def required_signal_power(power, n=1, confidence=0.99):
"""
required_signal_power(power, n=1, confidence=0.99):
Return the required power of a signal that will cause
at least a power 'power' in a power spectrum a fraction
'confidence' of the time. This is the inverse of
equation 16 in Groth, 1975, with solves for P_signal.
If called with 'power' = P_detect the result is
the search sensitivity. If called with 'power' = P_max,
then the result is the upper limit on the signal power
in the power spectrum.
"""
prob = 1.0 - confidence
def func(x, power=power, prob=prob, n=n):
return prob_power_series(power, x, n) - prob
def dfunc(x, power=power, n=n):
return power_probability(power, x, n)
P_signal = newton_raphson(func, dfunc, 0.0001, 100.0)
return P_signal
def required_signal_power(power, n=1, confidence=0.99):
"""
required_signal_power(power, n=1, confidence=0.99):
Return the required power of a signal that will cause
at least a power 'power' in a power spectrum a fraction
'confidence' of the time. This is the inverse of
equation 16 in Groth, 1975, with solves for P_signal.
If called with 'power' = P_detect the result is
the search sensitivity. If called with 'power' = P_max,
then the result is the upper limit on the signal power
in the power spectrum.
"""
prob = 1.0 - confidence
def func(x, power=power, prob=prob, n=n):
return prob_power_series(power, x, n) - prob
def dfunc(x, power=power, n=n):
return power_probability(power, x, n)
P_signal = newton_raphson(func, dfunc, 0.0001, 100.0)
return P_signal
def measure_phase(profile, template, sigma, fwhm):
"""
measure_phase(profile, template, sigma, fwhm):
TOA measurement technique from J. H. Taylor's talk
_Pulsar_Timing_and_Relativistic_Gravity_. Routine
takes two profiles, the first measured and the
second a high S/N template and determines the phase
offset of 'profile' from 'template'. Both profiles
must have the same number of points. 'sigma' denotes
the RMS noise level of the 'profile'. 'fwhm' is the
approximate width of the template pulse (0-1). The phase
returned is cyclic (i.e. from 0-1). The routine
returns a tuple comtaining (tau, tau_err, b, b_err, a).
Where 'tau' is the phase, 'B' is the scaling factor,
and 'a' is the DC offset. The error values are
estimates of the 1 sigma errors.
"""
from simple_roots import newton_raphson
N = len(profile)
if not (N == len(template)):
print "Lengths of 'profile' and 'template' must"
print " be equal in measure_phase()."
return 0.0
ft = rfft(profile)
p0 = ft[0].real
# Nyquist freq
ft[0] = complex(ft[0].imag, 0.0)
P_k = abs(ft)
frotate(P_k, len(ft), 1)
Theta_k = np.arctan2(-ft.imag, ft.real)
frotate(Theta_k, len(ft), 1)
ft = rfft(template)
s0 = ft[0].real
# Nyquist freq
ft[0] = complex(ft[0].imag, 0.0)
S_k = abs(ft)
frotate(S_k, len(ft), 1)
Phi_k = np.arctan2(-ft.imag, ft.real)
frotate(Phi_k, len(ft), 1)
# Estimate of the noise sigma (This needs to be checked)
# Note: Checked 10 Jul 2000. Looks OK.
sig = sigma * np.sqrt(N)
k = np.arange(len(ft), dtype='d') + 1.0
def fn(tau, k=k, p=P_k, s=S_k, theta=Theta_k, phi=Phi_k):
# Since Nyquist freq always has phase = 0.0
k[-1] = 0.0
return np.add.reduce(k * p * s *
np.sin(phi - theta + k * tau))
def dfn(tau, k=k, p=P_k, s=S_k, theta=Theta_k, phi=Phi_k):
# Since Nyquist freq always has phase = 0.0
k[-1] = 0.0
return np.add.reduce(k * k * p * s *
np.cos(phi - theta + k * tau))
numphases = 200
ddchidt = np.zeros(numphases, 'd')
phases = np.arange(numphases, dtype='d') / \
float(numphases-1) * TWOPI - PI
for i in np.arange(numphases):
ddchidt[i] = dfn(phases[i])
maxdphase = phases[np.argmax(ddchidt)] + \
0.5 * TWOPI / (numphases - 1.0)
# Solve for tau
tau = newton_raphson(fn, dfn, maxdphase - 0.5 * fwhm * TWOPI,
maxdphase + 0.5 * fwhm * TWOPI)
# Solve for b
c = P_k * S_k * np.cos(Phi_k - Theta_k + k * tau)
d = np.add.reduce(S_k**2.0)
b = np.add.reduce(c) / d
# tau sigma
tau_err = sig * np.sqrt(1.0 / (2.0 * b *
np.add.reduce(k**2.0 * c)))
# b sigma (Note: This seems to be an underestimate...)
b_err = sig * np.sqrt(1.0 / (2.0 * d))
# Solve for a
a = (p0 - b * s0) / float(N)
return (tau / TWOPI, tau_err / TWOPI, b, b_err, a)
def measure_phase(profile, template, sigma, fwhm):
"""
measure_phase(profile, template, sigma, fwhm):
TOA measurement technique from J. H. Taylor's talk
_Pulsar_Timing_and_Relativistic_Gravity_. Routine
takes two profiles, the first measured and the
second a high S/N template and determines the phase
offset of 'profile' from 'template'. Both profiles
must have the same number of points. 'sigma' denotes
the RMS noise level of the 'profile'. 'fwhm' is the
approximate width of the template pulse (0-1). The phase
returned is cyclic (i.e. from 0-1). The routine
returns a tuple comtaining (tau, tau_err, b, b_err, a).
Where 'tau' is the phase, 'B' is the scaling factor,
and 'a' is the DC offset. The error values are
estimates of the 1 sigma errors.
"""
from simple_roots import newton_raphson
N = len(profile)
if not (N == len(template)):
print "Lengths of 'profile' and 'template' must"
print " be equal in measure_phase()."
return 0.0
ft = rfft(profile)
p0 = ft[0].real
# Nyquist freq
ft[0] = complex(ft[0].imag, 0.0)
P_k = abs(ft)
frotate(P_k, len(ft), 1)
Theta_k = Num.arctan2(-ft.imag, ft.real)
frotate(Theta_k, len(ft), 1)
ft = rfft(template)
s0 = ft[0].real
# Nyquist freq
ft[0] = complex(ft[0].imag, 0.0)
S_k = abs(ft)
frotate(S_k, len(ft), 1)
Phi_k = Num.arctan2(-ft.imag, ft.real)
frotate(Phi_k, len(ft), 1)
# Estimate of the noise sigma (This needs to be checked)
# Note: Checked 10 Jul 2000. Looks OK.
sig = sigma * Num.sqrt(N)
k = Num.arange(len(ft), dtype='d') + 1.0
def fn(tau, k=k, p=P_k, s=S_k, theta=Theta_k, phi=Phi_k):
# Since Nyquist freq always has phase = 0.0
k[-1] = 0.0
return Num.add.reduce(k * p * s * Num.sin(phi - theta + k * tau))
def dfn(tau, k=k, p=P_k, s=S_k, theta=Theta_k, phi=Phi_k):
# Since Nyquist freq always has phase = 0.0
k[-1] = 0.0
return Num.add.reduce(k * k * p * s * Num.cos(phi - theta + k * tau))
numphases = 200
ddchidt = Num.zeros(numphases, 'd')
phases = Num.arange(numphases, dtype='d') / \
float(numphases-1) * TWOPI - PI
for i in Num.arange(numphases):
ddchidt[i] = dfn(phases[i])
maxdphase = phases[Num.argmax(ddchidt)] + \
0.5 * TWOPI / (numphases - 1.0)
# Solve for tau
tau = newton_raphson(fn, dfn, maxdphase - 0.5 * fwhm * TWOPI,
maxdphase + 0.5 * fwhm * TWOPI)
# Solve for b
c = P_k * S_k * Num.cos(Phi_k - Theta_k + k * tau)
d = Num.add.reduce(S_k**2.0)
b = Num.add.reduce(c) / d
# tau sigma
tau_err = sig * Num.sqrt(1.0 / (2.0 * b * Num.add.reduce(k**2.0 * c)))
# b sigma (Note: This seems to be an underestimate...)
b_err = sig * Num.sqrt(1.0 / (2.0 * d))
# Solve for a
a = (p0 - b * s0) / float(N)
return (tau / TWOPI, tau_err / TWOPI, b, b_err, a)
if (0):
# Use the following to generate the xs
from simple_roots import newton_raphson
rval = 0.0
rs = Num.arange(n+1, dtype=Num.float)/n
xs = Num.zeros(n+1, dtype=Num.float)
def func(x):
return Num.sin(2.0*Num.pi*x)/(2.0*Num.pi) + x - rval
def dfunc(x):
return Num.cos(2.0*Num.pi*x) + 1
for (ii, rval) in enumerate(rs[:-1]):
if (ii==n/2):
xs[ii] = 0.5
else:
xs[ii] = newton_raphson(func, dfunc, 0.0, 1.0)
xs[0] = 0.0
xs[n] = 1.0
cPickle.dump(xs, file("cosine_rand.pickle", "w"), 1)
else:
pfile = os.path.join(os.environ['PRESTO'],
"lib", "python", "cosine_rand.pickle")
xs = cPickle.load(file(pfile))
def cosine_rand(num):
"""cosine_rand(num): Return num phases that are randomly distributed
as per a sinusoid with maximum at phase=0 (0 < phase < 1).
"""
rands = n*Num.random.random(num)
indices = rands.astype(Num.int)
fracts = rands-indices