Signals and Systems

10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot

Imron Rosyadi

10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot

In previous sections, we learned about the Z-transform, its ROC, and how to find the inverse transform. Now, we’re going to connect the Z-transform back to the frequency domain in a very intuitive way: geometrically evaluating the Fourier Transform from the pole-zero plot. This method offers great insight into the frequency response characteristics of discrete-time systems.

Connection to the Fourier Transform

Recall:

• The Z-transform \(X(z)\) reduces to the Discrete-Time Fourier Transform (DTFT) \(X(e^{j\omega})\) when \(z\) is evaluated on the unit circle (\(|z|=1\)).
• This is valid if and only if the ROC of \(X(z)\) includes the unit circle, ensuring the DTFT converges.

Geometric Evaluation:

• Similar to the continuous-time case (Laplace to Fourier on \(j\omega\)-axis), we can geometrically evaluate the DTFT from the pole-zero plot.
• Instead of evaluating on the imaginary axis, we evaluate on the unit circle in the \(z\)-plane.
• We consider vectors from poles and zeros to points on the unit circle.

Connection to the Fourier Transform

General Form of Rational \(H(z)\): For a rational system function \(H(z)\), it can be written as: \[ H(z) = M \frac{\prod_{k=1}^M (z-z_k)}{\prod_{l=1}^N (z-p_l)} \] Where \(z_k\) are the zeros and \(p_l\) are the poles.

When evaluated on the unit circle, \(z=e^{j\omega}\): \[ H(e^{j\omega}) = M \frac{\prod_{k=1}^M (e^{j\omega}-z_k)}{\prod_{l=1}^N (e^{j\omega}-p_l)} \]

• The magnitude \(|H(e^{j\omega})|\) is the product of lengths of zero vectors divided by the product of lengths of pole vectors (scaled by \(|M|\)).
• The phase \(\angle H(e^{j\omega})\) is the sum of angles of zero vectors minus the sum of angles of pole vectors (plus \(\angle M\)).

Let’s start by recalling the fundamental link: the DTFT is the Z-transform evaluated on the unit circle. This means that if we want to understand the frequency response of a system, we look at how its Z-transform behaves along the unit circle.

The geometric evaluation method provides a visual and intuitive way to do this. For any rational system function, we can express it in terms of its poles and zeros. The magnitude and phase of the frequency response at any given frequency \(\omega\) can then be determined by the lengths and angles of vectors drawn from each pole and zero to the point \(e^{j\omega}\) on the unit circle.

10.4.1 First-Order Systems

Impulse Response: \[ h[n]=a^{n} u[n] \] Z-Transform: \[ H(z)=\frac{1}{1-a z^{-1}}=\frac{z}{z-a}, \quad|z|>|a| \] For the DTFT to converge, we require \(|a|<1\), so the ROC includes the unit circle.

Frequency Response: \[ H(e^{j\omega})=\frac{1}{1-a e^{-j \omega}} \] Pole-Zero Plot: - Pole: at \(z=a\) - Zero: at \(z=0\)

Geometric Evaluation:

• \(|\mathbf{v}_1|\) (from zero at origin to \(e^{j\omega}\)) has constant length 1.
• \(\angle \mathbf{v}_1 = \omega\).
• \(|\mathbf{v}_2|\) (from pole at \(a\) to \(e^{j\omega}\)) changes with \(\omega\). Its minimum length occurs when \(e^{j\omega}\) is closest to the pole, i.e., at \(\omega=0\) if \(a\) is real and positive.

Let’s consider a simple first-order system. The Z-transform has a pole at \(z=a\) and a zero at \(z=0\). For the DTFT to exist, the pole must be inside the unit circle, meaning \(|a|<1\).

To evaluate the frequency response \(H(e^{j\omega})\), we draw vectors from the pole and zero to a point \(e^{j\omega}\) on the unit circle. The zero at the origin always contributes a constant magnitude of 1 and a phase equal to \(\omega\). The pole’s contribution varies as we move around the unit circle. When \(e^{j\omega}\) is close to the pole, the vector length from the pole is small, leading to a peak in the magnitude response.

First-Order System: Magnitude & Phase Response

Magnitude Response:

• \(|H(e^{j\omega})| = \frac{|e^{j\omega}|}{|e^{j\omega}-a|} = \frac{1}{|e^{j\omega}-a|}\)
• When \(a\) is real and \(0<a<1\):
  • Minimum pole vector length occurs at \(\omega=0\) (where \(e^{j\omega}=1\)).
  • Maximum \(|H(e^{j\omega})|\) at \(\omega=0\).
  • Monotonically decreases as \(\omega\) increases from \(0\) to \(\pi\).

Phase Response:

• \(\angle H(e^{j\omega}) = \angle(e^{j\omega}) - \angle(e^{j\omega}-a) = \omega - \angle(e^{j\omega}-a)\)
• Starts at \(0\) for \(\omega=0\).
• Changes monotonically as \(\omega\) increases from \(0\) to \(\pi\).

First-Order System: Magnitude & Phase Response

viewof a_val_first_order = Inputs.range([-0.9, 0.9], {value: 0.5, step: 0.1, label: "Pole location 'a' (-0.9 to 0.9)"});

This interactive plot lets you see how the magnitude and phase response of a first-order system change as you move the pole location ‘a’.

Observe:

• When ‘a’ is positive, the magnitude peaks at \(\omega=0\) (DC). As ‘a’ approaches 1, the peak becomes sharper.
• When ‘a’ is negative, the magnitude peaks at \(\omega=\pi\) (Nyquist frequency). As ‘a’ approaches -1, the peak becomes sharper.
• The phase response shows a smooth transition across frequencies.

The closer the pole is to the unit circle, the more pronounced its effect on the frequency response. This is a fundamental principle in filter design.

10.4.2 Second-Order Systems

Impulse Response: (for \(0<r<1, 0 \le \theta \le \pi\)) \[ h[n]=r^{n} \frac{\sin (n+1) \theta}{\sin \theta} u[n] \] Frequency Response: \[ H(e^{j\omega})=\frac{1}{1-2 r \cos \theta e^{-j \omega}+r^{2} e^{-j 2 \omega}} \] System Function \(H(z)\): \[ H(z)=\frac{1}{1-(2 r \cos \theta) z^{-1}+r^{2} z^{-2}} \] Poles:

• \(z_1 = r e^{j\theta}\)
• \(z_2 = r e^{-j\theta}\) (complex conjugate pair)
• These poles are at radius \(r\) and angles \(\pm \theta\).

Zeros: Double zero at \(z=0\).

Geometric Evaluation:

• Since there is a double zero at the origin, \(|\mathbf{v}_1|^2 = 1^2 = 1\).
• Magnitude peaks when \(e^{j\omega}\) is close to either pole.
• The peak occurs near \(\omega = \theta\) (angle of the poles).

Second-order systems are crucial because they can exhibit resonant behavior. Their Z-transforms typically have a pair of complex conjugate poles. For stability, these poles must be inside the unit circle (\(r<1\)).

The geometric evaluation here involves three vectors: one from the double zero at the origin, and one from each complex pole. The zero at the origin still contributes a constant magnitude of 1. The magnitude response will have peaks at frequencies close to the angles of the poles. This is because when \(e^{j\omega}\) aligns with the angle of a pole, the distance from that pole to \(e^{j\omega}\) becomes minimal, leading to a large magnitude.

Second-Order System: Magnitude & Phase Response

Magnitude Response:

• \(|H(e^{j\omega})| = \frac{1}{|e^{j\omega}-re^{j\theta}||e^{j\omega}-re^{-j\theta}|}\)
• Peaks occur when \(e^{j\omega}\) is closest to the poles, i.e., when \(\omega \approx \pm \theta\).
• As \(r \rightarrow 1\), the poles move closer to the unit circle, making the peaks sharper and higher (more resonant).

Phase Response:

• \(\angle H(e^{j\omega}) = 2\omega - (\angle(e^{j\omega}-re^{j\theta}) + \angle(e^{j\omega}-re^{-j\theta}))\)
• Exhibits a sharp change around the peak frequency \(\theta\).

Second-Order System: Magnitude & Phase Response

viewof r_val_second_order = Inputs.range([0.1, 0.99], {value: 0.95, step: 0.01, label: "Pole Radius 'r' (0.1 to 0.99)"});
viewof theta_val_second_order = Inputs.range([0, 3.14], {value: 3.14/4, step: 3.14/100, label: "Pole Angle 'θ' (0 to π)"});

This interactive plot for a second-order system allows you to manipulate the pole radius ‘r’ and angle ‘theta’.

• Radius ‘r’: As ‘r’ approaches 1, the poles move closer to the unit circle. This results in a sharper and higher peak in the magnitude response, indicating a more resonant filter. This is a key concept in designing band-pass or band-stop filters.
• Angle ‘θ’: The pole angle ‘θ’ directly controls the center frequency of this peak. If ‘θ’ is small, the peak is near DC. If ‘θ’ is near \(\pi\), the peak is near the Nyquist frequency.

Observe how the phase response also changes dramatically around the resonant frequency, exhibiting a sharp transition.

Pole-Zero Placement and Frequency Response

Impact of Pole Location:

• Distance from origin (\(r\)):
  • Poles closer to the unit circle (\(r \approx 1\)) lead to sharper peaks (resonance) in the magnitude response.
  • Poles closer to the origin (\(r \approx 0\)) lead to flatter frequency responses and faster decaying impulse responses.
• Angle (\(\theta\)):
  • The angle of the pole determines the frequency at which the magnitude response will peak.
  • Poles on the positive real axis (\(\theta=0\)) affect DC (\(\omega=0\)).
  • Poles on the negative real axis (\(\theta=\pi\)) affect Nyquist frequency (\(\omega=\pi\)).
  • Poles at other angles affect intermediate frequencies.

Impact of Zero Location:

• Zeros on the unit circle (\(|z_k|=1\)) create “notches” or “nulls” in the magnitude response, where \(|H(e^{j\omega})|\) goes to zero. These are useful for creating band-stop filters.
• Zeros closer to the origin reduce the overall magnitude response in certain frequency ranges.

Tip

Filter Design Principle: By strategically placing poles and zeros in the \(z\)-plane, we can shape the frequency response of a discrete-time filter to achieve desired characteristics (e.g., low-pass, high-pass, band-pass, band-stop).

This slide summarizes the direct relationship between pole-zero locations and the resulting frequency response. This geometric understanding is incredibly powerful for filter design.

• Poles attract the magnitude response: close poles mean high magnitude.
• Zeros repel the magnitude response: close zeros mean low (or zero) magnitude.

By manipulating ‘r’ and ‘theta’ of poles and zeros, we can design filters to pass or reject specific frequency bands. This is the essence of digital filter design, a core application of the Z-transform.

Conclusion: Geometric Evaluation

Key Takeaways:

• The DTFT is the Z-transform evaluated on the unit circle (\(|z|=1\)).
• Geometric evaluation provides an intuitive way to understand frequency response.
• \(|H(e^{j\omega})|\) is proportional to the product of zero vector lengths divided by pole vector lengths.
• \(\angle H(e^{j\omega})\) is the sum of zero vector angles minus the sum of pole vector angles.

Applications in ECE:

• Filter Design: Directly visualize and understand how pole/zero placement shapes the frequency response of digital filters (e.g., low-pass, high-pass, band-pass, notch filters).
• System Analysis: Gain insight into system behavior, such as resonance (poles near unit circle) or nulls (zeros on unit circle).
• Stability: A stable LTI system has all its poles strictly inside the unit circle. This is visually evident in the pole-zero plot.

Important

The pole-zero plot is arguably the most insightful representation of a discrete-time system, providing a holistic view of its stability, causality, and frequency response characteristics.

To conclude, the geometric evaluation of the Fourier transform from the pole-zero plot is not just a mathematical exercise; it’s a fundamental design tool. It allows engineers to intuitively understand and manipulate the characteristics of discrete-time systems and filters. Whether you’re designing an audio equalizer, a control system, or a communication receiver, a strong grasp of pole-zero placement is invaluable.