Signals & Systems Lecture 16

Properties of Z Transform

1. Linearity Property

$$a{x}_1(n)+b{x}_2(n)\stackrel{Z}{\to }a{X}_1(z)+b{X}_2(z)$$

Where:

• $a$ and $b$ are constants
• $x_1(n)$ and $x_2(n)$ are signals with Z-Transforms $X_1(z)$ and $X_2(z)$, respectively.

2. Multiplication by $a^n$ Property

$$a^{n}x(n)\stackrel{Z}{\to }X\left(\frac{z}{a}\right)$$

3. Time Shifting Property

For a given signal $x(n)$ with Z-Transform $X(z)$ and region of convergence (ROC), shifting the signal by $k$ units in the time domain results in:

$$x(n-k)\stackrel{Z}{\to }{z}^{-k}X(z)$$

The ROC remains the same, except for any potential points where $z=0$ or $z=\infty$ might be included or excluded, depending on the nature of the shift. This property indicates that a time shift by $k$ units results in the multiplication of the Z-Transform by $z^{-k}$ in the frequency domain.

4. Region of Convergence (ROC) Calculations

The Region of Convergence (ROC) defines where the Z-Transform converges. The ROC is determined by the nature of the signal, including its stability and causality.

1. ROC for Finite-Length Signals

For finite-duration signals, the Z-Transform converges for all $z\ne 0$:

$$\text{ROC: }z\ne 0$$

2. ROC for Right-Sided Signals (Causal Signals)

For causal signals (right-sided sequences), the Z-Transform converges outside a circle of radius $r_0$:

$$\text{ROC: }|z|>r_0$$

Where $r_0$ is the magnitude of the largest pole.

3. ROC for Left-Sided Signals (Anti-Causal Signals)

For anti-causal signals (left-sided sequences), the Z-Transform converges inside a circle of radius $r_1$:

$$\text{ROC: }|z|<r_1$$

Where $r_1$ is the magnitude of the smallest pole.

4. ROC for Two-Sided Signals

For two-sided signals, the ROC is the ring between the poles, defined by $r_1$ and $r_0$:

$$\text{ROC: }{r}_1<|z|<r_0$$

5. ROC for Exponential Signals

For a signal of the form $x(n)=a^{n}u(n)$ (where $u(n)$ is the unit step function), the ROC is:

$$\text{ROC: }|z|>|a|$$

6. Special Considerations for ROC

• Causal Systems: The ROC must include $z=\infty$, ensuring system stability.
• Anti-Causal Systems: The ROC must include $z=0$.
• Stability: A system is BIBO stable if the ROC includes the unit circle $|z|=1$.

Summary of ROC Calculation Process

1. Identify whether the signal is right-sided, left-sided, or two-sided.
2. Locate the poles of the Z-Transform.
3. Determine the ROC based on the nature of the signal:
   • Right-sided: ROC is outside the largest pole.
   • Left-sided: ROC is inside the smallest pole.
   • Two-sided: ROC is between two poles.

Example Calculation of ROC

For a causal signal $x(n)=2^{n}u(n)$, the Z-Transform is:

$$X(z)=\frac{1}{1-2z^{-1}}$$

The ROC is determined by $|z|>2$ because the system is right-sided, and convergence occurs for values of $z$ outside the pole at $z=2$.

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Questions & Answers

Question 1

$x(n)=u(n+5)$ Applying Z Transform $Z[u(n)]=\frac{z}{z-1};|Z|>1$ By Time Shifting Property of Z Transform

$$Z\left[u(n+5)\right]=z^5\left[\frac{z}{z-1}\right]$$

References

• Date: 2024.10.22
• Time: 10:21