---
id: wiki-2026-0508-signal-processing-foundations
title: Signal Processing Foundations
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [DSP, Digital Signal Processing, Fourier Analysis]
duplicate_of: none
source_trust_level: A
confidence_score: 0.92
verification_status: applied
tags: [signal-processing, dsp, fourier, filter, sampling-theorem]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: scipy / numpy / torchaudio
---

# Signal Processing Foundations

## 매 한 줄
> **"매 signal processing 은 시간/공간 신호의 frequency / scale 분해와 그 위의 transform"**. Fourier (1822) 의 heat-equation series 부터 매 Cooley–Tukey FFT (1965), 그리고 2026 의 audio LLM (Whisper-large-v4, MoonShine), neural-vocoder (BigVGAN-2), brain–computer-interface (Neuralink N1) 까지의 backbone.

## 매 핵심

### 매 핵심 정리
- **Sampling theorem (Nyquist)**: bandwidth B 의 신호는 `f_s ≥ 2B` 면 perfect reconstruct.
- **Convolution theorem**: time-domain convolution ↔ frequency-domain product.
- **Parseval**: energy 는 time/freq 에서 동일.
- **Uncertainty**: Δt · Δf ≥ 1/(4π) — 매 wavelet/STFT trade-off 의 근원.

### 매 Filters
- **FIR**: stable, linear-phase 가능, 매 group-delay 일정.
- **IIR**: 매 efficient 하지만 phase 비선형 + stability 주의.
- **Linear phase 의 필요**: 매 audio crossover, ECG.
- **Adaptive (LMS, RLS, Kalman)**: 매 noise-cancelling, echo.

### 매 Transforms
- **DFT/FFT**: 매 stationary 분석.
- **STFT**: 매 short-time spectrum (mel-spec backbone).
- **Wavelet (DWT)**: 매 multiscale.
- **Hilbert**: 매 instantaneous amplitude/phase.
- **Cepstrum**: 매 pitch / formant.

### 매 응용
1. Audio gen: BigVGAN-2 vocoder 의 multi-scale STFT loss.
2. ASR: Whisper-large-v4 의 80-mel + log-magnitude.
3. EEG/BCI: bandpower in θ/α/β/γ.
4. Radar / lidar: matched filter, FMCW range-doppler.

## 💻 패턴

### FFT-based PSD (Welch)
```python
from scipy.signal import welch

def psd(x, fs):
    f, p = welch(x, fs=fs, nperseg=1024, noverlap=512, window="hann")
    return f, p
```

### STFT + log-mel spectrogram
```python
import torchaudio

mel = torchaudio.transforms.MelSpectrogram(
    sample_rate=16000, n_fft=400, hop_length=160, n_mels=80,
    f_min=0, f_max=8000, power=2.0,
)
log_mel = lambda wav: (mel(wav) + 1e-6).log()
```

### FIR low-pass design (firwin)
```python
from scipy.signal import firwin, filtfilt

def lpf(x, fs, cutoff_hz, taps=129):
    b = firwin(taps, cutoff_hz, fs=fs, window="hamming")
    return filtfilt(b, [1.0], x)
```

### Butterworth IIR (zero-phase)
```python
from scipy.signal import butter, sosfiltfilt

def bandpass(x, fs, lo, hi, order=4):
    sos = butter(order, [lo, hi], btype="band", fs=fs, output="sos")
    return sosfiltfilt(sos, x)
```

### Wavelet denoise (PyWavelets)
```python
import pywt, numpy as np

def wavelet_denoise(x, wavelet="db4", level=4):
    coeffs = pywt.wavedec(x, wavelet, level=level)
    sigma = np.median(np.abs(coeffs[-1])) / 0.6745
    thr = sigma * np.sqrt(2 * np.log(len(x)))
    coeffs[1:] = [pywt.threshold(c, thr, mode="soft") for c in coeffs[1:]]
    return pywt.waverec(coeffs, wavelet)[:len(x)]
```

### Hilbert envelope
```python
from scipy.signal import hilbert

def envelope(x):
    a = hilbert(x)
    return np.abs(a), np.unwrap(np.angle(a))
```

### Resampling (polyphase)
```python
from scipy.signal import resample_poly

def to_16k(x, fs):
    from math import gcd
    g = gcd(fs, 16000)
    return resample_poly(x, 16000 // g, fs // g)
```

### CQT / chroma (music-domain)
```python
import librosa  # 0.11.x
chroma = librosa.feature.chroma_cqt(y=wav, sr=sr, n_chroma=12, hop_length=512)
```

## 매 결정 기준

| 상황 | Approach |
|---|---|
| Stationary spectrum | FFT / Welch PSD |
| Time-varying | STFT / Mel |
| Transient / multiscale | Wavelet |
| Pitch / formant | Cepstrum / autocorr |
| Linear-phase audio | FIR (filtfilt) |
| Real-time low-latency | IIR (biquad cascade) |
| Resampling | Polyphase (resample_poly) |

**기본값**: ML feature 는 log-mel spectrogram, 분석은 Welch PSD, denoise 는 wavelet / RNNoise, real-time 은 SOS biquad.

## 🔗 Graph
- 부모: [[Linear-Algebra-Foundations]] · [[Probability Theory|Probability-Theory-Foundations]]
- 변형: [[Entropy in Information Theory|Information Theory]] · [[Signal in Noise]]
- 응용: [[Kalman-Filter-and-State-Tracking]] · [[Computer_Vision]]
- Adjacent: [[Roughness (그래픽 및 물리)]] · [[Cross-Frequency Coupling (CFC)]]

## 🤖 LLM 활용
**언제**: audio/EEG pipeline, vocoder loss design, sensor preprocessing, time-series feature engineering.
**언제 X**: pure tabular data, NLP token streams (use sequence models directly).

## ❌ 안티패턴
- **Aliasing 무시**: anti-alias LPF 없이 downsample → 매 spectral fold-back.
- **Window 의 leakage 미고려**: rectangular window → sidelobe 심함, Hann/Blackman 사용.
- **filtfilt 를 real-time 에 사용**: 매 non-causal — production 은 단방향 IIR + group-delay 보정.
- **dB 의 reference 누락**: dBFS / dBSPL / dBm 명시 필수.

## 🧪 검증 / 검토
- Verified (Oppenheim & Schafer "Discrete-Time Signal Processing" 3rd ed.; Mallat "Wavelet Tour" 3rd ed.; torchaudio docs 2.5).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Nyquist/Parseval/uncertainty, FIR/IIR/wavelet/STFT patterns, 2026 audio-LLM context |