信号与系统基本概念
f(t) y(t) f(k) y(k) h(t)
一. 常用信号
ε(t) δ(t) cos(ωt+Ф) est
ε(k) δ(k) cos(ωk+Ф) ak
二. 信号常用运算
x(t)=x1(t)+x2(t) x(t)=x1(t)-x2(t) x(t)=x1(-t) x(t)=x1(t-t0) x(t)=x1(at)
x(t)=x1(at-t0) x(t)=dx1(t)/dt
ex1:
y(t)=(t+2)*(ε(t+2)-ε(t)) y(1-2t)=?
h(k)
esk
x(k)=x1(k)+x2(k) x(k)=x1(k)-x2(k) x(k)=x1(-k) x(k)=x1(k-k0) x(k)=x1(ak) x(k)=x1(ak-k0) x(k)=x1(k)-x1(k-1)
+2ε(t)-2ε(t-2) 三. 周期信号与非周期信号
f(t+T)=f(t) ex2:
四. 奇偶函数
x(-t)=x(t) x(-t)=-x(t)
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;
f(n+N)=f(n)
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*x1(k)+a2*x2(k)
2.时不变与时变系统
时不变 x(t-t0)
ex3:
y(t)=x(t)*cos (ωCt) 线性? 时不变?
If x(t)= a1*x1(t)+a2*x2(t)
Then
y(t)=(a1*x1(t)+a2*x2(t)) *cos (ωCt)
= a1*x1(t) *cos (ωCt)+ a2*x2(t)* cos (ωCt) =a1*y1(t)+a2*y2(t) 线性 x(k-k0)
y(t-t0) y(k-k0)
a1*y1(t)+a2*y2(t) a1*y1(k)+a2*y2(k)
if x(t)=x1(t-t0) y(t)=x(t)*cos (ωCt)
= x1(t-t0) *cos (ωCt)
y1(t-t0)=x1(t-t0)*cos (ωCt-ωCt0) y(t)!= y1(t-t0) 时变
3.因果与非因果系统 1) 2) LTI h(t)=0 t<0 h(k)=0 k<0 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)
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)ε(k) H(z)=(1+1/3z)/(1-1/2z)
ROC |z|>1/2
-1
-1
k
3.稳定与非稳定系统 1)
BIBO
x(t) 有界 x(k) 有界
y(t) 有界 y(k) 有界
2） LTI
∫-∞|h(τ)|/dτ<+∞
+∞
∑-∞|h(k)|<+∞
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)ε(k) H(z)=(1+1/3z)/(1-1/2z)
ROC |z|>1/2
-1
-1
k
