import numpy as np
import matplotlib.pyplot as plt
from funcoes import u, conv_disc
t = np.arange(-2, 12)
x = (u(t) - u(t - 4)) * t + (u(t - 4) - u(t - 8)) * (8 - t)
h = np.zeros_like(t)
h[t == 0] = 1
h[t == 1] = -1
y = conv_disc(x, h, t)
plt.figure()
plt.subplot(1, 2, 1)
plt.stem(t, x)
plt.xticks(t)
plt.grid()
plt.legend(['x[n]'])
plt.xlabel('n')
plt.axis([t.min(), t.max(), -.5, 4.5])
plt.subplot(1, 2, 2)
plt.stem(t, h)
plt.xticks(t)
plt.grid()
plt.legend(['h[n]'])
plt.xlabel('n')
plt.axis([t.min(), t.max(), -1.5, 1.5])
plt.figure()
plt.stem(t, y)
plt.xticks(t)
ax_x = sp[0].axes[0] ax_y = sp[0].axes[1] plotit(title='x(t)', ax=ax_x, x=t, y=x_quizz(t), axis=axis, xticks=xticks, wait=False) plotit(title='y(t)', ax=ax_y, x=t, y=x_quizz(t+2), axis=axis, xticks=xticks, wait=False) plt.gcf().set_size_inches(7.41, 2.45) sp = plt.subplots(1, 2) ax_x = sp[0].axes[0] ax_y = sp[0].axes[1] plotit(title='x(t)', ax=ax_x, x=t, y=x_quizz(t), axis=axis, xticks=xticks, wait=False) plotit(title='y(t)', ax=ax_y, x=t, y=x_quizz(-t+2), axis=axis, xticks=xticks, wait=False) plt.gcf().set_size_inches(7.41, 2.45) sp = plt.subplots(1, 2) ax_x = sp[0].axes[0] ax_y = sp[0].axes[1] plotit(title='x(t)', ax=ax_x, x=t, y=x_quizz(t), axis=axis, xticks=xticks, wait=False) plotit(title='y(t)', ax=ax_y, x=t, y=x_quizz(-t-2), axis=axis, xticks=xticks, wait=False, ) plt.gcf().set_size_inches(7.41, 2.45) sp = plt.subplots(1, 1) ax_x = sp[0].axes[0] x = np.zeros_like(t) plotit(title='x(t)', ax=ax_x, x=t, y=x, axis=axis, xticks=xticks, wait=False, dirac=[[1, 2],[2, 1]]) plt.gcf().set_size_inches(7.41, 2.45) sp = plt.subplots(1, 1) ax_x = sp[0].axes[0] x = u(t) + u(t-2) - 2*u(t-3) plotit(title='x(t)', ax=ax_x, x=t, y=x, axis=axis, xticks=xticks, wait=False) plt.gcf().set_size_inches(7.41, 2.45)
def x_quizz(t):
return (2/3)*(u(t)-u(t-3)) * t
def x_def(t):
return (1 - 1e-3/np.pi * t) * (u(t)-u(t-1e3*np.pi))
def h_def(t):
return .5 * (u(t) - u(t - 2))
def meuh(t):
return u(t) - u(t - 2)
import matplotlib.pyplot as plt
def pixaxis(ax):
ticks_loc = ax.get_xticks().tolist()
ax.set_xticks(ax.get_xticks().tolist())
ax.set_xticklabels(["{:.0f}".format(x / np.pi) + 'π' for x in ticks_loc])
# Questão 1a
sp = plt.subplots(1, 2)
ax_x = sp[0].axes[0]
ax_y = sp[0].axes[1]
t = np.arange(-3, 16, .01)
xtick = np.arange(-3, 16)
axis = [-1, 15, -.5, 10.2]
x = u(t)
h = 2*u(t) - u(t-3) - u(t-6)
y = conv(x, h, t)
plotit(x=t, y=x, ax=ax_x, axis=axis, xticks=xtick)
plotit(x=t, y=y, ax=ax_y, axis=axis, wait=False, save=False, xticks=xtick)
sp = plt.subplots(1, 1)
ax = sp[0].axes[0]
plotit(x=t, y=y, ax=ax, axis=[-1, 15, -1, 10], xlabel='t [s]', ylabel='y(t)', xticks=xtick, linewidth=3)
# Questão 1b
x = impulse(t)+impulse(t-1)+impulse(t-2)
y = conv(x, h, t)
sp = plt.subplots(1, 2)
ax_x = sp[0].axes[0]
ax_y = sp[0].axes[1]
from graficos import myplot as plotit from funcoes import u, window from scipy.signal import step, lti import numpy as np import matplotlib.pyplot as plt sp = plt.subplots(2, 1) ax_x = sp[0].axes[0] ax_y = sp[0].axes[1] axis = [-1, 2.7, -.2, 1.2] t = np.arange(-1, 2.7, .01) x = u(t) y = .5 * x plotit(ax=ax_x, x=t, y=x, axis=axis, wait=False) plotit(ax=ax_y, x=t, y=y, axis=axis, wait=False) sp = plt.subplots(2, 1) ax_x = sp[0].axes[0] ax_y = sp[0].axes[1] axis = [-1, 2.7, -.2, 1.2] t = np.arange(-1, 2.7, .01) x = u(t) sys = lti([20], [1, 5, 40]) t_step, y = step(sys) plotit(ax=ax_x, x=t, y=x, axis=axis, wait=False) plotit(ax=ax_y, x=t_step, y=y, axis=axis, wait=False, hold=True) plotit(ax=ax_y, x=[t[0], 0], y=[0, 0], axis=axis, wait=False, hold=True)
import matplotlib.pyplot as plt
def pixaxis(ax):
ticks_loc = ax.get_xticks().tolist()
ax.set_xticks(ax.get_xticks().tolist())
ax.set_xticklabels(["{:.0f}".format(x / np.pi) + 'π' for x in ticks_loc])
# Questão 1
sp = plt.subplots(1)
ax_x = sp[0].axes[0]
t = np.arange(-3, 6, .01)
xtick = np.arange(-3, 7)
axis = [-1, 6, -.2, 2.2]
h = (-t + 2) * (u(t)-u(t-2))
plotit(x=t, y=h, ax=ax_x, axis=axis, xticks=xtick, yticks=[0, 1, 2], xlabel='t [s]', ylabel='h(t)', linewidth=3)
# Questão 1a
sp = plt.subplots(1)
ax_x = sp[0].axes[0]
x = u(t)
y = conv(x, h, t)
plotit(x=t, y=y, ax=ax_x, axis=axis, xticks=xtick, xlabel='t [s]', ylabel='y(t)', linewidth=3)
# Questão 1b
sp = plt.subplots(1)
ax_x = sp[0].axes[0]
x = u(t) - u(t-3)
y = conv(x, h, t)
plotit(x=t, y=y, ax=ax_x, axis=axis, xticks=xtick, xlabel='t [s]', ylabel='y(t)', linewidth=3)
def get_delay_secs(self):
return self.delay
def get_product(self):
return self.x * self.h_func(-t + self.delay)
def get_y_trunc(self):
idx_now = np.argmin(np.abs(t - self.delay))
t_trunc = self.t[:idx_now]
y_trunc = self.y[:idx_now]
return np.array([t_trunc, y_trunc])
t = np.arange(-10, 10, .001)
x = (-t + 1) * (u(t) - u(t - 1))
c = Convolver(x, h_def, t, .1)
tmin = -5
tmax = 5
xticks = np.arange(-10, 10, 1)
def on_key_press(event):
# print(format(event.key))
if event.key == 'right':
if c.direction == 'stop':
c.direction = 'right'
else:
c.direction = 'stop'
elif event.key == 'left':
if c.direction == 'stop':
def h_def(t):
return u(t) - u(t - 4)
def realpole(w, w0):
magnitude = u(w) - u(w-w0) + \
(w0/w) * (u(w-w0))
def pixaxis(ax):
ticks_loc = ax.get_xticks().tolist()
ax.set_xticks(ax.get_xticks().tolist())
ax.set_xticklabels(["{:.0f}".format(x / np.pi) + 'π' for x in ticks_loc])
def square(t, t0, t_init, min, max, duty):
t_per = np.mod(t - t_init, t0) / t0
x = np.zeros_like(t)
x[np.where(np.abs(t_per) <= duty)] = max
x[np.where(np.abs(t_per) > duty)] = min
return x
# espectro do sinal exemplo
sp = plt.subplots(1, 1)
ax_abs = sp[0].axes[0]
w = np.arange(-4 * np.pi, 4.001 * np.pi, 1e-3)
h = u(w + 2 * np.pi) - u(w - 2 * np.pi)
xtick = np.arange(w[0], w[-1], np.pi)
plotit(ax=ax_abs,
x=w,
y=h,
axis=[w[0], w[-1], -.1, 1.1],
axes=False,
save=False,
title='$H(\omega)$',
xlabel='$\omega$',
xticks=xtick,
wait=True)
pixaxis(ax_abs)
ax.set_xticklabels(["{:.0f}".format(x / np.pi) + 'π' for x in ticks_loc])
def square(t, t0, t_init, min, max, duty):
t_per = np.mod(t - t_init, t0) / t0
x = np.zeros_like(t)
x[np.where(np.abs(t_per) <= duty)] = max
x[np.where(np.abs(t_per) > duty)] = min
return x
# Questao SLIT
sp = plt.subplots(1, 1)
ax_abs = sp[0].axes[0]
t = np.arange(-1, 8, 1e-3)
h = np.exp(-1 * t) * u(t) - np.exp(-2 * t) * u(t)
plotit(ax=ax_abs,
x=t,
y=h,
axes=True,
axis=[t[0], t[-1], -0.1, h.max() * 1.1],
save=False,
title='h(t)',
xlabel='t',
wait=False)
sp = plt.subplots(1, 1)
ax_abs = sp[0].axes[0]
t = np.arange(-1, 7, 1e-3)
h = .5 * np.exp(.1 * t) * u(t) - np.exp(-1 * t) * u(t)
plotit(ax=ax_abs,
def get_product(self):
return self.x * self.h_func(-t + self.delay)
def get_y_trunc(self):
idx_now = np.argmin(np.abs(t - self.delay))
t_trunc = self.t[:idx_now]
y_trunc = self.y[:idx_now]
return np.array([t_trunc, y_trunc])
t = np.arange(-10, 10, .01)
#x = gaussian(t, 2, 1.5)
#x = np.zeros_like(t)
#for k in range(20):
# x += u(t-k)
x = u(t) - u(t - 2) + 2 * (u(t - 4) - u(t - 7))
def ret(t):
return u(t) - u(t - 1)
def meuh(t):
return u(t) - u(t - 2)
c = Convolver(x, meuh, t, .5)
#c = Convolver(x, u, t, .5)
tmin = -5
tmax = 5
xticks = np.arange(-10, 10, 1)
def x_def(t):
return (1 - 1/(100*np.pi) * t) * (u(t)-u(t-100*np.pi))
def ret(t):
return u(t) - u(t - 1)
sp = plt.subplots(1, 1) ax_x = sp[0].axes[0] t = np.arange(-.1, .1, 1e-5) axis = [-.02, .08, -.1, 1.1] # Onda quadrada x = square(t, .005, -.0025 / 2, 0., 1., .5) plotit(ax=ax_x, x=t, y=x, axis=axis, axes=True, title='x(t)', save=False) h = x_RC(t, 1, 10e-3) y = conv(x, h, t) ax_x = plt.subplots(1, 1)[0].axes[0] plotit(ax=ax_x, x=t, y=y, axis=axis, axes=True, title='y(t)', save=False) # Pulso x = u(t + .0025 / 2) - u(t - .0025 / 2) ax_x = plt.subplots(1, 1)[0].axes[0] plotit(ax=ax_x, x=t, y=x, axis=axis, axes=True, title='x(t)', save=False) h = x_RC(t, 1, 10e-3) y = conv(x, h, t) ax_x = plt.subplots(1, 1)[0].axes[0] plotit(ax=ax_x, x=t, y=y, axis=axis, axes=True, title='y(t)', save=False) # Sequencia com período crescente sp = plt.subplots(3, 1) ax_x = sp[0].axes[0] ax_abs = sp[0].axes[1] ax_ang = sp[0].axes[2] w0_0 = 2 * np.pi / .005 n = np.arange(-50, 50) for i in np.arange(0, 10, 1):
def h_def(t):
R = 2
L = 1
return np.exp(-t * R / L) * u(t) / L
def x_RC(t, R, C):
return np.exp(-t / (R * C)) * u(t) / (R * C)
def h_def(t):
return 2*u(t) - u(t-3) - u(t-6)
positive_samples = int(np.floor(t[-1] / width))
for i in range(1, positive_samples):
x_hat = x_hat + r_(t - i*width, width) * x[t0 + i*width_s]
return x_hat
def x_RLi(t, R, L):
return np.exp(-t*R/L)*u(t) / L
sp = plt.subplots(1, 2)
ax_x = sp[0].axes[0]
ax_y = sp[0].axes[1]
t = np.arange(-3, 10, .01)
xtick = np.arange(-3, 10)
axis = [-1, 5, -.5, 2]
h = impulse(t) / 2
x = u(t)
y = conv(x, h, t)
plotit(x=t, y=x, ax=ax_x, axis=axis, xticks=xtick)
plotit(x=t, y=y, ax=ax_y, axis=axis, wait=True, save=True, xticks=xtick)
x = impulse(t)
y = conv(x, h, t)
plotit(x=t, y=x, ax=ax_x, axis=axis, dirac=[[0], [1]], xticks=xtick)
plotit(x=t, y=y, ax=ax_y, axis=axis, save=True, dirac=[[0], [.5]], xticks=xtick)
x = impulse(t) + 2 * impulse(t - 1.5) - .5*impulse(t - 3.)
y = conv(x, h, t)
plotit(x=t, y=x, ax=ax_x, axis=axis, dirac=[[0, 1.5, 3.0], [1, 2, -.5]], xticks=xtick)
plotit(x=t, y=y, ax=ax_y, axis=axis, save=True, dirac=[[0, 1.5, 3], [.5, .75, -.25]], xticks=xtick)
h = x_RLi(t, R=2, L=1)
x = u(t)
y = conv(x, h, t)
def get_delay_secs(self):
return self.delay
def get_product(self):
return self.x * self.h_func(-t + self.delay)
def get_y_trunc(self):
idx_now = np.argmin(np.abs(t - self.delay))
t_trunc = self.t[:idx_now]
y_trunc = self.y[:idx_now]
return np.array([t_trunc, y_trunc])
t = np.arange(-10, 10, .001)
x = t * (u(t) - u(t-1))
c = Convolver(x, h_def, t, .2)
tmin = -5
tmax = 5
xticks = np.array([0, 1, 2])
def on_key_press(event):
# print(format(event.key))
if event.key == 'right':
if c.direction == 'stop':
c.direction = 'right'
else:
c.direction = 'stop'
elif event.key == 'left':
if c.direction == 'stop':
def x_RLi(t, R, L):
return np.exp(-t*R/L)*u(t) / L
from graficos import myplot as plotit
from funcoes import u, window, integrate, conv, movavgc, movavgpast, derivate, impulse
import numpy as np
import matplotlib.pyplot as plt
def piyaxis(ax):
ticks_loc = ax.get_yticks().tolist()
ax.set_yticks(ax.get_yticks().tolist())
ax.set_yticklabels(["{:.3f}".format(y / np.pi) + 'π' for y in ticks_loc])
# Questao 2b
sp = plt.subplots(1, 1)
ax_x = sp[0].axes[0]
w = np.arange(-30, 30, .1)
xtick = np.arange(w[0], w[-1], 5)
ytick = np.linspace(0, np.pi/20, 3)
resp = (np.pi/20) * (u(w+20)-u(w+15)+u(w-15)-u(w-20))
axis = [w[0], w[-1], -.1*np.max(resp), 1.1*np.max(resp)]
plotit(x=w, y=resp, ax=ax_x, axis=axis, xticks=xtick, yticks=ytick, xlabel='$\omega$', title='$Y(\omega)$')
piyaxis(ax_x)
from graficos import myplot as plotit
from funcoes import u, window, integrate, conv, movavgc, movavgpast, derivate, exp_u, sinc
import numpy as np
import matplotlib.pyplot as plt
t = np.arange(-1, 5, .01)
h = u(t) - u(t - 1)
x = np.cos(3 * np.pi * t) * u(t)
ya = conv(x, h, t)
x = np.cos(2 * np.pi * t) * u(t)
yb = conv(x, h, t)
plt.subplot(3, 1, 1)
plt.plot(t, h)
plt.subplot(3, 1, 2)
plt.plot(t, x)
plt.subplot(3, 1, 3)
plt.plot(t, ya)
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, ya)
plt.grid()
plt.xlabel('t [s]')
plt.ylabel('Questão (1a)')
plt.legend(['y(t)'])
plt.subplot(2, 1, 2)
plt.plot(t, yb)
plt.grid()
plt.xlabel('t [s]')
plt.ylabel('Questão (1b)')
def xfunc(t):
x = (np.exp(-2 * t) - np.exp(-15 * t)) * u(t)
x = 2 * x / x.max()
return x
def x_t(t):
return u(t) - u(t - 1)
axis=[t[0], t[-1], -.2, 1.2],
wait=False,
linewidth=2)
sp = plt.subplots(1, 1)
ax_x = sp[0].axes[0]
t = np.arange(-2.2, 1.8, .001)
x = np.exp(t)
plotit(ax=ax_x, x=t, y=x, axis=[t[0], t[-1], -.5, 6], wait=False, linewidth=2)
# Causais e não causais
sp = plt.subplots(1, 1)
ax_x = sp[0].axes[0]
axis = [-1, 2.7, -.2, .8]
t = np.arange(-1, 5, .01)
x = u(t)
sys = lti([20], [1, 5, 40])
t_step, y = step(sys, T=t * (t >= 0))
plotit(ax=ax_x, x=t_step, y=y, axis=axis, wait=False, hold=True)
plotit(ax=ax_x, x=[t[0], 0], y=[0, 0], axis=axis, wait=False, hold=True)
plt.gcf().set_size_inches(6.4, 3.05)
sp = plt.subplots(1, 1)
ax_x = sp[0].axes[0]
plotit(ax=ax_x, x=t[:-50], y=y[50:], axis=axis, wait=False, hold=True)
plt.gcf().set_size_inches(6.4, 3.05)
# Quizz: analógico / digital
plt.figure()
plt.subplot(1, 2, 1)
t = np.arange(-2, 5, .01)
t = np.arange(-.1, .1, 1e-5)
axis = [-.02, .02, -.02, 1.02]
# Onda quadrada
x = square(t, .01, 0., .4, 1., .5)
plotit(ax=ax_x,
x=t,
y=x,
axis=axis,
axes=True,
xlabel='t [s]',
title='x(t)',
save=False)
sp = plt.subplots(1, 1)
ax_x = sp[0].axes[0]
w = np.arange(-150 * np.pi, 150 * np.pi, .5)
axis = [-150 * np.pi, 150 * np.pi, -.2, 3.2]
# Onda quadrada
x = 3 * (u(w + 100 * np.pi) - u(w + 80 * np.pi) + u(w - 80 * np.pi) -
u(w - 100 * np.pi))
plotit(ax=ax_x,
x=w,
y=x,
axis=axis,
axes=True,
xlabel='ω [rad/s]',
title='|Y(ω)|',
save=False,
xticks=np.array([-100, -80, 0, 80, 100]) * np.pi)
pixaxis(ax_x)
