Signals and Systems

3.4 Convergence of the Fourier Series

Imron Rosyadi

Signals and Systems

3.4 Convergence of the Fourier Series

Welcome to this lecture on the convergence of the Fourier Series. Today, we’ll explore the conditions under which a periodic signal can be accurately represented by its Fourier series, and delve into some interesting phenomena that arise when approximating discontinuous signals. This is a critical topic for understanding the practical applications and limitations of Fourier analysis in ECE.

Approximating Signals: Euler, Lagrange, and Fourier

• Early mathematicians like Euler and Lagrange struggled with Fourier series for discontinuous signals.

  • Fourier, however, maintained that even discontinuous signals like square waves could be represented.

• We approximate a periodic signal \(x(t)\) using a finite sum of harmonically related complex exponentials: \[ x_{N}(t)=\sum_{k=-N}^{N} a_{k} e^{j k \omega_{0} t} \]

• The approximation error is defined as: \[ e_{N}(t)=x(t)-x_{N}(t) \]

Historically, the idea of representing a discontinuous function as a sum of continuous sinusoids was quite controversial. Euler and Lagrange, pillars of classical analysis, found this problematic. Fourier, with his more applied perspective, pushed the boundaries, asserting the validity of such representations for a much wider class of signals. This finite sum, \(x_N(t)\), is our practical tool for approximation, and understanding its behavior as N grows is key to convergence.

Minimizing Approximation Error

• To quantify the approximation’s quality, we use the energy in the error over one period: \[ E_{N}=\int_{T}\left|e_{N}(t)\right|^{2} d t \]

• Key Result: The coefficients \(a_k\) that minimize this energy \(E_N\) are precisely the Fourier series coefficients: \[ a_{k}=\frac{1}{T} \int_{T} x(t) e^{-j k \omega_{0} t} d t \]

Tip

This means that truncating the Fourier series provides the “best” approximation in a least-squares sense for a finite number of terms.

Minimizing the energy of the error is a common and powerful criterion in signal processing, often leading to optimal solutions. The fact that the Fourier series coefficients naturally emerge as the minimizers here is a fundamental and elegant result, strengthening the importance of the Fourier series. As we increase N, adding more terms, the energy in the error, E_N, will decrease. For a convergent series, E_N will approach zero as N approaches infinity.

When Does a Fourier Series Converge?

• Not all periodic signals have a valid Fourier series representation.
  • The integral for \(a_k\) might diverge, or the infinite series for \(x(t)\) might not converge to \(x(t)\).
• Fortunately, most practical ECE signals do have Fourier series representations.
• We’ll discuss two main classes of conditions guaranteeing convergence:
  1. Finite Energy over a Period (Square Integrability)
  2. Dirichlet Conditions (Point-wise Convergence)

While Fourier’s claim that any periodic signal could be represented was a bit ambitious, his intuition was largely correct for engineering applications. Pathological signals that don’t converge are rarely encountered in practice. Understanding these conditions helps us appreciate the robustness of Fourier analysis and its applicability to a vast range of real-world signals, including those with discontinuities.

Condition 1: Finite Energy (Square Integrability)

• A periodic signal \(x(t)\) has a Fourier series representation if it has finite energy over a single period: \[ \int_{T}|x(t)|^{2} d t<\infty \]
• Guarantees:
  • Coefficients \(a_k\) are finite.
  • The energy in the approximation error \(E_N\) converges to 0 as \(N \rightarrow \infty\). \[ \int_{T}|e(t)|^{2} d t=0 \quad \text{where} \quad e(t)=x(t)-\sum_{k=-\infty}^{+\infty} a_{k} e^{j k \omega_{0} t} \]

Important

This means \(x(t)\) and its infinite Fourier series representation are indistinguishable from an energy perspective, even if they differ at isolated points. Physical systems respond to signal energy.

This condition is perhaps the most fundamental for engineers because it directly relates to the physical reality of signals. If the difference between a signal and its Fourier series has zero energy, then any practical system responding to energy will treat them as identical. This is a very strong form of convergence for engineering purposes. Most signals we encounter, like square waves, triangle waves, and even pulse trains, satisfy this condition.

Condition 2: Dirichlet Conditions

• A set of conditions, developed by P. L. Dirichlet, that guarantee point-wise convergence.
• This means \(x(t)\) equals its Fourier series representation except at isolated discontinuities.
• At discontinuities, the series converges to the average of the values on either side.

While finite energy convergence is great for overall system response, sometimes we need to know that the series converges to the exact value of the signal at every point. This is where the Dirichlet conditions come in. They are more restrictive but provide a stronger guarantee about the specific values of the signal. Let’s look at each of these conditions.

Dirichlet Condition 1: Absolute Integrability

• Over any period, \(x(t)\) must be absolutely integrable: \[ \int_{T}|x(t)| d t<\infty \]
• Ensures: Each coefficient \(a_k\) will be finite. \[ \left|a_{k}\right| \leq \frac{1}{T} \int_{T}|x(t)| d t \]
• Violation Example: \(x(t) = 1/t\) for \(0 < t \leq 1\), periodic with \(T=1\).
  • The integral \(\int_0^1 (1/t) dt\) diverges (see Figure 3.8a in textbook).

This condition prevents signals from having “infinite area” over a period, which would make the Fourier coefficients infinite. The example \(x(t)=1/t\) blows up at \(t=0\), making its absolute integral infinite. Such signals are generally not physically realizable or encountered in typical ECE contexts.

Dirichlet Condition 2: Bounded Variation

• In any finite interval of time, \(x(t)\) must have a finite number of maxima and minima during any single period.
• Violation Example: \(x(t) = \sin(2\pi/t)\) for \(0 < t \leq 1\), periodic with \(T=1\).
  • This function has an infinite number of oscillations (maxima and minima) as \(t \rightarrow 0\) (see Figure 3.8b in textbook).

This condition essentially means the signal can’t oscillate infinitely fast or have an infinite number of “wiggles” within a finite time. The example given, \(\sin(2\pi/t)\), illustrates this perfectly. As t approaches zero, the argument of the sine function goes to infinity, causing infinitely many oscillations. Again, this is a highly pathological case not typically seen in practical signals.

Dirichlet Condition 3: Finite Discontinuities

• In any finite interval of time, there are only a finite number of discontinuities.
• Furthermore, each of these discontinuities must be finite.
• Violation Example: A signal with an infinite number of increasingly smaller sections, each introducing a discontinuity (see Figure 3.8c in textbook).

Note

Most physically generated or processed signals in ECE satisfy all three Dirichlet conditions.

This condition is about the “smoothness” of the signal in terms of its jumps. A signal cannot have infinitely many sudden jumps within a finite period. The example describes a signal that gets progressively more complex with an infinite number of steps, violating this condition. In engineering, we generally deal with signals that have a finite number of step changes or impulses within any given time frame.

Practical Implications of Dirichlet Conditions

For Signals Satisfying Dirichlet Conditions:

• Fourier series converges and equals \(x(t)\) everywhere except at isolated discontinuities.
• At discontinuities, the series converges to the average value of the signal on either side.

Why it matters for ECE:

• Signals differing only at isolated points have identical integrals.
• They behave identically under convolution.
• Therefore, they are considered equivalent for LTI system analysis.

This is a crucial takeaway for our field. Even with discontinuities, the Fourier series representation is incredibly robust. The differences are so localized that they don’t affect integral properties, which are fundamental to how LTI systems process signals (e.g., convolution involves integration). From a system’s perspective, these signals are effectively the same.

The Gibbs Phenomenon: An Introduction

• Discovered by Michelson (1898) and explained by Gibbs (1899).
• It describes the peculiar behavior of the Fourier series approximation near discontinuities.
• Observation: When approximating a discontinuous signal (like a square wave) with a finite Fourier series, ripples and overshoot occur at the discontinuities.

Michelson, a physicist, built a harmonic analyzer and was puzzled by the results when feeding it a square wave. He thought his machine was broken! He contacted Josiah Gibbs, a prominent mathematical physicist, who then provided the full explanation. This phenomenon highlights a key characteristic of approximating discontinuous functions with smooth sinusoids.

Visualizing the Gibbs Phenomenon

• Consider a symmetric square wave.
• We’ll observe its finite Fourier series approximation, \(x_N(t)\), as \(N\) increases.
• Notice the overshoot and ripples near the edges of the square wave.

viewof N = Inputs.range([1, 100], {step: 2, value: 5, label: "Number of terms (N)"})

Here we have an interactive plot. You can adjust the number of terms, N, using the slider. As you increase N, you’ll see the approximation get closer to the square wave. However, pay close attention to the regions around the discontinuities, where the signal jumps. You’ll notice persistent ripples and an overshoot, regardless of how large N becomes. This is the essence of the Gibbs phenomenon. This visualization directly relates to Figure 3.9 in your textbook.

Understanding the Gibbs Phenomenon

• Overshoot: For a discontinuity of unity height, the peak amplitude of the ripple is approximately 1.09 (an overshoot of ~9%).
  • This overshoot does not decrease with increasing \(N\).
• Ripples: As \(N\) increases, the ripples become compressed closer to the discontinuity.
  • The width of the ripples decreases, but their peak amplitude remains constant.
• Convergence at a point: For any fixed \(t_1\), \(x_N(t_1)\) will converge to \(x(t_1)\) as \(N \rightarrow \infty\) (or to the average at a discontinuity).
  • However, the closer \(t_1\) is to a discontinuity, the larger \(N\) must be for the error to be acceptably small.

It’s crucial to distinguish between point-wise convergence and the behavior of the peak overshoot. While the Fourier series does converge to the correct value at any given point (including the midpoint of a discontinuity), the maximum error, the overshoot, persists in the vicinity of the discontinuity. This means that even with millions of terms, you’ll still see that 9% overshoot, just squeezed into a smaller and smaller region around the jump. This is why it’s a “phenomenon.”

Practical Significance of Gibbs Phenomenon

• In real-world applications, if a truncated Fourier series \(x_N(t)\) is used to approximate a discontinuous signal:
  • Expect high-frequency ripples and overshoot near discontinuities.
• Engineering Consideration:
  • Choose a sufficiently large \(N\) so that the total energy in these ripples becomes insignificant.
  • While the peak overshoot remains, the energy contained within the shrinking ripple region diminishes.

Caution

For applications sensitive to peak values (e.g., driving an amplifier to saturation), the Gibbs phenomenon can be a critical design consideration.

The Gibbs phenomenon is not a failure of Fourier series, but rather a characteristic of approximating discontinuous signals with continuous functions. It means that if your system is sensitive to instantaneous peak values, you need to be aware that a Fourier series approximation will likely exceed the original signal’s amplitude at the edges. However, for most applications that depend on average power or integrated energy, the effect becomes negligible with enough terms, as the ripples become very narrow.

Conclusion: Convergence and Practicality

• Most practical signals in ECE have Fourier series representations.
  • Guaranteed by finite energy or Dirichlet conditions.
• Fourier series provides the best least-squares approximation for a finite number of terms.
• The Gibbs Phenomenon highlights a limitation for discontinuous signals:
  • Persistent overshoot and ripples near discontinuities, even as \(N \rightarrow \infty\).
  • Ripples compress, but peak amplitude remains constant (for a given discontinuity height).
• Despite Gibbs, Fourier series remains an indispensable tool for analyzing and synthesizing signals in ECE.

To wrap up, the Fourier series is an incredibly powerful tool in signals and systems. It allows us to decompose complex periodic signals into simpler, harmonically related components. While convergence is generally guaranteed for signals we encounter in practice, understanding the nuances like the Gibbs phenomenon is key to applying these tools effectively and interpreting their results correctly in engineering contexts. This knowledge helps us design better filters, understand system responses, and accurately model signal behavior.