DSP 101
M9 · L1
Module 9 — Spectral Analysis
Power Spectral Density
The essential tool for characterizing random signals — describing how power is distributed across frequencies.
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DSP 101
M9 · L1
The Core Idea
Power vs. Frequency
Random signals don't have a single DFT — they look different every time. PSD captures the average power at each frequency across all realizations.
- DFT gives amplitude/phase for deterministic signals
- PSD gives average power distribution for random signals
- Always real-valued and non-negative
- Units: watts/Hz or V²/Hz (or dBm/Hz in practice)
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DSP 101
M9 · L1
Wiener-Khinchin Theorem
PSD = DTFT of Autocorrelation
The PSD is the Fourier transform of the autocorrelation sequence — a profound link between time-domain correlation and spectral structure.
Wiener-Khinchin
S_{xx}(\omega) = \sum_{k=-\infty}^{\infty} R_{xx}[k]\, e^{-j\omega k}
Intuition
Short autocorrelation → broad flat PSD. Slowly decaying autocorrelation → low-frequency concentration
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DSP 101
M9 · L1
Power Relationship
Area Under PSD = Total Power
Integrating the PSD over all frequencies recovers the total signal power — linking spectral and time-domain representations.
R[0]
Total Power
∫S(ω)
PSD Integral
=
Always Equal
- dB scale: 10·log₁₀(S(f)) — shows weak and strong components
- Flat PSD → white noise (all frequencies equal power)
- Peaked PSD → colored noise or tonal signal
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DSP 101
M9 · L1
Two Conventions
One-Sided vs. Two-Sided
Real signals have symmetric PSD (S(ω) = S(−ω)), so both conventions carry the same information — but values differ.
- Two-sided: covers [−π, π] — area = total power
- One-sided: covers [0, π] — values doubled (G = 2S)
- Spectrum analyzers and pwelch() default to one-sided
- Complex (I/Q) signals require two-sided PSD
- 3 dB mismatch when conventions are confused!
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DSP 101
M9 · L1
PSD Estimation
The Periodogram
The simplest PSD estimate from finite data: squared DFT magnitude divided by N. Simple but fundamentally flawed.
Periodogram
\hat{S}(\omega) = \frac{1}{N}\left|\sum_{n=0}^{N-1} x[n]\, e^{-j\omega n}\right|^2
Critical weakness
Variance stays ~100% regardless of data length. More data = more frequency resolution, not less noise
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DSP 101
M9 · L1
Real-World Use
Where PSD Matters
- Noise floor: thermal noise at −174 dBm/Hz at 290 K
- Signal detection: spectral peaks above noise floor
- Wireless channel: frequency-selective fading profile
- Audio: room resonances, instrument harmonics
- Vibration: mechanical resonances, imbalance detection
- EEG: brain rhythm bands (δ, θ, α, β, γ waves)
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DSP 101
M9 · L1
Key Takeaways
What You Learned
- PSD = average power distribution across frequencies for random signals
- Wiener-Khinchin: PSD is DTFT of autocorrelation sequence
- Integrating PSD = total power (area = R_xx[0])
- One-sided doubles two-sided values; 3 dB error if confused
- Periodogram is simple but high-variance — next lesson: Welch's method
- White noise has flat PSD; thermal floor ≈ −174 dBm/Hz at 290 K
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