- 3441 - FOURIER TRANSFORM MATH - in our daily lives? The Fourier transform is essentially the limit of the Fourier series of a function as the period approaches infinity. At the heart of all digital-based technology, it’s the next stop on our journey for those curious to understand the nature that our everyday objects are represented in mathematics.
------------- 3441 - FOURIER TRANSFORM MATH - in our daily lives?
- Jean-Baptiste Fourier left us a stark reminder to continuously turn to our connection with nature as a source of inspiration for knowledge. How does the temperature on a metallic plate distribution over time? How about at any given point on the plate? Here is his math:
-
- Since ancient times, the circle was placed on a pedestal as the simplest shape for abstract comprehension. A simple center point and a fixed-length radius/string was all needed, every point on the perimeter perfectly equidistant from the center.
-
- The key to understanding the “Fourier transform” and the “Discrete Fourier Transform” is our ancient desire to express everything in terms of circles. The genius connection the rest of this piece revolves around, the heart of Fourier’s observation, stems from the realization below:
-
------------------- From simple rotations in a circle can create trigonometric functions of sine & cosine.
-
- Jean-Baptiste Joseph Fourier (1768–1830 A.D.) was far from the first person to realize this. However, he was the first to cleverly note that multiple simple waves, either sine or cosine, could be added to perfectly duplicate any type of periodic function.
-
- He derived the ingenious method that reverse-engineered his observation: the Fourier Series setup and the required Fourier analysis is the process necessary to uncover all the sine and cosine waves that converge to a targeted function.
-
- The analysis consists of deriving the coefficients (radius of) and frequencies (“speed” of rotation on) of the many circles whose summation mimics that of any generic periodic function.
-
- The Fourier Series is the circle and wave-equivalent of the “Taylor Series“. The Fourier Series is simply a long, intimidating function that breaks down any periodic function into a simple series of sine and cosine waves. Almost any function can be expressed as a series of sine and cosine waves created from rotating circles.
-
- Every circle rotating translates to a simple sin or cosine wave. It’s application to non-periodic functions is through the Fourier Transform. It has become one of the principal methods of analysis for mathematical physics, engineering, and signal processing.
-
- The Fourier Series a key underpinning to any and all digital signal processing. While the original theory of Fourier Series applies to periodic functions occurring in a natural wave motion, such as with light and sound, its generalizations, relate to significantly wider settings, such as the time-frequency analysis underlying the recent theories of wavelet analysis & local trigonometric analysis.
-
- Jean Baptiste Joseph Fourier (1768−1830) first introduced the idea that any periodic function can be represented by a series of sines and cosines waves in 1828. The “Analytical Theory of Heat” is a result of arriving at the answer to a particular heat equation.
-
- From the heat equation, Fourier evolved his findings to develop the Fourier Series. The Fourier Series has increased in importance, particularly in the digital age. From creating the base for physics such as Brownian motion, for finance such as in the “Black-Scholes equations“, or for electrical engineering such as in digital processing, Fourier’s work has only grown in both theoretical & practical applications.
-
- Unlike the “Fourier Transform“, a “Fourier Series” cannot be applied to general functions, they can only converge to “periodic functions“. To convergence to simple sine and cosine waves, three specific criteria must be met. Known as the “Dirichlet conditions“, named after one Peter Gustav Lejeune Dirichlet, all three conditions must be met for a periodic function with some period-length 2L:
-
----------------------- It has a finite number of discontinuities within the period 2L
-
---------------------- It has a finite average value in the period 2L
-
---------------------- It has a finite number of positive and negative maxima and minima
-
- Does the function have bounded variation? If f(x) is periodic over the length of some period 2L, and checks-off each condition list above, then the Fourier Series guarantees that some mix of cosine and sine waves can replicate f(x).
-
- An infinite series of numbers either goes to infinite or converges to a number, the same way an infinite series of expressions (either polynomials or trigonometric) either goes to infinite or converges to a function (or shape).
-
- Conversely, if we’re given a shape, we can approximate its function by creating an infinite series of varying sin and cosine waves. The Fourier Series is simply a function that’s described and derived by a literal summation of waves and constants.
-
- The “Fourier Analysis” is simply the actual process of reverse-engineering, or constructing from scratch (sin and cos) a period function. The goal is to solve for coefficients a0, an & bn.
-
-------------------- f(x) = Avg. Function Value + Sine/Cosine Waves Series
-
- The first part of the Fourier Series, the leading division that includes the coefficient a0 is simply the average value of the function; more specifically, it’s the net area between −L and L, divided by 2L (the period of the function).
-
- The second part of the equation, notated with a sigma/series symbol, represents the literal summation of different cosine and sines waves that should converge to target function. Both trigonometric functions are carried to the nth-degree in the series. For this second half of the equation, the challenge is to solve for an and bn.
-
- Fourier Analysis starts to solve for our target coefficients (a0, an, bn). There’s a standard setup for deriving all three coefficients as well as multiple short-cuts. Solving for a0, an, bn is straight-forward, yet far from simple. All three coefficients are solved through the following integrals:
-
--------------------- All three coefficients given assume a set period of 2π
-
--------------------- Solving for a0, the average value
-
- The first term on the left, a0 is at times referred to as the “average value” coefficient for that very reason. It is simply an integral of the function we’re attempting to replicate over it’s fixed period.
-
-------------------- Solving For aN— Summation of Cosine Waves
-
- aN is the leading coefficient for the cosine waves in our series; our goal is to figure out how this coefficient behaves at different values in the series.
-
--------------------- Solving For bN— Summation of Sine Waves
-
- Conversely, bN is the leading coefficient for the sine waves in our series; our goal here is to again figure out how this coefficient behaves at different values in the series.
-
- aN & bN are essentially the varying “weights” of their respective waves. They provide us with an approximation of which wave we’re “mixing” in most for any given series.
-
- Most Fourier Series are drastically reduced in their complexity early-on; based on the symmetry of the target function f(x), whether the function is even or odd, we can usually eliminate at least one of the coefficients. A function, relative to its symmetry across the origin or the y-axis, can be considered even or odd:
-
----------------- F(x) is even if F(-x)=F(x), such as Cosine(x), F(x)*Cosine(x), etc…
-
----------------- F(x) is odd if F(-x)=-F(x), such as Sine(x), F(x)*Sine(x), etc…
-
- Start a Fourier analysis by first checking whether F(x), the function or shape we’re approximating, is odd, even or neither. If a function is odd or even, we’re in luck. To recall some basic calculus, let’s remind ourselves what happens when we integrate either of the two trigonometric functions over some fixed period:
-
--------------------- The integral of Cos(x) from -L to L is 0
-
--------------------- The integral of Sin(x) from -L to L is also 0
-
- With both sets of facts above, it’s clear now how a function’s symmetry drastically reduces its complexity for a Fourier Analysis; basically, in most, not all, problems that we encounter, the Fourier coefficients a0, aN, bN become zero after integration.
-
- With knowledge of even and odd functions, a zero coefficient is predicted without performing the integration, leading us to, essentially, a powerful shortcut.
-
- A function F(x) is said to be even if F(-x) = F(x) for all values of x; therefore, the graph of an even function is always symmetrical about the y-axis ( it is a mirror image).
-
- For example, the graph of the function:
-
-------------------------- F(x) = Cos(πx)
-
- The symmetrical across the y-axis. If a function is even, then it follows that the integral part of solving for bN, no matter the nth-term of bN, is also equal to zero. Therefore, we can safely eliminate the bN part of our original series, leaving us with the truncated Fourier Series of an even function. Known as a “Half-Range Fourier Cosine Series“.
-
- An even function has only cosine terms in its Fourier expansion: the key to understanding this and the following shortcut is the simple reminder that every Fourier Series setup starts with both a sine & a cosine function.
-
- A function F(x) is said to be odd if F(-x) = -F(x) for all values of x; therefore, the graph of an odd function is always symmetrical about the origin (it’s unchanged when flipped over the x-axis and y-axis).
-
----------------------------- F(x) = sin(πx):
-
- The above is symmetrical across the origin. If a function is odd, then it follows that the integral of the series terms including aN , no matter the nth-term of aN, is also equal to zero. Therefore, we can safely eliminate the aN part of our original series, leaving us with the truncated Fourier Series of an odd function; known as a Half-Range Fourier Sine Series.
-
- Odd functions include extraneous information that helps us eliminate an additional term: a0. If a function is symmetrical across the origin, then this means that the area above the x-axis is equal to the area below the x-axis; which means that the average value of the function, our a0 term, is also equal to zero. Therefore, for a Half-Range Fourier Sine Series, we can safely eliminate both our first time a0 and our cosine term as such:
-
- Fourier expansion replicates a square-wave that oscillates from troughs of -1 to crests of 1 with a period of 2π; analyzing the function from -π to π-.
-
- The very first step to setting up a Fourier Series is not to jump into the setup, but rather to check if the target function displays either type of symmetry; looking at the graph, it’s pretty clear that it is indeed symmetrical around the origin. Therefore, the function we’re working with is odd.
-
- That tiny piece of analysis drastically reduces the complexity and required steps to complete our Fourier Series. Since we know it’s an odd-function, this means we can treat
the f(t) on the right side is simply the value of the shape / function we’re approximating.
-
- In this particular example, the value of the function f(t) is piecewise: from -π to 0, f(t) = -1; from 0 to π, f(t) = 1. Therefore, if we split bN to two different integrations, (-π,0) to (0,π), we can simply substitute the f(t) variable with either -1 or 1:
-
- Values of n to analyze patterns that’ll hint to the convergence of our coefficient bN. Start by writing out n = 1: For the first value of n = 1, our coefficient of bN converges to the fraction 4/π. We’ll now repeat this process for four additional values of n in hopes of noticing a pattern:
-
- All even values of bN converge to zero, while all the odd values of bN converge to:
4 / n*π.
-
- With bN solved, we can now plug the coefficient back into our Half-Range Fourier Sine Series that we setup above.
-
- This is a bit convoluted, however, it’s already perfectly accurate: the Fourier Series on the right indeed converges to our target square-wave.
-
- The Fourier Series is a way of representing periodic functions as an infinite sum of simpler sine & cosine waves. From signal processing to approximation theory to partial differential equations, it’s hard to overstate just how intricately the Fourier Series is tied with physics phenomena, anything with an identifiable pattern can be described with varying sin & cosine waves.
-
- The scope of our Fourier Series is limited compared to its successor, the Fourier Transform. The Fourier Series is used to represent a periodic function by a discrete sum, while the Fourier Transform is used to represent a general, non-periodic function.
-
- The Fourier transform is essentially the limit of the Fourier series of a function as the period approaches infinity. At the heart of all digital-based technology, it’s the next stop on our journey for those curious to understand the nature that our everyday objects are represented in mathematics.
-
February 2, 2022 FOURIER TRANSFORM MATH - in our daily lives? 3441
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ---
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
----------------------------- Friday, February 4, 2022 ---------------------------
No comments:
Post a Comment