数字通信 proakis 第四章课件
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s f
fc
fc
A
Bs
Bs
f
Bs f c
Confidential LinkAir Confidential
3
2. The analytic signals of s(t)
s ( f ) 2 u( f ) s( f )
s f
s t s ( f )e j 2 ft df
Rl f Sl f H l f
where
rl t sl (t ) * hl t
Thus, for mathematics convenience, we shall deal only with the transmission of equivalent low-pass signals through equivalent low-pass channels
Representation of band-pass signals and systems
Signal space representations Representation of digitally modulation signals Spectral characteristics of digitally modulation signals
with zero mean and power spectral density nn f and the power spectral density is assumed to be zero outside of an interval of frequencies central around fc
H f h t e j 2 ft df
2 Re hl t e j 2 fct e j 2 ft df
With the identity: Re
1 * 2
hl t e j 2 fct hl* t e j 2 fct e j 2 ft dt Hf hl t e
n t is said to be narrowband band-pass process
7
sl f
2A
fc
s f
fc
A
Bs
Bs
f
Confidential
LinkAir Confidential
8
6. The energy of s t
s 2 t dt
Re s t e
l
j 2 f c t
dt
2
With the identity
j 2 f c t
h t j h t 2hl t e j 2 fct
h t 2 Re hl t e j 2 fct
Confidential LinkAir Confidential
12
3. The relation of H ( f ) and H l f
t is called the envelop of s t
t is called the phase of
Confidential
s t
6
LinkAir Confidential
5. The relation of s( f ) and sl f
s f s t e j 2 ft df
s t Re sl t e j 2 fct
Confidential
LinkAThe other representation of band-pass signals
Definite:
sl t t e
Confidential LinkAir Confidential
14
As
Sl f f c H l* f f c 0 Sl* f f c H l f f c 0
R f
1 Sl f f c H l f f c Sl* f f c H l* f f c 2 1 Rl f f c Rl* f f c 2
r t s t * h t s h t d
R f S f H f
and
1 S f Sl f f c Sl* f f c 2
H f H l f f c H l* f f c
Digital Communications
Haitao liu Civil Aviation University of China， Communication Engineering , 2006
Chapter 4 Characterization Of Communication Signals and Systems
Confidential
LinkAir Confidential
9
4-1-2 Representation of band-pass systems
Hf
A
fc
fc
f
the bandpass system
h t the impulse response of bandpass system
Confidential
LinkAir Confidential
11
2. The definition of equivalent system
Hl f H f fc
hl t h t e j 2 fct
h t hl t e
Re sl t e j 2 fct e j 2 ft df
With the identity: Re 1 * 2
1 s f sl t e j 2 fct sl* t e j 2 fct e j 2 ft dt 2 1 1 * j 2 f f c t j 2 f f t sl t e dt sl t e c dt 2 2 * 1 j 2 f f c t j 2 f f c t sl t e dt sl t e dt 2 1 sl f f c sl* f f c 2 Confidential LinkAir Confidential
j t
Then
s t Re sl t e j 2 fct
j t Re t e e j 2 fct j (2 f c t t ) Re t e
s t t cos 2 f ct t
2A
fc
2u ( f )s f e j 2 ft df F 1 2u ( f ) F 1 s( f )
F 1 2u ( f ) t j
1 t
(t ) 1
1 j sgn( f ) t 1 ˆ s (t ) s(t ) Confidential t
signals
s t
sl t
h t
hl t
r t
rl t
r t s t * h t
rl t sl t * hl t ??
Proof:
r t Re rl t e j 2 fct
H f U f H f
h t F 1 H f 1 1 h t * t j 2 t 1 1 1 h t * t h t * j 2 2 t 1 h t j h t 2
Definite:
2A
sl f s f f c
sl t s t e j 2 fct s t js(t ) e j 2 fct
s t js (t ) sl t e j 2 fct
Confidential
LinkAir Confidential
2
4.1 Representation of band-pass signals and systems
4.1.1 Representation of band-pass signals
1. the band-pass signals s(t)
1 Re * 2
2 2 1 1 sl t dt sl t cos 4 f ct 2 t dt 2 2 2 1 sl t dt 2
The energy of s(t) is the one half of the energy of its equivalent low pass signal
F 1 s ( f ) s t
j s t s t t t 1 s t j s t t s t j s t
LinkAir Confidential
4
3. the equivalent low-pass signal of s(t)
Confidential
LinkAir Confidential
15
4-1-4 Representation of band-pass stationary stochastic
processes
1. Narrowband band-pass process
fc
n t is a sample function of wide-sense stationary stochastic process
j 2 f f c t
dt hl* t e
j 2 f f c t
dt
H l f f c H l* f f c
Confidential
LinkAir Confidential
13
4-1-3 Representation of a band-pass systems to a bandpass
Hf
the frequency response of bandpass system
As h(t) is real,
H * ( f ) H ( f ) or H ( f ) H * ( f )
Confidential LinkAir Confidential
10
1. The analytic signals of h(t)
fc
fc
A
Bs
Bs
f
Bs f c
Confidential LinkAir Confidential
3
2. The analytic signals of s(t)
s ( f ) 2 u( f ) s( f )
s f
s t s ( f )e j 2 ft df
Rl f Sl f H l f
where
rl t sl (t ) * hl t
Thus, for mathematics convenience, we shall deal only with the transmission of equivalent low-pass signals through equivalent low-pass channels
Representation of band-pass signals and systems
Signal space representations Representation of digitally modulation signals Spectral characteristics of digitally modulation signals
with zero mean and power spectral density nn f and the power spectral density is assumed to be zero outside of an interval of frequencies central around fc
H f h t e j 2 ft df
2 Re hl t e j 2 fct e j 2 ft df
With the identity: Re
1 * 2
hl t e j 2 fct hl* t e j 2 fct e j 2 ft dt Hf hl t e
n t is said to be narrowband band-pass process
7
sl f
2A
fc
s f
fc
A
Bs
Bs
f
Confidential
LinkAir Confidential
8
6. The energy of s t
s 2 t dt
Re s t e
l
j 2 f c t
dt
2
With the identity
j 2 f c t
h t j h t 2hl t e j 2 fct
h t 2 Re hl t e j 2 fct
Confidential LinkAir Confidential
12
3. The relation of H ( f ) and H l f
t is called the envelop of s t
t is called the phase of
Confidential
s t
6
LinkAir Confidential
5. The relation of s( f ) and sl f
s f s t e j 2 ft df
s t Re sl t e j 2 fct
Confidential
LinkAThe other representation of band-pass signals
Definite:
sl t t e
Confidential LinkAir Confidential
14
As
Sl f f c H l* f f c 0 Sl* f f c H l f f c 0
R f
1 Sl f f c H l f f c Sl* f f c H l* f f c 2 1 Rl f f c Rl* f f c 2
r t s t * h t s h t d
R f S f H f
and
1 S f Sl f f c Sl* f f c 2
H f H l f f c H l* f f c
Digital Communications
Haitao liu Civil Aviation University of China， Communication Engineering , 2006
Chapter 4 Characterization Of Communication Signals and Systems
Confidential
LinkAir Confidential
9
4-1-2 Representation of band-pass systems
Hf
A
fc
fc
f
the bandpass system
h t the impulse response of bandpass system
Confidential
LinkAir Confidential
11
2. The definition of equivalent system
Hl f H f fc
hl t h t e j 2 fct
h t hl t e
Re sl t e j 2 fct e j 2 ft df
With the identity: Re 1 * 2
1 s f sl t e j 2 fct sl* t e j 2 fct e j 2 ft dt 2 1 1 * j 2 f f c t j 2 f f t sl t e dt sl t e c dt 2 2 * 1 j 2 f f c t j 2 f f c t sl t e dt sl t e dt 2 1 sl f f c sl* f f c 2 Confidential LinkAir Confidential
j t
Then
s t Re sl t e j 2 fct
j t Re t e e j 2 fct j (2 f c t t ) Re t e
s t t cos 2 f ct t
2A
fc
2u ( f )s f e j 2 ft df F 1 2u ( f ) F 1 s( f )
F 1 2u ( f ) t j
1 t
(t ) 1
1 j sgn( f ) t 1 ˆ s (t ) s(t ) Confidential t
signals
s t
sl t
h t
hl t
r t
rl t
r t s t * h t
rl t sl t * hl t ??
Proof:
r t Re rl t e j 2 fct
H f U f H f
h t F 1 H f 1 1 h t * t j 2 t 1 1 1 h t * t h t * j 2 2 t 1 h t j h t 2
Definite:
2A
sl f s f f c
sl t s t e j 2 fct s t js(t ) e j 2 fct
s t js (t ) sl t e j 2 fct
Confidential
LinkAir Confidential
2
4.1 Representation of band-pass signals and systems
4.1.1 Representation of band-pass signals
1. the band-pass signals s(t)
1 Re * 2
2 2 1 1 sl t dt sl t cos 4 f ct 2 t dt 2 2 2 1 sl t dt 2
The energy of s(t) is the one half of the energy of its equivalent low pass signal
F 1 s ( f ) s t
j s t s t t t 1 s t j s t t s t j s t
LinkAir Confidential
4
3. the equivalent low-pass signal of s(t)
Confidential
LinkAir Confidential
15
4-1-4 Representation of band-pass stationary stochastic
processes
1. Narrowband band-pass process
fc
n t is a sample function of wide-sense stationary stochastic process
j 2 f f c t
dt hl* t e
j 2 f f c t
dt
H l f f c H l* f f c
Confidential
LinkAir Confidential
13
4-1-3 Representation of a band-pass systems to a bandpass
Hf
the frequency response of bandpass system
As h(t) is real,
H * ( f ) H ( f ) or H ( f ) H * ( f )
Confidential LinkAir Confidential
10
1. The analytic signals of h(t)