"""
A Digital Communications Synchronization
and PLLs Function Module
Copyright (c) March 2017, Mark Wickert
All rights reserved.
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modification, are permitted provided that the following conditions are met:
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"""
"""
Synchronization and PLLs
A collection of useful functions when studying PLLs
and synchronization and digital comm
Mark Wickert August 2014
"""
import numpy as np
import scipy.signal as signal
from . import digitalcom as dc
from .digitalcom import MPSK_bb
[docs]def NDA_symb_sync(z,Ns,L,BnTs,zeta=0.707,I_ord=3):
"""
zz,e_tau = NDA_symb_sync(z,Ns,L,BnTs,zeta=0.707,I_ord=3)
z = complex baseband input signal at nominally Ns samples
per symbol
Ns = Nominal number of samples per symbol (Ts/T) in the symbol
tracking loop, often 4
BnTs = time bandwidth product of loop bandwidth and the symbol period,
thus the loop bandwidth as a fraction of the symbol rate.
zeta = loop damping factor
I_ord = interpolator order, 1, 2, or 3
e_tau = the timing error e(k) input to the loop filter
Kp = The phase detector gain in the symbol tracking loop; for the
NDA algoithm used here always 1
Mark Wickert July 2014
Motivated by code found in M. Rice, Digital Communications A Discrete-Time
Approach, Prentice Hall, New Jersey, 2009. (ISBN 978-0-13-030497-1).
"""
# Loop filter parameters
K0 = -1.0 # The modulo 1 counter counts down so a sign change in loop
Kp = 1.0
K1 = 4*zeta/(zeta + 1/(4*zeta))*BnTs/Ns/Kp/K0
K2 = 4/(zeta + 1/(4*zeta))**2*(BnTs/Ns)**2/Kp/K0
zz = np.zeros(len(z),dtype=np.complex128)
#zz = np.zeros(int(np.floor(len(z)/float(Ns))),dtype=np.complex128)
e_tau = np.zeros(len(z))
#e_tau = np.zeros(int(np.floor(len(z)/float(Ns))))
#z_TED_buff = np.zeros(Ns)
c1_buff = np.zeros(2*L+1)
vi = 0
CNT_next = 0
mu_next = 0
underflow = 0
epsilon = 0
mm = 1
z = np.hstack(([0], z))
for nn in range(1,Ns*int(np.floor(len(z)/float(Ns)-(Ns-1)))):
# Define variables used in linear interpolator control
CNT = CNT_next
mu = mu_next
if underflow == 1:
if I_ord == 1:
# Decimated interpolator output (piecewise linear)
z_interp = mu*z[nn] + (1 - mu)*z[nn-1]
elif I_ord == 2:
# Decimated interpolator output (piecewise parabolic)
# in Farrow form with alpha = 1/2
v2 = 1/2.*np.sum(z[nn+2:nn-1-1:-1]*[1, -1, -1, 1])
v1 = 1/2.*np.sum(z[nn+2:nn-1-1:-1]*[-1, 3, -1, -1])
v0 = z[nn]
z_interp = (mu*v2 + v1)*mu + v0
elif I_ord == 3:
# Decimated interpolator output (piecewise cubic)
# in Farrow form
v3 = np.sum(z[nn+2:nn-1-1:-1]*[1/6., -1/2., 1/2., -1/6.])
v2 = np.sum(z[nn+2:nn-1-1:-1]*[0, 1/2., -1, 1/2.])
v1 = np.sum(z[nn+2:nn-1-1:-1]*[-1/6., 1, -1/2., -1/3.])
v0 = z[nn]
z_interp = ((mu*v3 + v2)*mu + v1)*mu + v0
else:
print('Error: I_ord must 1, 2, or 3')
# Form TED output that is smoothed using 2*L+1 samples
# We need Ns interpolants for this TED: 0:Ns-1
c1 = 0
for kk in range(Ns):
if I_ord == 1:
# piecewise linear interp over Ns samples for TED
z_TED_interp = mu*z[nn+kk] + (1 - mu)*z[nn-1+kk]
elif I_ord == 2:
# piecewise parabolic in Farrow form with alpha = 1/2
v2 = 1/2.*np.sum(z[nn+kk+2:nn+kk-1-1:-1]*[1, -1, -1, 1])
v1 = 1/2.*np.sum(z[nn+kk+2:nn+kk-1-1:-1]*[-1, 3, -1, -1])
v0 = z[nn+kk]
z_TED_interp = (mu*v2 + v1)*mu + v0
elif I_ord == 3:
# piecewise cubic in Farrow form
v3 = np.sum(z[nn+kk+2:nn+kk-1-1:-1]*[1/6., -1/2., 1/2., -1/6.])
v2 = np.sum(z[nn+kk+2:nn+kk-1-1:-1]*[0, 1/2., -1, 1/2.])
v1 = np.sum(z[nn+kk+2:nn+kk-1-1:-1]*[-1/6., 1, -1/2., -1/3.])
v0 = z[nn+kk]
z_TED_interp = ((mu*v3 + v2)*mu + v1)*mu + v0
else:
print('Error: I_ord must 1, 2, or 3')
c1 = c1 + np.abs(z_TED_interp)**2 * np.exp(-1j*2*np.pi/Ns*kk)
c1 = c1/Ns
# Update 2*L+1 length buffer for TED output smoothing
c1_buff = np.hstack(([c1], c1_buff[:-1]))
# Form the smoothed TED output
epsilon = -1/(2*np.pi)*np.angle(np.sum(c1_buff)/(2*L+1))
# Save symbol spaced (decimated to symbol rate) interpolants in zz
zz[mm] = z_interp
e_tau[mm] = epsilon # log the error to the output vector e
mm += 1
else:
# Simple zezo-order hold interpolation between symbol samples
# we just coast using the old value
#epsilon = 0
pass
vp = K1*epsilon # proportional component of loop filter
vi = vi + K2*epsilon # integrator component of loop filter
v = vp + vi # loop filter output
W = 1/float(Ns) + v # counter control word
# update registers
CNT_next = CNT - W # Update counter value for next cycle
if CNT_next < 0: # Test to see if underflow has occured
CNT_next = 1 + CNT_next # Reduce counter value modulo-1 if underflow
underflow = 1 # Set the underflow flag
mu_next = CNT/W # update mu
else:
underflow = 0
mu_next = mu
# Remove zero samples at end
zz = zz[:-(len(zz)-mm+1)]
# Normalize so symbol values have a unity magnitude
zz /=np.std(zz)
e_tau = e_tau[:-(len(e_tau)-mm+1)]
return zz, e_tau
[docs]def DD_carrier_sync(z,M,BnTs,zeta=0.707,type=0):
"""
z_prime,a_hat,e_phi = DD_carrier_sync(z,M,BnTs,zeta=0.707,type=0)
Decision directed carrier phase tracking
z = complex baseband PSK signal at one sample per symbol
M = The PSK modulation order, i.e., 2, 8, or 8.
BnTs = time bandwidth product of loop bandwidth and the symbol period,
thus the loop bandwidth as a fraction of the symbol rate.
zeta = loop damping factor
type = Phase error detector type: 0 <> ML, 1 <> heuristic
z_prime = phase rotation output (like soft symbol values)
a_hat = the hard decision symbol values landing at the constellation
values
e_phi = the phase error e(k) into the loop filter
Ns = Nominal number of samples per symbol (Ts/T) in the carrier
phase tracking loop, almost always 1
Kp = The phase detector gain in the carrier phase tracking loop;
This value depends upon the algorithm type. For the ML scheme
described at the end of notes Chapter 9, A = 1, K 1/sqrt(2),
so Kp = sqrt(2).
Mark Wickert July 2014
Motivated by code found in M. Rice, Digital Communications A Discrete-Time
Approach, Prentice Hall, New Jersey, 2009. (ISBN 978-0-13-030497-1).
"""
Ns = 1
Kp = np.sqrt(2.) # for type 0
z_prime = np.zeros_like(z)
a_hat = np.zeros_like(z)
e_phi = np.zeros(len(z))
theta_h = np.zeros(len(z))
theta_hat = 0
# Tracking loop constants
K0 = 1;
K1 = 4*zeta/(zeta + 1/(4*zeta))*BnTs/Ns/Kp/K0;
K2 = 4/(zeta + 1/(4*zeta))**2*(BnTs/Ns)**2/Kp/K0;
# Initial condition
vi = 0
for nn in range(len(z)):
# Multiply by the phase estimate exp(-j*theta_hat[n])
z_prime[nn] = z[nn]*np.exp(-1j*theta_hat)
if M == 2:
a_hat[nn] = np.sign(z_prime[nn].real) + 1j*0
elif M == 4:
a_hat[nn] = np.sign(z_prime[nn].real) + 1j*np.sign(z_prime[nn].imag)
elif M == 8:
a_hat[nn] = np.angle(z_prime[nn])/(2*np.pi/8.)
# round to the nearest integer and fold to nonnegative
# integers; detection into M-levels with thresholds at mid points.
a_hat[nn] = np.mod(round(a_hat[nn]),8)
a_hat[nn] = np.exp(1j*2*np.pi*a_hat[nn]/8)
else:
raise ValueError('M must be 2, 4, or 8')
if type == 0:
# Maximum likelihood (ML)
e_phi[nn] = z_prime[nn].imag * a_hat[nn].real - \
z_prime[nn].real * a_hat[nn].imag
elif type == 1:
# Heuristic
e_phi[nn] = np.angle(z_prime[nn]) - np.angle(a_hat[nn])
else:
raise ValueError('Type must be 0 or 1')
vp = K1*e_phi[nn] # proportional component of loop filter
vi = vi + K2*e_phi[nn] # integrator component of loop filter
v = vp + vi # loop filter output
theta_hat = np.mod(theta_hat + v,2*np.pi)
theta_h[nn] = theta_hat # phase track output array
#theta_hat = 0 # for open-loop testing
# Normalize outputs to have QPSK points at (+/-)1 + j(+/-)1
#if M == 4:
# z_prime = z_prime*np.sqrt(2)
return z_prime, a_hat, e_phi, theta_h
[docs]def time_step(z,Ns,t_step,Nstep):
"""
Create a one sample per symbol signal containing a phase rotation
step Nsymb into the waveform.
:param z: complex baseband signal after matched filter
:param Ns: number of sample per symbol
:param t_step: in samples relative to Ns
:param Nstep: symbol sample location where the step turns on
:return: the one sample per symbol signal containing the phase step
Mark Wickert July 2014
"""
z_step = np.hstack((z[:Ns*Nstep], z[(Ns*Nstep+t_step):], np.zeros(t_step)))
return z_step
[docs]def phase_step(z,Ns,p_step,Nstep):
"""
Create a one sample per symbol signal containing a phase rotation
step Nsymb into the waveform.
:param z: complex baseband signal after matched filter
:param Ns: number of sample per symbol
:param p_step: size in radians of the phase step
:param Nstep: symbol sample location where the step turns on
:return: the one sample symbol signal containing the phase step
Mark Wickert July 2014
"""
nn = np.arange(0,len(z[::Ns]))
theta = np.zeros(len(nn))
idx = np.where(nn >= Nstep)
theta[idx] = p_step*np.ones(len(idx))
z_rot = z[::Ns]*np.exp(1j*theta)
return z_rot
[docs]def PLL1(theta,fs,loop_type,Kv,fn,zeta,non_lin):
"""
Baseband Analog PLL Simulation Model
:param theta: input phase deviation in radians
:param fs: sampling rate in sample per second or Hz
:param loop_type: 1, first-order loop filter F(s)=K_LF; 2, integrator
with lead compensation F(s) = (1 + s tau2)/(s tau1),
i.e., a type II, or 3, lowpass with lead compensation
F(s) = (1 + s tau2)/(1 + s tau1)
:param Kv: VCO gain in Hz/v; note presently assume Kp = 1v/rad
and K_LF = 1; the user can easily change this
:param fn: Loop natural frequency (loops 2 & 3) or cutoff
frquency (loop 1)
:param zeta: Damping factor for loops 2 & 3
:param non_lin: 0, linear phase detector; 1, sinusoidal phase detector
:return: theta_hat = Output phase estimate of the input theta in radians,
ev = VCO control voltage,
phi = phase error = theta - theta_hat
Notes
-----
Alternate input in place of natural frequency, fn, in Hz is
the noise equivalent bandwidth Bn in Hz.
Mark Wickert, April 2007 for ECE 5625/4625
Modified February 2008 and July 2014 for ECE 5675/4675
Python version August 2014
"""
T = 1/float(fs)
Kv = 2*np.pi*Kv # convert Kv in Hz/v to rad/s/v
if loop_type == 1:
# First-order loop parameters
# Note Bn = K/4 Hz but K has units of rad/s
#fn = 4*Bn/(2*pi);
K = 2*np.pi*fn # loop natural frequency in rad/s
elif loop_type == 2:
# Second-order loop parameters
#fn = 1/(2*pi) * 2*Bn/(zeta + 1/(4*zeta));
K = 4 *np.pi*zeta*fn # loop natural frequency in rad/s
tau2 = zeta/(np.pi*fn)
elif loop_type == 3:
# Second-order loop parameters for one-pole lowpass with
# phase lead correction.
#fn = 1/(2*pi) * 2*Bn/(zeta + 1/(4*zeta));
K = Kv # Essentially the VCO gain sets the single-sided
# hold-in range in Hz, as it is assumed that Kp = 1
# and KLF = 1.
tau1 = K/((2*np.pi*fn)**2)
tau2 = 2*zeta/(2*np.pi*fn)*(1 - 2*np.pi*fn/K*1/(2*zeta))
else:
print('Loop type must be 1, 2, or 3')
# Initialize integration approximation filters
filt_in_last = 0; filt_out_last = 0;
vco_in_last = 0; vco_out = 0; vco_out_last = 0;
# Initialize working and final output vectors
n = np.arange(len(theta))
theta_hat = np.zeros_like(theta)
ev = np.zeros_like(theta)
phi = np.zeros_like(theta)
# Begin the simulation loop
for k in range(len(n)):
phi[k] = theta[k] - vco_out
if non_lin == 1:
# sinusoidal phase detector
pd_out = np.sin(phi[k])
else:
# Linear phase detector
pd_out = phi[k]
# Loop gain
gain_out = K/Kv*pd_out # apply VCO gain at VCO
# Loop filter
if loop_type == 2:
filt_in = (1/tau2)*gain_out
filt_out = filt_out_last + T/2*(filt_in + filt_in_last)
filt_in_last = filt_in
filt_out_last = filt_out
filt_out = filt_out + gain_out
elif loop_type == 3:
filt_in = (tau2/tau1)*gain_out - (1/tau1)*filt_out_last
u3 = filt_in + (1/tau2)*filt_out_last
filt_out = filt_out_last + T/2*(filt_in + filt_in_last)
filt_in_last = filt_in
filt_out_last = filt_out
else:
filt_out = gain_out;
# VCO
vco_in = filt_out
if loop_type == 3:
vco_in = u3
vco_out = vco_out_last + T/2*(vco_in + vco_in_last)
vco_in_last = vco_in
vco_out_last = vco_out
vco_out = Kv*vco_out # apply Kv
# Measured loop signals
ev[k] = vco_in
theta_hat[k] = vco_out
return theta_hat, ev, phi
[docs]def PLL_cbb(x,fs,loop_type,Kv,fn,zeta):
"""
Baseband Analog PLL Simulation Model
:param x: input phase deviation in radians
:param fs: sampling rate in sample per second or Hz
:param loop_type: 1, first-order loop filter F(s)=K_LF; 2, integrator
with lead compensation F(s) = (1 + s tau2)/(s tau1),
i.e., a type II, or 3, lowpass with lead compensation
F(s) = (1 + s tau2)/(1 + s tau1)
:param Kv: VCO gain in Hz/v; note presently assume Kp = 1v/rad
and K_LF = 1; the user can easily change this
:param fn: Loop natural frequency (loops 2 & 3) or cutoff
frequency (loop 1)
:param zeta: Damping factor for loops 2 & 3
:return: theta_hat = Output phase estimate of the input theta in radians,
ev = VCO control voltage,
phi = phase error = theta - theta_hat
Mark Wickert, April 2007 for ECE 5625/4625
Modified February 2008 and July 2014 for ECE 5675/4675
Python version August 2014
"""
T = 1/float(fs)
Kv = 2*np.pi*Kv # convert Kv in Hz/v to rad/s/v
if loop_type == 1:
# First-order loop parameters
# Note Bn = K/4 Hz but K has units of rad/s
#fn = 4*Bn/(2*pi);
K = 2*np.pi*fn # loop natural frequency in rad/s
elif loop_type == 2:
# Second-order loop parameters
#fn = 1/(2*pi) * 2*Bn/(zeta + 1/(4*zeta));
K = 4 *np.pi*zeta*fn # loop natural frequency in rad/s
tau2 = zeta/(np.pi*fn)
elif loop_type == 3:
# Second-order loop parameters for one-pole lowpass with
# phase lead correction.
#fn = 1/(2*pi) * 2*Bn/(zeta + 1/(4*zeta));
K = Kv # Essentially the VCO gain sets the single-sided
# hold-in range in Hz, as it is assumed that Kp = 1
# and KLF = 1.
tau1 = K/((2*np.pi*fn)^2);
tau2 = 2*zeta/(2*np.pi*fn)*(1 - 2*np.pi*fn/K*1/(2*zeta))
else:
print('Loop type must be 1, 2, or 3')
# Initialize integration approximation filters
filt_in_last = 0; filt_out_last = 0;
vco_in_last = 0; vco_out = 0; vco_out_last = 0;
vco_out_cbb = 0
# Initialize working and final output vectors
n = np.arange(len(x))
theta_hat = np.zeros(len(x))
ev = np.zeros(len(x))
phi = np.zeros(len(x))
# Begin the simulation loop
for k in range(len(n)):
#phi[k] = theta[k] - vco_out
phi[k] = np.imag(x[k] * np.conj(vco_out_cbb))
pd_out = phi[k]
# Loop gain
gain_out = K/Kv*pd_out # apply VCO gain at VCO
# Loop filter
if loop_type == 2:
filt_in = (1/tau2)*gain_out
filt_out = filt_out_last + T/2*(filt_in + filt_in_last)
filt_in_last = filt_in
filt_out_last = filt_out
filt_out = filt_out + gain_out
elif loop_type == 3:
filt_in = (tau2/tau1)*gain_out - (1/tau1)*filt_out_last
u3 = filt_in + (1/tau2)*filt_out_last
filt_out = filt_out_last + T/2*(filt_in + filt_in_last)
filt_in_last = filt_in
filt_out_last = filt_out
else:
filt_out = gain_out;
# VCO
vco_in = filt_out
if loop_type == 3:
vco_in = u3
vco_out = vco_out_last + T/2*(vco_in + vco_in_last)
vco_in_last = vco_in
vco_out_last = vco_out
vco_out = Kv*vco_out # apply Kv
vco_out_cbb = np.exp(1j*vco_out)
# Measured loop signals
ev[k] = vco_in
theta_hat[k] = vco_out
return theta_hat, ev, phi