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Abstract

This paper develops a rigorous functional-analytic framework for the MACD (Moving Average Convergence Divergence) indicator, a classical tool in technical analysis. We show that MACD, commonly defined as the difference between two moving averages, can be precisely interpreted as a phase-corrected, smoothed derivative operator. By analyzing nested and recursive moving averages, we establish that MACD is structurally equivalent to a band-pass filter and derive exact formulas expressing it as a finite difference of delayed and doubly averaged signals. We prove new operator identities demonstrating that MACD corresponds to the derivative of a phase-centered, double-smoothed average, appropriately delayed to correct for asymmetries introduced by causal averaging. This characterization unifies MACD with concepts from harmonic analysis and operator theory, providing a principled basis for understanding its role in signal detection, filtering, and trend analysis. The framework naturally generalizes to recursive decompositions, culminating in an expansion that expresses MACD as a weighted sum of delayed, smoothed derivatives, thereby revealing the true analytical structure underlying this widely used yet traditionally heuristic indicator.