02_1 Deterministic and Random Signal Analysis

The spectrum of a bandpass signal is Hermitian symmetric
X(f)
Since the spectrum is Hermitian
It is generally written as
x l(t) x(t)
带通信号
解析信号
X+(f)
x+(t)
X+(f)=X(f)u(f)
()2()
F t f πω↔−2
sg n ()t j ω
↔
sgn()f t
π↔X +(f )x +(t )
From x +(t ) to x l (t )
Define x l (t ) -等效低通，复包络Complex envelope , whose spectrum is given by X l ( f )This implies
From x (t ) to x l (t )带通信号低通表示法
As a result,
From x (t ) to x l (t )带通信号低通表示法
22()()()
()
()a rc ta n
()
x i q
x r t x t x t x t t x t θ=
+=0()cos(2())
x x r t f t t πθ+=02Re{()}
()j f t
l x t e x t π=等效低通表示法正交表示法三角函数表示法
00()cos(2)()sin(2)i q x t f t x t f t ππ−=复包络同相
分量
正交分量
包络
()
()()()()x j t l i q x x t x t jx t r t e
θ=+=
From X
( f ) to X( f )带通信号与等效低通信号频谱关系
l
From we obtain
Bandpass signal 带通信号
Analytic signal or pre-envelope x
(t) and X
+
等效低通信号或复包络
=+=
()()()(
x t x t jx t r
¾pre-envelope x
+
(t)
¾complex envelope x l(t)¾envelope
¾phase
()cos(2())
x x
r t f t t
πθ
+
2
Re{()}
()j f t
l
x t e
x tπ
=
00 ()cos(2)()sin(2) i q
x t f t x t f t
ππ
−
()
()()x j t
l x
x t r t eθ
=
= modulation
= demodulation
x t
()
The energy of a (complex) signal s We are interested in the connections
From Parseval’s Theorem we see Secondly
>>W and Thirdly, f
c
分量
Parseval’s relation immediately gives
?。