Lock a band, take a short lead, then gate empty readings

Two delay types on ordered prices

When a filter processes ordered prices, two delay types appear: group delay and phase delay. Group delay is the time lag, measured in bars, between an input waveform and the matching filter output. For a simple moving average that lag is the same at every frequency. Phase delay is that same lag written as a fraction of a cycle period, so a fixed group delay becomes a larger phase shift as the cycle shortens.

A five-bar simple moving average applies a two-bar group delay at all frequencies. That same two-bar group delay is a 36-degree phase delay versus a 20-bar cycle and a 72-degree phase delay versus a 10-bar cycle.

An exponential moving average has a large group delay at long periods that shrinks toward the Nyquist frequency. Both group delay and phase delay are therefore nonlinear across the spectrum.

Lock a band before the predictive stage

A negative-group-delay predictive stage requires band-limited input. Market series are treated as having essentially unlimited bandwidth, so a two-pole bandpass filter with an explicit center period is applied first. That bandpass isolates the chosen center period, treated as the dominant cycle whose peaks and valleys the later lead overlay is meant to mark. The predictive stage then receives the bandpass output instead of raw prices.

Negative group delay, in this construction, means the output leads selected passband components relative to the bandpass overlay. It does not imply a true look-ahead of unbounded market data.

Keep the requested lead short

The construction keeps the requested lead at no more than three bars. A longer lead raises output noise. Tightening bandpass bandwidth to cut that noise increases group and phase lag.

On the first bar the predictive-stage order is set to three times the chosen lead length. After that, the bandpass output is the only series fed into the predictive stage. The predictive stage sums multiple time-delayed feedback terms so components inside the bandpass passband show negative group delay relative to the bandpass output.

In a cyclic interval whose dominant period was about 20 bars, the predictive overlay moved ahead of the bandpass overlay at the peaks and valleys of the illustrated equity series.

Suppress empty-band readings

In a later trend interval the cycle overlay flipped against the price run because the passband held no cyclic content. An additional trend filter is required to suppress those empty-band readings so they are not treated as turning-point signals.