Digital Communication Formulas for GATE ECE Exam | Communication System - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


    
 
  
 
   
   
 
 
       
 
    
 
     
 
 
 
   
 
    
    
 
    
 
     
 
   
 
  
 
  
 
  
 
  
 
 
Digital Communication
 
 
Matched filter: 
 
?
 
impulse response  a(t) =   
 
 
( T –
 
t)
 
.  P(t) ?
 
i/p 
 
 
?
 
Matched filter o/p will be max at multiples of ‘T’ . So, sampling @ multiples of ‘T’ will give max SNR 
(2
nd
 
point )
 
?
 
matched filter is always causal  a(t) = 0 for  t < 0 
 
?
 
Spectrum of o/p signal of matched filter with the matched signal as i/p ie, except for a delay factor ; 
proportional to energy spectral density of i/p.
 
 
         Ø
 
( )
 
=  
   
(f) Ø(f) = Ø(f) Ø*(f) e
     
 
  
 
         Ø
 
( )
 
= |Ø( )|
 
 
e
      
 
? o/p signal of matched filter is proportional to shifted version of  auto correlation fine of i/p signal 
                 Ø
 
(t) = R
Ø
(t – T)
At t = T   Ø
 
(T) = R
Ø
(0)  ? which proves 2
nd
point 
Cauchy-Schwartz in equality :-
 |g
 
 
(t) g
 
(t) dt|
 
 
  
  =   g
 
 
(t)
 
  
dt   |g
 
(t)|
 
 
  
dt
If  g
 
(t) = c g
 
(t)  then equality holds otherwise  ‘<’ holds 
Raised Cosine pulses :
                              
P(t) = 
   (
  
 
)
(
  
 
)
. 
    (
   
 
)
    
 
   
 
P(f) =  
 
 
 
 
 
       | | = 
   
  
   cos
 
 
  
  
 | |  
   
  
     
   
  
  = | | = 
   
  
  | |  
   
  
 
 
? Bamdwidth of  Raised cosine filter   
 
= 
   
  
   ? Bit rate  
 
 
= 
  
 
   
a ? roll o   actor
  ? signal time period 
? For Binary PSK   
 
= Q  
 
  
 = Q   
  
 
 
 
 =  
 
 
erfc  
 
 
 
 
 
 .
? 4 PSK   
 
= 2Q   
  
 
 
 
  1 
 
 
     
  
 
 
 
  
      
Page 2


    
 
  
 
   
   
 
 
       
 
    
 
     
 
 
 
   
 
    
    
 
    
 
     
 
   
 
  
 
  
 
  
 
  
 
 
Digital Communication
 
 
Matched filter: 
 
?
 
impulse response  a(t) =   
 
 
( T –
 
t)
 
.  P(t) ?
 
i/p 
 
 
?
 
Matched filter o/p will be max at multiples of ‘T’ . So, sampling @ multiples of ‘T’ will give max SNR 
(2
nd
 
point )
 
?
 
matched filter is always causal  a(t) = 0 for  t < 0 
 
?
 
Spectrum of o/p signal of matched filter with the matched signal as i/p ie, except for a delay factor ; 
proportional to energy spectral density of i/p.
 
 
         Ø
 
( )
 
=  
   
(f) Ø(f) = Ø(f) Ø*(f) e
     
 
  
 
         Ø
 
( )
 
= |Ø( )|
 
 
e
      
 
? o/p signal of matched filter is proportional to shifted version of  auto correlation fine of i/p signal 
                 Ø
 
(t) = R
Ø
(t – T)
At t = T   Ø
 
(T) = R
Ø
(0)  ? which proves 2
nd
point 
Cauchy-Schwartz in equality :-
 |g
 
 
(t) g
 
(t) dt|
 
 
  
  =   g
 
 
(t)
 
  
dt   |g
 
(t)|
 
 
  
dt
If  g
 
(t) = c g
 
(t)  then equality holds otherwise  ‘<’ holds 
Raised Cosine pulses :
                              
P(t) = 
   (
  
 
)
(
  
 
)
. 
    (
   
 
)
    
 
   
 
P(f) =  
 
 
 
 
 
       | | = 
   
  
   cos
 
 
  
  
 | |  
   
  
     
   
  
  = | | = 
   
  
  | |  
   
  
 
 
? Bamdwidth of  Raised cosine filter   
 
= 
   
  
   ? Bit rate  
 
 
= 
  
 
   
a ? roll o   actor
  ? signal time period 
? For Binary PSK   
 
= Q  
 
  
 = Q   
  
 
 
 
 =  
 
 
erfc  
 
 
 
 
 
 .
? 4 PSK   
 
= 2Q   
  
 
 
 
  1 
 
 
     
  
 
 
 
  
      
    
 
  
 
   
   
 
 
   
 
    
       
 
 
 
   
    
 
 
 
 
   
 
 
  
 
      
  
 
 
 
      
     
 
FSK:-
 
For BPSK
 
   
 
 
= Q  
 
  
 
 
= Q  
 
 
 
 
 
 = 
 
 
 
erfc   
 
 
 
  
 
 
 
 
?
 
All signals have same energy (Const energy modulation )
 
 
?
 
Energy & min distance both can be kept constant while increasing no. of  points . But Bandwidth 
Compramised.
 
?
 
PPM is called as Dual of FSK .
 
?
 
For DPSK   
 
 
= 
 
 
 
e
  
 
/ 
 
 
 
? Orthogonal signals require factor of ‘2’ more energy to achieve same  
 
as anti podal signals
? Orthogonal signals are 3 dB poorer than antipodal signals. The 3dB difference is due to distance b/w 2 
points.
? For non coherent FSK   
 
= 
 
 
e
  
 
/ 
 
? FPSK & 4 QAM both have comparable performance .
? 32 QAM has 7 dB advantage over 32 PSK.
? Bandwidth of Mary PSK = 
 
 
 
= 
 
 
    
 
 
    ;  S = 
   
 
 
 
? Bandwidth of Mary FSK =  
 
  
 
= 
 
  
 
   
 
 
  ; S = 
   
 
 
 
? Bandwidth efficiency  S = 
 
 
 . 
.
? Symbol time  
 
=  
 
log
 
 
? Band rate  =