Z-Transform: Generalization of DTFT and ROC

Hello! Welcome to our next lesson.

In our previous session, we explored the periodicity of the Discrete-Time Fourier Transform (DTFT) in depth. We established that the DTFT, $X (e^{j ω})$ , is perfectly suited for analyzing signals that are absolutely summable ( $\sum_{n = - \infty}^{\infty} ∣ x [n] ∣ < \infty$ ). However, this condition excludes a large class of important signals, such as growing exponentials ( $x [n] = 2^{n} u [n]$ ) or even the simple unit step function, for which the DTFT sum does not converge.

Today, we will address this limitation by generalizing the DTFT. Our goal is to derive the z-transform, a more powerful tool that can represent a much broader family of signals. We will also define and explore the crucial concept of the Region of Convergence (ROC), the domain in which the z-transform exists, and see why it is an indispensable part of the transform's definition.

1. Generalizing the DTFT

Let's begin with the definition of the DTFT:

X (e^{j ω}) = n = - \infty \sum \infty x [n] e^{- j ω n}

The convergence problem arises from the sequence $x [n]$ itself. A clever way to force convergence is to multiply $x [n]$ by an exponentially decaying sequence before summing. Let's introduce a real, positive convergence factor, $r$ , and apply it as $r^{- n}$ . The DTFT of this new, weighted sequence, $x [n] r^{- n}$ , would be:

n = - \infty \sum \infty (x [n] r^{- n}) e^{- j ω n}

By grouping the terms raised to the power of $n$ , we get:

n = - \infty \sum \infty x [n] (r e^{j ω})^{- n}

This expression has a very elegant structure. We have essentially replaced the $e^{j ω}$ term from the DTFT with a more general complex variable, $z = r e^{j ω}$ . This $z$ is a complex number in polar form, with magnitude $r$ and angle $ω$ . Substituting $z$ gives us the definition of the bilateral z-transform:

X (z) = n = - \infty \sum \infty x [n] z^{- n}

This generalization should feel familiar. Given your background in continuous-time systems, you'll recognize the strong parallel to the relationship between the Fourier and Laplace transforms. The Continuous-Time Fourier Transform (CTFT) is a special case of the Laplace transform evaluated on the imaginary axis ( $s = j Ω$ ). Similarly, the DTFT is a special case of the z-transform evaluated on the unit circle, where $∣ z ∣ = 1$ (i.e., $r = 1$ ).

The following reading provides a concise summary of this relationship.

Part I 1 Relation to Discrete-Time Fourier Transform

The document 'Relation to Discrete-Time Fourier Transform' from MIT's MAS.160 course offers a direct and mathematically clear explanation of how the z-transform generalizes the DTFT.

Please read the first page of this PDF. Focus on how substituting z = re^{j\omega} into the z-transform definition reveals that it's the DTFT of a modified sequence, x[n]r^{-n}. Note the explicit analogy drawn with the Laplace and Fourier transforms.

To reinforce this fundamental idea, the following video explains the relationship visually and ties it into the broader family of transforms.

Relation between Laplace transform, Fourier transform, z-transform, DTFT, DFT and FFT

The video 'Relation between Laplace transform, Fourier transform, z-transform, DTFT, DFT and FFT' provides a clear walkthrough of how the z-transform generalizes the DTFT.

Watch the segment from 00:12:49 to 00:15:49. The presenter explicitly shows that by restricting the complex variable z to the unit circle (|z|=r=1), the z-transform formula reduces to the DTFT formula.

2. The Region of Convergence (ROC)

The z-transform is defined by an infinite series. As with any infinite series, we must ask: for which values of $z$ does this sum actually converge to a finite value? The set of all such values of $z$ in the complex plane is called the Region of Convergence (ROC). Without the ROC, the z-transform is ambiguous and incomplete.

Let's see why through some foundational examples.

Region of Convergence for the z-Transform

In this video, 'Region of Convergence for the z-Transform', Professor Barry Van Veen provides excellent, step-by-step derivations for the ROC of key signal types.

Please watch the following segments: Introduction (00:00 - 00:58): This introduces the concept of the ROC. Causal Signal (02:16 - 05:38): Follow the derivation for a right-sided exponential sequence, x[n]=\alpha^n u[n]. Pay close attention to how the geometric series convergence condition (|\alpha z^{-1}| < 1) defines the ROC as |z| > |\alpha|. Notice his point that the z-transform exists even for growing signals (|\alpha| \ge 1) where the DTFT fails. Anti-Causal Signal (05:38 - 09:33): Now watch the derivation for a left-sided sequence. The crucial takeaway here is that the final algebraic expression is identical to the causal case, but the ROC is now |z| < |\alpha|.

The Ambiguity of the Transform

As Professor Van Veen demonstrated, both the causal (right-sided) sequence $x_{1} [n] = a^{n} u [n]$ and the anti-causal (left-sided) sequence $x_{2} [n] = - a^{n} u [- n - 1]$ have a z-transform with the exact same algebraic expression:

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

However, their Regions of Convergence are completely different and, in fact, mutually exclusive:

For the causal signal $x_{1} [n] = a^{n} u [n]$ , the ROC is $∣ z ∣ > ∣ a ∣$ (the plane outside a circle of radius $∣ a ∣$ ).
For the anti-causal signal $x_{2} [n] = - a^{n} u [- n - 1]$ , the ROC is $∣ z ∣ < ∣ a ∣$ (the plane inside a circle of radius $∣ a ∣$ ).

A depiction of the pole-zero plot for $X (z) = \frac{z}{z - a}$ . The ROC for the causal signal $a^{n} u [n]$ is the shaded region outside the pole at $z = a$ (left). The ROC for the anti-causal signal $- a^{n} u [- n - 1]$ is the shaded region inside the pole (right). The algebraic expression for the transform is the same for both.

This is the most important concept in this lesson: The z-transform is uniquely defined only by the pair of its algebraic expression and its Region of Convergence. Stating one without the other is ambiguous.

Test your understanding!

A signal $x [n]$ has a z-transform $X (z) = \frac{1}{1 - 0.5 z ^{- 1}}$ . Can you uniquely determine the time-domain signal $x [n]$ from this information? If not, what are the two possible signals, and what additional information would you need to distinguish between them?

Show answer

No, the signal $x [n]$ cannot be uniquely determined. We need the ROC. There are two possibilities:

If the ROC is $∣ z ∣ > 0.5$ , the signal is causal and right-sided: $x [n] = (0.5)^{n} u [n]$ .
If the ROC is $∣ z ∣ < 0.5$ , the signal is anti-causal and left-sided: $x [n] = - (0.5)^{n} u [- n - 1]$ .

To distinguish between them, we must be given the ROC. For example, if we were also told that the DTFT of the signal exists, we would know the ROC must contain the unit circle. Since $∣ z ∣ > 0.5$ contains the unit circle but $∣ z ∣ < 0.5$ does not, we could uniquely identify the signal as the causal one.

3. Properties and Shapes of the ROC

The examples we've seen reveal a set of consistent properties that govern the structure of the ROC. These properties are extremely useful for inferring the nature of a signal directly from its transform's ROC.

The ROC is a ring or disk in the z-plane centered at the origin. For causal signals, the ring extends to infinity; for anti-causal signals, it extends inward to zero.
The DTFT of a sequence $x [n]$ converges if and only if the ROC of its z-transform includes the unit circle ( $∣ z ∣ = 1$ ). This is the condition for stability in LTI systems.
The ROC cannot contain any poles. Since the transform blows up at a pole, it by definition does not converge there. The boundaries of the ROC are therefore often defined by the locations of poles.
Right-sided sequences: If a sequence is right-sided (zero for $n < N_{0}$ ), its ROC is the exterior of a circle, in the form $∣ z ∣ > R_{ma x}$ , where $R_{ma x}$ is the magnitude of the largest pole.
Left-sided sequences: If a sequence is left-sided (zero for $n > N_{0}$ ), its ROC is the interior of a circle, in the form $∣ z ∣ < R_{min}$ , where $R_{min}$ is the magnitude of the smallest non-zero pole.
Two-sided sequences: A sequence that is infinite in both directions will have an ROC that is an annulus (a ring) bounded by poles, in the form $R_{1} < ∣ z ∣ < R_{2}$ . For the transform to exist, we must have $R_{1} < R_{2}$ .
Finite-length sequences: The ROC is the entire z-plane, except possibly at $z = 0$ or $z = \infty$ .

The following resources provide excellent textual references for these concepts.

Z-transform

The Wikipedia article on the Z-transform provides a formal definition of the ROC and walks through clear examples.

Please read the section 'Region of convergence' and its subsections (Example 1, 2, 3, and Examples conclusion). This text reinforces the video's derivations and explicitly states the key conclusion: 'the transform result alone is insufficient'.

Part I 1 Relation to Discrete-Time Fourier Transform

For a final summary, this document from MIT provides a handy list of the properties of the ROC.

Please read the section 'Properties of an ROC' on page 6. This list serves as an excellent summary of the rules governing the shape and meaning of the Region of Convergence.

Conclusion

In this lesson, we have taken a significant step forward from the DTFT to the more general z-transform. By introducing a complex variable $z = r e^{j ω}$ , we extended our analytical reach to a much wider class of signals, particularly those that are not absolutely summable.

Key Takeaways:

The z-transform, $X (z) = \sum_{n = - \infty}^{\infty} x [n] z^{- n}$ , is a generalization of the DTFT.
The DTFT is recovered from the z-transform by evaluating it on the unit circle, $z = e^{j ω}$ . This is analogous to how the CTFT is the Laplace transform on the $j Ω$ axis.
The Region of Convergence (ROC) is the set of points in the complex z-plane where the transform sum converges.
A z-transform is only uniquely defined by its algebraic expression AND its ROC. Different time-domain signals (e.g., causal vs. anti-causal) can have identical transform expressions.
The shape of the ROC indicates the nature of the signal: exterior of a circle for right-sided signals, interior for left-sided signals, and an annulus for two-sided signals.
The DTFT of a signal exists (and an LTI system is stable) if and only if the ROC of its z-transform includes the unit circle.

Preview of the Next Lesson:

We have now established that the DTFT is the z-transform on the unit circle. What does this mean graphically? In the next lesson, we will analyze the relationship between the z-plane and the DTFT frequency response. You will learn how the locations of a system's poles and zeros in the z-plane directly shape the magnitude and phase of its frequency response, providing a powerful and intuitive method for system analysis and design.

Recommended resources

1. Generalizing the DTFT

2. The Region of Convergence (ROC)

The Ambiguity of the Transform

3. Properties and Shapes of the ROC

Conclusion