# MATH310S26 - Day 14: Fourier Transform Pairs and the Dirac Delta Function ## Course Information **Course**: MATH310 - Introduction to Mathematical Modeling **Date**: February 18, 2026 (Day 14) - Happy Lunar New Year! 🐴 **Topics**: Dirac delta function, Fourier transforms of sinusoids, transform pair examples **Previous Lecture Day**: [[MATH310S26-Day12-Notes|Day 12 - From Fourier Series to Fourier Transform]] ## Administrative Notes ### Upcoming Schedule - **Monday, February 23**: Work day focused on [Fast Fourier Transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) code base - **Low Stakes Feedback #9**: Individual project specifications update - Focus: Where you are in your project, next steps, individual responsibilities - **Low Stakes Feedback #10**: Check on Fourier transforms (See below) - **AA03**: [Fourier series regression](https://en.wikipedia.org/wiki/Fourier_analysis) on global mean sea level data - Due: ~Friday, February 27 (flexible deadline) ### Project Timeline - **March 6**: Narrative outline due - **March 20**: Full test of version 1.0 with next steps outline ## Main Topics ### 1. The Fourier Transform Pair Revisited Building from [[MATH310S26-Day12-Notes|Day 12's derivation]], we have the complete transform pair: **Forward Transform**: $\hat{f}(\omega) = \mathcal{F}\{f\}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ **Inverse Transform**: $f(t) = \mathcal{F}^{-1}\{\hat{f}\}(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega t} d\omega$ **Physical Interpretation**: - $f(t)$: Amplitude as function of time (e.g., speaker cone position) - $\hat{f}(\omega)$: "Weights" or coefficients for each frequency $\omega$ - Transform pair: Decompose signal into frequencies ↔ Reconstruct from frequencies **Audio Analogy**: Your car stereo's bass/treble controls - Take audio signal $f(t)$ → Transform to $\hat{f}(\omega)$ - Modify $\hat{f}(\omega)$ (boost high frequencies, dampen low frequencies) - Transform back to get modified audio signal ### 2. The Challenge with Sinusoids **Problem**: Direct application of the forward transform to simple sinusoids fails! For $f(t) = \cos(\omega_0 t)$: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \cos(\omega_0 t) e^{-i\omega t} dt$ **Why this doesn't work**: - Cosine function doesn't decay at infinity - Integral diverges (doesn't converge) - Need a new mathematical framework **Connection to [[MATH310S26-Day12-Notes|Day 12]]**: This is why we needed to break away from [[MATH310S26-Day10-Notes|complex Fourier series]] - the limiting process revealed deeper issues with non-periodic functions. ### 3. The Dirac Delta Function: A "Tool of the Devil" #### Definition and Properties The [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) $\delta(t - t_0)$ is defined by three axiomatic properties: **Property 1: Ideal Localization** $\delta(t - t_0) = 0 \quad \text{whenever } t \neq t_0$ **Property 2: Normalization** $\int_{t_0 - \epsilon}^{t_0 + \epsilon} \delta(t - t_0) dt = 1, \quad \epsilon > 0$ **Property 3: Sifting Property** $\int_{t_0 - \epsilon}^{t_0 + \epsilon} \delta(t - t_0) f(t) dt = f(t_0)$ #### Physical and Mathematical Meaning **Thermometer Analogy** (Strichartz): - Old glass thermometer in turkey: doesn't measure temperature at a point - Actually measures accumulated temperature in small region around bulb - Delta function: idealization of "point measurement" - Physically impossible, but mathematically powerful **Geometric Interpretation**: - Function $f(t)$: 1-dimensional curve - Delta integral: extracts 0-dimensional point from 1-dimensional object - "Special tweezers" that pluck out specific values **Probability Connection**: - Discrete dice roll: uniform probability mass at {1,2,3,4,5,6} - Delta functions localize continuous distributions to discrete points - Transforms probability density functions → probability mass functions #### Mathematical Construction via Step Functions **Unit Step Function** (from differential equations): $\theta_c(t) = \begin{cases} 0 & \text{if } t < c \\ 1 & \text{if } t \geq c \end{cases}$ **Key Insight**: $\delta(t - c) = \frac{d}{dt}\theta_c(t)$ (distributional derivative) **Verification by Integration by Parts**: Starting with: $\int_{-\infty}^{\infty} \frac{d\theta_c}{dt} f(t) dt$ where $f(t)$ is a [test function](https://en.wikipedia.org/wiki/Test_function) Test function properties: - Smooth (infinitely differentiable) - Decays to 0 faster than any polynomial as $|t| \to \infty$ Using integration by parts: $\int_{-\infty}^{\infty} \frac{d\theta_c}{dt} f(t) dt = \left[\theta_c(t) f(t)\right]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} \theta_c(t) f'(t) dt$ - First term: $\theta_c(\pm\infty) \cdot f(\pm\infty) = 1 \cdot 0 = 0$ (test function decay) - Second term: $-\int_c^{\infty} 1 \cdot f'(t) dt = -[f(\infty) - f(c)] = f(c)$ **Result**: Confirms sifting property - delta function is the [distributional derivative](https://en.wikipedia.org/wiki/Distribution_(mathematical_analysis)) of the step function. ### 4. Fourier Transforms of Sinusoids Using Delta Functions #### Transform of Cosine For $f(t) = \cos(\omega_0 t)$, we postulate: $\hat{f}(\omega) = \frac{\sqrt{2\pi}}{2}[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$ **Verification via Inverse Transform**: $f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac{\sqrt{2\pi}}{2}[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] e^{i\omega t} d\omega$ Applying sifting property: - First delta: contributes $\frac{1}{2}e^{i\omega_0 t}$ when $\omega = \omega_0$ - Second delta: contributes $\frac{1}{2}e^{-i\omega_0 t}$ when $\omega = -\omega_0$ **Result**: $f(t) = \frac{1}{2}[e^{i\omega_0 t} + e^{-i\omega_0 t}] = \frac{1}{2}[(\cos(\omega_0 t) + i\sin(\omega_0 t)) + (\cos(\omega_0 t) - i\sin(\omega_0 t))]$ $= \cos(\omega_0 t) \quad \checkmark$ **Physical Interpretation**: - Energy ideally localized to exactly two frequencies: $\omega_0$ and $-\omega_0$ - All other frequencies have zero energy - Perfect "frequency selection" in an idealized stereo system #### Connection to Complex Fourier Series **General Pattern**: $\sum_{n=-\infty}^{\infty} c_n \delta(\omega - \omega_n)$ where each delta "pops off" when $\omega = \omega_n$ with coefficient $c_n$. **Inverse transform gives**: $f(t) = \frac{1}{\sqrt{2\pi}} \sum_{n=-\infty}^{\infty} c_n e^{i\omega_n t}$ **Setting** $\omega_n = \frac{n\pi}{L}$ **recovers the [[MATH310S26-Day10-Notes|complex Fourier series]]**! **Key Insight**: Fourier series are the special case where energy is ideally localized to discrete evenly spaced frequencies, giving periodic functions. The Fourier transform removes these "shackles" and allows analysis of any signal structure. ### 5. Transform of Sine Function #### Check Your Understanding Problem **Problem**: Find $\hat{f}(\omega)$ such that $\mathcal{F}^{-1}\{\hat{f}\}(t) = \sin(\omega_0 t)$ **Solution Strategy**: Cannot compute directly: $\mathcal{F}\{\sin(\omega_0 t)\}$ leads to divergent integral **From parity analysis**: - $\sin(\omega_0 t)$ is odd → transform should be odd - Only frequencies $\pm \omega_0$ present - But with **odd symmetry**: $\hat{f}(-\omega) = -\hat{f}(\omega)$ ### 6. Mathematical Framework and Limitations #### When Delta Functions Appear **Beyond audio applications**: - **CD/vinyl "pops"**: Delta function impurities in audio tracks - **Anomalous events**: Sudden spikes in time series data - **Discrete measurements**: Converting continuous to discrete sampling #### Functional Analysis Considerations **Resolution**: [Distribution theory](https://en.wikipedia.org/wiki/Distribution_(mathematics)) - Delta functions are not classical functions - They are **distributions** (generalized functions) - Only make sense inside integrals with test functions - Developed rigorously by [Laurent Schwartz](https://en.wikipedia.org/wiki/Laurent_Schwartz) **Connection to [[MATH310S26-Day5-Notes|eigenanalysis]]**: Like how we extended to complex eigenvalues, we extend function spaces to handle ideal localizations. ### 7. Comparison with Laplace Transform **Why Laplace transform is "nicer"** (from differential equations): $\mathcal{L}\{f(t)\} = \int_0^{\infty} f(t) e^{-st} dt$ **Key differences**: 1. **No imaginary unit**: $e^{-st}$ instead of $e^{-i\omega t}$ 2. **Exponential decay**: Controls behavior at infinity 3. **Semi-infinite domain**: $[0,\infty)$ instead of $(-\infty,\infty)$ **Result**: Common ODE functions transform without delta functions - exponential decay "bullies" any problematic behavior at infinity. **Fourier transform trade-off**: More general (handles any signal), but requires distribution theory for completeness. ## Check Your Understanding 1. **Sine transform**: Find $\hat{f}(\omega)$ for $f(t) = \sin(\omega_0 t)$ ## Key Takeaways 1. **Delta functions bridge discrete and continuous**: They allow Fourier transforms of periodic functions while maintaining the continuous frequency framework 2. **Ideal localization is unphysical but mathematically powerful**: We can't truly measure at a point, but the mathematical idealization enables precise frequency analysis 3. **Sinusoids require distribution theory**: The most basic functions for [[MATH310S26-Day7-Notes|Fourier analysis]] need the most sophisticated mathematical framework 4. **Transform pairs reveal signal structure**: Even when direct computation fails, we can postulate transforms based on symmetry and verify via inverse transformation 5. **Fourier transforms generalize Fourier series**: Delta functions show how discrete frequency series are special cases of continuous frequency analysis ## Mathematical Connections ### Previous Topics - [[MATH310S26-Day7-Notes|Fourier series introduction]]: Trigonometric basis functions - [[MATH310S26-Day10-Notes|Complex Fourier series]]: $e^{i\omega_n t}$ exponentials - [[MATH310S26-Day11-Notes|Sea level analysis]]: Practical frequency detection - [[MATH310S26-Day12-Notes|Fourier transform derivation]]: Continuous frequency limit ### Integration with Course Themes - [[MATH310S26-Day3-WorkdayMaterials (Two-by-two matrices, normal equations, and ordinary least squares)|Linear regression]] (Days 1-3): Projection onto basis functions - [[MATH310S26-Day5-Notes|Eigenanalysis]] (Days 4-6): Orthogonal decomposition concepts - **Mathematical modeling philosophy**: Using idealized tools (deltas) to understand real phenomena ### Looking Ahead - **Monday work day**: [Fast Fourier Transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) implementation - **Signal processing applications**: Filtering, denoising, frequency analysis - **Advanced topics**: [Convolution](https://en.wikipedia.org/wiki/Convolution), [sampling theory](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem), [wavelets](https://en.wikipedia.org/wiki/Wavelet) ## References - [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) - [Distribution theory](https://en.wikipedia.org/wiki/Distribution_(mathematics)) - [Test functions](https://en.wikipedia.org/wiki/Test_function) - [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) - [Laurent Schwartz](https://en.wikipedia.org/wiki/Laurent_Schwartz) (distribution theory founder) - [Robert Strichartz](https://en.wikipedia.org/wiki/Robert_Strichartz) (functional analysis) - [Integration by parts](https://en.wikipedia.org/wiki/Integration_by_parts) - [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula) - [Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform)