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DSP Learning Roadmap

A structured learning path for mastering Digital Signal Processing fundamentals through hands-on implementation in Mojo.

Phase 1: Foundations

dsplib.core — Core Math & Types

Learning Objectives:

  • Understand complex number representation and arithmetic
  • Learn polar ↔ cartesian coordinate conversion
  • Master Euler's formula: $e^{j\omega} = \cos(\omega) + j\sin(\omega)$

Topics:

  • Complex number struct (real, imaginary, magnitude, phase)
  • Complex addition, subtraction, multiplication, division
  • Complex conjugate and magnitude calculation
  • Polar form (r, θ) ↔ Cartesian form (x, y) conversion
  • Unit circle visualization
  • Phasor representation of sinusoidal signals

Exercises:

  • Implement complex exponential $e^{j\omega t}$
  • Visualize phasor rotation on unit circle
  • Verify Euler's identity: $e^{j\pi} = -1$

Phase 2: Signal Generation

dsplib.waves — Waveforms & Noise

Learning Objectives:

  • Generate common waveform types used in signal processing
  • Understand signal parameters: amplitude, frequency, phase, offset
  • Learn about different noise distributions and their properties

Topics:

  • Sine wave generation with SIMD acceleration
  • Cosine wave (phase-shifted sine)
  • Square wave (via sign function)
  • Sawtooth / triangle wave
  • Signal parameters: amplitude, frequency (Hz), phase, DC offset
  • Time array generation (sampling instants)
  • Signal composition (adding multiple frequencies)
  • Normal (Gaussian) noise generation
  • Uniform noise generation
  • Signal-to-Noise Ratio (SNR) concepts
  • Adding noise to clean signals

Exercises:

  • Generate and plot 440 Hz sine wave (musical A)
  • Create a chord (multiple frequencies simultaneously)
  • Generate white noise and visualize frequency spectrum
  • Add 20 dB SNR noise to a signal

Phase 3: Frequency Analysis

dsplib.fourier — Fourier Transforms

Learning Objectives:

  • Understand the relationship between time and frequency domains
  • Learn how DFT converts discrete signals to frequency representation
  • Visualize frequency content of signals

Topics:

  • Discrete Fourier Transform (DFT) — O(N²) implementation
  • Inverse DFT (IDFT) — signal reconstruction
  • [S] Complex exponential basis functions $W_N^{kn} = e^{-j2\pi kn/N}$
  • Magnitude and phase spectra
  • Power spectrum and spectral energy
  • Parseval's theorem (energy conservation)
  • Fast Fourier Transform (FFT) — O(N log N) implementation
  • Zero-padding for frequency resolution
  • Windowing functions (Hann, Hamming, Blackman)
  • Spectral leakage and windowing tradeoffs

Exercises:

  • DFT of a pure sine wave → verify single frequency peak
  • DFT of multi-tone signal → identify all frequency components
  • Compare DFT vs FFT computational complexity
  • Apply Hann window to reduce spectral leakage
  • Zero-pad signal to interpolate frequency bins

Phase 4: Filtering

dsplib.filters — Digital Filters

Learning Objectives:

  • Understand how filters modify signal frequency content
  • Learn FIR vs IIR filter characteristics
  • Implement convolution and difference equations

Topics:

  • Filter fundamentals: passband, stopband, cutoff frequency
  • Convolution theorem and linear convolution
  • FIR filters (Finite Impulse Response)
    • Moving average filter
    • Windowed-sinc filter
    • Design using frequency sampling
  • IIR filters (Infinite Impulse Response)
    • Difference equations
    • Direct Form I and II implementations
  • Biquad filters (second-order sections)
    • Low-pass, high-pass, band-pass, band-stop
  • Filter design basics
    • Butterworth (maximally flat magnitude)
    • Chebyshev (ripple in passband/stopband)
  • Frequency response analysis (magnitude & phase plots)
  • Group delay
  • Stability considerations for IIR filters

Exercises:

  • Implement 3-point moving average filter
  • Design low-pass FIR filter with specific cutoff
  • Create 2nd-order Butterworth low-pass
  • Plot frequency response (magnitude in dB, phase)
  • Filter noisy signal and compare SNR before/after

Phase 5: Advanced Topics & Applications

dsplib.applications — Real-World DSP

Learning Objectives:

  • Apply DSP techniques to practical problems
  • Understand real-time processing considerations
  • Combine multiple techniques for complete solutions

Topics:

  • Audio Processing

    • Simple audio effects (echo, reverb via delay)
    • Equalizer design using biquad filters
    • Dynamic range compression

  • Spectral Analysis

    • Short-Time Fourier Transform (STFT)
    • Spectrogram visualization
    • Harmonic analysis of musical instruments

  • Signal Reconstruction

    • Zero-order hold interpolation
    • Linear interpolation
    • Sinc interpolation / ideal low-pass reconstruction

  • Resampling

    • Upsampling (zero-insertion + interpolation)
    • Downsampling (decimation + anti-aliasing)
    • Sample rate conversion

  • Correlation & Detection

    • Cross-correlation
    • Auto-correlation
    • Matched filtering for signal detection

  • Modulation Basics

    • AM modulation/demodulation
    • Complex baseband representation

Exercises:

  • Build a simple 3-band audio equalizer
  • Create spectrogram of a musical recording
  • Implement sample rate converter (e.g., 44.1kHz → 48kHz)
  • Detect a known signal buried in noise using matched filtering

Suggested Project Order

Phase 1 → Phase 2 → Phase 3 → Phase 4 → Phase 5
   ↓         ↓          ↓          ↓          ↓
 core     waves      fourier    filters   applications

Each phase builds upon the previous. Skipping phases may leave gaps in understanding.

Quick Reference: DSP Pipeline

Signal → Generation → Analysis → Filtering → Applications
              ↓            ↓           ↓
          time-domain   frequency   frequency
                         domain      shaping

Most DSP work follows this pattern: generate/analyze signals in time domain, examine frequency content, apply filters to shape the spectrum, and build applications.