Complex Fourier Series Derivation

Hello! Welcome to your second lesson.

In our first lesson, we traced the historical journey of Fourier analysis, from a physics problem about heat to a revolutionary way of representing functions. We saw that Fourier's core idea was to build any periodic function from a sum of simple sine and cosine waves. The key to finding the "amount" of each wave needed—the coefficients $a_{n}$ and $b_{n}$ —was a mathematical property called orthogonality.

Today, we will formalize this idea and take it a step further. We'll move from the sine and cosine representation to the more elegant and powerful complex exponential Fourier series. Your background in electrical engineering, particularly with phasors and complex numbers for circuit analysis, will make this a very natural transition.

Our goal is to derive the formula for the complex Fourier series coefficients directly from the principle of orthogonality. This will not only give us a more compact notation but also simplify many of the proofs and applications we will encounter later.

1. From Real to Complex: A More Compact Form

Let's start with the real-valued Fourier series for a function $f (t)$ with period $T$ . Using the angular frequency $ω_{0} = 2 π / T$ , we can write the series as:

f (t) = \frac{a _{0}}{2} + n = 1 \sum \infty (a_{n} cos (n ω_{0} t) + b_{n} sin (n ω_{0} t))

This form is correct, but it requires two sets of coefficients, $a_{n}$ and $b_{n}$ . We can simplify this. Recall Euler's formula and the corresponding expressions for sine and cosine:

cos (θ) = \frac{e ^{j θ} + e ^{- j θ}}{2} and sin (θ) = \frac{e ^{j θ} - e ^{- j θ}}{2 j}

Substituting these into the Fourier series and rearranging the terms allows us to combine the sine and cosine components into a single sum of complex exponentials. The process involves some algebra to group terms with positive and negative exponents.

To see this algebraic manipulation performed step-by-step, the following video is quite clear.

Complex Fourier Series (fourier series engineering mathematics)

This video from the YouTube channel blackpenredpen shows how to substitute the sine and cosine terms with their complex exponential equivalents and regroup them into the standard complex Fourier series form.

Watch from 01:00 to 08:43. You don't need to memorize every step, but focus on the overall strategy: replace cosines and sines, collect terms for e^{jn\omega_0t} and e^{-jn\omega_0t}, and then cleverly combine the summations into one that runs from n=-\infty to +\infty.

After the algebraic rearrangement, we arrive at the elegant complex exponential form of the Fourier series:

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

Here, the coefficients $c_{n}$ are complex numbers that contain the amplitude and phase information for each frequency component $n ω_{0}$ . This single summation replaces the separate DC, cosine, and sine terms.

2. The Principle of Orthogonality for Complex Exponentials

The algebraic conversion we just saw shows that the real and complex forms are equivalent. However, to truly understand the complex Fourier series, it's best to derive it from first principles, just as Fourier did with sines and cosines. The foundation for this is the orthogonality of complex exponential functions.

In linear algebra, you know that two vectors are orthogonal if their dot product is zero. We can extend this concept to functions. We define an inner product of two complex functions, $g (t)$ and $h (t)$ , over an interval of one period $T$ as:

⟨ g (t), h (t)⟩ = \int_{T} g (t) h^{*} (t) d t

where $h^{*} (t)$ is the complex conjugate of $h (t)$ . Two functions are said to be orthogonal over the interval $T$ if their inner product is zero.

The complex Fourier series is built upon the set of basis functions ${ϕ_{n} (t) = e^{j n ω_{0} t}}$ for all integers $n$ . Let's prove that these basis functions are orthogonal.

Fourier Analysis

This document, 'Fourier Analysis' by J.A. Peacock, provides a clear, mathematical derivation of the orthogonality relation for complex exponentials. It's a foundational proof for everything that follows.

Please read Section 2.8, 'Complex Fourier Series', focusing on the derivation in and around Equation (2.46). This shows the calculation of the integral of the product of two basis functions, \phi_m(x) and \phi_n^*(x). Notice how the result is different for the cases when n=m and (n eq m).

As you saw in the reading, the inner product of two of our basis functions, $ϕ_{n} (t)$ and $ϕ_{m} (t)$ , is:

⟨ ϕ_{n} (t), ϕ_{m} (t)⟩ = \int_{0}^{T} e^{j n ω_{0} t} (e^{j m ω_{0} t})^{*} d t = \int_{0}^{T} e^{j (n - m) ω_{0} t} d t

This integral has two possible outcomes:

If $n \neq = m$ : The integral of the complex exponential over a whole number of its periods is zero. $\int_{0}^{T} e^{j (n - m) ω_{0} t} d t = [\frac{e ^{j (n - m) ω_{0} t}}{j ( n - m ) ω _{0}}]_{0}^{T} = \frac{e ^{j (n - m) 2 π} - e ^{0}}{j ( n - m ) ω _{0}} = \frac{1 - 1}{j ( n - m ) ω _{0}} = 0$
If $n = m$ : The exponent becomes zero, and the integrand is just $e^{0} = 1$ . $\int_{0}^{T} 1 d t = T$

We can summarize this fundamental result using the Kronecker delta, $δ_{nm}$ :

\int_{T} e^{j (n - m) ω_{0} t} d t = T δ_{nm}

This is the orthogonality condition for the complex exponential basis functions. It's the mathematical tool that allows us to isolate and calculate each Fourier coefficient.

3. Deriving the Fourier Coefficients

Now we can use the orthogonality property to find the formula for any coefficient $c_{n}$ . This is the central derivation of today's lesson. We start by assuming that our function $f (t)$ can be represented by the series:

f (t) = k = - \infty \sum \infty c_{k} e^{j k ω_{0} t}

(Here I've used $k$ as the summation index to avoid confusion with the specific index $n$ we want to find).

To isolate a single coefficient, say $c_{n}$ , we'll use the "sifting" property of orthogonal functions. The following video walks through this derivation beautifully.

Fourier Series: Complex Version! Part 1

This video by jen foxbot demonstrates the derivation of the Fourier series coefficients using the average value function and the orthogonality principle. It's a very clear, step-by-step approach.

Please watch from 08:44 to 16:05. The video first finds the DC coefficient C_0 as a special case, which is simply the average value of the function. Then, it derives the general formula for any coefficient C_n by multiplying by the complex conjugate exponential and integrating. Pay close attention to how the orthogonality principle makes all other terms in the infinite sum vanish.

Let's re-create the steps from the video with our notation.

Select the coefficient to find: We want to find $c_{n}$ .
Multiply by the conjugate basis function: We multiply both sides of the series equation by $ϕ_{n}^{*} (t) = e^{- j n ω_{0} t}$ : $f (t) e^{- j n ω_{0} t} = (k = - \infty \sum \infty c_{k} e^{j k ω_{0} t}) e^{- j n ω_{0} t} = k = - \infty \sum \infty c_{k} e^{j (k - n) ω_{0} t}$
Integrate over one period: Now, we integrate both sides over a full period $T$ . $\int_{0}^{T} f (t) e^{- j n ω_{0} t} d t = \int_{0}^{T} k = - \infty \sum \infty c_{k} e^{j (k - n) ω_{0} t} d t$
Swap the summation and integral: Assuming the series converges nicely (a topic for our next lesson), we can swap the operations: $\int_{0}^{T} f (t) e^{- j n ω_{0} t} d t = k = - \infty \sum \infty c_{k} (\int_{0}^{T} e^{j (k - n) ω_{0} t} d t)$
Apply the orthogonality property: The integral on the right is $T δ_{k n}$ . This means it is $0$ for every value of $k$ except when $k = n$ , in which case it is $T$ . The infinite sum collapses to a single term! $\int_{0}^{T} f (t) e^{- j n ω_{0} t} d t = c_{n} \cdot T$
Solve for $c_{n}$ : Finally, we isolate $c_{n}$ by dividing by $T$ . $c_{n} = \frac{1}{T} \int_{0}^{T} f (t) e^{- j n ω_{0} t} d t$

This is the fundamental formula for calculating the complex Fourier series coefficients.

Test your understanding!

In the derivation, we multiplied the series by $e^{- j n ω_{0} t}$ . What would have happened if we had multiplied by $e^{j n ω_{0} t}$ instead? Would the "sifting" still work?

Show answer

No, it would not have worked as cleanly. If we multiplied by $e^{j n ω_{0} t}$ , the exponent inside the integral would become $e^{j (k + n) ω_{0} t}$ . Due to orthogonality, this integral is zero unless $k + n = 0$ , or $k = - n$ . So, instead of isolating the coefficient $c_{n}$ , the process would have isolated the coefficient $c_{- n}$ . Multiplying by the complex conjugate $e^{- j n ω_{0} t}$ is what ensures the exponents subtract ( $k - n$ ), leading to the crucial condition $k = n$ for a non-zero result.

Conclusion

We have now established the pair of equations that form the foundation of Fourier series analysis in its most common form.

The Synthesis Equation (building the signal): $f (t) = n = - \infty \sum \infty c_{n} e^{j n ω_{0} t}$
The Analysis Equation (finding the components): $c_{n} = \frac{1}{T} \int_{T} f (t) e^{- j n ω_{0} t} d t$

The analysis equation projects the function $f (t)$ onto each basis function to find the component $c_{n}$ . The synthesis equation then reconstructs the original function by summing up all the scaled basis functions.

Key Takeaways:

The real (sine/cosine) Fourier series can be algebraically converted into the more compact complex exponential form.
The set of functions ${e^{j n ω_{0} t}}$ for integer $n$ forms an orthogonal basis set over any interval of length $T = 2 π / ω_{0}$ .
This orthogonality allows us to derive the analysis equation, which isolates any Fourier coefficient $c_{n}$ by multiplying the signal by the corresponding conjugate basis function and integrating over one period.

Preview of the Next Lesson:
We've made a powerful assumption in this lesson: that our periodic function can be represented by this infinite series. Fourier's contemporaries were highly skeptical of this, especially for functions with discontinuities. In the next lesson, we will investigate the conditions under which this assumption is valid. We will study the Dirichlet conditions, which formally state the requirements for a function's Fourier series to converge.

Recommended resources

1. From Real to Complex: A More Compact Form

2. The Principle of Orthogonality for Complex Exponentials

3. Deriving the Fourier Coefficients

Conclusion