What is the Z-transform and why is it used in DSP?

The Z-transform converts discrete-time signals to the complex z-domain: X(z) = sum of x[n]*z^(-n). It is the discrete-time equivalent of the Laplace transform. Uses: Analyzing discrete-time systems and stability (poles inside unit circle = stable), Designing digital filters (transfer function H(z)), and Solving difference equations. The unit circle z = e^(jw) corresponds to the frequency response. Z-transform properties (linearity, time shift, convolution) parallel Laplace transform properties, enabling systematic discrete-time system analysis.