The purpose of this note is to describe very briefly a machine[2] which
has been implemented to recover the noise-perturbed binary waveform.
A simplified block diagram of the machine is shown in Figure 1. The
experimental machine has been designed to operate on signals of 10³
samples duration.
[2] The operation of this machine is described in substantially greater
detail in J. J. Spilker, Jr., D. D. Luby, R. D. Lawhorn, “Adaptive
Binary Waveform Detection,” Philco Western Development Laboratories,
Communication Sciences Department, TR #75, December 1963.
Each analog input sample enters the machine at left and may either
contain a signal sample plus noise or noise alone. In order to permit
digital operation in the machine, the samples are quantized in a
symmetrical three-level quantizer. The samples are then converted
to vector form, _e.g._, the previous 10³ samples form the vector
components. A new input vector, ⮕Y⁽ⁱ⁾, is formed at each sample instant.
Define the signal sample values as s₁, s₂, ..., sₙ. The observed vector
Y⁽ⁱ⁾ is then either (a) perfectly centered signal plus noise, (b)
shifted signal plus noise, or (c) noise alone.
{ (s₁, s₂, ..., sₙ) + (n₁, n₂, ..., nₙ) (a)
(Y⁽ⁱ⁾)ᵗ = { (0, ..., s₁, s₂, ..., sₙ₋ⱼ) + (n₁, n₂, ..., nₙ) (b)
{ (0 ... 0) + (n₁, n₂, ..., nₙ) (c)
At each sample instant, two measurements are made on the input
vector, an energy measurement ‖Y⁽ⁱ⁾‖² and a polarity coincidence
cross-correlation with the present estimate of the signal vector stored
in memory. If the weighted sum of the energy and cross-correlation
measurements exceeds the present threshold value Γᵢ, the input vector
is accepted as containing the signal (properly shifted in time), and
the input vector is added to the memory. The adaptive memory has 2^{Q}
levels, 2^{Q-1} positive levels, 1 zero level and 2^{Q-1}-1 negative
levels. New contributions are made to the memory by normal vector
addition except that saturation occurs when a component value is at the
maximum or minimum level.
The acceptance or rejection of a given input vector is based on a
hypersphere decision boundary. The input vector is accepted if the
weighted sum γᵢ exceeds the threshold Γᵢ
γᵢ = Y⁽ⁱ⁾∙M⁽ⁱ⁾ + α‖Y⁽ⁱ⁾‖² ⩾ Γᵢ.
[Illustration: Figure 1—Block diagram of the adaptive binary waveform
detector]
Geometrically, we see that the input vector is accepted if it falls on
or outside of a hypersphere centered at ⮕C⁽ⁱ⁾ = -⮕M⁽ⁱ⁾/2α having radius
squared
Γ⁽ⁱ⁾ ‖M⁽ⁱ⁾‖²
[r⁽ⁱ⁾]² = ——— + —————— .
α (2α)²
Both the center and radius of this hypersphere change as the machine
adapts. The performance and optimality of hypersphere-type decision
boundaries have been _discussed in related work_ by Glaser[3] and
Cooper.[4]
[3] F. M. Glaser, “Signal Detection by Adaptive Filters,” _IRE Trans.
Information Theory_, pp. 87-90; April 1961.
[4] P. W. Cooper, “The Hypersphere in Pattern Recognition,”
_Information and Control_, pp. 324-346; December 1962.
The threshold value, Γᵢ, is adapted so that it increases if the
memory becomes a better replica of the signal with the result that γᵢ
increases. On the other hand, if the memory is a poor replica of the
signal (for example, if it contains noise alone), it is necessary that
the threshold decay with time to the point where additional acceptances
can modify the memory structure.
The experimental machine is entirely digital in operation and, as
stated above, is capable of recovering waveforms of up to 10³ samples
in duration. In a typical experiment, one might attempt to recover
an unknown noise-perturbed, pseudo-random waveform of up to 10³ bits
duration which occurs at random intervals. If no information is
available as to the signal waveshape, the adaptive memory is blank at
the start of the experiment.
In order to illustrate the operation of the machine most clearly, let
us consider a repetitive binary waveform which is composed of 10³ bits
of alternate “zeros” and “ones.” A portion of this waveform is shown in
Figure 2a. The waveform actually observed is a noise-perturbed version
of this waveform shown in Figure 2b at-6 db signal-to-noise ratio. The
exact sign of each of the signal bits obviously could not be accurately
determined by direct observation of Figure 2b.
[Illustration: (a) Binary signal]
[Illustration: (b) Binary signal plus noise