信号与系统(英文)chapter 1-信号与系统

1.5.2 Discrete-Time Complex Exponential and Sinusoidal signals(sequences)（序列）: n
x[n] = Cα
Ⅰ.Real Exponential Signals: if C﹑a are real Ⅱ. General Complex Exponential Signals * both C and a are complex numbers
x[n] = x[n + N] for all values of n. In this case, we say that x(t) (x[n]) is periodic with period T(N).
Example 1.1 Determine the fundamental period of the signal x(t) = 2cos(10πt+1)-sin(4πt-1).
1.5 SEVERAL BASIC SIGNALS
1.5.1 Continuous-time Complex Exponential （复 指数函数） and Sinusoidal Signals（正弦信号） at complex exponential signals:
x(t ) = Ce
Ⅰ.Real Exponential Signals: if C﹑a are real Ⅱ. Periodic Complex Exponential and Sinusoidal Signals : * if C is real and a purely imaginary jω 0 t periodic complex exponential
∫
T
−T
x(t ) dt
+N
2
1 2 P∞ = lim x[n] ∑ N →∞ 2N + 1 n =− N
in continuous time and discrete time, respectively.
With these definitions, we can identify three important classes of signals:
n1
E ∞ = lim ∫
T
T → ∞ −T
x (t ) dt = ∫
2
n=− N
+∞
−∞
+∞
x (t ) dt
2
2
E ∞ = lim
N →∞
∑ x[n]
+N
2
= ∑ x[n]
−∞
The time-average power over an infinite interval:
1 P∞ = lim T →∞ 2T
x(t ) = Ce
Ⅲ.General Complex Exponential Signals: * both C and a are complex numbers C = C e jθ , a = r + j ω 0
at rt rt
Ce = C e cos(ω 0 t + θ ) + j C e sin(ω 0 t + θ )
1.2 CLASSIFICATIONS OF SIGNALS
1.2.1 Continuous-time and Discrete-time Signals （连续时间和离散时间信号） • continuous-time signals’ independent variable is continuous : x(t)=e t • discrete-time signals are defined only at discrete times : x[n] =2n
1.2.3 Determinate and Random Signals （确定性信号和随机信号） • A determinate signal ——x(t) • A random signal —— cannot find a function to represent it
1.2.4
Energy and Power Signals
x(-t/3+2) 1 0 1 2
t
x(t+2) 1 -2/3 -1/3 0
(a)
x(t/3+2) 1 -2 -1 1 0 4/3 5/3 2 0 x(t)
x(-t/3) 1 -6 -5 -4 0 x(t) x(t/3) 1 0 4 5 6 1 0 4/3 5/3 2
1.4 Even-odd Decomposition of a Signal x (t) = Ev{x(t)} + Od {x(t)}
Representing Signals Graphically
x(t)
0
t
(a)
x[n] x[0] -3 -1 -2 x[-1]
0 1 2 3 4 5
x[4]
-4
n
(b)
Figure 1.1 Graphical representations of (a) continuous-time and (b) discrete-time signals
（能量信号和功率信号）
The instantaneous power is The total energy over time interval (t1,t2) is The average power in this interval is
t2
1 2 p(t) = v(t)i(t) = v (t) R p ( t ) dt =
CHAPTER 1 Signals and Systems
1.1 Definition and Mathematical Representation of Signals
• Signals: physical phenomena or physical quantities, which change with time or space. • Functions of one or more independent variables. example: x(t)
C = C e jθ , α = α e jω
n n
0
Cα = C α cos(ω 0 n + θ ) + j C α sin(ω 0 n + θ )
n
α >1
0 <α <1
α < −1
Figure 1.4 the real exponential signal
−1 < α < 0
x[n] = Cα
2 1
Similarly, the total energy in a discrete-time signal x[n] over the time interval n1 ≤ n ≤ n2 : n x [ n ] 2 ∑
2
The total energy over an infinite interval is, in continuous time, in discrete time,
• Energy signals • Power signals • Signals with neither finite total energy nor finite average power
1.3 Transformations of the Independent Variable of Signals
ω0
)]
• 2π = N
ω0
0
is a rational number, the fundamental period 2π M N= M
•
2π
ω0
is an irrational number, the sinusoidal will not be periodic at all.
ω6
15π cos( n) 8
Figure 1.6 Two periodic sinusoidal sequence and an aperiodic sinusoidal sequence
n
α >1
n
α <1
n
Figure 1.5
(a) Growing sinusoidal sequence (b) Decaying sinusoidal sequence
1.5.3 Periodicity Property of Discrete-Time Complex Exponentials（离散正弦序列的周期性）
x(t)
± C e rt
x(t)
Fig. 1.3 (a) Growing sinusoidal signal, x(t ) = Ce rt cos(ω 0 t + θ ) r>0; （增长的） rt x ( t ) = Ce cos(ω 0 t + θ ) (b) Decaying sinusoid , r<0. （衰减的）
From trigonometry, we know that the fundamental period of cos(10πt+1) is T1=1/5, and sin(4πt-1) is T2=1/2. What about the fundamental period of x(t)? If there is a rational T, and it is the lowest common multiple（最小公倍数） of T1 and T2, then we say that x(t) is periodic with fundamental period T, or else, x(t) is aperiodic. For the x(t) in this example, the lowest common multiple of 0.2 and 0.5 is unit 1, and it is rational, so that the fundamental period of x(t) is 1.