MICROPROCESSORS

Dr. Hugh Blanton

ENTC 4337/ENTC 5337

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Z Transform, Sampling and Definition 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.

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Z Transform, Sampling and Definition 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.

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Z Transform, Sampling and Definition 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.

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• A mathematical representation of the sampled signal is shown below.

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• 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.

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• 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.

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• 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.

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• Taking the Laplace transform of the sampled signal using the integral definition and the properties of the delta function results in the following

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• The Laplace transform has the Laplace variable s occurring in the exponent and can be awkward to handle.

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• A much simpler expression results if the following substitutions are made

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• The definition of the Z Transform is

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• If the sampling time T is fixed then the Z Transform can also be written

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• If the sampling time T is fixed then the Z Transform can also be written

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• 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.

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Z Transform, Step and Related Functions Z Transform, Step and Related Functions

• The definition of the Z transform is shown below.

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• 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.

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• 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:-

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• A Step function delayed by 1 sampling interval.

• The Z transform is:

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• 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-1X(z) where X(z) is the Z transform of x(k).

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• For a kT interval delay of the step function the Z transform is multiplied by z-k

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• The equation for a ramp and its samples are shown below:

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• The Z transform of the ramp is given by:

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• Multiplying by z-1 gives:

• Subtracting the last 2 equations give:

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• Rearranging the expression for the Z transform gives the final expression for the Z transform of the ramp as:

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Z Transform Table Z Transform Table

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Z Transform Table Z Transform Table

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Z Transform Table Z Transform Table

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Z Transform Table Z Transform Table