MICROPROCESSORS Dr. Hugh Blanton ENTC 4337/ENTC 5337

Dr. Blanton - ENTC Introduction 2 / 30 Z Transform, Sampling and Definition Most real signals are analog and in order to utilize the processing power of modern digital processors it is necessary to convert these analog signals into some form which can be stored and processed by digital devices.

Dr. Blanton - ENTC Introduction 3 / 30 Z Transform, Sampling and Definition The standard method is to sample the signal periodically and digitize it with an A to D converter using a standard number of bits 8, 16 etc. Digital signal processing is primarily concerned with the processing of these sampled signals.

Dr. Blanton - ENTC Introduction 4 / 30 Z Transform, Sampling and Definition The diagram below illustrates the situation. The blue line shows the analog signal while the red lines shows the samples arising from periodic sampling at intervals T.

Dr. Blanton - ENTC Introduction 5 / 30 A mathematical representation of the sampled signal is shown below.

Dr. Blanton - ENTC Introduction 6 / 30 This is equivalent to modulating a train of delta functions by the analog signal. The delta function effectively "filters" out the values of the signal at times corresponding to the zeros in the argument of the delta function.

Dr. Blanton - ENTC Introduction 7 / 30 This process is also referred to as "ideal" sampling since it results in sampled signals of "zero" width, "infinite" height, magnitude x*(t) and whose spectrum is perfectly periodic.

Dr. Blanton - ENTC Introduction 8 / 30 The previous equation is equivalent to the following since the delta function has the effect of making x(t) nonzero only at times t = kT.

Dr. Blanton - ENTC Introduction 9 / 30 Taking the Laplace transform of the sampled signal using the integral definition and the properties of the delta function results in the following

Dr. Blanton - ENTC Introduction 10 / 30 The Laplace transform has the Laplace variable s occurring in the exponent and can be awkward to handle.

Dr. Blanton - ENTC Introduction 11 / 30 A much simpler expression results if the following substitutions are made

Dr. Blanton - ENTC Introduction 12 / 30 The definition of the Z Transform is

Dr. Blanton - ENTC Introduction 13 / 30 If the sampling time T is fixed then the Z Transform can also be written

Dr. Blanton - ENTC Introduction 14 / 30 If the sampling time T is fixed then the Z Transform can also be written

Dr. Blanton - ENTC Introduction 15 / 30 The final result is a polynomial in Z. The Z Transform plays a similar role in the processing of sampled signals as the Laplace transform does in the processing of continuous signals.

Dr. Blanton - ENTC Introduction 16 / 30 Z Transform, Step and Related Functions The definition of the Z transform is shown below.

Dr. Blanton - ENTC Introduction 17 / 30 The step function is defined as: and is shown graphically below. A continuous step function shown above is plotted in blue and the sampled step in red.

Dr. Blanton - ENTC Introduction 18 / 30 When a step function is sampled, each sample has a constant value of 1. The Z Transform can be written as a sum of terms as indicated below. The expression for X(z) is a geometric series which converges if |z| > 1 to:-

Dr. Blanton - ENTC Introduction 19 / 30 A Step function delayed by 1 sampling interval. The Z transform is:

Dr. Blanton - ENTC Introduction 20 / 30 This can be summed to give the Z transform of the delayed step. The Z transform of x(k-1) can be written as z -1 X(z) where X(z) is the Z transform of x(k).

Dr. Blanton - ENTC Introduction 21 / 30 For a kT interval delay of the step function the Z transform is multiplied by z -k

Dr. Blanton - ENTC Introduction 22 / 30 The equation for a ramp and its samples are shown below:

Dr. Blanton - ENTC Introduction 23 / 30 The Z transform of the ramp is given by:

Dr. Blanton - ENTC Introduction 24 / 30 Multiplying by z -1 gives: Subtracting the last 2 equations give:

Dr. Blanton - ENTC Introduction 25 / 30 Rearranging the expression for the Z transform gives the final expression for the Z transform of the ramp as:

Dr. Blanton - ENTC Introduction 26 / 30 Z Transform Table

Dr. Blanton - ENTC Introduction 27 / 30 Z Transform Table

Dr. Blanton - ENTC Introduction 28 / 30 Z Transform Table

Dr. Blanton - ENTC Introduction 29 / 30 Z Transform Table