Definitions of z-Transforms

$$H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$$

is the bilateral (2-sided) z-transform. Its inverse z-transform is defined as:

$$Z^{-1}[H(z)] = h[n] = \frac{1}{2 \pi j} \oint H(z) z^{n-1} dz$$

which is a counterclockwise contour integral along a closed path in the $z$-plane. We will see how to take inverse z-transforms using tables and partial fraction expansion.

We can also define a unilateral z-transform as

$$H_u(z) = \sum_{n=0}^{\infty} h[n] z^{-n}.$$

IMPORTANT: The textbook uses the notation $Z_b[f[n]]$ and $F_b(z)$ to denote the two-sided or bilateral z- Transform of a function $f(n)$, but since we will only look at the bilateral transform (and NOT the unilateral), we will drop the $b$ subscript here.

Bilateral z-Transform

Since our focus is on the Bilateral z-Transform, let's jump ahead to that section, and then we'll come back to some of the earlier sections.

The bilateral (2-sided) z-transform.

$$H_b(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$$

is for 2-sided, anticausal, and causal signals (i.e. all signals).

Note that, whereas for Laplace Transform we considered where the integral converges, here we consider where the sum converges.

We must consider the Region of Convergence (ROC) of the z-transform for the bilateral z-transform because left-sided and right-sided time functions will have the same z-transform and only the ROC will distinguish between the two possible time functions.

Remember:

$$\sum_{i=0}^{\infty} a^i = {1 \over 1-a}, \quad \vert a\vert<1$$

You'll use this a lot!

 

Find the z transforms of

$$x_1[n] = a^n u[n] \quad \mbox{and} \quad x_2[n] = -(a^n) u[-n-1]$$

and plot the ROCs and pole/zero diagrams.

We see that we must specify the ROC for the bilateral z- transform to be unique.

Definitions and Regions of Convergence

• $x[n]$ is right-sided if $x[n] = 0, n < n_0$

• $x[n]$ is left-sided if $x[n] = 0, n > n_0$

We can write

$$X(z) = \cdots + x[-2] z^2 + x[-1] z + x[0] + x[1] z^{-1} +x[2] z^{-2} + \cdots$$

1. We've seen that right-sided signals have an ROC of the form $\vert z\vert > r_{max}$, i.e.,
   

   Examine for right-sided $x[n]$
   

   $$X(z) = \sum_{n=n_0}^{\infty} x[n] z^{-n}$$
   
   
   
   
   $$X(z) = \sum_{n=n_0}^{\infty} x[n] \left( {1 \over z} \right)^n$$
   
   

   As $n \rightarrow \infty$, need $(1/z)^n \rightarrow 0$ for sum to converge.

   This will happen for values of $z$ outside rather than inside the pole, i.e. $\vert z\vert > r_{max}.$

   What about $z=\infty$ ?

   If $x[n]$ is not causal but is still right-sided, e.g. $x[n]=u[n+1]$, then
   

   $$X(z)=\sum_{n=-1}^{\infty} z^{-n} = z + \sum_{n=0}^{\infty} z^{-n}$$
   
   

   Will not converge at $z=\infty$, and we won't include it in the ROC.

   Thus we can tell if a system is causal from the ROC of the z-transform of its impulse response.

   $$\begin{eqnarray*} \vert z\vert > r_{max} & \Rightarrow & {\rm CAUSAL} \\ \inf... ...ert > r_{max} &\Rightarrow & \mbox{right-sided but not causal} \end{eqnarray*}$$
   
   

2. Left-sided signals have ROC of form $\vert z\vert < r_{min}$, i.e.,
   

   Examine for left-sided $x[n]$
   

   $$X(z) = \sum_{n=-\infty}^{n_0} x[n] z^{-n}$$
   
   

   As $n \rightarrow -\infty$, need $(1/z)^n \rightarrow 0$ or $z^\infty \rightarrow 0$

   This happens for values of z inside rather than outside the poles.

   What about $z=0$ ?

   If $x[n]$ is left-sided but not strictly anticausal

   ($x[n] = 0$ for $n > n_0 > 0$ but $x[n_0] \ne 0$)

   e.g. $x[n]=u[-n+1]$, then
   

   $$X(z) = \sum_{n=-\infty}^1 z^{-n} = z^{-1} + \sum_{n=0}^\infty z^{n}$$
   
   
   does not converge at $z=0$ so don't include $z=0$ in the ROC.

3. 2-sided signals have ROC of the form
   
   
   

4. Finite Duration $x[n]$ has ROC of entire z-plane except possibly $z=0$ or $z=\infty$
   
   $$\delta[n-1] \leftrightarrow z^{-1}, \vert z\vert >0$$
   
   
   
   
   $$\delta[n+1] \leftrightarrow z, \vert z\vert < \infty$$
   
   

FACT: An ROC must contain the unit circle for stability - this holds for causal, anticausal, and two-sided signals.

Find the z-Transform of $x[n] = a^{\vert n\vert}$ for $\vert a\vert<1$.

Find the z-Transform of

$$x[n] = 3^n u[-n-1] + 4^n u[-n-1].$$

Find the z-transform of $\frac{1}{2}\delta[n-1]+ 3 \delta[n+1]$.

What is its ROC?

Find the z-transform of

$$x[n] = (.5)^n u[n-1] + 3^n u[-n-1].$$

Would this system be BIBO stable?

Find the z-transform of $x[n] = r^{n} \sin (bn) u[n]$ using Euler's rule.

Note that the previous examples are for a unilateral z-transform but if you add a $u[n]$ to all the time functions, you will get the same answer as for the bilateral transform. You can derive all these transform pairs for practice in taking z-transforms.

Insights from the Pole-Zero Plot and ROC

Things that you can tell about a signal from its pole-zero plot (and ROC):

• When the ROC includes the unit circle, then the signal is absolutely summable. (If the signal is an impulse response $\Rightarrow$ the system is stable.)
  
  
• A pole on the positive real axis corresponds to a simple decaying or growing function (of form $a^n$ for a pole at $z=a$).
  
  
  
  
• Poles off the positive real axis correspond to an oscillating signal where the frequency of oscillation is the angle from the positive real axis. (Poles on the negative real axis have an angle of $\pi$, so the frequency of oscillation is $\pi$, as in $(-1)^n$.) When the poles are ...
  • on the unit circle $\Rightarrow$ sinusoidal functions with constant amplitude
  • not on the unit circle $\Rightarrow$ sinusoidal functions with a decaying (or growing) envelope (rate of decay/growth depends on the distance from the pole to the origin).
  
  
• Poles and zeroes must come in complex conjugate pairs for the signal to be real (consequence of the z-transform property: $x[n]^* \longleftrightarrow X^*(z^*)$).