Power Spectral Density | DSP-101

What is Power Spectral Density?

When we analyze a signal in the frequency domain using the DFT, we see the amplitude and phase of each frequency component. But for random or stochastic signals — noise, speech, vibration, wireless channels — the concept of a deterministic spectrum breaks down. These signals are never repeated twice, and their DFT looks different every time.

What remains stable across realizations is the statistical distribution of power across frequencies. This is captured by the Power Spectral Density (PSD): a function S(f) that describes how much signal power, on average, is concentrated near each frequency. Think of it as the "average energy landscape" of a random process.

PSD answers questions that a single DFT cannot: What frequency band dominates a noise floor? How wide is an interference signal? Where does the channel attenuate signal power? These are fundamental questions in communications, audio engineering, radar, and biomedical signal processing.

The Wiener-Khinchin Theorem

The mathematical foundation of PSD is the Wiener-Khinchin theorem, which links the PSD to the autocorrelation function of a signal. For a wide-sense stationary (WSS) random process x[n], the autocorrelation sequence is:

Autocorrelation Sequence

R_{xx}[k] = E\bigl[x[n]\,x^*[n-k]\bigr]

R_xx[k] measures how similar the signal is to a time-shifted version of itself. For k=0 it equals the total signal power. E[·] denotes statistical expectation.

The Wiener-Khinchin theorem states that the PSD is the Discrete-Time Fourier Transform (DTFT) of the autocorrelation sequence:

Wiener-Khinchin Theorem

S_{xx}(\omega) = \sum_{k=-\infty}^{\infty} R_{xx}[k]\, e^{-j\omega k}

S_xx(ω) is the PSD as a function of angular frequency ω ∈ [−π, π]. It is real-valued and non-negative everywhere — it represents power, not amplitude.

This theorem is profound: it tells us that the spectral structure of a random signal is completely encoded in its autocorrelation. A signal with a narrow autocorrelation (decorrelates quickly) has a broad, flat PSD — it is spectrally white. A signal with a slowly decaying autocorrelation has most of its power concentrated at low frequencies.

Units and Interpretation

PSD has units of power per unit frequency. In continuous time, this is watts/Hz (or equivalently, V²/Hz for voltage signals). In discrete time, the frequency axis is normalized to the interval [−π, π] (in radians/sample) or equivalently [−f_s/2, f_s/2] (in Hz). The key relationship is:

Total Power from PSD

P = R_{xx}[0] = \frac{1}{2\pi}\int_{-\pi}^{\pi} S_{xx}(\omega)\, d\omega

Integrating (or summing) the PSD over all frequencies recovers the total signal power — the area under the PSD curve equals R_xx[0], the zero-lag autocorrelation.

In practice, PSD is often displayed on a decibel scale (dB/Hz) to show both strong and weak spectral components simultaneously. A noise floor 60 dB below the signal peak would be invisible on a linear scale but clearly visible in dB. The conversion is: S_dB(f) = 10 · log₁₀(S(f)).

One-Sided vs. Two-Sided PSD

For a real-valued signal x[n], the PSD is symmetric: S_xx(ω) = S_xx(−ω). This means that the PSD for negative frequencies contains exactly the same information as for positive frequencies. Two conventions therefore exist:

Two-Sided PSD

Defined over [−π, π] or [−f_s/2, f_s/2]
Total area = total power (correct integration)
Used in theoretical derivations and complex signals
Required for complex baseband signals (I/Q data)
Default in most mathematical treatments

One-Sided PSD

Defined over [0, π] or [0, f_s/2]
Values doubled to preserve total power: G(f) = 2S(f)
More intuitive for real-world measurements
Used by spectrum analyzers and measurement tools
Default in MATLAB's pwelch() with default settings

Always Check Which Convention Is Used

When comparing PSD values from different tools or textbooks, confirm whether the one-sided or two-sided convention is in use. A 3 dB discrepancy often traces back to this confusion. MATLAB's pwelch() returns a one-sided PSD by default; Python's scipy.signal.welch() also returns one-sided by default.

One-sided PSD = 2 × two-sided PSD (at each positive frequency)

Estimating PSD from Finite Data

The true PSD is a theoretical quantity defined by an expectation over all realizations of a random process. In practice, we only have a finite data record x[0], x[1], …, x[N−1]. The simplest estimator is the periodogram:

The Periodogram

\hat{S}(\omega) = \frac{1}{N}\left|\sum_{n=0}^{N-1} x[n]\, e^{-j\omega n}\right|^2

X_N(ω) is the DFT of the N-sample window. Dividing by N normalizes by data length. This gives an asymptotically unbiased estimate but with high variance.

The periodogram has a critical weakness: its variance does not decrease as N increases. No matter how much data you collect, the periodogram remains noisy — each spectral estimate fluctuates by roughly 100% around its expected value. For reliable PSD estimates, we need more sophisticated approaches.

Practical Applications

PSD is not merely theoretical — it is the fundamental tool for spectral characterization in virtually every signal processing domain:

Application	What PSD Reveals	Key Parameter
Noise characterization	Thermal noise floor, colored noise shape	Noise power density (dBm/Hz)
Signal detection	Spectral peaks above noise floor	SNR per frequency bin
Wireless channel	Frequency-selective fading profile	Coherence bandwidth
Audio / acoustics	Room resonances, instrument tones	Spectral centroid, bandwidth
Vibration analysis	Mechanical resonances, imbalance	Peak frequencies, RMS per band
Biomedical (EEG)	Brain rhythm bands (δ, θ, α, β, γ)	Band power ratios

A classic example is white noise: a random signal whose samples are uncorrelated. Its autocorrelation is a scaled impulse (R_xx[k] = σ² δ[k]), and its PSD is flat: S_xx(ω) = σ² for all ω. This is why thermal noise is called "white" — by analogy with white light containing all visible frequencies equally.

White Noise and the Noise Floor

In communications receiver design, the thermal noise spectral density is N₀/2 watts/Hz (two-sided), where N₀ = kT (Boltzmann's constant × temperature). At room temperature (T = 290 K), N₀ ≈ −174 dBm/Hz — the universal noise floor limit. Any signal component must be above this floor to be detectable without extensive averaging.

Thermal noise floor: N₀ = kT ≈ −174 dBm/Hz at 290 K

Key Takeaways

Power Spectral Density (PSD) describes how signal power is distributed across frequencies — the essential tool for characterizing random or stochastic signals.
The Wiener-Khinchin theorem links PSD to autocorrelation: S_xx(ω) = DTFT{R_xx[k]}. PSD is real-valued and non-negative everywhere.
Integrating the PSD over all frequencies gives the total signal power: P = R_xx[0].
For real signals, PSD is symmetric about zero frequency. One-sided PSD doubles the two-sided values to cover only positive frequencies.
The periodogram (|DFT|²/N) is the simplest PSD estimator but has high variance that does not decrease with more data.
White noise has a flat PSD — equal power at every frequency. Colored noise has a frequency-dependent shape.
PSD is used across noise analysis, signal detection, wireless channels, audio, vibration, and biomedical applications.