Arrhythmia is an abnormal rhythm of the heart which leads to sudden death. Among these arrhythmias, some are shockable, and some are non-shockable arrhythmias with external defibrillation. The automated external defibrillator (AED) is used as the automated arrhythmia diagnosis system and requires an accurate and rapid decision to increase the survival rate. Therefore, a precise and quick decision by the AED has become essential in improving the survival rate. This paper presents an arrhythmia diagnosis system for the AED by engineering methods and generalized function theories. In the arrhythmia diagnosis system, the proposed wavelet transform with pseudo-differential like operators-based method effectively generates a distinguishable scalogram for the shockable and non-shockable arrhythmia in the abnormal class signals, which leads to the decision algorithm getting the best distinction. Then, a new quality parameter is introduced to get more details by quantizing the statistical features on the scalogram. Finally, design a simple AED shock and non-shock advice method by following this information to improve the precision and rapid decision. Here, an adequate topology (metric function) is adopted to the space of the scatter plot, where we can give different scales to select the best area of the scatter plot for the test sample. As a consequence, the proposed decision method gives the highest accuracy and rapid decision between shockable and non-shockable arrhythmias. The proposed arrhythmia diagnosis system increases the accuracy to 97.98%, with a gain of 11.75% compared to the conventional approach in the abnormal class signals. Therefore, the proposed method contributes an additional 11.75% possibility for increasing the survival rate. The proposed arrhythmia diagnosis system is general and could be applied to distinguish different arrhythmia-based applications. Also, each contribution could be used independently in various applications.