Chapter 1 : Introduction of Signals and Spectra¶
Example 1.1 Page No : 12¶
In [5]:
import math
from scipy.integrate import quad
#page 12
#problem 1.1
#Given signal u = 2*math.exp(-3*t)
#Since the function integral does not accept %inf as limit we need to use approximation by changing variables.
#First the signal is to be expressed in terms of 'x'.
def Signal(x):
return 2*math.exp(-3*x);
#We then substitute x = math.tan(z), and then express the given signal wrt 'z' and not 'x'.
def Gmodified(z):
x = math.tan(z);
return (Signal(x))**2/(math.cos(z))**2;
E = quad(Gmodified,0,math.atan(10))[0]
# Results
print 'The energy of this signal is %.3f'%E
Example 1.2 Page No : 12¶
In [6]:
import math
from scipy.integrate import quad
#Given signal u = 2*math.sin(0.5*math.pi*t)
#Since u is periodic, averaging over -infinity to + infinity will give the same result as averaging over -2 to 2, where 4 is the time period.
t0 = -2;
t1 = 2;
# Calculations
def f1(t):
return (2*math.sin(0.5*math.pi*t))**2
E = quad(f1,t0,t1)[0]
# Results
print 'The power of the signal is ',E
Example 1.3 Page No : 18¶
In [7]:
%matplotlib inline
from matplotlib.pyplot import plot,subplot,xlabel,ylabel
import math
from numpy import arange,zeros
#page 18
#problem 1.3
#u1(T) vs T
T = arange(-5,5+0.0082,0.0082)
u1 = zeros(len(T))
u1[T <= 0] = 0;
u1[T>0] = 1;
xlabel('T');
ylabel('u(T)')
subplot(131);
plot(T,u1);
#u2(T-t) vs T
#Shifting the given signal by t units to the right, we get
#Let us assume the amount of time to be shited is 3 units
t = 3;
T = arange(-5,5+0.0082,0.0082)
u2 = zeros(len(T))
u2[T<= t] = 0;
u2[T>t] = 1;
xlabel('T');
ylabel('u(T - t)')
subplot(132);
plot(T,u2);
#u(t - T) = u(-(T - t))
T = arange(-5,5+0.0082,0.0082)
u3 = zeros(len(T))
u3[T>= t] = 0;
u3[T<t] = 1;
xlabel('T');
ylabel('u(t - T)')
subplot(133);
plot(T,u3);
Example 1.4 Page No : 18¶
In [1]:
%matplotlib inline
from matplotlib.pyplot import plot,subplot,xlabel,ylabel
import math
from numpy import arange,zeros
#u1(t)
t = arange(-5,5+0.0082,0.0082)
u1 = zeros(len(t))
u1[t<= 0] = 0;
u1[t>0] = 1;
xlabel('t');
ylabel('u(t)')
subplot(131);
plot(t,u1);
#u2(t-T)
#Shifting the given signal by t units to the right, we get
#Let us assume the amount of time to be shited is 3 units
T = 3;
t = arange(-5,5+0.0082,0.0082)
u2 = zeros(len(t))
u2[t<= T] = 0;
u2[t>T] = 1;
xlabel('t');
ylabel('u(t-T)')
subplot(132);
plot(t,u2);
#u(t) - u(t - T)
t = arange(-5,5+0.0082,0.0082)
u3 = u1 - u2;
xlabel('t');
ylabel('u(t) - u(t-T)')
subplot(133);
plot(t,u3);
Example 1.5 Page No : 18¶
In [10]:
%matplotlib inline
import math
from scipy.integrate import quad
from numpy import arange,zeros
from matplotlib.pyplot import subplot,xlabel,ylabel,plot
#page 18
#problem 1.5
#V1(t) = u(t) - u(t - 5)
t = arange(-5,5+0.1,0.1)
V1 = zeros(len(t))
V1[t<= 0] = 0;
V1[t>0] = 1;
xlabel('t');
ylabel('V1(t)')
subplot(121);
plot(t,V1);
#V2(t) = 2*t*(u(t) - u(t - 3))
t = arange(0,3+0.1,3);
V2 = 2*t;
xlabel('t');
ylabel('V2(t)')
subplot(122);
plot(t,V2);
#Autocorrelation R12(0) = R
def f2(t):
return 2*t
R = quad(f2,0,3)[0]
def f3(t):
return 1
E1 = quad(f3,0,5)[0]
#In the textbook, E2 has been computed as 18 instead of 36
def f4(t):
return 4*t**2
E2 = quad(f4,0,3)[0]
c = R/(E1*E2)**0.5;
print 'The correlation term R12(0) is ',R
print 'The autocorrelation term R1(0) is ',E1;
print 'The autocorrelation term R2(0) is ',E2;
Example 1.6 Page No : 19¶
In [11]:
%matplotlib inline
import math
from scipy.integrate import quad
from numpy import arange,zeros
from matplotlib.pyplot import subplot,xlabel,ylabel,plot
#V1(t) = u(t) - u(t - 5)
t = arange(-5,5+0.1,.1)
V1 = zeros(len(t))
V1[t<= 0] = 0;
V1[t>0] = 1;
xlabel('t');
ylabel('V1(t)')
subplot(121);
plot(t,V1);
#V2(t) = 2*t*(u(t) - u(t - 3))
t = arange(0,3+0.1,.1)
V2 = 2*t;
xlabel('t');
ylabel('V2(t)')
subplot(122);
plot(t,V2);
#Autocorrelation R12(1) = Ra
#The range is t = 0 to 2, as signal V2(t) has been shifted left by one unit, V2(t-1)
def f5(t):
return 2*(t+1)
Ra = quad(f5,0,2)[0]
print 'The correlation term R12(1) is ',Ra
#Autocorrelation R12(-1) = Rb
#The range is t = 1 to 4, as signal V2(t) has been shifted right by one unit, V2(t+1)
def f6(t):
return 2*(t-1)
Rb = quad(f6,1,4)[0]
print 'The correlation term R12(-1) is ',Rb
