def test_error_bounded_addition():
a = error_bounded(1., interval(-0.1, 0.3))
b = error_bounded(1., interval(-0.3, 0.2))
c = a + b
assert c.number == 2.
assert math.isclose(c.error_bounds.lower_bound, -0.4)
assert math.isclose(c.error_bounds.upper_bound, 0.5)
d = a + 2
assert d.number == 3.
assert d.error_bounds == a.error_bounds
e = 1 + b
assert e.number == 2.
assert e.error_bounds == b.error_bounds
f = error_bounded(interval(-1, 1), interval(-0.1, 0.2))
g = a + f
assert g.number == interval(0, 2)
assert g.error_bounds == interval(-0.2, 0.5)
with FixedFormatArithmeticLogicUnit(
format_=Q(7), rounding_method=nearest_integer
):
h = (a + (-0.5)) + fixed(0.2)
assert h.number == fixed(0.7)
assert a.error_bounds in h.error_bounds
def computation_error_bounds(self, input_ranges, state_ranges):
state_ranges = state_ranges or []
domain = self.algorithm.define(
inputs=tuple(error_bounded(fixed(r)) for r in input_ranges),
states=tuple(error_bounded(fixed(r)) for r in state_ranges),
parameters=tuple(self.parameters))
error_bounds = []
unit = ProcessingUnit.active()
with unit.trace() as trace:
scope = dict(domain)
compare = type(unit).compare
for change in self.algorithm.step(scope):
if len(trace) > 0:
if any(func is not compare for func, *_ in trace):
for value in change.values():
if isinstance(value, tuple):
error_bounds.extend(elem.error_bounds
for elem in value)
else:
error_bounds.append(value.error_bounds)
trace.clear()
scope.update(change)
scope.update(domain)
if not error_bounds:
return interval(0)
if len(error_bounds) == 1:
return error_bounds[0]
return interval(
lower_bound=np.c_[[eb.lower_bound for eb in error_bounds]],
upper_bound=np.c_[[eb.upper_bound for eb in error_bounds]])
def test_symbolic_expression():
with FixedFormatArithmeticLogicUnit(
format_=Q(7), allows_overflow=False,
rounding_method=nearest_integer) as alu:
x, y, z, w = sympy.symbols('x y z w')
expr = ((x + 0.5 * y) * z) - w
subs = {x: fixed(0.25), y: fixed(0.1), z: 0.25, w: 0.5}
result = expr.subs(subs)
assert math.isclose(result, -0.425, abs_tol=2**alu.format_.lsb)
def test_nonassociative_expression():
with MultiFormatArithmeticLogicUnit(
wordlength=8,
allows_overflow=False,
allows_underflow=True,
rounding_method=nearest_integer,
) as alu:
x, y, z, w = sympy.symbols('x y z w')
expr = x + y + z + w
subs = {
x: fixed(2**-5, format_=Q(2, 5)),
y: fixed(2**-3, format_=Q(3, 4)),
z: fixed(2**-1, format_=Q(4, 3)),
w: fixed(2, format_=Q(5, 2))
}
result0 = nonassociative(variant=0)(expr).subs(subs)
result1 = nonassociative(variant=1)(expr).subs(subs)
assert math.isclose(result0, result1, abs_tol=2**result0.format_.lsb)
assert result0 != result1
def test_mixed_symbols():
with FixedFormatArithmeticLogicUnit(format_=Q(7), allows_overflow=False):
a = sympy.sympify(fixed(0.5))
b = sympy.sympify(error_bounded(0.25))
assert (a * b).number == fixed(0.125)
def main():
random.seed(7)
np.random.seed(7)
fs = 200 # Hz
fo = 40 # Hz
rp = 1 # dB
rs = 80 # dB
fpass = fo - 5 # Hz
fstop = fo + 5 # Hz
# N, fo = signal.cheb1ord(fpass, fstop, rp, rs, fs=fs)
# N, fo = signal.cheb2ord(fpass, fstop, rp, rs, fs=fs)
# N, fo = signal.buttord(fpass, fstop, rp, rs, fs=fs)
N, fo = signal.ellipord(fpass, fstop, rp, rs, fs=fs)
wp = 2 * fs * np.tan(np.pi * fo / fs)
# prototype = signal.lti(*signal.cheb1ap(N, rp))
# prototype = signal.lti(*signal.cheb2ap(N, rs))
# prototype = signal.lti(*signal.buttap(N))
prototype = signal.lti(*signal.ellipap(N=N, rp=rp, rs=rs))
model = lowpass_to_lowpass(prototype, wo=wp)
model = discretize(model, dt=1 / fs)
print(model)
# Use PSD output to white noise input PSD ratio as response
window = signal.get_window('blackman', 256)
psd = functools.partial(signal.welch,
scaling='density',
window=window,
fs=fs)
input_range = interval(lower_bound=-0.5, upper_bound=0.5)
input_noise_power_density = 0.0005
input_noise = np.random.normal(scale=np.sqrt(input_noise_power_density *
fs / 2),
size=512)
assert np.max(input_noise) < ProcessingUnit.active().rinfo().max
assert np.min(input_noise) > ProcessingUnit.active().rinfo().min
input_noise = np.array([fixed(n) for n in input_noise])
_, outputs = model.output(input_noise.astype(float), t=None)
output_noise = outputs.T[0]
freq, output_noise_power_density = psd(output_noise)
expected_response = 10 * np.log10(output_noise_power_density /
input_noise_power_density + 1 / inf)
# Take quantization noise into account
noise_floor = -6.02 * (ProcessingUnit.active().wordlength - 1) - 1.76
expected_response = np.maximum(expected_response, noise_floor)
# Formulate GP problem
toolbox = solvers.gp.formulate(
prototype,
transforms=[
functools.partial(lowpass_to_lowpass, wo=wp),
functools.partial(discretize, dt=1 / fs)
],
evaluate=functools.partial(
evaluate,
input_range=input_range,
input_noise=input_noise,
input_noise_power_density=input_noise_power_density,
psd=psd,
expected_response=expected_response),
weights=(1., 1., 1., 1., -1., -1., -1., -1.),
forms=[DirectFormI, DirectFormII],
variants=range(1000),
dtype=fixed,
tol=1e-6)
# Solve GP problem
only_visualize = False
if not only_visualize:
with multiprocessing.Pool() as pool:
try:
toolbox.register('map', pool.map)
stats = deap.tools.Statistics(
key=lambda code: code.fitness.values)
stats.register('avg', np.mean, axis=0)
stats.register('med', np.median, axis=0)
stats.register('min', np.min, axis=0)
pareto_front = deap.tools.ParetoFront()
population = toolbox.population(512)
population, logbook = solvers.gp.nsga2(population,
toolbox,
mu=512,
lambda_=128,
cxpb=0.5,
mutpb=0.05,
ngen=25,
stats=stats,
halloffame=pareto_front,
verbose=True)
finally:
toolbox.register('map', map)
with open('front.pkl', 'wb') as f:
pickle.dump(pareto_front, f)
with open('logbook.pkl', 'wb') as f:
pickle.dump(logbook, f)
else:
with open('front.pkl', 'rb') as f:
pareto_front = pickle.load(f)
with open('logbook.pkl', 'rb') as f:
logbook = pickle.load(f)
codes = []
criteria = []
for code in pareto_front:
crt = Criteria(*code.fitness.values)
if crt.frequency_response_error >= inf:
continue
if crt.stability_margin <= 0:
continue
if crt.overflow_margin < 0:
continue
if crt.underflow_margin < 0:
continue
codes.append(code)
criteria.append(crt)
frequency_response_error = np.array(
[crt.frequency_response_error for crt in criteria])
arithmetic_snr = np.array([crt.arithmetic_snr for crt in criteria])
stability_margin = np.array([crt.stability_margin for crt in criteria])
memory_size = np.array([crt.size_of_memory for crt in criteria])
memory_size_in_bytes = memory_size * ProcessingUnit.active().wordlength / 8
plt.figure()
gen, med = logbook.select('gen', 'med')
crt = Criteria(*np.array(med).T)
def fit_most_within_unit_interval(values):
values = np.asarray(values)
med = np.median(values)
mad = np.median(np.absolute(values - med))
if not mad:
return values
return (values - med) / (4. * mad)
plt.plot(gen,
fit_most_within_unit_interval(crt.overflow_margin),
label='Med[$M_o$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.underflow_margin),
label='Med[$M_u$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.stability_margin),
label='Med[$M_s$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.arithmetic_snr),
label='Med[$SNR_{arit}$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.frequency_response_error),
label='Med[$E_2$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.number_of_adders),
label='Med[$N_a$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.number_of_multipliers),
label='Med[$N_m$]')
plt.plot(gen,
fit_most_within_unit_interval(crt.size_of_memory),
label='Med[$N_e$]')
plt.ylabel('Normalized evolution')
plt.xlabel('Generations')
plt.ylim([-10, 10])
plt.legend(loc='upper right')
plt.savefig('population_evolution.png')
plt.figure()
plt.plot(*logbook.select('gen', 'nunfeas'))
plt.ylabel('Unfeasibles')
plt.xlabel('Generations')
plt.savefig('unfeasibles.png')
def pareto2d(ax, x, y):
ax.scatter(x, y, marker='o', facecolors='none', edgecolors='r')
idx = np.argsort(x)
ax.plot(x[idx], y[idx], 'k--')
fig = plt.figure()
ax = fig.add_subplot(3, 1, 1)
idx = solvers.gp.argnondominated(-frequency_response_error,
stability_margin)
pareto2d(ax, frequency_response_error[idx], stability_margin[idx])
ax.set_ylabel('Margin $M_s$')
ax.invert_xaxis()
ax = fig.add_subplot(3, 1, 2)
idx = solvers.gp.argnondominated(-frequency_response_error, arithmetic_snr)
pareto2d(ax, frequency_response_error[idx], arithmetic_snr[idx])
ax.set_ylabel('$SNR_{arit}$ [dBr]')
ax.invert_xaxis()
ax = fig.add_subplot(3, 1, 3)
idx = solvers.gp.argnondominated(-frequency_response_error,
-memory_size_in_bytes)
pareto2d(ax, frequency_response_error[idx], memory_size_in_bytes[idx])
ax.set_xlabel('Error $E_2$ [dBr]')
ax.set_ylabel('Total $N_e$ @ Q15 [bytes]')
ax.invert_xaxis()
ax.invert_yaxis()
plt.savefig('pareto_fronts.png')
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
idx = solvers.gp.argnondominated(-frequency_response_error, arithmetic_snr,
-memory_size_in_bytes)
scatter = ax.scatter(frequency_response_error[idx],
arithmetic_snr[idx],
memory_size_in_bytes[idx],
c=stability_margin[idx],
cmap='jet_r')
fig.colorbar(scatter, ax=ax, label='Margin $M_s$')
ax.set_xlabel('Error $E_2$ [dBr]')
ax.set_ylabel('$SNR_{arit}$ [dBr]')
ax.set_zlabel('Total $N_e$ @ Q15 [bytes]')
ax.invert_xaxis()
ax.invert_zaxis()
plt.savefig('pareto_sampling.png')
index = np.argsort(frequency_response_error)[0]
implement = toolbox.compile(codes[index])
optimized_implementation_a = implement(prototype)
index = np.argsort(frequency_response_error)[3]
implement = toolbox.compile(codes[index])
optimized_implementation_b = implement(prototype)
print(memory_size_in_bytes[np.argsort(frequency_response_error)])
print(stability_margin[np.argsort(frequency_response_error)])
print(arithmetic_snr[np.argsort(frequency_response_error)])
pretty(optimized_implementation_a).draw('optimized_a.png')
pretty(optimized_implementation_b).draw('optimized_b.png')
func = signal_processing_function(optimized_implementation_a)
output_noise = func(input_noise).T[0].astype(float)
_, output_noise_power_density = psd(output_noise)
optimized_implementation_a_response = \
10 * np.log10(output_noise_power_density / input_noise_power_density + 1/inf)
func = signal_processing_function(optimized_implementation_b)
output_noise = func(input_noise).T[0].astype(float)
_, output_noise_power_density = psd(output_noise)
optimized_implementation_b_response = \
10 * np.log10(output_noise_power_density / input_noise_power_density + 1/inf)
show_biquad_cascade = True
if show_biquad_cascade:
biquad_cascade = series_diagram([
DirectFormI.from_model(signal.dlti(
section[:3], section[3:], dt=model.dt),
dtype=fixed)
for section in signal.zpk2sos(model.zeros, model.poles, model.gain)
],
simplify=False)
pretty(biquad_cascade).draw('biquad.png')
func = signal_processing_function(biquad_cascade)
output_noise = func(input_noise).T[0].astype(float)
_, output_noise_power_density = psd(output_noise)
biquad_cascade_response = \
10 * np.log10(output_noise_power_density / input_noise_power_density + 1/inf)
plt.figure()
plt.plot(freq, expected_response, label='Model')
if show_biquad_cascade:
plt.plot(freq, biquad_cascade_response, label='Biquad cascade')
pass
plt.plot(freq,
optimized_implementation_a_response,
label='Optimized realization A')
plt.plot(freq,
optimized_implementation_b_response,
label='Optimized realization B')
plt.xlabel('Frecuency $f$ [Hz]')
plt.ylabel('Response $|H(f)|$ [dBr]')
plt.legend()
plt.savefig('response.png')
plt.show()
def test_literal_conversion():
with FixedFormatArithmeticLogicUnit(
format_=Q(7), rounding_method=nearest_integer) as alu:
assert Fixed(0.25) == sympy.sympify(fixed(0.25))