Page 21 - Artificial Intelligence in the Age of Neural Networks and Brain Computing

8 CHAPTER 1 Nature’s Learning Rule: The Hebbian-LMS Algorithm

Lucky’s work turned out to be of extraordinary significance. He was using an
adaptive algorithm to adjust the weights of a transversal digital filter for data trans-
mission over telephone lines. The invention of his adaptive equalizer ushered in the
era of high speed digital data transmission.
Telephone channels ideally would have a bandwidth uniform from 0 Hz to
3 kHz, and a linear phase characteristic whose slope would correspond to the bulk
delay of the channel. Real telephone channels do not respond down to zero fre-
quency, are not flat in the passband, do not cut off perfectly at 3 kHz, and do not
have linear phase characteristics. Real telephone channels were originally designed
for analog telephony, not for digital data transmission. These channels are now used
for both purposes.
Binary data can be sent by transmitting sharp positive and negative impulses into
the channel. A positive pulse is a ONE, a negative pulse is a ZERO. If the channel
were ideal, each impulse would cause a sinc function response at the receiving end
of the channel. When transmitting data pulses at the Nyquist rate for the channel, a
superposition of sinc functions would appear at the receiving end. By sampling or
strobing the signal at the receiving end at the Nyquist rate and adjusting the timing
of the strobe to sample at the peak magnitude of a sinc function, it would be possible
to recover the exact binary data stream as it was transmitted. The reason is that when
one of the sinc functions has a magnitude peak, all the neighboring sinc functions
would be having zero crossings and would not interfere with the sensing of an indi-
vidual sinc function. There would be no “intersymbol interference,” and perfect
transmission at the Nyquist rate would be possible (assuming low noise, which is
quite realistic for land lines).
The transfer function of a real telephone channel is not ideal and the impulse
response is not a perfect sinc function with uniformly spaced zero crossings. At
the Nyquist rate, intersymbol interference would happen. To prevent this, Lucky’s
idea was to filter the received signal so that the transfer function of the cascade of
the telephone channel and an equalization filter at the receiving end would closely
approximate the ideal transfer function with a sinc-function impulse response. Since
every telephone channel has its own “personality” and can change slowly over time,
the equalizing filter would need to be adaptive.
Fig. 1.5 shows a block diagram of a system that is similar to Lucky’s original
equalizer. Binary data are transmitted at the Nyquist rate as positive and negative
pulses into a telephone channel. At the receiving end, the channel output is inputted
to a tapped delay line with variable weights connected to the taps. The weighted
signals are summed. The delay line, weights, and summer comprise an adaptive
transversal filter. The weights are given initial conditions. All weights are set to
zero except for the first weight, which is set to the value of one. Initially, there is
no filtering and, assuming that the telephone channel is not highly distorting, the
summed signal will essentially be a superposition of sinc functions separated with
Nyquist spacing. At the times when the sinc pulses have peak magnitudes, the
quantized output of the signum will be a binary sequence that is a replica of the
transmitted binary data. The quantized output will be the correct output sequence.