通信原理(英文版)

6
【Example 2.4】Find the waveform and the frequency spectral density of a sample function. Solution: The definition of the sample function is
sin t Sa ( t ) t
d(t)
1
(f)
0
t
0
f
meaning of d function: It is a pulse with infinite height, infinitesimal width, and unit area. Sa(t) has the following property:
Physical
F ( ) lim
/2 / 2
cos 0 te
jt
sin[( 0 ) / 2] sin[( 0 ) / 2] dt lim 2 ( ) / 2 ( ) / 2 0 0


The frequency spectral density of d(t):
( f ) d (t )e
jt
d (t ) 0
t 0

dt 1 d (t )dt 1

7
d(t)
and its frequency spectral density:
f (t ) f (t 1) t
1
Its frequency spectrum:
1 C ( jn 0 ) T0

T0 / 2
T0 / 2
s(t )e
jn 0t
dt sin(t )e
0
j 2nt
2 dt (4n 2 1)
f(t)
0, u (t ) 1, 当t 0, 当t 0
1 0
Fig. 2.2.6 Unit step functionu(t) Nhomakorabea= d(t)
Difference
t
between frequency spectral density S(f) of energy signal and frequency spectrum of periodic power signal: S(f ) － continuous spectrum； C(jn0) － discrete Unit of S(f ): V/Hz； Unit of C(jn0): V Amplitude of S(f ) at a frequency point ＝ infinitesimal

of d(t)
f (t 0 ) f (t )d (t t 0 )dt
d(t) is an even function:
d ( t ) d ( t )
f (t 0 ) f (t )d (t 0 t )dt



d(t) is the derivative of unit step function:
C( jn0 ) 1 T0

T0 / 2
T0 / 2
s( t )e jn0 t dt
where 0 = 2 / T0 = 2f0 ∵ C(jn0) is a complex function, ∴ C(jn0) = |Cn|ejn where |Cn| － amplitude of the component with frequency nf0 n － phase of the component with frequency nf0 Fourier series of signal s(t):
the frequency spectral density Sa(t) is:
sin t jt Sa( ) e dt t 0

1 1 else
From the above equation, we see that Sa() is a gate function. 【Example 2.5】Find the unit impulse function and its frequency spectral density. Solution: Unit impulse function is usually called d function d(t). Its definition is d (t ) dt 1

The inverse Fourier transform of S() is the original signal:
s(t ) S ( )e jt d


【Example 2.3】Find the frequency spectral density of a rectangular pulse. Solution: Let the expression of the rectangular pulse be
1 g (t ) 0 t /2 t /2
Then its frequency spectral density is its Fourier transform:
G( )
/2
/ 2
e jt dt
1 j / 2 j / 2 sin( / 2) (e e ) j / 2
t
－0
0
(b) 频谱密度
0
(a) 波形
Introducing
d(t), the concept of frequency spectral density
10
can be generalized to power signal.
Energy
spectral density
E s (t )dt
( 0 ) ( 0 ) lim Sa Sa 2 2 2
Referencing eq.(2.2-19), the above equation can be written as: F ( ) [d ( 0 ) d ( 0 )]
3
/2
Frequency spectrum figure
4
【Example 2.2】Find the frequency spectrum of a sinusoidal wave after full-wave rectification. Solution：Assume the expression of the signal is f (t ) sin(t ) 0 t 1

2.1.2 Energy signals and power signals


1 T /2 2 P lim s (t )dt , then P = 0 for energy signal. Average power: T T T / 2
For power signal: P 0, i.e., power signal has infinite duration. Energy signal has finite energy, but its average power equals 0. Power signal has finite average power, but its energy equals infinity.1
Chapter 2 Signals
2.1 Classification of Signals
2.1.1 Deterministic signals and random signals

What is deterministic signal? What is random signal?
Signal power: Let R = 1, then P = V2/R = I2R = V2 = I2 Signal energy：Let S represent V or I，if S varies with time，then S can be rewritten as s(t), Hence, the signal energy E = s2(t)dt Energy signal satisfies 0 E s 2 (t )dt

Let the energy of an energy signal s(t) be E, then the energy of the signal is decided by 2
If its frequency spectral density is S(f ), then from Parseval’s theorem we have
t


k


Sa ( kt ) dt 1
t
When k , amplitude , and the zero-spacing of the waveform 0， k Hence, d (t ) lim Sa ( kt )
k
t

8
Characterisitics
9
【Example 2.6】Find the frequency spectral density of a cosinusoidal wave with infinite length. Solution: Let the expression of a cosinusoidal wave be f (t) = cos0t, then according to eq. (2.2-10), F() can be written as
V f (t ) 0 / 2 t / 2 / 2 t (T / 2) t