MATH 414 Lecture 14

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Fourier Transforms

In fourier series, we construct Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as a $2a$ -periodic function.

Analysis: $\alpha _{n}={\frac {1}{2a}}\,\int _{-a}^{a}f(x)\,\mathrm {e} ^{-{\frac {\pi \,n\,i}{a}}}\,\mathrm {d} x$
Synthesis: $f(x)=\sum _{n=-\infty }^{\infty }\alpha _{n}\,\mathrm {e} ^{i\,{\frac {\pi \,n}{a}}\,x}$

Fundamental Frequency

$\nu _{fund}={\frac {1}{2a}}$ (natural)
$\omega _{circular}=\omega _{1}={\frac {2\pi }{\mbox{period}}}={\frac {\pi }{a}}$ (circular frequency; this is what we will be using)
$\omega _{n}=n\,\omega _{1}$

All allowed frequencies are integer multiples of the fundamental frequency.

Gap

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{n+1} - \omega_n = \left( n + 1 \right) \, \omega_1 - n \, \omega_1 = \omega_1}

Spacing gets smaller as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} gets larger

What happens as spacing goes to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} ? We approach a continuum of frequencies.

Derivation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_n = \frac{\pi \, n}{a}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} F(\lambda) &= \int_{-a}^{a} f(x) \, \mathrm{e}^{-i \, \lambda \, x} \,\mathrm{d}x \\ F \left( \frac{n \, \pi}{a} \right) &= \int_{-a}^{a} f(x) \, \mathrm{e}^{-\frac{i \, \pi \, n}{a} \, x} \,\mathrm{d}x \\ \alpha_n &= \frac{1}{2a} \, F \left( \frac{n \, \pi}{a} \right) \end{align}}

Now

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sum_{n=-\infty}^\infty F(\lambda_n) \, \frac{1}{2a} \, \mathrm{e}^{i \, \lambda_n \, x}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \lambda = \lambda_{n+1} - \lambda_n = \frac{\pi (n+1)}{a} - \frac{\pi \, n}{a} = \frac{\pi}{a}} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{1}{2} \sum_{n=-\infty}^\infty F(\lambda_n) \, \mathrm{e}^{i \, \lambda_n \, x} \, \Delta \lambda}

This is a riemann sum for

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} \, \int_{-\infty}^\infty F(\lambda) \, \mathrm{e}^{i \, \lambda \, x} \,\mathrm{d}x}

As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \to \infty} , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f(x) &= \frac{1}{2\pi} \, \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(t) \, \mathrm{e}^{-i \, \lambda \, t} \,\mathrm{d}t \right) \mathrm{e}^{i \, \lambda \, x} \,\mathrm{d}x \\ &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{f}(\lambda) \, \mathrm{e}^{i \, \lambda \, x} \,\mathrm{d}x \end{align}}

This is our synthesis step, and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) \, \mathrm{e}^{i \, \lambda \, x} \,\mathrm{d}x}

is our corresponding analysis step.

Definition

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{F} \left[ f \right] \left( \lambda \right) = \hat{f}(\lambda) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} f(x) \, \mathrm{e}^{-i \, \lambda \, x} \,\mathrm{d}x \end{align}}

Example

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} 1 & x \in \left[ -\pi, \pi \right] \\ 0 & \mbox{otherwise} \end{cases}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{f}(\lambda) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\pi}^{\pi} \mathrm{e}^{i \, \lambda \, x} \\ &= \frac{2}{\sqrt{2\pi}} \, \left( \frac{ \mathrm{e}^{i \, \lambda \, \pi} - \mathrm{e}^{-i \, \lambda \, \pi}}{2i\lambda} \right) \\ &= \sqrt{\frac{2}{\pi}} \, \frac{\sin{(\lambda \, \pi)}}{\lambda} \\ \hat{f}(0) &= \sqrt{2\pi} \end{align}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{sinc}(z) := \frac{\sin{(\pi \, z)}}{\pi \, z}} . In this example, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f} = \sqrt{2\pi} \, \mathrm{sinc}(\lambda)}

Inverse Fourier Transform

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}^{-1}[\hat{f}](x) = \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \hat{f}(\lambda) \, \mathrm{e}^{i \lambda x} \,\mathrm{d}\lambda}

A Quartet of Functions

Function	Fourier Transform
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(x)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(-\lambda)}

When you know one transform, you know another.