Notes on the Fourier Transform - Eli Bendersky's website
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The Fourier series is a great tool for analyzing periodic functions. But<br>what about functions that don’t repeat? We’ve<br>seen<br>that we can compute Fourier series for a non-periodic function defined<br>on a finite interval, as long as we don’t care about its behavior beyond<br>that interval.
Let’s extend this idea to functions that never repeat; that is,<br>non-periodic functions defined on the interval (-\infty,\infty).
Visualizing Fourier series for non-repeating functions
To motivate the subject ahead, let’s look back at the example used in<br>the earlier post about Fourier<br>series:
\[t(x)=<br>\begin{cases}<br>x & 0 \leq x \leq 1 \\<br>2-x & 1<br>With an odd extension into [-2,0]. In that post, to make the<br>Fourier series work, we assumed t(x) keeps repeating with a<br>period 2L=4 on the entire x axis. Here, let’s face the<br>reality that it does not - in fact - repeat, and observe how our Fourier<br>series work out.
Recall that the Fourier series approximating t(x) are the sine<br>series (since it’s an odd function):
\[t(x)=\frac{8}{\pi^2}\bigg[ sin\frac{\pi x}{2}-\frac{1}{3^2} sin\frac{3\pi x}{2}+\frac{1}{5^2}sin\frac{5\pi x}{2}-\cdots\bigg]\]<br>The following visualization is interactive. By default, it shows<br>t(x) (with its odd extension) and no Fourier series<br>approximation. We’ll proceed by a series of steps and observe the<br>outcome:
n (terms in the Fourier series)
x min
x max
Step 1 : set to some non-zero number; already at 3, the<br>approximation is very good.
The frequency spacing is \frac{\pi}{L} (this is the coefficient<br>of x in the sines). Note that the Fourier series repeats every<br>2L, as expected.
Step 2 : increase L to 6. This means our series are<br>constructed assuming t(x) has a period of 12, not 4. Note how<br>the Fourier series look now - they repeat every 12, and they don’t match<br>t(x) as well as before. We can increase to a higher<br>number to make the match better. As L grows, the spacing between<br>adjacent frequencies decreases.
Step 3 : increase L to 10. We no longer see the repetitions,<br>so feel free to increase the values of x min and x max until you do.<br>Note again that we need to add more and more coefficients to match<br>t(x) better with this larger L, and the spacing adjacent<br>frequencies grows smaller.
Increasing L means our function repeats at larger and larger<br>intervals. The logical conclusion of this progression is to ask - what<br>happens if the function never repeats, meaning<br>L\rightarrow\infty? While not mathematically rigorous, the<br>visual experiment here lets us make some conjectures: we’ll likely need<br>an infinite number of coefficients for a good approximation, and<br>moreover, the spacing between these coefficients will tend to zero.
In other words, instead of a discrete set of coefficients, we’ll end up<br>with a continuous line, or function. The function produced by this<br>process is the Fourier transform of t(x), and the next section<br>shows its mathematical derivation.
Fourier series with L\rightarrow\infty leading to Fourier transform
In these notes, we’ll be using the complex exponential formulation of<br>Fourier series:
\[f(x)=\sum_{n=-\infty}^{\infty}C_n\cdot e^{in\pi x/L}\]<br>With:
\[C_n=\frac{1}{2L}\int_{-L}^{L}f(x)e^{-in\pi x/L}dx\]<br>We’re interested in a non-periodic defined on the interval<br>(-\infty,\infty). So we’ll be exploring the above equations for<br>L\rightarrow\infty.
First, let’s make a slight change of notation. Instead of writing<br>formulae in terms of the period (2L), we’ll be using the n-th<br>harmonic angular frequency w_n:
\[w_n=\frac{n\pi}{L}\]<br>So we can slightly rewrite our series as:
\[f(x)=\sum_{n=-\infty}^{\infty}C_n\cdot e^{i w_n x}=\sum_{n=-\infty}^{\infty}C_n\cdot e^{i\cdot n \Delta w x}\]<br>Using \Delta w as the difference between two consecutive<br>frequencies:
\[\Delta w=w_n-w_{n-1}=\frac{n\pi}{L}-\frac{(n-1)\pi}{L}=\frac{\pi}{L}\]<br>Using this notation, C_n is expressed as:
\[C_n=\frac{\Delta w}{2\pi}\int_{-\pi/\Delta w}^{\pi/\Delta w}f(x)e^{-i\cdot n \Delta w x}dx\]<br>So far there are no new insights here, just some new notation. Now we’re<br>going to use it to facilitate the next step.
Since L\rightarrow \infty, then \Delta w\rightarrow 0.<br>Let’s calculate the limit of the Fourier series representation of<br>when \Delta w\rightarrow 0:
\[f(x)=\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}C_n\cdot e^{i\cdot n \Delta w x}\]<br>And substitute the latest C_n into this equation, changing its<br>dummy integration variable from x to t to avoid<br>confusion [1]
\[f(x)=\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}\left[\frac{\Delta w}{2\pi}\int_{-\pi/\Delta w}^{\pi/\Delta w}f(t)e^{-i\cdot n \Delta w t}dt\right]\cdot e^{i\cdot n \Delta w x}\]<br>Reordering slightly, and also replacing n\Delta w by w_n<br>in the complex exponents:
\[f(x)=\frac{1}{2\pi}\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}\left[\int_{-\pi/\Delta w}^{\pi/\Delta w}f(t)e^{-i\cdot w_n t}dt\right]\cdot e^{i\cdot...