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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  = 
        
   
 
 
      
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FAQs on Digital Communication Formulas for GATE ECE Exam - Communication System - Electronics and Communication Engineering (ECE)

1. What are the key formulas related to digital communication that are important for the GATE ECE exam in Electronics and Communication Engineering (ECE)?
Ans. Some important formulas related to digital communication for the GATE ECE exam are: 1. Shannon's Capacity Formula: C = B * log2(1 + (S/N)), where C is the channel capacity in bits per second, B is the bandwidth in hertz, S is the signal power, and N is the noise power. 2. Nyquist Bit Rate Formula: R = 2B * log2(M), where R is the bit rate, B is the bandwidth, and M is the number of signal levels or symbols. 3. Hartley's Law Formula: C = 2B * log2(M), where C is the channel capacity, B is the bandwidth, and M is the number of signal levels or symbols. 4. Power Spectral Density (PSD) Formula: PSD = (1/T) * |X(f)|^2, where PSD is the power spectral density, T is the time period, and X(f) is the Fourier transform of the signal. 5. Bit Error Rate (BER) Formula: BER = (1/2) * erfc(sqrt(Eb/No)), where BER is the bit error rate, Eb is the bit energy, and No is the noise power spectral density.
2. How is Shannon's capacity formula used in digital communication and why is it important for the GATE ECE exam?
Ans. Shannon's capacity formula is used to calculate the maximum data rate that can be transmitted through a channel without error. It takes into account the bandwidth of the channel, the signal power, and the noise power. This formula is important for the GATE ECE exam as it helps in determining the maximum achievable data rate and understanding the limitations of a communication channel.
3. What is the significance of Nyquist bit rate formula in digital communication and how is it relevant for the GATE ECE exam?
Ans. The Nyquist bit rate formula is used to calculate the maximum bit rate that can be achieved in a communication system without inter-symbol interference. It considers the bandwidth of the channel and the number of signal levels or symbols used. This formula is important for the GATE ECE exam as it helps in understanding the theoretical limit of the bit rate and the impact of the channel bandwidth and signal levels on the transmission capacity.
4. How does Hartley's law formula contribute to digital communication and why is it relevant for the GATE ECE exam?
Ans. Hartley's law formula is used to calculate the maximum data rate that can be transmitted through a channel, similar to the Nyquist bit rate formula. However, it considers the logarithmic relationship between the channel capacity, bandwidth, and signal levels. This formula is important for the GATE ECE exam as it provides an alternative approach to calculate the channel capacity and understand the impact of signal levels on the transmission capacity.
5. What is the significance of the power spectral density formula in digital communication and how is it useful for the GATE ECE exam?
Ans. The power spectral density formula is used to analyze the power distribution of a signal in the frequency domain. It helps in understanding the signal characteristics, noise properties, and channel capacity. This formula is important for the GATE ECE exam as it enables the analysis of signal quality, noise effects, and the capacity of a communication channel.
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