信号与系统基本概念
一.常用信号
ε(t) δ(t) cos(ωt+Ф) e st
ε(k) δ(k) cos(ωk+Ф) a k e sk
二.信号常用运算
x(t)=x1(t)+x2(t) x(k)=x1(k)+x2(k)
x(t)=x1(t)-x2(t) x(k)=x1(k)-x2(k)
x(t)=x1(-t) x(k)=x1(-k)
x(t)=x1(t-t0) x(k)=x1(k-k0)
x(t)=x1(at) x(k)=x1(ak)
x(t)=x1(at-t0) x(k)=x1(ak-k0)
x(t)=dx1(t)/dt x(k)=x1(k)-x1(k-1)
ex1:
y(t)=(t+2)*(ε(t+2)-ε(t)) +2ε(t)-2ε(t-2)
y(1-2t)=?
三.周期信号与非周期信号
f(t+T)=f(t) f(n+N)=f(n)
ex2:
f(k)=cos(2k)
g(k)=cos(π/3k)+cos(π/4k)
周期信号?
f(k): N=2π/2=π
g(k): N=m1*N1=m2*N2
N1=2π/(π/3)=6
N2=8;
N=m1*6=8*m2
N=m1*3=4*m2
m1=4 m2=3
N=4*6=24;
四.奇偶函数
x(-t)=x(t)
x(-t)=-x(t)
五.系统分类
LTI----线性时不变系统
1.线性与非线性系统
线性:
零状态下:
a1*x1(t)+a2*x2(t) a1*y1(t)+a2*y2(t)
a1*x1(k)+a2*x2(k) a1*y1(k)+a2*y2(k)
2.时不变与时变系统
时不变
x(t-t0) y(t-t0)
x(k-k0) y(k-k0)
ex3:
y(t)=x(t)*cos (ωC t)
线性? 时不变?
If x(t)= a1*x1(t)+a2*x2(t)
Then
y(t)=(a1*x1(t)+a2*x2(t)) *cos (ωC t)
= a1*x1(t) *cos (ωC t)+ a2*x2(t)* cos (ωC t)
=a1*y1(t)+a2*y2(t)
线性
if x(t)=x1(t-t0)
y(t)=x(t)*cos (ωC t)
= x1(t-t0) *cos (ωC t)
y1(t-t0)=x1(t-t0)*cos (ωC t-ωC t0)
y(t)!= y1(t-t0)
时变
3.因果与非因果系统
1)
y(t)=f(x(t))
y(t) 仅与现在和过去的x值有关(x(t-τ) τ>=0)
y(k)=f(x(k))
y(k) 仅与现在和过去的x值有关(x(k-n) n>=0)
2)
LTI
h(t)=0 t<0
h(k)=0 k<0
3)
LTI
H(s) ROC: Right-half plane
H(z) ROC: Exterior of a Circle (+∞)
H(s) rational
ROC: Right-half plane to the rightmost pole H(z) rational
ROC: Exterior of a Circle
outside the rightmost pole
the order of numerator
<= the order of denominator
ex4:
y(n) =x(n)+1/3x(n-1)
h(n)=(1/2) kε(k)
H(z)=(1+1/3z-1)/(1-1/2z -1)
ROC |z|>1/2
3.稳定与非稳定系统
1)
BIBO
x(t) 有界y(t) 有界
x(k) 有界y(k) 有界
2）
LTI
∫-∞+∞|h(τ)|/dτ<+∞
∑k=-∞+∞|h(k)|<+∞
3)
LTI
H(s) ROC Include jw axis
H(Z) ROC Include |z|=1
Rational & Causal
H(s) Poles lie in left-half of s-plane
=real part of poles <0
H(Z) Poles lie inside unit circle
= |pi|<1
ex5:
y(n) =x(n)+1/3x(n-1)
h(n)=(1/2) kε(k)
H(z)=(1+1/3z-1)/(1-1/2z -1)
ROC |z|>1/2。