DFT, DTFT, and Z-Transform Relationship

1. From the z-Transform to the DTFT: A Walk on the Unit Circle

Let's start by recalling the definitions of the (bilateral) z-transform and the DTFT for a sequence $x[n]$:

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

DTFT:
$X(e^{j\omega })={\sum }_{n=-\infty }^{\infty }x[n]e^{-j\omega n}$

Comparing these two equations reveals a direct and profound relationship. If we take the z-transform expression $X(z)$ and restrict the complex variable $z$ to have the specific form $z=e^{j\omega }$, the equation becomes identical to the DTFT.

What does this substitution mean geometrically? The complex variable $z$ can be any point in the complex plane. However, the expression $z=e^{j\omega }$ represents a complex number with a magnitude of 1 (since $|e^{j\omega }|=1$) and a phase angle of $\omega$. As the real-valued frequency $\omega$ varies from $0$ to $2\pi$, the point $z$ traces a circle of radius 1 centered at the origin of the z-plane. This is the unit circle.

Therefore, the DTFT is simply the z-transform evaluated on the unit circle.

The Importance of the Region of Convergence (ROC)

This relationship holds if and only if the z-transform converges on the unit circle. As you'll recall from our study of the z-transform, a transform is only defined within its Region of Convergence (ROC). This leads to a critical insight:

A discrete-time signal has a well-defined, convergent DTFT if and only if the ROC of its z-transform includes the unit circle.

This connects directly to system stability. For a causal LTI system, stability requires that the impulse response $h[n]$ be absolutely summable. This is precisely the condition for the existence of its DTFT. In the z-domain, this corresponds to the ROC of the system's transfer function $H(z)$ including the unit circle. This is why we check if all poles are inside the unit circle for a causal, stable system.

2. The Full Picture: From the z-Transform to the DFT

We now have the two key links in our chain of reasoning:

1. From Previous Lesson: The DFT is the DTFT sampled at $N$ discrete frequencies ${\omega }_k=\frac{2\pi }{N}k$.
2. From This Lesson: The DTFT is the z-transform evaluated on the unit circle, $z=e^{j\omega }$.

Combining these, we can conclude that the DFT must be the z-transform evaluated at $N$ specific points on the unit circle. Let's formalize this.

The DFT coefficients $X[k]$ are obtained by evaluating the DTFT at $\omega ={\omega }_k$:
$X[k]=X(e^{j\omega }){|}_{\omega =\frac{2\pi }{N}k}=X(e^{j\frac{2\pi }{N}k})$

Since $X(e^{j\omega })$ is just $X(z)$ with $z=e^{j\omega }$, this means:
$X[k]=X(z){|}_{z=e^{j\frac{2\pi }{N}k}}$

This is our final, unified relationship. The $N$-point DFT is simply the sequence of values you get by evaluating the signal's z-transform at the $N$ complex Nth-roots of unity, which are $N$ equally spaced points on the unit circle.

The following reading provides a clear, step-by-step development that formally connects these three transforms. It starts by showing why the DFT is insufficient for analyzing certain systems (like IIR filters), which motivates the z-transform, and then circles back to show how the DFT and DTFT are recovered from it.

3. A Visual Hierarchy

We can summarize these relationships with the following hierarchy, moving from the most general to the most specific and computationally practical:

1. The z-Transform: $X(z)$

   • Domain: The entire complex z-plane. It is a function of a complex variable $z$.
   • Concept: The most general frequency-domain representation for discrete-time signals and systems. Its ROC is crucial for determining properties like stability and causality. It can represent signals for which the DTFT does not converge.

2. The Discrete-Time Fourier Transform (DTFT): $X(e^{j\omega })$

   • Domain: A path along the unit circle in the z-plane. It is a continuous, periodic function of the real frequency variable $\omega$.
   • Relationship: It is a special case of the z-transform, obtained by setting $z=e^{j\omega }$.
   • Concept: The "true" theoretical spectrum of a discrete-time signal. It exists only if the z-transform's ROC includes the unit circle.

3. The Discrete Fourier Transform (DFT): $X[k]$

   • Domain: A set of $N$ discrete points on the unit circle in the z-plane. It is a discrete sequence of $N$ complex numbers.
   • Relationship: It is a special case of the DTFT (and therefore the z-transform), obtained by sampling at frequencies ${\omega }_k=\frac{2\pi }{N}k$, which corresponds to evaluating the z-transform at $z_k=e^{j\frac{2\pi }{N}k}$.
   • Concept: The practical, computable version of the Fourier transform for finite-length signals, forming the basis of FFT algorithms.

Conclusion

In this lesson, we have unified the three fundamental tools for discrete-time frequency analysis. You now have a clear mental model that organizes these concepts into a coherent hierarchy.

Key Takeaways:

• The z-transform is the most general representation, defined over the complex z-plane.
• The DTFT is a special case of the z-transform, corresponding to its evaluation on the unit circle ($z=e^{j\omega }$). It represents the continuous spectrum of a signal and exists only if the ROC includes the unit circle.
• The DFT is a special case of the DTFT, obtained by taking $N$ uniform samples. This is equivalent to evaluating the z-transform at $N$ equally spaced points on the unit circle ($z_k=e^{j\frac{2\pi }{N}k}$).

This unified view is not just an academic exercise; it provides a deep understanding of what you are actually computing when you use an FFT algorithm. You are taking a sparse sample of the signal's z-transform on the unit circle.

Preview of the Next Lesson:

Our realization that the DFT is a sampling of the true spectrum raises an important question: what are the practical consequences of this sampling? What happens if a signal's frequency components don't perfectly align with our DFT sample points? This leads to two important artifacts in practical spectral analysis. In our next lesson, we will analyze spectral leakage and the picket-fence effect in DFT-based spectral analysis, which are direct results of the finite-length, sampled nature of the DFT.