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 2Q levels, 2Q-1 positive levels, 1 zero level and 2Q-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.
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
| - M⁽ⁱ⁾ | |
| C⁽ⁱ⁾ = | —— |
| 2α |
having radius squared