Num.int32)
cand_symbols[cand_symbols > 26] = 26
for ii in [26, 25, 24, 23, 22, 21, 20]:
inds = Num.nonzero(cand_symbols == ii)[0]
ppgplot.pgpt(cand_ts[inds], cand_DMs[inds], ii)
# Now fill the infomation area
ppgplot.pgsvp(0.05, 0.95, 0.87, 0.97)
ppgplot.pgsch(1.0)
ppgplot.pgmtxt('T', 0.5, 0.0, 0.0,
"Single pulse results for '%s'" % short_filenmbase)
ppgplot.pgsch(0.8)
# first row
ppgplot.pgmtxt('T', -1.1, 0.02, 0.0, 'Source: %s'%\
info.object)
ppgplot.pgmtxt('T', -1.1, 0.33, 0.0, 'RA (J2000):')
ppgplot.pgmtxt('T', -1.1, 0.5, 0.0, info.RA)
ppgplot.pgmtxt('T', -1.1, 0.73, 0.0, 'N samples: %.0f' % orig_N)
# second row
ppgplot.pgmtxt('T', -2.4, 0.02, 0.0, 'Telescope: %s'%\
info.telescope)
ppgplot.pgmtxt('T', -2.4, 0.33, 0.0, 'DEC (J2000):')
ppgplot.pgmtxt('T', -2.4, 0.5, 0.0, info.DEC)
ppgplot.pgmtxt('T', -2.4, 0.73, 0.0, 'Sampling time: %.2f \gms'%\
(orig_dt*1e6))
# third row
if info.instrument.find("pigot") >= 0:
instrument = "Spigot"
else:
instrument = info.instrument
ppgplot.pgmtxt('T', -3.7, 0.02, 0.0, 'Instrument: %s' % instrument)
if (info.bary):
ppgplot.pgmtxt('T', -3.7, 0.33, 0.0,
'MJD\dbary\u: %.12f' % info.epoch)
else:
ppgplot.pgmtxt('T', -3.7, 0.33, 0.0,
'MJD\dtopo\u: %.12f' % info.epoch)
ppgplot.pgmtxt('T', -3.7, 0.73, 0.0, 'Freq\dctr\u: %.1f MHz'%\
((info.numchan/2-0.5)*info.chan_width+info.lofreq))
ppgplot.pgiden()
ppgplot.pgend()
import numpy as np
import presto
import os
import matplotlib.pyplot as plt
print "Testing FFT stuff...",
N = 20
x = np.random.standard_normal(N)
nx = presto.rfft(presto.rfft(x, -1), 1)
assert(np.allclose(x, nx, atol=1e-6))
print "success"
print "Testing FFTW call...",
cx = np.random.standard_normal(N).astype(np.complex64)
ncx = np.array(cx, copy=1)
presto.fftwcall(cx, -1)
presto.fftwcall(cx, 1)
assert(np.allclose(cx/N, ncx, atol=1e-6))
print "success"
print "Testing tablesixstepfft call...",
cx = np.random.standard_normal(N).astype(np.complex64)
ncx = np.array(cx, copy=1)
presto.tablesixstepfft(cx, -1)
presto.tablesixstepfft(cx, 1)
assert(np.allclose(cx/N, ncx, atol=1e-6))
print "success"
print "Testing reading infiles...",
x = presto.read_inffile("1937_DM71.02_zerodm.inf", verbose=False)
assert(x.telescope=="GBT")
def corr(profile, template):
ft = rfft(template)
fp = rfft(profile)
return rfft(ft*fp, 1)
r = N / 4.0 # average freq over "observation"
#r = N/4.0 + 0.5 # average freq over "observation"
rint = num.floor(r)
dr = 1.0 / 32.0
dz = 0.18
np = 512 # number of pixels across for f-fdot image
z = 0.0 # average fourier f-dot
w = 0.0 # fourier freq double deriv
noise = 0.0
#noise = 4*num.random.standard_normal(N)
us = num.arange(N, dtype=num.float64) / N # normalized time coordinate
r0 = r - 0.5 * z + w / 12.0 # Make symmetric for all z and w
z0 = z - 0.5 * w
phss = 2.0 * num.pi * (us * (us * (us * w / 6.0 + z0 / 2.0) + r0))
ft = presto.rfft(num.cos(phss) + noise)
ffdot = presto.ffdot_plane(ft, rint - np / 2 * dr, dr, np, 0.0 - np / 2 * dz,
dz, np)
pffdot = presto.spectralpower(ffdot.flat)
theo_max_pow = N**2.0 / 4.0
frp = max(pffdot) / theo_max_pow # Fraction of recovered power
print("Fraction of recovered signal power = %f" % frp)
# print "Raw power should be ~%.2e" % theo_max_pow
pffdot = pffdot / theo_max_pow
pffdot.shape = (np, np)
rs = num.arange(np) * dr - np / 2 * dr
zs = num.arange(np) * dz - np / 2 * dz
rgx = num.asarray([rs[0], rs[np - 1]])
rgy = num.asarray([zs[0], zs[np - 1]])
freqcut = pffdot[np / 2, :]
fdotcut = pffdot[:, np / 2]
r = N/4.0 # average freq over "observation"
#r = N/4.0 + 0.5 # average freq over "observation"
rint = num.floor(r)
dr = 1.0/32.0
dz = 0.18
np = 512 # number of pixels across for f-fdot image
z = 10.0 # average fourier f-dot
w = -40.0 # fourier freq double deriv
noise = 0.0
noise = 1.0*norm(N)
us = num.arange(N, dtype=num.float64) / N # normalized time coordinate
r0 = r - 0.5 * z + w / 12.0 # Make symmetric for all z and w
z0 = z - 0.5 * w
phss = 2.0 * num.pi * (us * (us * (us * w/6.0 + z0/2.0) + r0))
ft = presto.rfft(num.cos(phss)+noise)
ffdot = presto.ffdot_plane(ft, rint-np/2*dr, dr, np, 0.0-np/2*dz, dz, np)
pffdot = presto.spectralpower(ffdot.flat)
theo_max_pow = N**2.0/4.0
frp = max(pffdot) / theo_max_pow # Fraction of recovered power
print "Fraction of recovered signal power = %f" % frp
a = time.clock()
[maxpow, rmax, zmax, rd] = presto.maximize_rz(ft, r+norm(1)[0]/5.0,
z+norm(1)[0], norm=1.0)
print "Time for rz:", time.clock()-a
print r, rmax, z, zmax, theo_max_pow, maxpow
a = time.clock()
[maxpow, rmax, zmax, wmax, rd] = presto.maximize_rzw(ft, r+norm(1)[0]/5.0,
z+norm(1)[0],
w+norm(1)[0]*5.0,
norm=1.0)
r = N / (4 * numharm) + uniform(0.0, 1.0,
1)[0] # average freq over "observation"
z = uniform(-100, 100, 1)[0] # average fourier f-dot
w = uniform(-600, 600, 1)[0] # fourier freq double deriv
# w = 0.0 # fourier freq double deriv
data = np.zeros_like(us)
for ii in range(numharm):
rh = r * (ii + 1)
zh = z * (ii + 1)
wh = w * (ii + 1)
r0 = rh - 0.5 * zh + wh / 12.0 # Make symmetric for all z and w
z0 = zh - 0.5 * wh
phss = 2.0 * np.pi * (us * (us * (us * wh / 6.0 + z0 / 2.0) + r0))
data += np.cos(phss)
data += noiseamp * norm(N)
ft = presto.rfft(data)
offset = uniform(-1.0, 1.0, 3) * np.array([0.5, 2.0, 20.0]) / numharm
a = time.clock()
if (numharm > 1):
[maxpow, rmax, zmax, rds] = presto.maximize_rz_harmonics(ft,
r + offset[0],
z + offset[1],
numharm,
norm=1.0)
else:
[maxpow, rmax, zmax, rd] = presto.maximize_rz(ft,
r + offset[0],
z + offset[1],
norm=1.0)
def main():
parser = OptionParser(usage)
parser.add_option("-x", "--xwin", action="store_true", dest="xwin",
default=False, help="Don't make a postscript plot, just use an X-window")
parser.add_option("-p", "--noplot", action="store_false", dest="makeplot",
default=True, help="Look for pulses but do not generate a plot")
parser.add_option("-m", "--maxwidth", type="float", dest="maxwidth", default=0.0,
help="Set the max downsampling in sec (see below for default)")
parser.add_option("-t", "--threshold", type="float", dest="threshold", default=5.0,
help="Set a different threshold SNR (default=5.0)")
parser.add_option("-s", "--start", type="float", dest="T_start", default=0.0,
help="Only plot events occuring after this time (s)")
parser.add_option("-e", "--end", type="float", dest="T_end", default=1e9,
help="Only plot events occuring before this time (s)")
parser.add_option("-g", "--glob", type="string", dest="globexp", default=None,
help="Process the files from this glob expression")
parser.add_option("-f", "--fast", action="store_true", dest="fast",
default=False, help="Use a faster method of de-trending (2x speedup)")
parser.add_option("-b", "--nobadblocks", action="store_false", dest="badblocks",
default=True, help="Don't check for bad-blocks (may save strong pulses)")
parser.add_option("-d", "--detrendlen", type="int", dest="detrendfact", default=1,
help="Chunksize for detrending (pow-of-2 in 1000s)")
(opts, args) = parser.parse_args()
if len(args)==0:
if opts.globexp==None:
print full_usage
sys.exit(0)
else:
args = []
for globexp in opts.globexp.split():
args += glob.glob(globexp)
useffts = True
dosearch = True
if opts.xwin:
pgplot_device = "/XWIN"
else:
pgplot_device = ""
fftlen = 8192 # Should be a power-of-two for best speed
chunklen = 8000 # Must be at least max_downfact less than fftlen
assert(opts.detrendfact in [1,2,4,8,16,32])
detrendlen = opts.detrendfact*1000
if (detrendlen > chunklen):
chunklen = detrendlen
fftlen = int(next2_to_n(chunklen))
blocks_per_chunk = chunklen / detrendlen
overlap = (fftlen - chunklen)/2
worklen = chunklen + 2*overlap # currently it is fftlen...
max_downfact = 30
default_downfacts = [2, 3, 4, 6, 9, 14, 20, 30, 45, 70, 100, 150, 220, 300]
if args[0].endswith(".singlepulse"):
filenmbase = args[0][:args[0].rfind(".singlepulse")]
dosearch = False
elif args[0].endswith(".dat"):
filenmbase = args[0][:args[0].rfind(".dat")]
else:
filenmbase = args[0]
# Don't do a search, just read results and plot
if not dosearch:
info, DMs, candlist, num_v_DMstr = \
read_singlepulse_files(args, opts.threshold, opts.T_start, opts.T_end)
orig_N, orig_dt = int(info.N), info.dt
obstime = orig_N * orig_dt
else:
DMs = []
candlist = []
num_v_DMstr = {}
# Loop over the input files
for filenm in args:
if filenm.endswith(".dat"):
filenmbase = filenm[:filenm.rfind(".dat")]
else:
filenmbase = filenm
info = infodata.infodata(filenmbase+".inf")
DMstr = "%.2f"%info.DM
DMs.append(info.DM)
N, dt = int(info.N), info.dt
obstime = N * dt
# Choose the maximum width to search based on time instead
# of bins. This helps prevent increased S/N when the downsampling
# changes as the DM gets larger.
if opts.maxwidth > 0.0:
downfacts = [x for x in default_downfacts if x*dt <= opts.maxwidth]
else:
downfacts = [x for x in default_downfacts if x <= max_downfact]
if len(downfacts) == 0:
downfacts = [default_downfacts[0]]
if (filenm == args[0]):
orig_N = N
orig_dt = dt
if useffts:
fftd_kerns = make_fftd_kerns(default_downfacts, fftlen)
if info.breaks:
offregions = zip([x[1] for x in info.onoff[:-1]],
[x[0] for x in info.onoff[1:]])
outfile = open(filenmbase+'.singlepulse', mode='w')
# Compute the file length in detrendlens
roundN = N/detrendlen * detrendlen
numchunks = roundN / chunklen
# Read in the file
print 'Reading "%s"...'%filenm
timeseries = Num.fromfile(filenm, dtype=Num.float32, count=roundN)
# Split the timeseries into chunks for detrending
numblocks = roundN/detrendlen
timeseries.shape = (numblocks, detrendlen)
stds = Num.zeros(numblocks, dtype=Num.float64)
# de-trend the data one chunk at a time
print ' De-trending the data and computing statistics...'
for ii, chunk in enumerate(timeseries):
if opts.fast: # use median removal instead of detrending (2x speedup)
tmpchunk = chunk.copy()
tmpchunk.sort()
med = tmpchunk[detrendlen/2]
chunk -= med
tmpchunk -= med
else:
# The detrend calls are the most expensive in the program
timeseries[ii] = scipy.signal.detrend(chunk, type='linear')
tmpchunk = timeseries[ii].copy()
tmpchunk.sort()
# The following gets rid of (hopefully) most of the
# outlying values (i.e. power dropouts and single pulses)
# If you throw out 5% (2.5% at bottom and 2.5% at top)
# of random gaussian deviates, the measured stdev is ~0.871
# of the true stdev. Thus the 1.0/0.871=1.148 correction below.
# The following is roughly .std() since we already removed the median
stds[ii] = Num.sqrt((tmpchunk[detrendlen/40:-detrendlen/40]**2.0).sum() /
(0.95*detrendlen))
stds *= 1.148
# sort the standard deviations and separate those with
# very low or very high values
sort_stds = stds.copy()
sort_stds.sort()
# identify the differences with the larges values (this
# will split off the chunks with very low and very high stds
locut = (sort_stds[1:numblocks/2+1] -
sort_stds[:numblocks/2]).argmax() + 1
hicut = (sort_stds[numblocks/2+1:] -
sort_stds[numblocks/2:-1]).argmax() + numblocks/2 - 2
std_stds = scipy.std(sort_stds[locut:hicut])
median_stds = sort_stds[(locut+hicut)/2]
print " pseudo-median block standard deviation = %.2f" % (median_stds)
if (opts.badblocks):
lo_std = median_stds - 4.0 * std_stds
hi_std = median_stds + 4.0 * std_stds
# Determine a list of "bad" chunks. We will not search these.
bad_blocks = Num.nonzero((stds < lo_std) | (stds > hi_std))[0]
print " identified %d bad blocks out of %d (i.e. %.2f%%)" % \
(len(bad_blocks), len(stds),
100.0*float(len(bad_blocks))/float(len(stds)))
stds[bad_blocks] = median_stds
else:
bad_blocks = []
print " Now searching..."
# Now normalize all of the data and reshape it to 1-D
timeseries /= stds[:,Num.newaxis]
timeseries.shape = (roundN,)
# And set the data in the bad blocks to zeros
# Even though we don't search these parts, it is important
# because of the overlaps for the convolutions
for bad_block in bad_blocks:
loind, hiind = bad_block*detrendlen, (bad_block+1)*detrendlen
timeseries[loind:hiind] = 0.0
# Convert to a set for faster lookups below
bad_blocks = set(bad_blocks)
# Step through the data
dm_candlist = []
for chunknum in xrange(numchunks):
loind = chunknum*chunklen-overlap
hiind = (chunknum+1)*chunklen+overlap
# Take care of beginning and end of file overlap issues
if (chunknum==0): # Beginning of file
chunk = Num.zeros(worklen, dtype=Num.float32)
chunk[overlap:] = timeseries[loind+overlap:hiind]
elif (chunknum==numchunks-1): # end of the timeseries
chunk = Num.zeros(worklen, dtype=Num.float32)
chunk[:-overlap] = timeseries[loind:hiind-overlap]
else:
chunk = timeseries[loind:hiind]
# Make a set with the current block numbers
lowblock = blocks_per_chunk * chunknum
currentblocks = set(Num.arange(blocks_per_chunk) + lowblock)
localgoodblocks = Num.asarray(list(currentblocks -
bad_blocks)) - lowblock
# Search this chunk if it is not all bad
if len(localgoodblocks):
# This is the good part of the data (end effects removed)
goodchunk = chunk[overlap:-overlap]
# need to pass blocks/chunklen, localgoodblocks
# dm_candlist, dt, opts.threshold to cython routine
# Search non-downsampled data first
# NOTE: these nonzero() calls are some of the most
# expensive calls in the program. Best bet would
# probably be to simply iterate over the goodchunk
# in C and append to the candlist there.
hibins = Num.flatnonzero(goodchunk>opts.threshold)
hivals = goodchunk[hibins]
hibins += chunknum * chunklen
hiblocks = hibins/detrendlen
# Add the candidates (which are sorted by bin)
for bin, val, block in zip(hibins, hivals, hiblocks):
if block not in bad_blocks:
time = bin * dt
dm_candlist.append(candidate(info.DM, val, time, bin, 1))
# Prepare our data for the convolution
if useffts: fftd_chunk = rfft(chunk, -1)
# Now do the downsampling...
for ii, downfact in enumerate(downfacts):
if useffts:
# Note: FFT convolution is faster for _all_ downfacts, even 2
goodchunk = fft_convolve(fftd_chunk, fftd_kerns[ii],
overlap, -overlap)
else:
# The normalization of this kernel keeps the post-smoothing RMS = 1
kernel = Num.ones(downfact, dtype=Num.float32) / \
Num.sqrt(downfact)
smoothed_chunk = scipy.signal.convolve(chunk, kernel, 1)
goodchunk = smoothed_chunk[overlap:-overlap]
#hibins = Num.nonzero(goodchunk>opts.threshold)[0]
hibins = Num.flatnonzero(goodchunk>opts.threshold)
hivals = goodchunk[hibins]
hibins += chunknum * chunklen
hiblocks = hibins/detrendlen
hibins = hibins.tolist()
hivals = hivals.tolist()
# Now walk through the new candidates and remove those
# that are not the highest but are within downfact/2
# bins of a higher signal pulse
hibins, hivals = prune_related1(hibins, hivals, downfact)
# Insert the new candidates into the candlist, but
# keep it sorted...
for bin, val, block in zip(hibins, hivals, hiblocks):
if block not in bad_blocks:
time = bin * dt
bisect.insort(dm_candlist,
candidate(info.DM, val, time, bin, downfact))
# Now walk through the dm_candlist and remove the ones that
# are within the downsample proximity of a higher
# signal-to-noise pulse
dm_candlist = prune_related2(dm_candlist, downfacts)
print " Found %d pulse candidates"%len(dm_candlist)
# Get rid of those near padding regions
if info.breaks: prune_border_cases(dm_candlist, offregions)
# Write the pulses to an ASCII output file
if len(dm_candlist):
#dm_candlist.sort(cmp_sigma)
outfile.write("# DM Sigma Time (s) Sample Downfact\n")
for cand in dm_candlist:
outfile.write(str(cand))
outfile.close()
# Add these candidates to the overall candidate list
for cand in dm_candlist:
candlist.append(cand)
num_v_DMstr[DMstr] = len(dm_candlist)
if (opts.makeplot):
# Step through the candidates to make a SNR list
DMs.sort()
snrs = []
for cand in candlist:
if not Num.isinf(cand.sigma):
snrs.append(cand.sigma)
if snrs:
maxsnr = max(int(max(snrs)), int(opts.threshold)) + 3
else:
maxsnr = int(opts.threshold) + 3
# Generate the SNR histogram
snrs = Num.asarray(snrs)
(num_v_snr, lo_snr, d_snr, num_out_of_range) = \
scipy.stats.histogram(snrs,
int(maxsnr-opts.threshold+1),
[opts.threshold, maxsnr])
snrs = Num.arange(maxsnr-opts.threshold+1, dtype=Num.float64) * d_snr \
+ lo_snr + 0.5*d_snr
num_v_snr = num_v_snr.astype(Num.float32)
num_v_snr[num_v_snr==0.0] = 0.001
# Generate the DM histogram
num_v_DM = Num.zeros(len(DMs))
for ii, DM in enumerate(DMs):
num_v_DM[ii] = num_v_DMstr["%.2f"%DM]
DMs = Num.asarray(DMs)
# open the plot device
short_filenmbase = filenmbase[:filenmbase.find("_DM")]
if opts.T_end > obstime:
opts.T_end = obstime
if pgplot_device:
ppgplot.pgopen(pgplot_device)
else:
if (opts.T_start > 0.0 or opts.T_end < obstime):
ppgplot.pgopen(short_filenmbase+'_%.0f-%.0fs_singlepulse.ps/VPS'%
(opts.T_start, opts.T_end))
else:
ppgplot.pgopen(short_filenmbase+'_singlepulse.ps/VPS')
ppgplot.pgpap(7.5, 1.0) # Width in inches, aspect
# plot the SNR histogram
ppgplot.pgsvp(0.06, 0.31, 0.6, 0.87)
ppgplot.pgswin(opts.threshold, maxsnr,
Num.log10(0.5), Num.log10(2*max(num_v_snr)))
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCLNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "Signal-to-Noise")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Number of Pulses")
ppgplot.pgsch(1.0)
ppgplot.pgbin(snrs, Num.log10(num_v_snr), 1)
# plot the DM histogram
ppgplot.pgsvp(0.39, 0.64, 0.6, 0.87)
# Add [1] to num_v_DM in YMAX below so that YMIN != YMAX when max(num_v_DM)==0
ppgplot.pgswin(min(DMs)-0.5, max(DMs)+0.5, 0.0, 1.1*max(num_v_DM+[1]))
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "DM (pc cm\u-3\d)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Number of Pulses")
ppgplot.pgsch(1.0)
ppgplot.pgbin(DMs, num_v_DM, 1)
# plot the SNR vs DM plot
ppgplot.pgsvp(0.72, 0.97, 0.6, 0.87)
ppgplot.pgswin(min(DMs)-0.5, max(DMs)+0.5, opts.threshold, maxsnr)
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "DM (pc cm\u-3\d)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "Signal-to-Noise")
ppgplot.pgsch(1.0)
cand_ts = Num.zeros(len(candlist), dtype=Num.float32)
cand_SNRs = Num.zeros(len(candlist), dtype=Num.float32)
cand_DMs = Num.zeros(len(candlist), dtype=Num.float32)
for ii, cand in enumerate(candlist):
cand_ts[ii], cand_SNRs[ii], cand_DMs[ii] = \
cand.time, cand.sigma, cand.DM
ppgplot.pgpt(cand_DMs, cand_SNRs, 20)
# plot the DM vs Time plot
ppgplot.pgsvp(0.06, 0.97, 0.08, 0.52)
ppgplot.pgswin(opts.T_start, opts.T_end, min(DMs)-0.5, max(DMs)+0.5)
ppgplot.pgsch(0.8)
ppgplot.pgbox("BCNST", 0, 0, "BCNST", 0, 0)
ppgplot.pgmtxt('B', 2.5, 0.5, 0.5, "Time (s)")
ppgplot.pgmtxt('L', 1.8, 0.5, 0.5, "DM (pc cm\u-3\d)")
# Circles are symbols 20-26 in increasing order
snr_range = 12.0
cand_symbols = (cand_SNRs-opts.threshold)/snr_range * 6.0 + 20.5
cand_symbols = cand_symbols.astype(Num.int32)
cand_symbols[cand_symbols>26] = 26
for ii in [26, 25, 24, 23, 22, 21, 20]:
inds = Num.nonzero(cand_symbols==ii)[0]
ppgplot.pgpt(cand_ts[inds], cand_DMs[inds], ii)
# Now fill the infomation area
ppgplot.pgsvp(0.05, 0.95, 0.87, 0.97)
ppgplot.pgsch(1.0)
ppgplot.pgmtxt('T', 0.5, 0.0, 0.0,
"Single pulse results for '%s'"%short_filenmbase)
ppgplot.pgsch(0.8)
# first row
ppgplot.pgmtxt('T', -1.1, 0.02, 0.0, 'Source: %s'%\
info.object)
ppgplot.pgmtxt('T', -1.1, 0.33, 0.0, 'RA (J2000):')
ppgplot.pgmtxt('T', -1.1, 0.5, 0.0, info.RA)
ppgplot.pgmtxt('T', -1.1, 0.73, 0.0, 'N samples: %.0f'%orig_N)
# second row
ppgplot.pgmtxt('T', -2.4, 0.02, 0.0, 'Telescope: %s'%\
info.telescope)
ppgplot.pgmtxt('T', -2.4, 0.33, 0.0, 'DEC (J2000):')
ppgplot.pgmtxt('T', -2.4, 0.5, 0.0, info.DEC)
ppgplot.pgmtxt('T', -2.4, 0.73, 0.0, 'Sampling time: %.2f \gms'%\
(orig_dt*1e6))
# third row
if info.instrument.find("pigot") >= 0:
instrument = "Spigot"
else:
instrument = info.instrument
ppgplot.pgmtxt('T', -3.7, 0.02, 0.0, 'Instrument: %s'%instrument)
if (info.bary):
ppgplot.pgmtxt('T', -3.7, 0.33, 0.0, 'MJD\dbary\u: %.12f'%info.epoch)
else:
ppgplot.pgmtxt('T', -3.7, 0.33, 0.0, 'MJD\dtopo\u: %.12f'%info.epoch)
ppgplot.pgmtxt('T', -3.7, 0.73, 0.0, 'Freq\dctr\u: %.1f MHz'%\
((info.numchan/2-0.5)*info.chan_width+info.lofreq))
ppgplot.pgiden()
ppgplot.pgend()
def corr(profile, template):
ft = rfft(template)
fp = rfft(profile)
return rfft(ft * fp, 1)
import numpy as np
import presto
import os
import matplotlib.pyplot as plt
print "Testing FFT stuff...",
N = 20
x = np.random.standard_normal(N)
nx = presto.rfft(presto.rfft(x, -1), 1)
assert (np.allclose(x, nx, atol=1e-6))
print "success"
print "Testing FFTW call...",
cx = np.random.standard_normal(N).astype(np.complex64)
ncx = np.array(cx, copy=1)
presto.fftwcall(cx, -1)
presto.fftwcall(cx, 1)
assert (np.allclose(cx / N, ncx, atol=1e-6))
print "success"
print "Testing tablesixstepfft call...",
cx = np.random.standard_normal(N).astype(np.complex64)
ncx = np.array(cx, copy=1)
presto.tablesixstepfft(cx, -1)
presto.tablesixstepfft(cx, 1)
assert (np.allclose(cx / N, ncx, atol=1e-6))
print "success"
print "Testing reading infiles...",
x = presto.read_inffile("1937_DM71.02_zerodm.inf", verbose=False)
assert (x.telescope == "GBT")
for n in range(numtrials):
r = N/(4*numharm) + uniform(0.0, 1.0, 1)[0] # average freq over "observation"
z = uniform(-100, 100, 1)[0] # average fourier f-dot
w = uniform(-600, 600, 1)[0] # fourier freq double deriv
w = 0.0 # fourier freq double deriv
data = np.zeros_like(us)
for ii in range(numharm):
rh = r * (ii + 1)
zh = z * (ii + 1)
wh = w * (ii + 1)
r0 = rh - 0.5 * zh + wh / 12.0 # Make symmetric for all z and w
z0 = zh - 0.5 * wh
phss = 2.0 * np.pi * (us * (us * (us * wh/6.0 + z0/2.0) + r0))
data += np.cos(phss)
data += noiseamp * norm(N)
ft = presto.rfft(data)
offset = uniform(-1.0, 1.0, 3) * np.array([0.5, 2.0, 20.0]) / (0.5 * numharm)
a = time.clock()
if (numharm > 1):
[maxpow, rmax, zmax, rds] = presto.maximize_rz_harmonics(ft, r+offset[0],
z+offset[1], numharm,
norm=1.0)
else:
[maxpow, rmax, zmax, rd] = presto.maximize_rz(ft, r+offset[0],
z+offset[1],
norm=1.0)
rztime += time.clock() - a
rzerrs[n] = (maxpow/numharm - theo_max_pow) / theo_max_pow, rmax - r, zmax - z
