- Sinusoidal signal
- Index signal
Sinusoidal signal
Define a continuous sinusoidal signal:
$x(t)=Acos(\omega_0 t+\phi)$
Wherein, A is the amplitude, $ \ omega_0 $ on frequency, $ \ phi $ is a phase
drawing a continuous sinusoidal signal python Examples (Note that the computer is stored in discrete digital herein is able to draw a continuous sinusoidal signal of plot points and accuracy because looks like a continuous, amplified actually discrete):
x = np.arange(0,10,0.01) omega = 1 phi = 1 y = np.sin (omega * x + phi) plt.plot(x,y) plt.xlim ((0.10)) plt.grid ()
Properties of sinusoidal signal:
a) periodically:
$x(t)=x(t+ T_0)$==>$Acos[\omega_0+\phi]=Acos[\omega_0+\omega_0 T_0 +\phi]$
$ \ Omega_0 T_0 = 2 \ pi m $, where m is an integer: $ T_0 = \ frac {2 \ pi m} {\ omega_0} $ => cycle: $ \ frac {2 \ pi m} {\ omega_0} $.
b) transfer and phase change is equivalent to the time
$ Acos [\ omega_0 (t + t_0)] = Acos [\ omega_0 t + \ omega_0 t_0] $, where $ \ omega_0 t_0 $ a phase change
$Acos[\omega_0 (t+t_0) + \phi]=Acos[\omega_0 t+\omega_0 t_0 \phi]$
c) parity
Even function $ x (t) = x (-t) $
Odd function $ x (t) = - x (-t) $
Discrete sine signal definition:
$x[n]=Acos(\omega_0 n+\phi)$
Wherein, A is the amplitude, $ \ omega_0 $ on frequency, $ \ phi $ is the phase.
Examples of discrete sine signal is plotted python
x = np.arange(0,10,0.1) omega = 1 phi = 1 y = np.sin (omega * x + phi) plt.plot(x,y,'o') plt.xlim ((0.10)) plt.grid ()
Of course, discrete and continuous nature of the same, to mention a few examples:
a) Transfer of phase change is equivalent to the time
$ Acos [\ omega_0 (n + n_0)] = Acos [\ omega_0 n + \ omega_0 n_0] $, 其中 $ n_0 = \ Delta \ phi $.
b) the discrete signal, phase shift => time change? ? ?
Note that the phase change where $ \ Delta \ phi $ necessarily be divisible by $ \ omega_0 $
c) periodically:
$\Omega_0 N = 2\pi m$ => $N = \frac{2\pi m}{\Omega_0}$
Continuous signals and discrete signal difference
a) $ x (t) = Acos (\ omega_0 t + \ phi) $, any $ \ $ omega O reflects periodicity.
b) $ x [n] = Acos (\ Omega_0 n + \ phi) $, $ N = \ frac {2 \ pi m} {\ Omega_0} $ holds only the integer of.
Index signal
Continuous exponential signal is defined:
$x(t)=C e^{at}$
Where, C and a are real numbers. $ A> 0 $ when plotted as curve
x = np.arange(0,10,0.01) C = 1 a = 1 y = C*np.exp(a*x) plt.plot(x,y) plt.xlim ((0.10)) plt.grid ()
Discrete exponential signal definition:
$x[n]=C e^{\beta n}= C \alpha^{n}$
C and $ \ alpha $ are real numbers
x = np.arange(0,10,0.1) C = 1 a = 1 y = C*np.exp(a*x) plt.plot(x,y,'o') plt.xlim ((0.10)) plt.grid ()
当$\alpha <0 and \left | a \right | < 1 $
x = np.arange(-10,2,1) C = -1 a = -0.5 y = np.power (a, x) plt.plot(x,y,'o') plt.xlim ((- 10.2)) plt.grid ()
At this time, if a similar $ x [n] = C e ^ {\ beta n} = C \ alpha ^ {n} $, to write such equations, the complex appeared.
Complex: $ X (T) = C {AT ^} $ E, where C is a complex, and then
a) $C = \left | a \right | e^{j\theta}$,
b) $ a= r+j\omega_0$,
c) $ x (t) = \ left | C \ right | e ^ {j \ theta} e ^ {(r + j \ omega_0) t} = \ left | C \ right | e ^ {rt} e ^ { j (\ omega_0 t + \ theta)} $, where Euler's formula:
$e^{j(\omega_0 t+ \theta)}=cos(\omega_0 t + \theta) + j sin( \omega_0 t + \theta)$
Of course, there can be written in discrete form:
$e^{j(\Omega_0 n+ \theta)}=cos(\Omega_0 n + \theta) + j sin( \Omega_0 n + \theta)$
And Euler's equation, this time the complex exponential function appears periodically.