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# Z Transforms - PowerPoint PPT Presentation

Z Transforms. Another French Mathematician?. Z Transforms. The Z transform is to discrete time systems as the Laplace transform is to continuous time systems. The Fourier transform is the Laplace transform evaluated on the j w axis.

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### Z Transforms

Another French Mathematician?

The Z transform is to discrete time systems as the Laplace transform is to continuous time systems.

The Fourier transform is the Laplace transform evaluated on the jw axis.

The discrete Fourier transform is the Z transform evaluated on a unit circle in the Z transform plane.

As with the Laplace and Fourier transforms, the Z transform only if there is a region in the z plane where the Z transform sum converges.

The Z transform is a linear operation:

To prove this:

The Z transform also has a time shifting property:

This is easily proven:

Changing variables:

There is a discrete time unit step function, u(n). This is analogous to the continuous time unit step, u(t):

The discrete time unit step is:

We can write a causal signal either this way:

Or like this:

Let’s derive some Z transform pairs. First, though, we need several formulas dealing with geometric series.

Here’s a finite geometric series:

Multiplying both sides by a:

If we subtract the last equation from the preceding one:

Solving for SN-1:

Note that a can be real, imaginary, or complex, but cannot be equal to 1.

Now consider an infinite geometric series:

For this to converge, aN must approach zero as N approaches infinity. In other words,

If this condition is met,

if a is complex, it may be written as

In which case r < 1 is equivalent to

Now let’s find the Z transform of the discrete time impulse:

So

Now let’s try the shifted impulse, d(n-k):

Now, let’s take the causal series:

This converges if

Using the formula for an infinite geometric series, and substituting cz-1 for a:

Now let’s try this one:

Using Euler’s formula:

for c in

But we can substitute

This gives us:

But this can be rewritten as:

And this can be simplified as:

So we have this transform pair:

A similar derivation yields this:

Uniqueness “Problem”

Recall that this is a transform pair:

Now, consider the “anticausal twin” of cnu(n):

This can be rewritten as:

So we can derive its Z transform:

But this can also be written as:

So:

Using the geometric series formula:

We can rearrange this a bit more:

But, we already derived this:

The only difference between the Z transform pairs for the causal series and it’s anticausal twin is the region of convergence - so we need to know which series we’re dealing with. Normally it will be the causal one.

Examples:

For a continuous time system, the transfer function is:

For a continuous time system, the transfer function is:

We have two ways to find the transfer function:

1. Take the Z transforms of the input and output sequences X(z) and Y(z), then divide Y(z) by X(z)

2. Take the Z transform of the system impulse response sequence:

Here’s a difference equation for a generalized discrete time system:

Here’s its Z transform, by inspection:

Rearranging,

So the transfer function is:

This can also be rewritten as:

This is similar to the way we wrote the transfer function of a continuous time system. The lk’s and pk’s are the zeros and poles, respectively.

Since the coefficients (ak and bk) are real numbers, the poles and zeros must either be real, or complex conjugate pairs.

Since the feedback coefficients (ak) of an FIR system are all zero, the denominator of its transfer function is 1. Therefore, an FIR system has no poles. It may be referred to as an all-zero system.

Let’s talk about the impulse response for a minute. Suppose we have a discrete-time system with an impulse response h(n), and we drive it with an impulsive input:

Then the response is:

Or, in the Z transform domain,