时域信号分析小结与复习(1) 2000/10/12 基本概念 典型基本信号: f()=A(-∞<t<+∞) ●f(t)=Acos(Ot+q)(-∞<t<∞) f(t=Ae <t< (t>0) f()=(0)=105(a < (t=0) 6(t)dt=1 单位冲激信号的性质: f(t)6(t)=f(o)6(t) 2.f(1)(t-t)=f(t)o(t-t) f(to(dt=f(o) 4.f(0)6(t-10)t=f() 6.d(1) d() d(rdr=u(t) 7.o1(1)=∑6(t-n7)
时域信号分析小结与复习（1） 2000/10/12 基本概念 一、典型基本信号： ⚫ f (t) = A (− t +) ⚫ f (t) = Acos( t + ) (− t ) ⚫ ( ) = (− ) + f t Ae t jt ⚫ = = = 0 ( 0) 0.5 ( 0) 1 ( 0) ( ) ( ) t t t f t u t ⚫ = = = = − ( ) 1 0 ( 0) ( 0) ( ) ( ) t dt t t t f t 单位冲激信号的性质： 1. f (t) (t) = f (o) (t) 2. ( ) ( ) ( ) ( ) 0 0 0 f t t −t = f t t −t 3. − f (t) (t)dt = f (0) 4. − ( ) ( − ) = ( ) 0 0 f t t t dt f t 5. (t) = (−t) 6. dt du t t ( ) ( ) = − = t ( )d u(t) 7. =− = − n T (t) (t nT)
f()6()=∑f(Om7)(-m7) f(t)=δ(n)-d6(t) 冲激偶的性质: "(t)=-6”(-) 2.o()dt=0 (rdr=d(t) 4.f(1)o"(t)=f(o)5”()-f(o)6(t) 5.f()δ(-t0)=f(t0)(t-t0)-f(t0)05(-10) 6.「f(n)6'( 7.上f()6(t-t0=-f() 8.Jf(b8(=(-1yf(o Sgm(1)={0(=0) Sgn(t)=l(t)-l(-1)=2l(t)-1 R(t=tu(t) 0(t<0) t(t≥0) R()=(x) R(t) S(r)dI dR(o) =
=− = − n f (t) T (t) f (nT) (t nT) ⚫ dt d t f t t ( ) ( ) ( ) ' = = 冲激偶的性质： 1. '(t) = − '(−t) 2. − '(t)dt = 0 3. − = t '( )d (t) 4. f (t) '(t) = f (o) '(t) − f '(o) (t) 5. ( ) '( ) ( ) '( ) '( ) ( ) 0 0 0 0 0 f t t −t = f t t −t − f t t −t 6. − f (t) '(t)dt = − f '(0) 7. − ( ) '( − ) = − '( ) 0 0 f t t t dt f t 8. − ( ) ( ) = (−1) (0) (n) (n) (n) f t t dt f ⚫ Sgn（t） − = = 1 ( 0) 0 ( 0) 1 ( 0) ( ) t t t Sgn t ⚫ Sgn(t) = u(t) − u(−t) = 2u(t) −1 ⚫ R(t） = = ( 0) 0 ( 0) ( ) ( ) t t t R t tu t R t u d − ( ) = ( ) − − R(t) = ( )d ( ) ( ) u t dt dR t =
d-R(=8(t ●单边指数信号f(t)=Aelu(t)=Ael(t) t=T f()=Ae=0.3684 双边指数信号f()=Aeah=Ae ●复指数信号f(t)=Ae=Ae0 2. s=0 f(0=Ae 3. S=J@ f(t)=Ae/= a(cos ot+jsin ot) 4、s=σ+jiof(1)= allots=Ae( cos ot+ isin a) sin t Sa(t) f(t) Sa(t) 偶函数 lim f(t)=lim ont t→0 3.当t=kz(k=士1±2+3…)时,f()==0 4. lim f(t)= lim sin t 信号的时域分解 1.分解成无穷多个阶跃的连续和
( ) ( ) 2 2 t dt d R t = ⚫ 单边指数信号 ( ) ( ) ( ) 1 f t Ae u t Ae u t t t − − = = t f (t) Ae 0.368A 1 1 = = = = − ⚫ 双边指数信号 t t f t Ae Ae 1 ( ) − − = = ⚫ 复指数信号 st j t f t Ae Ae ( ) ( ) + = = 1. s = 0 f (t) = A 2. t s f t Ae = ( ) = 3. s j f (t) Ae A(cos t jsin t) j t = = = + 4、 ( ) (cos sin ) ( ) s j f t Ae Ae t i t j t t = + = = + + ⚫ Sa(t) ( ) sin ( ) Sa t t t f t = = 1. 偶函数 2. 1 sin lim ( ) lim 0 0 = = → → t t f t t t 3. 当 t = k (k = 1,2,3) 时， 0 sin ( ) = = t t f t 4. 0 sin lim ( ) = lim = → → t t f t t t 5. = − dt t sin t 二、信号的时域分解 1. 分解成无穷多个阶跃的连续和
f()≈f(0)n()+∑∫'(△u(t-k△r)△ f(1)=f(0)u(0)+|f"((t-r)dr 2.分解成无穷多个冲激的连续和 f()≈∑f(k△z)(-Az)△ k=-∞ f(r)= f(rs(t-rdr 3.分解成直流分量和交流分量之和(略) 4.分解成偶分量和奇分量之和(略) ●正交函数与正交函数集 1.两个矢量正交条件:A1·A2=0, 2.两个实函数正交 f,(of2(t)dt f()f2(1)dt=0 f2(t)2 3.n个实函数正交{g()n={g1()!2()g3()…gn(l)} gi(7)8 k(i=j) 4.任意函数可一用完备正交集来表示 f()=∑Cg1(
=− + − k f (t) f (0)u(t) f '(k )u(t k ) f (t) = f (0)u(t) + f '( )u(t − )d − 2. 分解成无穷多个冲激的连续和 − =− f (t) f (k ) (t k ) k − f (t) = f ( ) (t − )d 3. 分解成直流分量和交流分量之和(略) 4. 分解成偶分量和奇分量之和（略） ⚫ 正交函数与正交函数集 1．两个矢量正交条件： A1 • A2 = 0， 0 2 2 1 2 12 = • = A A A C 2．两个实函数正交 = 2 1 1 ( ) 2 ( ) 0 t t f t f t dt ， 0 [ ( )] ( ) ( ) 2 1 2 1 2 2 1 2 12 = = t t t t f t dt f t f t dt C 3．n 个实函数正交 g (t) g1 (t), g2 (t), g3 (t), , gn (t) i n = = = 2 1 ( ) 0 ( ) ( ) ( ) t t i i j K i j i j g t g t dt 4．任意函数可一用完备正交集来表示 = = 1 ( ) ( ) i i i f t C g t
例1-1:已知信号波形,写出信号表达式 f()=t(1)-(t-1)u(t-1) f(t)=δ(1)-d(t-1)+d(t-2)-d(t-3) 例1-2:已知信号的数学表达式求信号波形。 1。f(1)=ecos(4xt)u(t-1)-u(t-2 信号窗为区间[1,2] cos(4x)的频率为2Hz,周期为05s 在[1,2]内应有2个余弦波 例13()2=1-2[(+2)-(-2) 例1-4.画出下列函数波形f()=∑snm(t-n)(t-n)
例 1-1：已知信号波形，写出信号表达式。 f (t) = tu(t) − (t −1)u(t −1) f (t) = (t) − (t −1) + (t − 2) − (t − 3) 例 1-2：已知信号的数学表达式求信号波形。 1。 ( ) = cos(4 )[ ( −1) − ( − 2] − f t e t u t u t t 信号窗为区间 [1, 2 ] cos(4 t) 的频率为 2Hz，周期为 0.5s 在 [1, 2 ] 内应有 2 个余弦波 1 2 例 1-3 ( 2) ( 2) 2 ( ) 1 + − − = − u t u t t f t 1 2 例 1-4. 画出下列函数波形 = = − − 0 ( ) sin[ ( )] ( ) n f t t n u t n