Constructing frequency response from a three-stage filter cascade

An oscillator written as a construction problem

A three-stage market oscillator can be read as an explicit construction problem rather than as a finished plotting recipe. In the archive workflow, the composite indicator is built as three cascaded elementary filters. Because the arrangement is a cascade, each stage output becomes the next stage input, and the composite transfer function is the product of the three stage transfer functions.

TradersWeek editorial reading: once that product is written down, power and phase curves can be tabulated first and used as a quantitative baseline before the same cascade is applied to ordered price, volume, or breadth observations.

The differenced weighted moving average

The first stage is a differenced weighted moving average. A weighted moving average is a finite-lookback smoother whose coefficients depend on a chosen length and that can be differenced before later stages. Its transfer function uses a lookback length L and a scale D equal to 2 divided by L plus one.

The finite sum of backward-shift terms inside that moving-average stage is a geometric series with a closed analytic form. The stage therefore does not have to be left as an open sum when the later stages are attached.

Two exponential-smoothing stages

Each later stage is exponential smoothing: a recursive one-pole smoother that blends the current observation with its own lagged output through a single constant. Each of those stages has a first-order transfer function whose numerator is the smoothing constant and whose denominator is one minus one-lag feedback of one minus that constant.

The product after all three stages

After the three stages are multiplied, the composite transfer function has a scale equal to the product of the two smoothing constants and D. It also has a second-order denominator whose coefficients are built from those same constants.

A transfer function is a mapping from an ordered input series to a filter output, written with backward-shift operators and later evaluated as a function of frequency. The composite mapping stays in that operator form until a frequency is chosen.

Unit-delay substitution at a chosen frequency

Frequency response at a chosen frequency F is obtained by unit-delay substitution. The unit delay is replaced with cosine of w minus j sine of w, where w is F converted from degrees to radians. That replacement turns the transfer function into a complex value at the chosen frequency.

Power and phase of the composite response

Power at frequency F is the squared magnitude of the complex response. That power-spectrum value describes how strongly the frequency is passed. Phase in degrees is the combination of three arctangents scaled by 180 over pi, which is the phase-spectrum reading after the stage arguments are combined.

When tabulated power is normalized so the peak equals one, the constant squared scale factor can be omitted from the power formula. The shape of the power-spectrum curve is then enough to compare frequencies.