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This paper primarily discusses the measurement of partial discharge (PD) phenomena and clustering in the defect pattern of a cross-linked polyethylene power cable joint. First, a high-speed data acquisition and pretreatment were performed for PD electrical signals at a sampling rate of 20 MS/s. The crucial characteristic signals were reversed to reduce the calculated amount of noise. A characteristic matrix was created according to the resulting dynamic error of chaos synchronization. The characteristic parameters were extracted using the fractal theory. Finally, the extension theory was used to develop a diagnostic system and anti-interference test. A comparison with the existing Hilbert-Huang transform (HHT) method revealed that the two characteristics extracted from the chaos synchronization results using the fractal theory were recognized at a higher pattern recognition rate by employing the extension theory. The proposed method can extract crucial information concerning PD as a defect in power cable joints.