Dirichlet Conditions and Fourier Series Convergence

Hello! Welcome to your third lesson.

In our last session, we derived the powerful complex exponential form of the Fourier series. We now have two fundamental equations: one to synthesize a signal from its complex exponential components, and one to analyze a signal to find those components.

Synthesis: f (t) = n = - \infty \sum \infty c_{n} e^{j n ω_{0} t} Analysis: c_{n} = \frac{1}{T} \int_{T} f (t) e^{- j n ω_{0} t} d t

However, our derivation rested on a big assumption: that any periodic function can be represented by this infinite sum. This was a point of major debate in the early 19th century. Joseph Fourier boldly claimed it was true for any function, but mathematicians like Lagrange were skeptical, believing it could only work for continuous functions.

Today, we will address this question directly. Our goal is to analyze the convergence of the Fourier series. We will explore the set of sufficient conditions, laid out by the German mathematician Peter Dirichlet, that guarantee a function's Fourier series representation will converge. Understanding these conditions is crucial because it tells us the rules of the road for applying Fourier series to the types of signals you'll encounter in engineering, which often include sharp edges and discontinuities.

1. The Conditions for Convergence

What properties must a function have for us to be confident its Fourier series exists and converges? Dirichlet established a set of conditions that are sufficient for this. While they aren't strictly necessary in all theoretical cases, they cover virtually every signal that can be physically generated or processed.

A great, concise overview of these conditions is provided in the following video.

Electrical Engineering: Ch 18: Fourier Series (10 of 35) The Dirichlet Conditions

This short video from Michel van Biezen's Electrical Engineering series clearly lists the four conditions commonly referred to as the Dirichlet conditions.

Watch the entire video (it's less than 2 minutes). Pay attention to the four distinct requirements for a function f(t).

Let's formalize and expand upon the points from the video. The conditions can be grouped into what are often called the "weak" and "strong" Dirichlet conditions.

The Dirichlet Conditions for Fourier Series Convergence — A summary of the Dirichlet conditions for the existence and convergence of a Fourier Series. These conditions ensure a periodic function is 'well-behaved' enough for the series to be a valid representation.

Weak Dirichlet Condition:
The function $f (t)$ must be absolutely integrable over one period.

\int_{T} ∣ f (t) ∣ d t < \infty

This is the most fundamental requirement. Why? It guarantees that the Fourier coefficients $c_{n}$ are finite. If the coefficients could be infinite, the series representation would be meaningless. We can quickly prove this by taking the magnitude of the analysis equation:

∣ c_{n} ∣ = \frac{1}{T} \int_{T} f (t) e^{- j n ω_{0} t} d t \leq \frac{1}{T} \int_{T} ∣ f (t) ∣∣ e^{- j n ω_{0} t} ∣ d t

Since $∣ e^{- j n ω_{0} t} ∣ = 1$ , we have:

∣ c_{n} ∣ \leq \frac{1}{T} \int_{T} ∣ f (t) ∣ d t

If the integral is finite, then every coefficient $∣ c_{n} ∣$ must also be finite. A function like $f (t) = 1/ t$ on the interval $[0, T]$ would violate this condition.

Strong Dirichlet Conditions:
These two conditions, in addition to the weak condition, ensure that the series converges pointwise (a concept we'll explore next). Within any one period, the function $f (t)$ must have:

A finite number of maxima and minima. This rules out functions that oscillate infinitely fast, like $sin (1/ t)$ near $t = 0$ .
A finite number of finite discontinuities. This means the function can have jumps (like a square wave), but it can't have an infinite number of them, nor can the jumps be of infinite magnitude.

Almost every signal you'll work with in signal processing—square waves, sawtooth waves, rectified sine waves, etc.—satisfies these conditions.

2. What "Convergence" Means: Two Perspectives

The question "does the series converge?" is more subtle than it first appears. It's important to distinguish between two types of convergence, which this next reading explains well.

Convergence of Fourier Series

This article from LibreTexts, 'Convergence of Fourier Series,' provides the mathematical distinction between energy convergence and pointwise convergence. It also touches on the historical Fourier-Lagrange debate.

Please read the 'Introduction' and the main 'Dirichlet Conditions' sections. Focus on understanding the difference between the two approaches to convergence discussed and how the Dirichlet conditions relate to them.

Let's summarize the two key ideas from the reading. Let $f_{N} (t)$ be the partial sum of the Fourier series up to the $N$ -th harmonic:

f_{N} (t) = n = - N \sum N c_{n} e^{j n ω_{0} t}

Energy Convergence (Convergence in $L^{2}$ ): This means the energy of the error between the true function and its $N$ -term approximation approaches zero as $N$ goes to infinity.
$N \to \infty lim \int_{T} ∣ f (t) - f_{N} (t) ∣^{2} d t = 0$
This is a very practical type of convergence for engineering, as many systems are defined by their response to signal energy. The weak Dirichlet condition is sufficient to guarantee energy convergence.
Pointwise Convergence: This is the more intuitive idea. Does the value of the approximation $f_{N} (t)$ approach the value of the function $f (t)$ at every single point $t$ ?
$N \to \infty lim f_{N} (t) = f (t) (at a specific value of t)$
This is not guaranteed by energy convergence alone. For pointwise convergence, we need the strong Dirichlet conditions.

The answer to Fourier's original problem lies here: even for discontinuous functions, the series converges in the energy sense. And with the strong Dirichlet conditions met, it also converges pointwise, but with a special behavior at the discontinuities themselves.

3. The Fourier Convergence Theorem

So, if a function satisfies the Dirichlet conditions, what value does its Fourier series converge to at any given point? The answer depends on whether the function is continuous at that point.

The following video provides an excellent explanation and visualization of this principle, known as the Fourier Convergence Theorem.

Fourier Series - the Fourier Convergence Theorem

This video by Sarah Schott explains the Fourier Convergence Theorem, clearly demonstrating what happens at points of continuity versus points of discontinuity. It also shows a powerful application of this theorem.

Please watch from the beginning to 04:56. The video covers the statement of the theorem and then applies it to graph the Fourier series for a discontinuous function. Notice how the series' graph differs from the original function's graph precisely at the jumps.

As the video explained, the convergence behavior is split into two cases:

Case 1: At a point of continuity. If $f (t)$ is continuous at a point $t_{0}$ , the Fourier series converges to the value of the function itself:
$lim_{N \to \infty} f_{N} (t_{0}) = f (t_{0})$
Case 2: At a point of discontinuity. If $f (t)$ has a finite jump discontinuity at a point $t_{0}$ , the Fourier series does not converge to either value. Instead, it converges to the arithmetic mean (the average) of the values on either side of the jump.
$lim_{N \to \infty} f_{N} (t_{0}) = \frac{f ( t _{0}^{-} ) + f ( t _{0}^{+} )}{2}$
where $f (t_{0}^{-})$ is the limit as $t$ approaches $t_{0}$ from the left, and $f (t_{0}^{+})$ is the limit from the right. For a typical square wave that jumps from -1 to +1 at $t = 0$ , the series will converge to $\frac{- 1 + 1}{2} = 0$ right at the jump.

Test your understanding!

Consider a periodic sawtooth wave defined over one period $[- π, π]$ by $f (t) = t$ . This function is then repeated with period $T = 2 π$ .

Does this function satisfy the Dirichlet conditions?
At $t = π /2$ , what value will its Fourier series converge to?
The function has jump discontinuities at $t = π, 3 π, \dots$ and $t = - π, - 3 π, \dots$ . What value will the Fourier series converge to at $t = π$ ?

Show answer

Yes. It is absolutely integrable ( $\int_{- π}^{π} ∣ t ∣ d t = π^{2} < \infty$ ), has no internal maxima or minima (only at the endpoints), and has one finite discontinuity per period (the jump from $π$ down to $- π$ ).
At $t = π /2$ , the function is continuous. The value is $f (π /2) = π /2$ . The series will converge to $π /2$ .
At $t = π$ , there is a jump. The limit from the left is $f (π^{-}) = π$ . The limit from the right is the start of the next cycle, so $f (π^{+}) = - π$ . The series will converge to the average: $\frac{π + ( - π )}{2} = 0$ .

4. A Closer Look: The Gibbs Phenomenon

Knowing the series converges to the midpoint at a jump is one thing, but how the partial sums behave near that jump is another fascinating story. The convergence is not a gentle settling. Instead, the partial sums exhibit an overshoot that does not go away.

Gibbs Phenomenon in Fourier Series Approximation — Fourier series approximation of a discontinuous function for an increasing number of terms (M in the plot). Note the persistent overshoots at the discontinuities. This is the Gibbs Phenomenon.

This behavior is called the Gibbs Phenomenon. As you add more and more terms to the Fourier series partial sum ( $f_{N} (t)$ ):

The approximation gets better and better across the continuous parts of the function.
Near the discontinuity, the approximation develops a sharp overshoot, followed by rapid ringing.
As $N \to \infty$ , this overshoot does not die down in amplitude. It always overshoots the jump by about 9% of the total jump height.
What does change is that the region containing the overshoot gets squeezed infinitely close to the point of discontinuity.

So, while the series converges pointwise to the midpoint right at the jump, the function representing the partial sum never becomes a perfect replica. This overshoot is a fundamental property of representing a discontinuity with a sum of continuous sinusoids. It's an important effect to be aware of in signal processing, as it is conceptually related to the "ringing" artifacts you see from sharp filters.

Conclusion

Today we've added a crucial layer of rigor to our understanding of Fourier series. We've moved from simply using the formulas to understanding when and how they are valid.

Key Takeaways:

The Dirichlet Conditions (absolutely integrable, finite maxima/minima, finite number of finite discontinuities) are a set of sufficient criteria ensuring a Fourier series will converge.
At points where the function is continuous, the series converges to the function's value.
At jump discontinuities, the series converges to the average of the values on either side of the jump.
The Gibbs Phenomenon describes the characteristic overshoot and ringing of the Fourier series partial sums near a discontinuity, which persists even as the number of terms approaches infinity.

Preview of the Next Lesson:
Our entire discussion so far has been about periodic signals. But what about signals that are not repetitive, like a single rectangular pulse or a decaying exponential? To handle these non-periodic signals, we need to extend our framework. In the next lesson, we will do this by considering a periodic signal and letting its period $T$ go to infinity. This limiting process will transform the discrete Fourier series into the continuous Fourier Transform, a cornerstone of modern signal processing.

Recommended resources

1. The Conditions for Convergence

2. What "Convergence" Means: Two Perspectives

3. The Fourier Convergence Theorem

4. A Closer Look: The Gibbs Phenomenon

Conclusion