The Z-transform is a mathematical tool used in signal processing and control theory to analyze discrete-time signals. It converts a sequence of numbers (or samples) into a complex frequency domain representation. This transformation helps engineers and scientists understand the behavior of systems, making it easier to design filters and controllers. In essence, the Z-transform takes a time-domain signal and expresses it in terms of its frequency components. By using the Z-transform, one can manipulate and solve problems related to digital systems, stability, and filter design more efficiently, leading to better performance in applications like audio processing and communications.