Thursday, October 30, 2008

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
  • Ak 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 Bk for even and Ak for odd
    • Multiply Ak with cosine wave and Bk with sin wave: modulate
  • To demodulate see slides, involved multiplying whole signal by two trig identities which will give you Ak and Bk 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
    • SNRdb = 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

Thursday, October 2, 2008

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 kth 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 (Aout/Ain) 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