Signal Processing Foundations

Week 1

Course Overview & Foundations

Why DSP matters for AI, the mathematics of sine waves and Fourier synthesis, and building your first audio pipeline in Python.

• Why Signal Processing? The End-to-End AI Pipeline
• Signals as Math: Sine Wave Anatomy & Fourier Synthesis
• App: Pitch Synthesizer — hearing equations come alive

Week 2

Sampling & Aliasing

What it means to sample, the Nyquist-Shannon theorem, and how a wrong sample rate breaks audio and sensor data.

• Discrete signals: what it means to sample
• Nyquist-Shannon Theorem & Aliasing
• App: Choosing sample rate in librosa & resampling for ML

Week 3

LTI Systems & 1D Convolution

Linearity, time-invariance, impulse responses, and the sliding-window convolution mechanic.

• LTI Systems: Linearity, Time-Invariance & Impulse Response h[n]
• 1D Convolution x[n]∗h[n]: the sliding window mechanic
• App: Echo & reverb synthesis, noise-cancelling filters

Week 4

Discrete Fourier Transform

Time-domain vs. frequency-domain, the DFT summation formula and its inverse, with DTMF tone decoding as a live application.

• Time Domain vs. Frequency Domain — what the DFT reveals
• The DFT Equation: from summation to spectrum & back (IDFT)
• App: DTMF tone decoding — how a phone knows which key you pressed

Week 5

Fast Fourier Transform

Why the naïve DFT fails at scale, Cooley-Tukey's divide-and-conquer butterfly, and building a real-time spectrum analyzer.

• Why naïve DFT fails at scale: the O(N²) bottleneck
• Cooley-Tukey & the Butterfly: achieving O(N log N)
• App: Real-time audio spectrum analyzer with scipy.fft

Week 6

Spectral Analysis & Denoising

Identifying noise in the frequency domain, window functions to reduce spectral leakage, and an FFT denoising pipeline for ECG signals.

• Noise in the frequency domain: what a corrupted signal looks like
• Window functions & spectral leakage — choosing the right shape
• App: FFT denoising pipeline for ECG signals & audio restoration

Week 7

Digital Filters (FIR / IIR)

Windowed-sinc FIR design, IIR difference equations and z-domain poles/zeros, and band-pass filtering for bio-signals.

• FIR Filters: windowed-sinc design & why linear phase matters
• IIR Filters: difference equations & z-domain poles/zeros
• App: EEG/ECG band-pass filtering & audio equalizer

Week 9

Time-Frequency Analysis (STFT)

Why global Fourier fails on non-stationary signals, the STFT 2D time-frequency map, and speaker diarization in production.

• Why the global Fourier transform fails for non-stationary signals
• STFT: windowing, overlap-add & the 2D time-frequency map
• App: Speaker diarization & music onset detection

Week 10

Spectrograms & Mel-Scale

Reading time-frequency maps, the logarithmic Mel scale and human auditory perception, and building a full librosa feature pipeline.

• From STFT to spectrogram: reading time-frequency maps
• Mel scale, mel-filterbanks & human auditory perception
• App: librosa pipeline — raw audio → mel-spectrogram → speech model

Week 11

Wavelet Transform & Multi-Resolution

STFT's fixed-resolution problem, the DWT Multi-Resolution Analysis with approximation and detail coefficients, and ECG denoising with PyWavelets.

• STFT's fixed-resolution problem & the wavelet as a solution
• DWT & Multi-Resolution Analysis: approximation + details
• App: ECG denoising & image compression with PyWavelets

Week 12

2D Signals & Image Processing

Images as 2D discrete signals, the 2D DFT magnitude spectrum and phase, and MRI k-space reconstruction.

• Images as 2D discrete signals: pixels, matrices & spatial frequency
• 2D Fourier Transform: magnitude spectrum & phase interpretation
• App: MRI k-space reconstruction & texture-based defect detection

Week 13

2D Convolution & Kernels

Sliding kernel mechanics with zero-padding, Gaussian vs. Sobel/Laplacian operators, and how kernels become CNN feature detectors.

• 2D Convolution: sliding kernel mechanics & zero-padding rules
• Gaussian blur vs. Sobel/Laplacian — smoothing vs. edge sharpening
• App: Kernels as CNN feature detectors — bridging to deep learning

Week 14

PCA, Augmentation & GSP

Eigenvector-based dimensionality reduction, signal and image data augmentation strategies, and Graph Signal Processing on 3D point clouds.

• PCA: eigenvectors, variance capture & dimensionality reduction
• Data augmentation strategies for signal & image datasets
• App: Graph Laplacian & 3D point-cloud processing with PyGSP