A discrete-time signal refers to a signal that is only defined at a discrete time, referred to as a discrete signal, or a sequence. Discrete sequences are usually denoted by x(n), and the arguments must be integers.
Waveform plotting of discrete-time signals uses the stem function in matplotlib.pyplot. Due to the limited number of matrix elements, it can only represent a sequence of finite length within a certain time range; and for an infinitely long sequence, it can only be represented within a certain time range.
1. Unit pulse train
The unit pulse sequence (n), also known as the unit sample sequence, is defined as
It should be noted that the unit impulse sequence is not a simple discrete sampling of the unit impulse function, it takes a definite value of 1 at n=0.
The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n0=0
n1=-12
n2=12
L=n2-n1+1
n=np.linspace(n1,n2,L)
a1 = np.zeros(L)#只有n=0时， a1=1；当n≠0时，a1=0
a1[n0-n1] = 1
plt.ylim(-0.1,1.2)
plt.xlim(-15,15)
plt.stem(n,a1)# 画棉棒
plt.title('δ(n)')
plt.grid('on')# 显示网格线
plt.show()
np.linspace(start, stop, num, endpoint, retstep, dtype)
star and stop are the start and end positions, both scalars
num is the total number of interval points including start and stop, the default is 50
endpoint is a bool value, when it is False, the last point will be removed to calculate the interval
rest is a bool value, and it is True The data list and the interval value will be returned at the same time.
The dtype defaults to the type of the input variable. After the type is given, the generated array type will be converted to the target type.
The program running result is shown in the figure below.
2. Unit step sequence u(n)
The unit step sequence u(n) is defined as
The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n0=0
n1=-10
n2=15
L=n2-n1+1
n=np.linspace(n1,n2,L)
a2 = np.zeros(n2-n1+1)
a2[(n0-n1):] = 1# 只有n>=0时， x=1；当n<0时，x=0
plt.ylim(-0.1,1.2)
plt.xlim(-12,18)
plt.stem(n,a2)
plt.title('u(n)')
plt.grid('on')
plt.show()
The program running result is shown in the figure below.
3. Rectangular sequence RN(n)
The rectangular sequence RN (n) is defined as
The rectangular sequence has an important parameter, which is the sequence width N. The relationship between R (n) N and u(n) is:
The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n0=0
n1=-5
n2=10
N=5
L=n2-n1+1
n=np.linspace(n1,n2,L)
a3 = np.zeros(L)
a3[(n0-n1):(n0-n1+N)] = 1
plt.ylim(-0.1,1.2)
plt.stem(n,a3)
plt.title('R5(n)')
plt.grid('on')
plt.show()
The program running result is shown in the figure below.
4. Unilateral index series
A one-sided index series is defined as
The value range of the one-sided index sequence n is n 0 .
[Example 1-4] Draw unilateral index series respectively
waveform diagram.
The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n0=0
n1=-5
n2=20
L=n2-n1+1
n=np.linspace(n1,n2,L)
un = np.zeros(n2-n1+1)
un[(n0-n1):] = 1
a1=1.2
a2=-1.2
a3=0.8
a4=-0.8
x1=(a1**n)*un
x2=(a2**n)*un
x3=(a3**n)*un
x4=(a4**n)*un
plt.figure('signal',figsize=(16,12))
plt.subplot(221)
plt.stem(n,x1)
plt.xlabel('n')
plt.ylabel('x(n)')
plt.title('(1.2)^n')
plt.grid('on')
plt.subplot(222)
plt.stem(n,x2)
plt.xlabel('n')
plt.ylabel('x(n)')
plt.title('(-1.2)^n')
plt.grid('on')
plt.subplot(223)
plt.stem(n,x3)
plt.xlabel('n')
plt.ylabel('x(n)')
plt.title('(0.8)^n')
plt.grid('on')
plt.subplot(224)
plt.stem(n,x4)
plt.xlabel('n')
plt.ylabel('x(n)')
plt.title('(-0.8)^n')
plt.grid('on')
plt.show()
The program running result is shown in the figure below.
It can be seen from the figure that when |a|>1, the unilateral index sequence diverges; when |a|<1, the sequence converges. When a > 0, the sequence takes positive values; when a < 0, the sequence value swings positive and negative.
5. Sine sequence
A sine sequence is defined as
The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n0=0
n1=-30
n2=30
L=n2-n1+1
PI=np.pi
w=1/6
n=np.linspace(n1,n2,L)
x=np.sin(w*PI*n)
plt.stem(n,x)
plt.ylim(-1.1,1.1)
plt.xlim(-30,30)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('sin(πn/6)')
plt.grid('on')
plt.show()
The program running result is shown in the figure below.
6. Complex exponential sequence
A complex exponential sequence is defined as
Draw 
the curves of the real part, imaginary part, modulus and phase angle of the complex exponential sequence changing with time, and observe its time-domain characteristics. The program code example is as follows:
import numpy as np
import matplotlib.pyplot as plt
n1=0
n2=30
L=n2-n1+1
n=np.linspace(n1,n2,L)
A=2
a=-1/10
PI=np.pi
b=PI*1/6
c=complex(a,b)
x=A*np.exp(c*n)
plt.figure('signal',figsize=(12,10))
plt.subplot(2,2,1)
plt.stem(n,x.real)
plt.title('Re[x]')
plt.grid('on')
plt.subplot(2,2,2)
plt.stem(n,x.imag)
plt.title('Im[x]')
plt.grid('on')
plt.subplot(2,2,3)
plt.stem(n,np.abs(x))
plt.title('|x|')
plt.grid('on')
plt.subplot(2,2,4)
plt.stem(n,np.angle(x))
plt.title('arg[x]')
plt.grid('on')
plt.show()
The program running result is shown in the figure below.
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