contents

  • signal picks up noise while propagating in the channel
  • it also get distorted as the channel acts as some sort of filter,
    • that is not necessary lowpass or linear-phase
  • interference occurs as well

  • there might be parts of the channel that might assumed to be usable and actually not

  • the receiver has to deal with a copy of the transmitted signal
    • very far from the idealized version used in the math for designing the transmitter
  • adaptive filtering techniques enable digital receivers to cope with the distortions and the noise introduced by the channel
    • topics: advanced signal processing classes
  • this is an overview
    • your ADSL receiver for instance
    • allows high data rates

receiver design

  • following is the sound made by a dial-up internet modem
    • when connecting to the internet
  • for graphical analysis of this sound, refer to receiver schematic below

qam-receiver

fig: QAM receiver signal flow schematic

  • baseband complex samples \( \hat{b}[n] \) are plotted on the complex plane
  • if the input at the receiver is a signal \( \hat{s}[n] = \cos(( \omega_c + \omega_0 ) n ) \)
    • the obtained baseband for this is \( \hat{b}[n] = e^{j\omega_0 n} \)
    • so they point on the unit circle on the argand plane
    • the angle between successive points will be \( \omega_0 \)

pilot tones

  • the receiver sends pilot tones
  • pilot tones are simple sinusoids used to probe the channel
    • channel probing

  • used to gauge the response at particular frequencies

  • some components
    1. many sinusoids
      • which have abrupt phase changes
      • phase reversals are used as time markers
      • to estimate propagation delay of channel
    2. training sequence
      • known sequence is sent my transmitter
      • the receiver uses channel response to this known sequence to train an equalizer to offset channel effects
    3. handshake procedure between transmitter and receiver
      • just before core information transmission begins
      • low bitrate QAM transmission using only four points
        • 2 bits per symbol
        • parameters exchange of speed, constellation size etc
        • since only 4 points constellation,
          • so even in noisy conditions ensure vital information exchange
    4. data transmission proper

receiver function

  • challenges faced at the receiver and measures taken to offset each challenge
    • interference
      • handshake and line probing
    • propagation delay
      • delay estimation
    • linear distortion
      • adaptive equalization
    • clock drifts between the receiver and the transmitter
      • timing recovery
      • advanced topic

main challenges

  • challenge distortion

  • time-varying discrepancies in clocks \( T_s^{\prime} = T_s \)

  • the channel is approximated as a linear filter in the continuous-time domain to begin analysis
    • \( D(j\omega) \): filter response
    • the filter is assumed to introduce all distortion and delays
  • signal at receiver end: \( \hat{s}(t) \)
    • delayed and distorted version of transmitted signal
  • clock of transmitter: \( T_s \)
  • clock of receiver: \( T_s^{\prime} \)
    • no guarantee these two are synchronized

fig: DAC at transmitter and ADC at the receiver

delay compensation

  • assuming the following are in sync
    • clock of transmitter: \( T_s \)
    • clock of receiver: \( T_s^{\prime} \)
  • channel introduces a delay of \(d\) seconds
    • channel is a simple delay block
    • \( \hat{s}(t)=s(t-d) \rightarrow D(j\Omega) = e^{-j\Omega d} \)
  • we can write \( d = (b + \tau) T_s \) with \( b \in \mathbb{N} \) and \( \vert \tau \vert < \frac{1}{2} \)
  • \(b\) is the bulk delay

  • \(\tau\) is the fractional delay

  • bulk delay is simple to tackle
    • also called the integer delay
    • they are simply the delay that the channel adds to the signal
    • this does not sift the peaks of the data with respect to the sampling interval
  • discontinuities in pilot tones help figure out bulk delay
    • impulses cannot be used as they are full band and get filtered out
  • the fractional delay is more involved
    • it shifts the peaks with respect to the sampling intervals
    • interpolation is used to get the fractional delay compensation
  • transmit \( b[n] = e^{j\omega_0 n} \)
    • \( s[n] = \cos((\omega_c + \omega_)) n ) \)
  • receive \( \hat{s}[n] = \cos((\omega_c + \omega_0) (n - b - \tau)) \)
  • after demodulation and bulk delay offset
    • \( \hat{b}[n] = e^{j\omega_0(n-\tau)} \)
  • multiply by known frequency
    • \( \hat{b}[n]e^{-j\omega_0 n} = e^{-j \omega_0 \tau} \)
  • after offsetting bulk delay
    • \( \hat{s}[n] = s(n-\tau) T_s \)
  • subsample values need to be computed
  • in theory, compensate with a sinc fractional delay
    • \( h[n] = sinc(n=\tau) \)
  • in practice use lagrange approximation
    • practical application of lagrange polynomials
  • lagrange approximation is around \( n \)
    • to compute \(x(n + \tau) \) with \( \vert \tau \vert < \frac{1}{2} \)

\[ \begin{align} x_L(n;t) & = \sum_{k = - N}^{N} x[n-k] L_k^{(N)} (t) \
L_k^{(N)} (t) & = \prod_{i = -N; i \neq k}^{n} \frac{t-i}{k-i} \
\text{ where } & = k = -N, \ldots , N \
\end{align} \]

  • \( x(n+\tau) \approx x_L(n;\tau) \)
  • so, in summary
    • estimate the delay \(\tau\)
    • compute the \( 2N + 1 \) lagrangian coefficients
    • filter with the resulting FIR

adaptive equalization

  • measure to compensate for distortion
  • let the channel distortion be \( D(z) \)
    • \( E(z) \) is the equalizer compensation to offset channel distortion
  • in theory \( E(z) = 1/D(z) \)
  • but \(D(z)\) is not known

  • \( D(z) \) may change over time during transmission

  • hence the equalization \(E(z\) needs to adapt continuously
  • following is the schematic of an adaptive equalizer

adaptive-equalizer

fig: core adaptive equalizer schematic

  • the filter coefficient changes in time based on the error
    • obtained from the output with the transmitted signal
  • the exact signal is sent by the transmitter
    • the receiver has the same copy to get the adaptive equalizer started
    • this is a bootstrapping technique
  • there are some symbols that are common to both the transmitter and receiver together
    • this is called a training sequence
    • handshake 4 point QAM
    • the equalizer is initialized with this shared symbol set

adaptive-equalizer

fig: adaptive equalizer schematic in the big picture

  • this process of bootstrapping is not error free
    • but a generally good place to get started
  • details of adaptive signal processing is an advanced topic
    • needs more research, reading and understanding

adsl

  • ADSL: asymmetric digital subscriber line
  • adsl receives signals on a copper wire channel
  • DSLAM: digital subscriber line access multiplier

abstract-view

fig: telephone network overview

  • last mile: copper wire connecting the home modem to the exchange (CO - central office)
  • copper wire has a large bandwidth
  • POTS: plain old telephone system
  • the (A)symmetry in the bandwidth is the A of the ADSL

adsl-bandwidth

fig: adsl channel - copper wire bandwidth

channel propagation challenges

  1. attenuation: the uneven curve across the bandwidth
    • physical wire imperfections
    • parasitic capacitance
  2. electrical interference: large grey blog in a specific frequency region
    • running the vacuum for instance raises the noise floor of the copper channel
  3. localized radio interference: the impulse at a specific frequency
    • ship-to-shore communications: 0 - 100 kHz
    • airplane communications: 100 - 500 kHz
    • AM radio band: 500 kHz +

adsl-bandwidth-issues

fig: adsl channel propagation challenges

  • the channel is divided into independent sub-channels
  • different channels are treated separately
    • localized treatment in the receiver across all bands
    • the cleanest channels are used to send maximum data

subchannel structure

  • allocate N sub-channels over the total positive bandwidth
  • equal sub-channel bandwidth \( \frac{F_{max}}{N} \)
  • equally spaced sub-channels with center frequency \( \frac{kF_{max}}{N} \)
    • \( k = 0, \ldots, N - 1 \)
digital design
  • pick \(F_s = 2 F_{max} \)
    • \( F_{max} \) is high now
  • center frequency for each subchannel
    • \( \omega_k - 2\pi \frac{kF_{max}/N}{F_s} = \frac{2\pi}{2N}k \)
  • bandwidth of each sub-channel \( \frac{2\pi}{2N} \)
  • to send symbols over a subchannel
    • upsampling factor \( K \geq 2N \)

adsl

fig: subchannels of the adsl channel (N = 3)

  • QAM modem is added on each channel
  • decide on constellation size independently for each channel
    • clean channel gets high numbered constellation
    • noisy channel gets low numbered constellation
  • noisy or forbidden sub-channels send zeros
  • the structure of the communication scheme is sent to the receiver from the transmitter
    • part of handshake procedure
  • classic modulation scheme is applied in each channel as per below schematic

adsl

fig: modem on each sub-channel

  • the receiver modem bank has several modems in parallel
  • each channel has two unique attributes
    • frequency of modulation
    • mappers symbols series

adsl

fig: modem bank

discrete multitone modulation

  • the modem banks maybe seen as oscillators whose output is summed to obtain a signal
    • each oscillator is scaled with an amplitude
    • and phase offset
    • this bank is run for N samples to get the signal
    • these are constant through the generation

adsl-bandwidth

fig: oscillator bank paradigm for modem bank

  • in the modem scenario, the amplitude and phase change at every sample
    • they embed the complex symbol sequence
  • with the discrete multitone modulation, adsl may be implemented with a simple inverse FFT
    • provided that the symbols can be help constant during the whole upsampling event
    • the modem structure can be mapped to the inverse DFT structure if this is done
  • the ADSL trick:
    • instead of using a god lowpass filter use a the 2N-tap interval indicator \[ h[n] = \Bigg( \begin{matrix} 1 & \text{ for } 0 \leq n \leq 2N \\ 0 & \text{ otherwise } \end{matrix} \Bigg) \
      \]

adsl

fig: modem on each sub-channel

  • oscillator in the modulator runs freely
  • with simplification, in each chunk of 2N samples, the symbol is kept constant

adsl-bandwidth

fig: simplification of subchannel modem

  • aggregate bandpass signal is calculated by \[ \begin{align} c[n] & = \sum_{k = 0 }^{N-1} a_k[ \lfloor \frac{n}{2N} \rfloor] e^{j \frac{2\pi}{2N} nk} \
    & = 2N * IDFT_{2N} \{ [ a_0[m] \text{ } a_1[m] \text{ } \ldots a_{N-1}[m] 0 \text{ } 0 \text{ } \ldots ] \} [n] \
    m & = \lfloor \frac{n}{2N} \rfloor \
    \end{align} \]

adsl-bandwidth

fig: simplified subchannels’ modem bank

  • final goal: calculate \(s[n]\) \[ s[n] = Re\{ c[n] \} = \frac{( c[n] + c^*[n] )}{2} \]

\[ IDFT \{ [ x_0 \text{ } x_1 \text{ } \ldots \text{ } x_{N-2} \text{ } x_{N-2} ] \}^* = IDFT \{ [ x_0 \text{ } x_1 \text{ } \ldots \text{ } x_{N-2} \text{ } x_{N-2} ]^* \} \]

\[ c[n] = 2N * IDFT\{ [ a_0[m] \text{ } a_1[m] \text{ } \ldots \text{ } a_{N-1}[m] \text{ } 0 \text{ } 0 \ldots 0 ] \} [n] \]

  • hence, since baseband always has read valued symbols

\[ s[n] = N * IDFT\{ [ 2a_0[m] \text{ } a_1[m] \text{ } \ldots \text{ } a_{N-1}[m] \text{ } a_{N-1}^* [m] \text{ } a_{N-2}^* [m] \ldots a_{1}^* [m] ] \} [n] \]

ADSL schematic

adsl-bandwidth

fig: simplified subchannels’ modem bank

ADSL specs

  • \(F_{max} = 1104 kHz\)
  • N = 256
  • each QAM can send 0 - 15 bit per symbol
  • forbidden channels 0 to 7
    • dedicated to voice
  • channels 7 - 31: upstream data
  • max theoretical throughput: 14.9 Mbps (downstream)

references