Lecture 9

Phase Shift Keying

 

• Baseband transmission – the transmission of digital information over a low pass / low frequency channel.
  
• Use nyquist for low pass channel
  
  • Represent binary information with a polar NRZ
    
  • Multiply X₁(t) with sine wave generator
    
  • Which transforms
    

Into

• To demodulate
  
  • Multiply PSK signal with sine wave generator
    
  • Output can be expressed as
    

• Filter out the high frequency component
  
• A_k is retrieved
  
• Trig identities
  
  • 2cos² (x)= 1 + cos(2x)
    
  • 2sin²(x)= 1 – sin(2x)
    
  • 2cos(x)sin(x) = 0 + sin(2x)
    

• After demodulation of this example we still only end up with W
  
• Our goal was 2W (nyquist)
  
• To fix this we put ASK and PSK together
  
• Send two signals simultaneously on same carrier
  
• F₁ = f
  
• F₂ = f + π/2 (shifted 90 degrees)
  
• To do this
  
  • Take the bitstream, split into two sequences f odd and even numbered bits
    
  • Determine the bipolar NRZ B_k for even and A_k for odd
    
  • Multiply A_k with cosine wave and B_k with sin wave: modulate
    
• To demodulate see slides, involved multiplying whole signal by two trig identities which will give you A_k and B_k independently
  

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
    
  • 
    

Lecture 7

Lecture 7

 

• Pulse Code Modulation
  
  • A way of digitizing voice signals
    
  • Voice signal is band limited to 4kHz ( sampling rate = 8 ksamples/s)
    
  • 8-bit nonuniform quantizer is used to quantize each sample (data rate = 64kbits/s)
    
  • It can be shown the SNR for PCM = (6m-10)dB
    
    
• How fast and reliable can a digital transmission occur through a channel?
  
• Depends on a number of factors:
  
  • Amount of energy present in the signal
    
  • Noise properties of the channel
    
  • Distance for the signal to propagate
    
  • Bandwidth(BW) of the transmission medium
    
• Bandwidth:
  
  • Determines the range of frequencies that can be transmitted through a channel
    
  • Consider a sinusoidal wave:
    
  • Frequency present in the wave = f₀Hz or 2πf₀ radians / s
    
• Effective bandwidth
  
  • Most energy
    
  • Wave of
    
• Square waves have infinite bandwidth, the k^th component kf = 1/k
  
  • Therefore most energy and amplitude is in the first few components
    
  • S(t) = A 4/π[
    
    • Square wave with infinite frequency
      
  • A bandwidth of 4Mhz has a data rate of 2Mbps
    
• Cos(Ѳ) = sin( Ѳ + π/2)
  
• Amplitude response A(f): is the ratio of the output amplitude to input amplitude (A_out/A_in) as a function of frequency
  
• Phase shift: is a variation in φ(f) as a function of frequency
  
• Signal power
  

 

• P power distributed across resistance R
  
• V voltage across resistance R
  
• Instantaneous power is proportional to s(t)²
  
• Average power from (t1, t2)
  
• Take P =
  
• Transmission impairment
  
  • Attenuation
    
    • When he signal falls off as a measure of distance
      
    • 
      

Lecture 6

Lecture 6

 

• Phase
  
  • 270 degrees = 3/2 pi
    
  • 360 degrees = 2 pi
    
  • 180 degrees = pi
    
  • 90 degrees = ½ pi
    
• Wavelength
  
  • Pi = vt
    
  • pi(f) = v
    
• periodic signals
  
  • with fundamental frequency of f₀ = 1/t Hz may be represented by the 'fourier series', defined as:
    
  • 
    
• Sampling
  
  • Obtain the value of signal every T seconds
    
    • Choice of T is determined b how fast a signal changes, it, the frequency of content of the signal
      
    • Nyquist sampling theorem says:
      
      • Sampling rate (1/T) >= maximum frequency in the signal
        
    • An analogue signal
      
      • Defined for all time can have any amplitude
        
    • Discrete time signal:
      
      • Defined for multiples of T can have any amplitude
        
      • Must be sure sampling rate is greater than maximum frequency of the signals
        
  • Quantization
    
    • Approximate signal to certain levels. Number of levels used to determine the resolution
      
    • Digital signal
      
      • Defined for multiples of T amplitude limited to a few levels