Much like Sir William Henry Bragg stated, often, the recipe for new discovery entails new light, so elements can be viewed in a fresh perspective.
This week on Tim’s Blackboard, I will start with the motivation for Fourier to introduce his series, follow by his unintentionally visit to the frequency domain, and end with how the new frequency domain view helps us understand the root cause of eye closure.
Fourier and “The Analytic Theory of Heat”
Whenever frequency domain is in a conversation, there is no escape from mentioning the name of this famous mathematician and physicist: Joseph Fourier.
Fig. 1: Jean-Baptiste Joseph Fourier. Image credit: https://commons.wikimedia.org
Although well-known for Fourier Series and Transform, in his 1800’s publications, the French-born scientist in Fig. 1 was originally analyzing heat flow.
To solve the heat equation in a metal plate, Fourier had the idea to decompose a complicated heat source as a linear combination of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigen-solutions. Nowadays, this superposition or linear combination is known as the Fourier Series .
Fourier Series: Unfamiliar Yet Familiar
Although trying to represent a complicated function with linear combinations of sine and cosines might sound foreign, the decomposition of a complicated element into simpler sub-elements is a familiar idea.
In his lecture on Fourier Series, MIT Professor Dennis Freeman cleverly illustrates the similarity between Fourier Series and the Cartesian representation of an arbitrary vector in 3D-space .
Fig. 2: An arbitrary vector in 3D-space.
Shown in Fig. 2 is an arbitrary vector in 3D-space. Without additional coordinate information, our view of the vector is a geometric one: a line. However, as soon as we place the vector in a coordinate system, the vector geometry translates to vector magnitudes and directions. In a Cartesian system, there are three different components: one in x-direction, one in y-direction and one in the z-direction, as demonstrated in Fig. 3.
Fig. 3: Representation of an arbitrary vector in 3D-space in Cartesian coordinates. The original vector is separated into three components of various magnitudes in different directions.
The concept of Fourier Series is extremely similar. In Fourier Series, one deconstructs a periodic function into sines and cosines of different frequencies. The different frequencies of cosines and sines are analogous to the different directions in the Cartesian coordinates.
Take a classic ideal square wave for example. Fig. 4 shows the comparison between representing a vector in 3D-space and expressing a square wave with Fourier Series.
Fig. 4: Comparison between a vector in 3D-space and an ideal square wave expressed in Fourier Series. The sine waves of different frequencies correspond to the different directions in Cartesian coordinate system.
It is important to note in Fig. 4, the “…” in the Fourier Series expression indicates an infinite sum of sines with only odd harmonics. Mathematically, we write
Unlike the vector in 3D-space, where only three magnitudes and directions are needed to recreate the vector, we need infinite number of magnitude and directions to truthfully represent the ideal square wave in Fourier Series.
But Tim, what if instead of infinite number of odd harmonics, I only have the first 10?
In ADS, there is a Vf_Square source that lets you experiment with the number of harmonics you desire to be in the Fourier Series. The result of the simulation is in Fig. 5.
Fig. 5: ADS simulation result of including only the first 10 odd harmonics in the square wave.
Stepping into Frequency Domain
Writing a function in the form of Fourier Series gives us a fresh perspective. Specifically, by looking at the Fourier Series construction of a function, we are able to visualize the frequency components present in the function and the strength of each frequency component.
Let’s revisit the ideal square wave expression, the Fourier Series shown below has both “direction” and “magnitude.”
Because the multiplication factor in front of ω0 indicates the frequency of the sine wave, we plot the factor, n, on the x-axis. For each nth harmonic, there is a specific magnitude that goes on the y-axis. Fig. 6 illustrates the parameters we are plotting.
Fig. 6: Illustration of what goes on a frequency domain plot. On the x-axis, we plot the harmonics, and we plot the magnitude on the y-axis.
Fig. 7 displays the log-log plot of frequency domain spectrum up to the 100th harmonic of the sine wave component that makes up the ideal square wave. The 1/n relationship of the magnitude and harmonic is made clear in a log-log plot.
Fig. 7: Frequency spectrum of an ideal square wave up to the 100th harmonic. The magnitude of the harmonics is inversely proportional to the order of each harmonic.
Extension of Fourier Series
Indeed, Fourier Series is very useful when it comes to representing a periodic waveform. Nonetheless, one major limitation of Fourier Series is the assumption of periodic waveform.
Let’s take the impulse response of a channel for example. Fig. 8 is the waveform of a channel we investigated in Dirac Delta Misnomer. The impulse response is NOT a periodic function. If I am interested in the impact of the channel on different frequency components, I would need a way to transform the aperiodic time domain response to the frequency domain.
Fig. 8: Time domain impulse response of a channel is not a periodic function.
In the next post, I will show that with the help of Fourier transform, an extension of Fourier Series, I can convert the time domain impulse response to the frequency domain insertion loss, as shown in Fig. 9.
Fig. 9: Frequency domain representation of the time domain impulse, also known as the insertion loss.
Because the insertion loss plot gives us valuable information on how each frequency component is affected by the channel, we can then identify the root cause of eye closure.
As Sir William Bragg points out, new discovery requires a new point of view. There is no doubt Fourier’s approach to the heat equation is a novel one.
By using Fourier Series, we examine the ideal square wave through the frequency domain looking glass. In the next weeks, we will see how we apply Fourier transform to understand the root cause of eye closure.
That's this week's Tim's Blackboard. See you in two weeks!
Experiment with the square wave source:
Wikipedia contributors, "Fourier series," 9 August 2017. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Fourier_series&oldid=786176863.
D. Freeman, "6.003 Signals and Systems," Massachusetts Institute of Technology: MIT OpenCourseWare, Fall 2011. [Online]. Available: https://ocw.mit.edu.