Transcript ppt

Dr. Ali Hussein Muqaibel
Electrical Engineering Department
King Fahd University of Petroleum & Minerals
g FM (t )  A  cos c t  k f a(t ) 
k f a(t )
1
Otherwise
Narrowband
Wideband
mp
 m p  m(t )  m p
m (t )
t
mˆ (t )
 m p  mˆ (t )  m p
–mp
t t t
t


g FM (t )  A  cos c t  k f  m( )d 



t


gˆ FM (t )  A  cos c t  k f  mˆ ( )d 



Nyquist sampling theorem
m(t) with a bandwidth of Bm (Hz), the
minimum sampling frequency is 2Bm
t = 1/2Bm
mp
m (t )
t=1/2Bm
t
mˆ ( t )
Frequency i
z(t)
–mp
t
t t t
t=1/2Bm
gFM(t)
t0
t
c  k f m p  i (t )  c  k f m p
z (t )  A  rect  2 Bm  t  t0    cos i t 
z (t )  A  rect  2 Bm  t  t0    cos i t 
Z ( ) 
A
4 Bm

   i
sinc 

 4 Bm
  j  i t0
   i

e

sinc



 4 Bm
  j  i t0 

e



 
sinc 

1  


  



i
i–4Bm
i+4Bm
|G F M (  )|
BWFM  2k f  m p  8 Bm
 2    8 Bm
...
...
...
...
(rad/s)

(rad/s)
 c– k fm p – 4  B m
 c – k fm p
It is not delta!
c
 c+ k fm p
 c+ k fm p + 4  B m
BW FM  2k f  m p  8 B m
 2    8 B m
(rad/s)
(rad/s)
Using the fact that f = kf mp/2 , the bandwidth in Hz becomes
BW FM 
2k f  m p
 4B m (Hz)
2
 2  f  4B m (Hz)
 2  f  2B m  (Hz)
In practice, this bandwidth is higher than the actual bandwidth of FM signals. Consider for
example narrowband FM. Using this formula for the bandwidth, we see that the bandwidth is
twice the actual bandwidth.
BW FM  2  f  2B m
(Hz)  2  f  B m  (Hz)
 2    2 B m  (rad/s)
BWFM  2  f  2 Bm
(Hz)  2  f  Bm  (Hz)
 2    2 Bm  (rad/s)
where
Bm = Bandwidth of the Message Signal m(t) in Hz,
and
 = kf mp  f = kf mp/2.
Special Case
For very wideband FM
For narrowband FM
f
Bm
Bm
BWFM  2f
(Hz)
f
BWFM  2Bm
(Hz)
For FM/PM, the modulation index or deviation ratio ,β, is defined as

f
Bm
Accordingly we may rewrite Carson's rule as
BW FM  2  f  B m  =2B m f / B m  1  2B m    1 (Hz)
.
Notice that the above results can be readily extended for PM:
i (t )  c  k p m (t )
  k p m p ,where m p =max  m (t ) 
BW PM
 k pmp

 2  f  B m   2 
 Bm 
2



Notice that
•BWFM depends on the max(m(t)).
•BWPM depends on the frequency content (change of m(t)) which is related to max  m (t )



On the text book do examples 5.3-5.4 & 5.5
(In the class we start 5.3 then continue 5.3 &
5.4 to see the effect of doubling the
amplitude of the message also examine the
effect of time expansion/compression. Do
example 5.5.
Historical Note: Read the historical Note
about Edwin H. Armstrong (1890-1954)
We started FM with the objective of saving
bandwidth but this out to be wrong. So why
FM what are the features that make FM
applicable?
1. FM can exchange bandwidth for quality

a.
b.
2.
3.
4.
Signal to Noise Ration (SNR) α (Transmission bandwidth)2
Recall that AM has limited bandwidth
FM has constant amplitude (fixed Power)
FM is Immune to nonlinearity.
……..