Description: The Z-Transform Theorem establishes a fundamental relationship between the Z-transform and the Fourier transform in the context of discrete signals. This theorem is crucial for the analysis and design of systems in the frequency domain, allowing the conversion of time-discrete signals to their representation in the frequency domain. The Z-transform, which is defined for discrete sequences, provides a way to analyze the stability and behavior of linear time-invariant (LTI) systems. Through this theorem, it can be established that the Z-transform of a discrete sequence converges to the Fourier transform when evaluated on the imaginary axis of the complex plane. This means that, under certain convergence conditions, frequency information from a signal can be extracted from its representation in the Z-domain. This theorem is not only fundamental in the field of signal processing but also has practical applications in various technological domains where understanding the relationship between these two transforms is essential for effective system design and analysis.
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