A colleague asked me this question while discussing about the Fourier transform of a signal from an European version of GPS called Galileo. My initial knee-jerk answer was that Fourier transform lists the frequencies that are important for our analysis. But that answer left a bad taste in my mouth. It felt like this tool was contrived to answers made up problems so that we can gloat on our own smug sense of knowledge. However, after a bit circuitous research that involved reading about an overweight French general, the inner workings of our ear and finally some philosophical jousting with simplicity, I feel comfortable in answering my colleague. But this route was not without any casualties. After this exercise, I feel like I have opened up a Pandora’s box of unanswered questions regarding simplicity and most importantly about our perceptions clouding reality. Let’s start this journey with Fourier transforms.
sinusoidal decomposition of a complicated signal
To understand any complicated signal, one of the first step is to generate the Fourier transform of that signal. Fourier transform is a mathematical function that decomposes a time varying signal, as shown in figure to the right, into several sinusoidal waves. These sinusoidal waves will have different frequency, amplitude and phases but when you add them all together, the original waveform is magically recreated. The fundamental idea here is complexity reduction by splitting a waveform into manageable chunks. For reasons that initially baffled me, the powers there be chose sinusoidal waves as this manageable chunk.
The story of the genesis of Fourier transform itself is quite interesting and is a prime example of “proof by intimidation“. Jean Baptiste Joseph Fourier, the man behind this tool, was an orphan who happened to be on the better side of the guillotine during French revolution and subsequently rose through the ranks of Napoleonic army to become the governor of southeastern France. He invented this technique while analyzing the flow of heat through metals. Perhaps influenced by his desire to design weaponry that would withstand the intense heat and pressure generated from gunpowder ignition. In his The Analytic Theory of Heat published in 1822, he posited that any function, continuous or discontinuous, could be broken down into a series of sinusoidal waves. Though his theory wasn’t fully correct nor anyone could visualize its practical purpose, no one dare challenge him lest their death wish was to piss of a middle aged portly French general. However, it turns out that nature is filled with sinusoidal waves. From ocean waves to clanking glasses to banjo strumming hippies, this mysterious waveform appears over and over again.
Sine waves in nature
Stretching a spring
Very early on it was noticed that when a spring was stretched or a tuning fork was bent or when pressure was applied to an elastic membrane such as a drumhead, the force that was trying to restore it to its natural resting state increased with the extent of stretch. For the picture shown to the left, say adding a mass m to the spring stretched it by
Sine waves in our ears
Our ears have a stretched membrane, like a drumhead, called the Basilar membrane. This membrane is inside a spiral bone cavity called cochlea and the nerve endings that provides us with the sensation of hearing is attached to this membrane. Whenever, sound waves cause this membrane to vibrate, an electrical pulse is sent to our brain which is interpreted as sound. Additionally, this membrane has an ability to differentiate various frequencies. This is made possible because of the gradual tapering of the Basilar membrane from the start of the spiral cavity to the end. This variation of thickness results in Basilar membrane having different stiffness constant k over its length. The thinner section at the start vibrates to high frequency sounds and as you wind up towards the end of this spiral cavity, it resonates with low bass sounds. The video below created by the Howard Hughes Medical institute Prof Jim Hudspeth clearly illustrates region within Basilar membrane that vibrates to various audio frequencies.
As shown in the video above, when a sinusoidal sound wave of particular frequency hits our ears, it excite only a very small section of the Basilar membrane. Consequently, a very small set of nerves in that regions is excited resulting in the least amount of auditory data being sent to the brain. In my opinion, our perception of simplicity is strongly tied to the amount of brain processing involved in a task. Naturally, a sinusoidal wave exciting a small set of nerves within our auditory framework would be perceived as the simplest wave by us. This explains why we use sinusoidal waves as the basic building block in Fourier transform. However, now I am left to wonder if our perception of simplicity is clouding an objective analysis of signal?
What is simplicity?
Maybe the following questions will help us better understand the loaded term “simplicity“. Why are symmetrical objects appealing to us? Why are we uncomfortable around strangers but happy to be with our friends (at least most :-)) ? Why do we feel learning a new task is complex but once we have mastered it, then it becomes second nature? Why do we feel the need to reach out to people who share same nationality, ethnicity, linguistic etc..? The answer that I can think of is Simplicity.
Our brain perceives simplicity as anything that doesn’t involve utilizing a large part of its processing capacity. Unfamiliarity, whether it is surroundings, objects or people, drives our brain into sensory overdrive. All our senses, whether we need for that task or not, are alert and are flooding huge amount of data to the brain. The brain is constantly trying to sift through this data deluge, looking for relevant patterns for storage, discarding redundant superficial data/senses and so on. The process of mastering a task is all about creating this sensory data filter to let learned patterns in while blocking out unwanted triggers. So it is natural for us to feel comfortable around sine waves and would want to see any complicated wave decomposed into multiple sinusoidal waves.
However, as we become complacent and start patting our backs on being able to break down a complicated signal into familiar series of sine waves, I am wondering whether our perceived sense of simplicity is hindering an objective analysis of this signal? Are we wasting time and effort in trying to come up with designs and manufacturing processes that operate within contrived parameters for a sinusoidal wave (for e.g. bandwidth requirement)? Seems like I am not the only one who had these thoughts and fears. Mathematicians who operate on a different frame of mind than engineers have been working on this problem quietly for a while. The generalized framework for studying non-sinusoidal building blocks is called the Hilbert space. A very good introduction to function space of which Hilbert space is a subset is available here.