# MATH310S26 - Day 17: Energy in Frequency Space, Sound Synthesis, and Spectrograms **Date**: February 25, 2026 **Topics**: Parseval's theorem, energy spectral density, waveform synthesis, time-frequency analysis via spectrograms **Previous Classes**: [[MATH310S26-Day16-WorkdayMaterials (FFT and frequency discovery)]], [[MATH310S26-Day15-Notes]], [[MATH310S26-Day14-Notes]] ## Administrative Notes - Guest presentation today: Former students (now seniors) presenting on "Whale Songs and Targeted Movement" - Low-stakes feedback will be a form + PDF with spectrogram interpretation questions - Canvas project updates returned to all but two students - Next Monday (March 2): Another work day planned ## Part 1: Fourier Transform Pairs and Energy ### Recall: The Fourier Transform Pair Starting with our fundamental transform relationships: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ $f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega t} d\omega$ Note the conjugate relationship: one uses $e^{-i\omega t}$ and the other uses $e^{i\omega t}$. ### Inner Product Interpretation The Fourier transform can be understood as an inner product in function space: $\hat{f}(\omega) \propto \langle f(t), e^{-i\omega t} \rangle$ This connects to the abstract notion of [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space), where functions behave like vectors in infinite-dimensional space, maintaining familiar properties from $\mathbb{R}^n$. ## Part 2: Physical Interpretation - Simple Harmonic Oscillators ### Key Points 1. **For each $\omega$**: We have a [simple harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator) 2. **Each oscillator has energy**: $E \propto$ amplitude² 3. **Complex amplitudes**: Energy is $E \propto |\hat{f}(\omega)|^2 = \hat{f}(\omega) \cdot \hat{f}^*(\omega)$ ### Total Energy and Parseval's Theorem To get the total energy across all frequencies: $E_{total} \propto \int_{-\infty}^{\infty} |\hat{f}(\omega)|^2 d\omega = ||\hat{f}||^2_{\mathbb{R}}$ This integral represents the **norm** of $\hat{f}$ on $\mathbb{R}$, connecting to [Parseval's theorem](https://en.wikipedia.org/wiki/Parseval%27s_theorem). **Physical contexts**: - **Sound**: Loudness per frequency - **Light**: Intensity per color (photon energies based on frequency) - **Data**: Energy per frequency component ## Part 3: Sinc Function and Energy Spectral Density ### Rectangular Pulse Transform (Review) Recall from [[MATH310S26-Day15-Notes]] that the rectangular pulse has Fourier transform: $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \cdot \frac{\sin(\omega L)}{\omega L}$ This is the [sinc function](https://en.wikipedia.org/wiki/Sinc_function), graphically appearing as:  ### From Amplitude to Energy: The Squaring Effect When we square $\hat{f}(\omega)$ to get energy spectral density $|\hat{f}(\omega)|^2$: - Values between 0 and 1 → get smaller - Values greater than 1 → get larger - Negative regions → become positive - Overall effect: Enhanced peak, suppressed sidelobes **Diffraction pattern connection**: Looking at $|\hat{f}(\omega)|^2$ from above (bright = high energy, dark = low energy) gives the classic diffraction pattern observed with lasers - bright central maximum with alternating bright/dark fringes. ## Part 4: Sound Synthesis and Waveforms Remember the flute video.  ### Classic Video Game Sound Synthesis Early video game systems (NES ~50KB storage) used simple waveforms for efficient sound generation. Thinking about these sounds we look at the following "simple sounds" whose original video can be found at [https://www.youtube.com/watch?v=TlzXxkftxvk](https://www.youtube.com/watch?v=TlzXxkftxvk) #### Pure Sine Wave  - Single frequency spike in spectrum - "Clinical" or "artificial" sound quality - Minimal storage requirements #### Square Wave  - Strong harmonics at odd multiples of fundamental - Harsh, "8-bit" sound character - Used for aggressive game sounds (death sequences shown from Super Mario) - Energy distributed significantly into harmonics #### Triangle Wave  - Harmonics present but decay more rapidly than square wave - Softer, less harsh than square wave - More "musical" while still computationally simple - Better approximation to natural instruments ### Harmonic Content and Timbre The distribution of energy across harmonics determines the [timbre](https://en.wikipedia.org/wiki/Timbre) or "color" of a sound: - **More harmonic energy** → Harsher, brighter sound - **Rapid harmonic decay** → Softer, mellower sound - **Pure fundamental only** → Artificial, sine-like quality Someone repurposed an organ to make these chipsounds.  ## Part 5: Spectrograms - Time-Frequency Analysis ### What is a Spectrogram? A [spectrogram](https://en.wikipedia.org/wiki/Spectrogram) represents how frequency content changes over time:  - **Horizontal axis**: Time - **Vertical axis**: Frequency - **Color/Brightness**: Energy/amplitude at that time-frequency point - Created by taking Fourier transforms of small time windows ([Short-time Fourier transform](https://en.wikipedia.org/wiki/Short-time_Fourier_transform)) ### Visualizing Musical Structures #### Example: Violin  Shows fundamental frequency and harmonic series characteristic of bowed string instruments. ### Applications: Music Recognition (Shazam) The app [Shazam](https://en.wikipedia.org/wiki/Shazam_(application)) works by: 1. Recording audio snippet 2. Computing spectrogram via FFT 3. Identifying peak patterns ("fingerprint") 4. Database lookup against known song fingerprints ## Part 6: Advanced Vocal Techniques ### Singer Examples Analyzed #### 1. Pure Sine (Synthesized)  Single horizontal line - one pure frequency maintained over time. #### 2. Square Wave (Synthesized)  Multiple horizontal lines showing fundamental and odd harmonics. #### 3. Triangle Wave (Synthesized)  Harmonics present but with decreasing intensity. #### 4. Polyphonic/Overtone Singing  - One person singing two notes simultaneously - Constant fundamental (drone) plus moving overtone melody - Achieved through precise control of vocal tract resonances - Demonstrated by Ana Maria (overtone singer) in class video #### 5. Deep Bass Singing  Singer attempting extremely low frequencies - fundamental and harmonics all descending together. #### 6. Yodeling  - Characteristic rapid jumps between frequencies - Abrupt transitions visible as discontinuous horizontal lines - Energy rapidly shifting between different frequency bands Original source of the last three videos.  - The following is a video I made showing the polyphonic singer with the spectrogram visualization.  - Lastly, here are the two spectrograms of our mario sounds. Can you spot the one that is from the 8-bit days? ![[Teaching/MATH310/MATH310S26/Lecture Notes/Media/Day17 Media/NES.png]] ![[Teaching/MATH310/MATH310S26/Lecture Notes/Media/Day17 Media/Wonder.png]] ## Part 7: Introduction to Signal Processing - RC Circuits ### The RC Circuit Differential Equation Consider a resistor-capacitor circuit with input signal $E(t)$: $R\dot{q} + \frac{1}{C}q = E(t)$ Where: - $R$ = resistance - $C$ = capacitance - $q$ = charge - $E(t)$ = external signal (forcing function) ### Low-Pass Filtering This [RC circuit](https://en.wikipedia.org/wiki/RC_circuit) acts as a **[low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter)**: - Attenuates high-frequency components - Allows low frequencies to "pass through" - In audio: Reduces treble, emphasizes bass - Makes piccolos quieter, tubas remain loud This connects our Fourier analysis to practical signal processing applications. ## Check Your Understanding Based on the quiz shown in the lecture materials: 1. **Fourier Transform Practice**: Calculate the Fourier transform of $f(t) = t$ on $(-1, 1)$ and zero otherwise. 2. **Spectrogram Interpretation**: Given spectrograms, describe: - What type of sound/signal created it - Key features (harmonics, frequency changes, energy distribution) - Whether the sound would be harsh/soft, simple/complex 3. **Energy and Squaring**: Explain why $|\hat{f}(\omega)|^2$ represents energy per frequency and how squaring affects the visual representation. ## Key Takeaways 1. **Energy in frequency space** is given by $|\hat{f}(\omega)|^2$, connecting to physical quantities like loudness and intensity 2. **Spectrograms** provide time-frequency analysis, revealing how frequency content evolves - crucial for music analysis, speech processing, and signal identification 3. **Simple waveforms** (sine, square, triangle) have characteristic harmonic signatures that determine their timbre and applications 4. **Signal processing** operations like filtering can be understood through their effect on frequency components 5. **Mathematical abstraction meets physical reality**: Fourier analysis provides a bridge between abstract function spaces and measurable physical phenomena ## Mathematical Connections - **Previous**: Built on [[MATH310S26-Day15-Notes]] (sinc function) and [[MATH310S26-Day16-WorkdayMaterials (FFT and frequency discovery)]] (FFT applications) - **Upcoming**: RC circuit analysis, convolution theorem, filtering applications - **Foundational**: Continues our progression from [[MATH310S26-Day7-Notes]] (Fourier series) through continuous transforms - **Applications**: Connects to signal processing, audio engineering, and pattern recognition