# MATH310S26 - Day 15: Fourier Transforms and Physical Interpretation ## Course Information **Course**: MATH310 - Introduction to Mathematical Modeling **Date**: February 20, 2026 (Day 15) **Topics**: Rectangular pulse transform, sinc function, physical interpretation of Fourier transforms **Previous Day**: [[MATH310S26-Day14-Notes|Day 14 - Fourier Transform Pairs and Delta Functions]] ## Administrative Notes - **Monday (February 23)**: Work day on [Fast Fourier Transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) implementation - **Low Stakes Feedback #9**: Individual project status update - **Low Stakes Feedback #10**: Fourier transform of sine function - **Name stickers**: Distributed for attendance tracking ## Main Topics ### 1. Fourier Transform Pair vs. Fourier Series #### Comparison and Connection **[[MATH310S26-Day10-Notes|Fourier Series]]** (Periodic): $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{i\omega_n t}$ - Restriction: $\omega_n = \frac{n\pi}{L}$ (discrete frequencies) - Forces periodicity: $f(t + 2L) = f(t)$ - Coefficients: $c_n = \frac{1}{2L}\int_{-L}^{L} f(t)e^{-i\omega_n t} dt$ **[[MATH310S26-Day12-Notes|Fourier Transform]]** (General): $f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega t} d\omega$ - All frequencies allowed: $\omega \in \mathbb{R}$ - No periodicity requirement - Transform: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt$ **Key Insight**: Can recover Fourier series by [[MATH310S26-Day14-Notes|Dirac localization]] of $\omega$ to $\omega_n$ ### 2. Example: Fourier Transform of Rectangular Pulse #### Function Definition $f(t) = \begin{cases} \frac{1}{2L} & |t| \leq L \\ 0 & |t| > L \end{cases}$ **Physical interpretation**: - Displacement of drumhead or speaker membrane - Water displacement by hand motion - Non-physical aspect: Instantaneous jumps (would require infinite acceleration) #### Transform Calculation Starting with: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ Since $f(t) = 0$ outside $[-L, L]$: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \cdot \frac{1}{2L} \int_{-L}^{L} e^{-i\omega t} dt$ Evaluating the integral: $= \frac{1}{2L\sqrt{2\pi}} \left[\frac{e^{-i\omega t}}{-i\omega}\right]_{-L}^{L}$ $= \frac{1}{2L\sqrt{2\pi}} \cdot \frac{1}{-i\omega}(e^{-i\omega L} - e^{i\omega L})$ Using $e^{i\theta} - e^{-i\theta} = 2i\sin(\theta)$: $= \frac{1}{2L\sqrt{2\pi}} \cdot \frac{2i\sin(\omega L)}{-i\omega}$ **Final Result**: $\boxed{\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \cdot \frac{\sin(\omega L)}{\omega L}}$ ### 3. The Sinc Function #### Definition and Properties The [sinc function](https://en.wikipedia.org/wiki/Sinc_function): $\text{sinc}(x) = \frac{\sin(x)}{x}$ **Properties of $\hat{f}(\omega)$**: 1. **Even symmetry**: $\hat{f}(-\omega) = \hat{f}(\omega)$ - Reflects that $f(t)$ is even in time domain - General principle: Symmetry is preserved under Fourier transform 2. **Decay at infinity**: $\lim_{\omega \to \pm\infty} \hat{f}(\omega) = 0$ - Numerator bounded: $|\sin(\omega L)| \leq 1$ - Denominator grows: $|\omega L| \to \infty$ 3. **Zeros**: $\hat{f}(\omega) = 0$ when $\sin(\omega L) = 0$ - Occurs at $\omega = \frac{n\pi}{L}$ for $n \neq 0$ 4. **Value at origin** (using [L'Hôpital's rule](https://en.wikipedia.org/wiki/L%27Hôpital%27s_rule)): $\lim_{\omega \to 0} \frac{\sin(\omega L)}{\omega L} = \lim_{\omega \to 0} \frac{L\cos(\omega L)}{L} = 1$ Therefore: $\hat{f}(0) = \frac{1}{\sqrt{2\pi}}$ #### Visual Description ("Squiddy") - Central peak at $\omega = 0$ with height $\frac{1}{\sqrt{2\pi}}$ - Oscillating side lobes with decreasing amplitude - Zeros at $\omega = \frac{n\pi}{L}$ for integer $n \neq 0$ - Even symmetry about vertical axis ### 4. Physical Interpretation #### Frequency Decomposition **What the transform tells us**: - $\hat{f}(0)$ largest → DC component (constant) dominates - Neighboring frequencies needed to create sharp edges - Higher frequencies contribute less (decay of sinc) - Zeros indicate frequencies not needed for reconstruction ### 5. Wave Diffraction Demonstration #### Laser Through Wire Experiment **Setup**: Laser pointer with copper wire across aperture **Classical expectation** (particles): - Two columns of deflected "peanuts" - Shadow region behind wire **Actual observation** (waves): - Central bright spot - Alternating bright/dark fringes - Pattern follows sinc function intensity **Connection to Fourier**: The [diffraction pattern](https://en.wikipedia.org/wiki/Diffraction) is the Fourier transform of the aperture function! #### Applications 1. **Water waves**: Passing through harbor opening 2. **Sound waves**: Stadium acoustics design - Avoid "dead spots" from destructive interference - Optimize speaker placement 3. **Optics**: [Single-slit diffraction](https://en.wikipedia.org/wiki/Diffraction#Single-slit_diffraction) - Intensity pattern is $|\text{sinc}|^2$ ### 6. Time-Frequency Duality #### Scaling Relationship **Question**: What happens to $\hat{f}(\omega)$ if we spread the rectangle (increase $L$)? **Time domain**: Rectangle gets wider **Frequency domain**: Sinc function gets narrower **Mathematical relationship**: - Width in time: $2L$ - First zero in frequency: $\omega = \frac{\pi}{L}$ - Product: $(2L) \cdot \frac{\pi}{L} = 2\pi$ (constant!) **[Uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle)** connection: - Cannot localize simultaneously in time and frequency - Narrower in time → broader in frequency - Broader in time → narrower in frequency ### 7. Connection to Harmonic Oscillator (Board Work) #### Energy Conservation Example For mass-spring system: $m\ddot{y} + ky = 0$ **Energy**: $E = \frac{1}{2}m\dot{y}^2 + \frac{1}{2}ky^2$ (constant) **Complex solution**: $y(t) = c_1\cos(\omega t) + c_2\sin(\omega t)$ where $\omega = \sqrt{\frac{k}{m}}$ **Fourier perspective**: Natural frequency $\omega$ appears as peaks in frequency spectrum ## Key Takeaways 1. **Sinc function emerges naturally** from rectangular pulses - fundamental in signal processing 2. **Wave nature revealed through diffraction** - Fourier transform predicts physical phenomena 3. **Time-frequency duality** - Localization in one domain means spreading in the other 4. **Physical interpretation matters** - Transform coefficients are amplitudes for wave reconstruction 5. **Symmetry preservation** - Even/odd functions maintain symmetry through transform ## Mathematical Connections ### Course Progression - [[MATH310S26-Day1-Notes|Days 1-3]]: Linear systems and projections - [[MATH310S26-Day5-Notes|Days 4-6]]: Eigenanalysis and orthogonal decomposition - [[MATH310S26-Day7-Notes|Days 7-8]]: Introduction to Fourier series - [[MATH310S26-Day10-Notes|Days 10-11]]: Complex form and applications - [[MATH310S26-Day12-Notes|Day 12]]: Breaking periodicity → Fourier transform - [[MATH310S26-Day14-Notes|Day 14]]: Delta functions and transform pairs - **Today**: First "normal" transform - rectangular pulse ### Looking Ahead - FFT algorithms for computational efficiency - [Convolution theorem](https://en.wikipedia.org/wiki/Convolution_theorem) - [Parseval's theorem](https://en.wikipedia.org/wiki/Parseval%27s_theorem) (energy conservation) - Applications to [signal processing](https://en.wikipedia.org/wiki/Signal_processing) and [image processing](https://en.wikipedia.org/wiki/Digital_image_processing) ## References - [Sinc function](https://en.wikipedia.org/wiki/Sinc_function) - [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) - [Diffraction](https://en.wikipedia.org/wiki/Diffraction) - [Time-frequency analysis](https://en.wikipedia.org/wiki/Time%E2%80%93frequency_analysis) - [Uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle)