In [1]:
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
%matplotlib inline
In [2]:
class Process:
"""Signal processing"""
rate = 1
class Gain(Process):
"""Apply a flat gain."""
def __init__(self, g):
self.gain = g
def __call__(self, x):
return x*self.gain
class US(Process):
"""Upsampler, zero stuffer."""
def __init__(self, r):
self.rate = r
def __call__(self, x):
return np.vstack((x*self.rate, np.zeros((self.rate - 1, len(x))))).T.ravel()
class DS(Process):
"""Downsampler"""
def __init__(self, r, prefer_mid=True):
self.r = r
self.rate = 1./r
self.prefer_mid = prefer_mid
def __call__(self, x):
if self.prefer_mid:
start = (len(x) - 1) // 2 % self.r
else:
start = 0
return x[start::self.r]
class FIR(Process):
"""Finite impulse response filter."""
def __init__(self, b, a=1):
b = np.atleast_1d(b)
# b[np.fabs(b) < 3e-7] = 0
self.b, self.a = b, a
def __call__(self, x):
return np.convolve(x, self.b)/self.a
def quantize(self, m):
self.b = np.round(self.b*m).astype(np.int32)
self.a = np.round(self.a*m).astype(np.int32)
return self
def freqz(self, worN=1024):
return signal.freqz(self.b, self.a, worN)
def plot_freq(self, ax, color=None, r=1, f=1, inset=True, worN=1024):
s, h = self.freqz(worN)
s /= np.pi*f
hh = np.absolute(h/r)
ax.plot(s, hh, color=color)
if inset:
ax.plot(s, np.fabs(1 - hh), color=color)
def plot_time(self, ax, color=None, r=1):
ax.step(np.arange(len(self.b))/r, self.b/self.a, color=color)
class LPF(FIR):
"""Lowpass FIR filter with Remez/Parks-McClellan"""
def __init__(self, n, wa, wb, k=(1., 1), g=(1., 0)):
b = signal.remez(
numtaps=n,
bands=(0, wa, wb, 1.),
desired=g,
weight=k,
fs=2,
grid_density=256)
FIR.__init__(self, b)
class HBF(LPF):
"""Half-band filter"""
def __init__(self, n, w):
LPF.__init__(self, 4*n - 1, w, 1 - w)
class FIR2(FIR):
"""Use firwin2 to construct a FIR"""
def __init__(self, n, w, g):
b = signal.firwin2(n, w, g)
FIR.__init__(self, b)
class FIRLS(FIR):
"""Use firls to construct a FIR"""
def __init__(self, n, w, g, weight=None):
b = signal.firls(n, w, g, weight)
FIR.__init__(self, b)
class CIC(FIR):
"""Cascaded integrator comb interpolator"""
# CIC: http://www.tsdconseil.fr/log/scriptscilab/cic/cic-en.pdf
# resampling: http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/01300_Resampling.pdf
def __init__(self, r, m, n):
# self.rate = r if r > 0 else -1/r
self.rate = 1 # use a US/DS before to implement the rate change
self.r = r # rate change
self.m = m # delay
self.n = n # filter order
b = 1
a = 1
for i in range(n):
b = np.convolve(np.ones(abs(r)*m, np.int32), b)
a *= abs(r)*m
FIR.__init__(self, b, a)
#b = 1 #[max(r, 1)] + [0]*(max(r, 1) - 1)
#b = b[::max(1, -r)]/max(1, -r)
def disabled__call__(self, x): # not yet working, instead defer to FIR
k = (1 << 30)/np.fabs(x).max()
x = (x*k).astype(np.int64)
if self.r > 0:
for i in range(self.n):
x = np.diff(np.r_[0, x], self.m)
x = US(self.r)(x)
for i in range(self.n):
x = np.cumsum(x)
else:
for i in range(self.n):
x = np.cumsum(x)
x = DS(-self.r)(x)
for i in range(self.n):
x = np.diff(np.r_[0, x], self.m)
return x/k
In [3]:
# CIC droop compensation filter
# frequency nodes
s = np.linspace(1e-9, .4*2*1.02, 31)
rm, n, k = 5, 5, 20/1.*2
# cic response
h = (np.sin(np.pi*rm*s/k)/(rm*np.sin(np.pi*s/k)))**n
plt.plot(s, 1/np.absolute(h), ".")
ciccomp = FIR(signal.firls(
9,
np.repeat(s, 2)[1:-1],
1/np.absolute(np.repeat(h, 2)[1:-1])
))
# ciccomp.b /= ciccomp.b.sum()
# include the final CIC gain and post-scaling here
ciccomp.b *= .9995*(2.**9/5**4)
ciccomp.quantize(1 << 16)
FIR(Gain(5**4/2.**9)(ciccomp([1.]))).plot_freq(plt.gca(), inset=False)
# ciccomp coefficients:
ciccomp.b, ciccomp.a
Out[3]:
(array([ 24, -85, 281, -1314, 55856, -1314, 281, -85, 24],
dtype=int32),
65536)
In [5]:
# Phaser interpolation filter, 20x
steps = [
# 1/10 samples per cycles
Gain(5**4/2.**9), # compensate ciccomp+cic+shift gain
# 1/10
ciccomp,
# 1/10
US(2), HBF(10, .4).quantize(1 << 18),
# 1/5
US(2), HBF(5, 1/4.).quantize(1 << 18),
# 2/5
US(5), CIC(5, 1, 5),
# 2/1
]
fig, ax = plt.subplots(1, 3, figsize=(25, 5))
x = np.ones(1)
ax[2].step(np.arange(len(x)), x)
r = 1.
colors = list("kbgrcm")
for step in steps:
r *= step.rate
x = step(x)
# print(step, x)
if hasattr(step, "plot_freq"):
color = colors.pop(0)
step.plot_freq(ax[0], r=1, color=color)
q = FIR(b=x)
q.plot_freq(ax[1], r=r, f=1/r, color=color, inset=True, worN=2048)
q.plot_time(ax[2], r=r, color=color)
ax[0].grid()
ax[0].grid(which="minor", alpha=.2)
ax[0].set_ylim(1e-5, 2)
ax[0].set_xlim(0, 1)
ax[0].set_yscale("log")
ax[0].set_xlabel("frequency (filter step output nyquist)")
ax[0].set_title("filter step response and ripple")
ax[1].grid()
ax[1].grid(which="minor", alpha=.2)
ax[1].set_ylim(1e-5, 2)
#ax[1].set_ylim(.9, 1.1)
ax[1].set_xlim(1e-2*r, r)
ax[1].set_xscale("log")
ax[1].set_title("cumulative filter step response and ripple")
ax[1].set_xlabel("frequency (overall input nyquist)")
ax[1].set_yscale("log")
ax[1].plot(np.array([1e-3, .4, np.nan, .6, 1e2])*2, [1e-2, 1e-2, np.nan, 1e-7**.5, 1e-7**.5])
ax[2].set_title("cumulative impulse response")
ax[2].set_xlabel("time (input samples)")
fig, ax = plt.subplots(1, 3, figsize=(25, 5))
# build a shaped pulse
a0 = np.ones(32)
a = CIC(6, 1, 5)(a0)
af = np.array([(.5, .4), (.5, -.15)])
t = np.arange(len(a))
y0 = sum(a*ai*np.exp(2j*np.pi*fi*t) for ai, fi in af)
# pass it through the filter
y = y0
for step in steps:
y = step(y)
t0 = 10*4
ax[2].plot(t*t0, a, t*t0, y0.real, np.arange(len(y))/r*t0, y.real)
m = 1 << 16
h = np.absolute(np.fft.fft(y, m)/r/len(a0))
s = np.fft.fftfreq(m)*r/t0
for i in range(2):
ax[i].plot(af[:, 1]/t0, np.absolute(af[:, 0]), "o", s, h)
ax[0].set_ylim(1e-7, 1)
ax[0].set_yscale("log")
ax[0].set_title("wide band amplitude response to shaped pulse")
ax[0].set_xlabel("frequency (f_DAC)")
ax[0].grid()
ax[0].grid(which="minor", alpha=.2)
ax[1].set_ylim(1e-4, 1)
ax[1].set_yscale("log")
ax[1].set_xlim(-1/t0, 1/t0)
ax[1].grid()
ax[1].grid(which="minor", alpha=.2);
ax[1].set_title("amplitude response to shaped pulse")
ax[1].set_xlabel("frequency (f_DAC)")
ax[2].set_title("shaped pulse (envelope blue, data orange, response green)")
ax[2].set_xlabel("time (1/f_DAC)")
Out[5]:
Text(0.5, 0, 'time (1/f_DAC)')
In [6]:
# individual filter step taps
for step in steps:
if hasattr(step, "b"):
print((list(step.b), step.a, len(step.b))) # print taps (a/b), and number of taps
else:
print(step.rate) # print rate change
1 ([24, -85, 281, -1314, 55856, -1314, 281, -85, 24], 65536, 9) 2 ([-167, 0, 428, 0, -931, 0, 1776, 0, -3115, 0, 5185, 0, -8442, 0, 14028, 0, -26142, 0, 82873, 131072, 82873, 0, -26142, 0, 14028, 0, -8442, 0, 5185, 0, -3115, 0, 1776, 0, -931, 0, 428, 0, -167], 262144, 39) 2 ([294, 0, -1865, 0, 6869, 0, -20436, 0, 80679, 131072, 80679, 0, -20436, 0, 6869, 0, -1865, 0, 294], 262144, 19) 5 ([1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1], 3125, 21)
In [7]:
# 14 bit quantization snr
1.67+6.02*14
Out[7]:
85.95
In [8]:
# overall scale
FIR(x)([1.]).sum()/20.
Out[8]:
0.9993545722817482
In [9]:
# passband ripple
plt.ylim(-.012, .005)
plt.xlim(0, (1<<16)/(2*r))
plt.plot(20*np.log10(np.absolute(np.fft.rfft(x/r, 1<<16))))
plt.plot(np.array([0, .4, np.nan, 0, .4])/r*(1<<16), 20*np.log10([1-1e-3, 1-1e-3, np.nan, 1, 1]))
# passband flatness
20*np.log10(1.001)
Out[9]:
0.008681549586371858
In [16]:
# group delay
np.argmax(np.fabs(x))/r
Out[16]:
16.25
In [17]:
plt.xscale("log")
plt.xlabel("f (f_DAC)")
plt.xlim(1e-3, .5)
plt.yscale("log")
plt.ylim(1e-5, 1.1)
plt.ylabel("amplitude response")
plt.grid(which="both")
plt.title("Phaser interpolation filter amplitude response (blue) and ripple (orange)")
FIR(x).plot_freq(plt.gca(), r=r, f=4)
In [12]:
# overall impulse response of 20x interpolation chain (aply 20x zero stuffing to your data!)
x
Out[12]:
array([-2.04411488e-12, -1.02205744e-11, -3.06617232e-11, -7.15440207e-11,
-1.43088041e-10, -2.47337900e-10, -3.78161252e-10, -5.21249294e-10,
-6.54116761e-10, -7.46101930e-10, -7.65840849e-10, -6.81267334e-10,
-4.59612973e-10, -6.74071216e-11, 5.29523092e-10, 1.32165932e-09,
2.25579201e-09, 3.23502038e-09, 4.11875242e-09, 4.72270494e-09,
4.90307190e-09, 4.55652448e-09, 3.62021106e-09, 2.07175719e-09,
-7.07343559e-11, -2.69990768e-09, -5.61915374e-09, -8.54261032e-09,
-1.10951621e-08, -1.28124405e-08, -1.33658618e-08, -1.25626264e-08,
-1.03457196e-08, -6.79391112e-09, -2.12175534e-09, 3.34559526e-09,
9.17238997e-09, 1.48122799e-08, 1.96083180e-08, 2.27929590e-08,
2.36176745e-08, 2.13529533e-08, 1.52883010e-08, 4.73224031e-09,
-1.09876892e-08, -3.24138554e-08, -5.99585331e-08, -9.39039051e-08,
-1.34402061e-07, -1.81474998e-07, -2.34462726e-07, -2.92023267e-07,
-3.52132658e-07, -4.12084945e-07, -4.68492191e-07, -5.17795175e-07,
-5.56263399e-07, -5.79995082e-07, -5.84917165e-07, -5.66785309e-07,
-5.22009438e-07, -4.47653740e-07, -3.41436669e-07, -2.01730942e-07,
-2.75635412e-08, 1.80865156e-07, 4.22181377e-07, 6.93837952e-07,
9.92114318e-07, 1.31211651e-06, 1.64649736e-06, 1.98545647e-06,
2.31674023e-06, 2.62564182e-06, 2.89500118e-06, 3.10834994e-06,
3.24991139e-06, 3.30460048e-06, 3.25802383e-06, 3.09647974e-06,
2.80489580e-06, 2.36682897e-06, 1.76446549e-06, 9.78620966e-07,
-1.12597063e-08, -1.20496730e-06, -2.58152830e-06, -4.09920487e-06,
-5.69549487e-06, -7.28713185e-06, -8.77902223e-06, -1.00642453e-05,
-1.10240532e-05, -1.15278709e-05, -1.14332963e-05, -1.06911942e-05,
-9.34569640e-06, -7.53420150e-06, -5.48737515e-06, -3.52914992e-06,
-1.92852121e-06, -8.99547296e-07, -6.01349314e-07, -1.13811125e-06,
-2.55907994e-06, -4.67536742e-06, -7.05995093e-06, -9.04767288e-06,
-9.73524088e-06, -7.98122773e-06, -2.96699490e-06, 5.80330745e-06,
1.85027405e-05, 3.49815768e-05, 5.47673001e-05, 7.67996476e-05,
9.94306096e-05, 1.20424430e-04, 1.36957605e-04, 1.45618885e-04,
1.43428807e-04, 1.27839694e-04, 9.67356554e-05, 4.84325881e-05,
-1.83218252e-05, -1.03681630e-04, -2.06702600e-04, -3.25342237e-04,
-4.56459771e-04, -5.95816161e-04, -7.38689545e-04, -8.79875243e-04,
-1.01368576e-03, -1.13395076e-03, -1.23401712e-03, -1.30787327e-03,
-1.35014919e-03, -1.35611645e-03, -1.32168819e-03, -1.24341910e-03,
-1.11935631e-03, -9.49039368e-04, -7.33500250e-04, -4.75263360e-04,
-1.78345528e-04, 1.51938937e-04, 5.08978250e-04, 8.84863143e-04,
1.27038687e-03, 1.65504521e-03, 2.02800231e-03, 2.37809066e-03,
2.69381114e-03, 2.96333299e-03, 3.17449379e-03, 3.31642950e-03,
3.37957439e-03, 3.35566109e-03, 3.23772056e-03, 3.02008211e-03,
2.70023665e-03, 2.27883672e-03, 1.75969648e-03, 1.14979168e-03,
4.59259715e-04, -2.98978290e-04, -1.10921744e-03, -1.95296910e-03,
-2.80896088e-03, -3.65313662e-03, -4.46137663e-03, -5.20949765e-03,
-5.87325285e-03, -6.42833188e-03, -6.85036080e-03, -7.11837671e-03,
-7.21482771e-03, -7.12557292e-03, -6.83988248e-03, -6.35043753e-03,
-5.65606680e-03, -4.76174654e-03, -3.67860060e-03, -2.42390037e-03,
-1.02106481e-03, 5.01807128e-04, 2.11193646e-03, 3.77186420e-03,
5.43945136e-03, 7.06787892e-03, 8.61108816e-03, 1.00237807e-02,
1.12614184e-02, 1.22802234e-02, 1.30371782e-02, 1.34961951e-02,
1.36281162e-02, 1.34107135e-02, 1.28286888e-02, 1.18736736e-02,
1.05481291e-02, 8.86534568e-03, 6.84944328e-03, 4.53537123e-03,
1.96890830e-03, -7.96730127e-04, -3.70089204e-03, -6.67548879e-03,
-9.64499512e-03, -1.25264491e-02, -1.52391708e-02, -1.77047619e-02,
-1.98471059e-02, -2.15923682e-02, -2.28689962e-02, -2.36179901e-02,
-2.37929035e-02, -2.33598430e-02, -2.22974686e-02, -2.05969933e-02,
-1.82675961e-02, -1.53364220e-02, -1.18485823e-02, -7.86715382e-03,
-3.47317984e-03, 1.24085134e-03, 6.17100615e-03, 1.12019058e-02,
1.62067262e-02, 2.10471982e-02, 2.55896001e-02, 2.97047580e-02,
3.32680453e-02, 3.61593834e-02, 3.82632410e-02, 3.94851875e-02,
3.97518922e-02, 3.90111250e-02, 3.72317561e-02, 3.44037559e-02,
3.05458800e-02, 2.57056691e-02, 1.99594488e-02, 1.34123299e-02,
6.19820836e-03, -1.53129923e-03, -9.60685540e-03, -1.78414019e-02,
-2.60301596e-02, -3.39506288e-02, -4.13872612e-02, -4.81314605e-02,
-5.39815818e-02, -5.87429323e-02, -6.22277705e-02, -6.42824656e-02,
-6.47874972e-02, -6.36574555e-02, -6.08410410e-02, -5.63210646e-02,
-5.01253227e-02, -4.23265966e-02, -3.30426530e-02, -2.24362439e-02,
-1.07151065e-02, 1.88263726e-03, 1.50884801e-02, 2.86035298e-02,
4.20985094e-02, 5.52137564e-02, 6.75980530e-02, 7.89086248e-02,
8.88111421e-02, 9.69797188e-02, 1.03096913e-01, 1.06896042e-01,
1.08161180e-01, 1.06727163e-01, 1.02479582e-01, 9.53547896e-02,
8.53624611e-02, 7.25855952e-02, 5.71805148e-02, 3.93768665e-02,
1.94776204e-02, -2.15595347e-03, -2.51042853e-02, -4.88895057e-02,
-7.29754455e-02, -9.67676357e-02, -1.19665074e-01, -1.41060224e-01,
-1.60339015e-01, -1.76880846e-01, -1.90058577e-01, -1.99303321e-01,
-2.04104433e-01, -2.04009520e-01, -1.98624432e-01, -1.87613268e-01,
-1.70743952e-01, -1.47888230e-01, -1.19021673e-01, -8.42236755e-02,
-4.36774566e-02, 2.32991454e-03, 5.34075173e-02, 1.09060580e-01,
1.68690481e-01, 2.31594746e-01, 2.97018659e-01, 3.64155258e-01,
4.32145338e-01, 5.00077450e-01, 5.66987902e-01, 6.31941022e-01,
6.94029153e-01, 7.52372662e-01, 8.06119932e-01, 8.54447366e-01,
8.96631465e-01, 9.32048824e-01, 9.60176134e-01, 9.80590186e-01,
9.92967863e-01, 9.97115296e-01, 9.92967863e-01, 9.80590186e-01,
9.60176134e-01, 9.32048824e-01, 8.96631465e-01, 8.54447366e-01,
8.06119932e-01, 7.52372662e-01, 6.94029153e-01, 6.31941022e-01,
5.66987902e-01, 5.00077450e-01, 4.32145338e-01, 3.64155258e-01,
2.97018659e-01, 2.31594746e-01, 1.68690481e-01, 1.09060580e-01,
5.34075173e-02, 2.32991454e-03, -4.36774566e-02, -8.42236755e-02,
-1.19021673e-01, -1.47888230e-01, -1.70743952e-01, -1.87613268e-01,
-1.98624432e-01, -2.04009520e-01, -2.04104433e-01, -1.99303321e-01,
-1.90058577e-01, -1.76880846e-01, -1.60339015e-01, -1.41060224e-01,
-1.19665074e-01, -9.67676357e-02, -7.29754455e-02, -4.88895057e-02,
-2.51042853e-02, -2.15595347e-03, 1.94776204e-02, 3.93768665e-02,
5.71805148e-02, 7.25855952e-02, 8.53624611e-02, 9.53547896e-02,
1.02479582e-01, 1.06727163e-01, 1.08161180e-01, 1.06896042e-01,
1.03096913e-01, 9.69797188e-02, 8.88111421e-02, 7.89086248e-02,
6.75980530e-02, 5.52137564e-02, 4.20985094e-02, 2.86035298e-02,
1.50884801e-02, 1.88263726e-03, -1.07151065e-02, -2.24362439e-02,
-3.30426530e-02, -4.23265966e-02, -5.01253227e-02, -5.63210646e-02,
-6.08410410e-02, -6.36574555e-02, -6.47874972e-02, -6.42824656e-02,
-6.22277705e-02, -5.87429323e-02, -5.39815818e-02, -4.81314605e-02,
-4.13872612e-02, -3.39506288e-02, -2.60301596e-02, -1.78414019e-02,
-9.60685540e-03, -1.53129923e-03, 6.19820836e-03, 1.34123299e-02,
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