Lecture 8

 

Lecture 8

Channel capacity

 

• Nyquist channel capacity
  
  • Binary transmission: 1 bit per pulse => transmission rate = 1 / duration of the pulse = 2Wbps
    
  • Number of bits represented by one pulse = log₂(M) = m
    
  • Nyquist signalling rate in bps = m * (1 / duration of the pulse) = 2mW
    
  • Data rate is linear in BW under no noise
    
    • C = 2wlog₂(M)bps
      
      • Where '2W' is the bandwidth
        
      • And M is the discrete signal (voltage levels)
        
    • Increasing m, increases the transmission rate. Is there an upper limit?
      
• Noise limits accuracy
  
  • Receiver makes decision based on transmitted pulse level + noise
    
  • Error rate depends on relative value of noise amplitude an spacing between signal levels
    
  • Large (positive or negative) noise values can cause wrong decision
    
• Noise distribution
  
  • Noise is characterized by probability density of amplitude samples
    
  • Likelihood that certain amplitude occurs
    
  • Thermal electronic noise is inevitable ( due to vibrations of electrons)
    
  • Noise distribution is Gaussian (bell shaped)
    
• Channel capacity
  
  • Is the maximum bit rate supported by a channel
    
  • Can the channel capacity C be made infinite by increasing m?
    
    • NO, there are other constraints introduced by noise and channel interference
      
• Shannon capacity
  
  • Data rate cs noise and error rate
    
  • Data rate DR (goes up) then bit time (goes down) and bit rate (goes up) and error rate (goes up)
    
  • Noise affects more bits
    
  • High data more errors for the same given noise
    
  • Signal power or strength vs noise power
    
    • Signal power / noise power
      
  • SNR_db = 10log₁₀{signal power / noise power}
    
  • By increasing m the difference between adjacent levels is reduced affecting SNR
    
  • Reduction in SNR affects the channel capacity (C).
    
  • Shannon channel capacity theorem provides an upper bound on the channel capacity in terms of bandwidth for a noisy channel
    
    • C = Wlog₂(1+SNR)bps
      
• Line coding
  
  • Converts a binary sequence into a digital signal
    
    • Unipolar NRZ(non return to zero): bit 1 is represented by a +A volts
      
    • Bit 0 is represented by 0 volts
      
  • Average transmitted power per pulse = ½ * A² + ½ * (0) = A²/2
    
  • Average value of a signal = A²/2 Volts
    
  • 
    
  • Polar NRZ : bit 1 is represented by +A/2 volts
    
  • Bit 0 is represented by –A/2 volts
    
  • Average transmitted power per pulse = ½ * (a/2)² + ½ * (-A/2)² = A² / 4
    
  • Half the power used as compared to unipolar NRZ with same distance between levels
    
  • Average value of signal = 0 volts
    
• NRZ – inverted
  
  • First bit 1 is represented by +A/2 volts
    
  • No change for bit 0; flip to the opposite voltage for the next bit 1
    
  • Average transmitted power per pulse = A²/4
    
  • Average value of signal = 0 volts
    
• Bipolar encoding
  
  • Bit 0 is represented by 0 volts
    
  • Bit 1 is represented by consecutive values of A/2 and –A/2
    
  • Average transmitted power per pulse = A² / 8
    
  • Average value of signal = 0 volts
    
• Manchester encoding
  
  • Bit 0 is the change from –A/2 to A/2
    
  • Bit 1 is the change from A/2 to –A/2
    
  • Got popular among Ethernet / token ring networks that were close together and cost was more important than bandwidth
    
  •