Digital Signal Processing and Machine Learning Roadmap: 5-Hub Learning Path with scipy.signal/numpy.fft/pywt/sklearn

Why a Roadmap

Digital signal processing (DSP) and machine learning (ML) each contain a vast number of methods, and learners often stall on where to start. In practice the pipeline is “sensor data → preprocessing (filter) → features → ML model”, so you need a path that interleaves the two.

This blog has built five aggregation hubs over the past cycles:

1. https://yuhi-sa.github.io/en/posts/20260521_bode_plot/1/ — Bode plot hub
2. https://yuhi-sa.github.io/en/posts/20260522_monte_carlo_optimization/1/ — Monte Carlo optimization hub
3. https://yuhi-sa.github.io/en/posts/20260522_filter_design_guide/1/ — Filter design guide hub
4. https://yuhi-sa.github.io/en/posts/20260524_time_frequency_guide/1/ — Time-frequency analysis hub
5. https://yuhi-sa.github.io/en/posts/20260525_ml_timeseries_guide/1/ — ML time-series hub

This article ties them together as a level- and goal-oriented learning roadmap.

1. Hub Overview

2. Learning Path Diagram

[Basic Math]
  │ complex numbers, linear algebra, probability
  ↓
[Fourier Analysis] ── FFT intro ──┐
  │                                │
  ↓                                ↓
[Bode Plot Hub]       [Filter Design Hub]
  │                                │
  └──────────┬─────────────────────┘
             ↓
   [Time-Frequency Hub]
             │
             ↓
   [ML Time-Series Hub] ← [Monte Carlo Opt Hub]
             │             (hyperparameter tuning)
             ↓
        [Applications]

3. Level-Based Tracks

Level 1: Complete Beginner

Goal: get comfortable with Python, numpy, matplotlib.

1. https://yuhi-sa.github.io/en/posts/20260223_matplotlib_tips/1/ — matplotlib essentials
2. https://yuhi-sa.github.io/en/posts/20210514_py_print/1/ — Python print overwriting
3. https://yuhi-sa.github.io/en/posts/20260225_fft/1/ — FFT introduction
4. https://yuhi-sa.github.io/en/posts/20260225_moving_average/1/ — moving averages
5. https://yuhi-sa.github.io/en/posts/20220206_ema/1/ — EMA frequency response

Estimated 2–4 weeks.

Level 2: With Math Basics

Goal: build intuition for frequency response and filter design.

1. https://yuhi-sa.github.io/en/posts/20260521_bode_plot/1/
2. https://yuhi-sa.github.io/en/posts/20260223_lowpass_filter/1/
3. https://yuhi-sa.github.io/en/posts/20260226_butterworth/1/
4. https://yuhi-sa.github.io/en/posts/20260314_chebyshev_filter/1/
5. https://yuhi-sa.github.io/en/posts/20260316_bessel_filter/1/
6. https://yuhi-sa.github.io/en/posts/20260226_fir_iir/1/
7. https://yuhi-sa.github.io/en/posts/20260522_filter_design_guide/1/

Estimated 4–8 weeks.

Level 3: Python-Comfortable

Goal: master time-frequency analysis for non-stationary signals.

1. https://yuhi-sa.github.io/en/posts/20260429_stft/1/
2. https://yuhi-sa.github.io/en/posts/20260226_wavelet/1/
3. https://yuhi-sa.github.io/en/posts/20260522_wavelet_packet/1/
4. https://yuhi-sa.github.io/en/posts/20260318_hilbert_transform/1/
5. https://yuhi-sa.github.io/en/posts/20260528_mode_decomposition/1/
6. https://yuhi-sa.github.io/en/posts/20260524_time_frequency_guide/1/

Estimated 8–12 weeks.

Level 4: Applied Practitioner

Goal: combine ML and stochastic optimization with DSP.

1. https://yuhi-sa.github.io/en/posts/20260228_timeseries_anomaly/1/
2. https://yuhi-sa.github.io/en/posts/20260226_kmeans_gmm/1/
3. https://yuhi-sa.github.io/en/posts/20260226_ensemble_learning/1/
4. https://yuhi-sa.github.io/en/posts/20260317_lstm_timeseries/1/
5. https://yuhi-sa.github.io/en/posts/20260525_ml_timeseries_guide/1/
6. https://yuhi-sa.github.io/en/posts/20260223_bayesian_optimization/1/
7. https://yuhi-sa.github.io/en/posts/20210329_cem/1/
8. https://yuhi-sa.github.io/en/posts/20260226_genetic_algorithm/1/
9. https://yuhi-sa.github.io/en/posts/20260215_mppi/1/
10. https://yuhi-sa.github.io/en/posts/20260522_monte_carlo_optimization/1/

Estimated 12–20 weeks.

4. Field-Specific Depth Order

• A. Frequency response → filter design: scipy.signal.TransferFunction → bode → butter → cheby1/2 → bessel → iirnotch.
• B. Fourier → Time-frequency: numpy.fft.fft → scipy.signal.windows → welch → stft → pywt.cwt → hilbert → PyEMD.EMD / vmdpy.VMD.
• C. State estimation: Kalman → EKF → UKF → Particle filter → RTS smoother.
• D. ML time-series: KMeans → GaussianMixture → RandomForestClassifier → GradientBoostingClassifier → keras.layers.LSTM → IsolationForest.
• E. Stochastic optimization: numpy.random → scipy.optimize.minimize → CEM → SA → GA → MPPI → Bayesian optimization (scikit-optimize).

5. Cross-Cutting Topics

• Feature engineering: STFT band-power, Wavelet energy, HHT instantaneous frequency as feature vectors for RandomForest classification.
• Anomaly detection: Notch + STFT preprocessing → IsolationForest scoring.
• Forecasting: SSA trend removal → LSTM short-term prediction.
• Hyperparameter tuning: Bayesian optimization for filter cutoff or LSTM hidden size.

See each hub’s “Applications” section and the integrated evaluation code in https://yuhi-sa.github.io/en/posts/20260525_ml_timeseries_guide/1/.

6. Learning Checklist (40 items)

Math basics

1. Physical meaning of \(e^{j\omega t}\)
2. Relation between Laplace and Fourier transforms
3. Transfer function of an LTI system
4. Parseval’s identity
5. Nyquist theorem

Bode / Filter design

6. Define cutoff at \(-3\) dB
7. Compare Butterworth and Chebyshev passband/stopband
8. Why Bessel has flat group delay
9. Chain scipy.signal.butter → lfilter
10. IIR vs FIR stability and linear phase
11. Notch \(Q\) vs bandwidth
12. Role of bilinear z-transform

Fourier / Time-frequency

13. Output length and frequency axis of numpy.fft.fft
14. Window functions vs spectral leakage
15. Units of PSD (V²/Hz)
16. Choose STFT frame and hop length
17. CWT vs DWT
18. Why Wavelet Packet has higher resolution than CWT
19. Hilbert transform and analytic signal
20. Mode mixing in EMD vs VMD
21. Choose \(L\) in SSA from period

ML time-series

22. K-means K selection (elbow / silhouette)
23. EM updates for GMM
24. RandomForest vs GBDT
25. LSTM gates (input/forget/output)
26. IsolationForest scoring
27. Avoid leakage in time-series train/val/test split
28. Kalman filter prediction / update
29. EKF vs UKF Jacobian-free advantage

Stochastic optimization

30. CEM elite rate
31. SA temperature schedule
32. GA crossover / mutation
33. MPPI as weighted CEM
34. BO acquisition functions (EI / PI / UCB)

Integration

35. Sensor → filter → feature → classifier pipeline
36. STFT features → IsolationForest
37. SSA trend removal then LSTM
38. Hyperparameter BO implementation
39. Multi-subplot visualization with matplotlib
40. Reproducibility with numpy.random.seed

7. Common Stumbling Blocks Q&A

Q1. FFT frequency axis is confusing. Use np.fft.fftfreq(N, d=1/fs) or np.fft.rfftfreq.

Q2. The filter rings. scipy.signal.butter expects normalized frequency relative to fs/2; pass Wn=cutoff/(fs/2) or use fs=fs kwarg.

Q3. LSTM does not learn. Most often due to missing scaling — apply StandardScaler first.

Q4. Cannot pick STFT window length. Aim for 5–10 periods of the target frequency: nperseg = int(5 / target_freq * fs).

Q5. EMD or VMD? VMD when mode count is known; EMD (EEMD) for adaptive exploration. See https://yuhi-sa.github.io/en/posts/20260528_mode_decomposition/1/.

Q6. Anomaly detection has false positives. Use Hilbert envelope, Wavelet energy, or STFT band averages so features reflect locality. Tune contamination of IsolationForest.

Q7. BO budget runs out. Switch acquisition from EI to UCB after 50 trials with no progress.

8. Related Articles

Five hubs:

• https://yuhi-sa.github.io/en/posts/20260521_bode_plot/1/
• https://yuhi-sa.github.io/en/posts/20260522_monte_carlo_optimization/1/
• https://yuhi-sa.github.io/en/posts/20260522_filter_design_guide/1/
• https://yuhi-sa.github.io/en/posts/20260524_time_frequency_guide/1/
• https://yuhi-sa.github.io/en/posts/20260525_ml_timeseries_guide/1/

Component articles:

• https://yuhi-sa.github.io/en/posts/20260225_fft/1/ / https://yuhi-sa.github.io/en/posts/20260228_fft_window_psd/1/ / https://yuhi-sa.github.io/en/posts/20260429_stft/1/
• https://yuhi-sa.github.io/en/posts/20260226_wavelet/1/ / https://yuhi-sa.github.io/en/posts/20260522_wavelet_packet/1/ / https://yuhi-sa.github.io/en/posts/20260318_hilbert_transform/1/ / https://yuhi-sa.github.io/en/posts/20260528_mode_decomposition/1/
• https://yuhi-sa.github.io/en/posts/20260226_butterworth/1/ / https://yuhi-sa.github.io/en/posts/20260314_chebyshev_filter/1/ / https://yuhi-sa.github.io/en/posts/20260316_bessel_filter/1/ / https://yuhi-sa.github.io/en/posts/20260226_fir_iir/1/
• https://yuhi-sa.github.io/en/posts/20260228_notch_filter/1/ / https://yuhi-sa.github.io/en/posts/20260312_bandpass_filter/1/ / https://yuhi-sa.github.io/en/posts/20260313_highpass_filter/1/ / https://yuhi-sa.github.io/en/posts/20260223_lowpass_filter/1/
• https://yuhi-sa.github.io/en/posts/20260224_kalman_filter/1/ / https://yuhi-sa.github.io/en/posts/20260224_ekf/1/ / https://yuhi-sa.github.io/en/posts/20260226_ukf/1/ / https://yuhi-sa.github.io/en/posts/20260223_particle_filter/1/ / https://yuhi-sa.github.io/en/posts/20260223_rts_smoother/1/
• https://yuhi-sa.github.io/en/posts/20260226_kmeans_gmm/1/ / https://yuhi-sa.github.io/en/posts/20260226_ensemble_learning/1/ / https://yuhi-sa.github.io/en/posts/20260317_lstm_timeseries/1/ / https://yuhi-sa.github.io/en/posts/20260228_timeseries_anomaly/1/
• https://yuhi-sa.github.io/en/posts/20260223_bayesian_optimization/1/ / https://yuhi-sa.github.io/en/posts/20210329_cem/1/ / https://yuhi-sa.github.io/en/posts/20260226_simulated_annealing/1/ / https://yuhi-sa.github.io/en/posts/20260226_genetic_algorithm/1/ / https://yuhi-sa.github.io/en/posts/20260215_mppi/1/