EC-4.6 The Z-Transform

Work through these

• Sequences instead of functions, and why a new transform is needed

  A sequence has values only at whole-numbered instants, so an integral over time has nothing to integrate and is replaced by a sum. The transform variable is complex for the same reason it was in the continuous case.

  NPTEL: Transform Techniques for Engineers · Course

• The transform and its region of convergence, which is now an annulus

  The sum converges in a ring of the complex plane rather than a half plane, and whether the ring includes the unit circle decides whether the sequence is stable. The same expression with a different ring is a different sequence.

• The delay property, and how a difference equation turns algebraic

  Delaying a sequence by one step multiplies its transform by a single factor, which converts a difference equation into a rational expression in one unknown. This is the exact counterpart of differentiation in the continuous case.

  NPTEL: Transform Techniques for Engineers · Course

• The relation to Laplace, and what the mapping does to the plane

  Sampling a continuous signal maps the left half plane into the inside of the unit circle and the imaginary axis onto the circle itself. The mapping is not one to one, which is where aliasing comes from in a later module.

• Inverting by partial fractions, and by long division

  Partial fractions give a closed form for the sequence, while long division gives the first few values directly and is often all a check needs. Knowing both means never being stuck when one is awkward.

• Poles inside the unit circle, and what that will mean later

  A sequence whose poles all lie inside the unit circle decays, and one with a pole outside grows without bound. The systems reading of that fact belongs to a later module and is named here rather than taken.