Power Spectral Density

Introduction

Power Spectral Density (PSD) describes how the power of a signal is distributed across different frequencies. While the Fourier Transform magnitude squared gives the energy spectral density for energy signals, many practical signals (like noise, speech in progress, or ongoing communication signals) have infinite total energy but finite average power. For these power signals, the PSD provides the meaningful frequency-domain characterization.

The PSD tells you how much average power exists per unit frequency bandwidth. If you measure the PSD of thermal noise and see it is flat across all frequencies, you know the noise power in any frequency band is proportional to that band's width. If you see a peak in the PSD of a vibration signal at 60 Hz, you know there is a strong periodic component at that frequency contributing significant power.

Understanding PSD is essential for practical engineering because most real-world signals of interest — communication waveforms, biomedical signals, seismic data, audio — are ongoing processes best characterized by their average power distribution rather than total energy.

Definition

For a power signal $x(t)$ with finite average power, the PSD is defined as:

$$S_{xx}(\omega) = \lim_{T \to \infty} \frac{1}{2T}|X_T(j\omega)|^2$$

where $X_T(j\omega)$ is the Fourier Transform of the signal truncated to $[-T, T]$.

The total average power is recovered by integrating the PSD: $$P_x = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}(\omega)d\omega$$

Units: The PSD has units of power per frequency (e.g., watts/Hz or V²/Hz). If $x(t)$ is measured in volts, then $S_{xx}$ has units V²/Hz.

Power in a frequency band: The power contained between frequencies $\omega_1$ and $\omega_2$ is: $$P_{[\omega_1,\omega_2]} = \frac{1}{2\pi}\int_{\omega_1}^{\omega_2}S_{xx}(\omega)\,d\omega + \frac{1}{2\pi}\int_{-\omega_2}^{-\omega_1}S_{xx}(\omega)\,d\omega$$

For real signals where $S_{xx}$ is even, this simplifies to: $$P_{[\omega_1,\omega_2]} = \frac{1}{\pi}\int_{\omega_1}^{\omega_2}S_{xx}(\omega)\,d\omega$$

Wiener-Khinchin Theorem

The PSD and autocorrelation function form a Fourier Transform pair:

$$S_{xx}(\omega) = \mathcal{F}\{R_{xx}(\tau)\} = \int_{-\infty}^{\infty} R_{xx}(\tau)e^{-j\omega\tau}d\tau$$

$$R_{xx}(\tau) = \mathcal{F}^{-1}\{S_{xx}(\omega)\} = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}(\omega)e^{j\omega\tau}d\omega$$

This fundamental result connects time-domain correlation structure to frequency-domain power distribution. Computing the PSD via the autocorrelation (correlogram method) is one of two standard estimation approaches.

Significance: The Wiener-Khinchin theorem tells us that knowing how correlated a signal is at various time lags is equivalent to knowing its power distribution across frequency. Signals with rapidly decaying autocorrelation (short memory) have wide-bandwidth PSD. Signals with slowly decaying autocorrelation (long memory) have narrow-bandwidth PSD concentrated at low frequencies.

Properties of PSD

1. Non-negative: $S_{xx}(\omega) \geq 0$ for all $\omega$ (power cannot be negative)
2. Real-valued: Even though $X(j\omega)$ is complex, $S_{xx}$ is always real
3. Even function: $S_{xx}(-\omega) = S_{xx}(\omega)$ for real signals
4. Area equals power: $P_x = \frac{1}{2\pi}\int S_{xx}(\omega)d\omega = R_{xx}(0)$
5. PSD of filtered signal: If $y = h*x$, then $S_{yy}(\omega) = |H(j\omega)|^2 S_{xx}(\omega)$
6. Linearity for uncorrelated inputs: If $x_1$ and $x_2$ are uncorrelated, $S_{x_1+x_2} = S_{x_1} + S_{x_2}$

Worked Example: PSD of a Filtered Signal

Problem: White noise with PSD $S_{nn}(\omega) = N_0/2$ passes through a first-order low-pass filter $H(j\omega) = \frac{1}{1+j\omega/\omega_c}$. Find the output PSD and total output power.

Solution: Using the filtering property: $$S_{yy}(\omega) = |H(j\omega)|^2 S_{nn}(\omega) = \frac{1}{1+(\omega/\omega_c)^2}\cdot\frac{N_0}{2}$$

$$S_{yy}(\omega) = \frac{N_0/2}{1+\omega^2/\omega_c^2}$$

The output PSD is no longer flat — it is shaped by the filter's magnitude response. High-frequency noise has been attenuated.

Total output power: $$P_y = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{N_0/2}{1+\omega^2/\omega_c^2}\,d\omega = \frac{N_0}{4\pi}\cdot\pi\omega_c = \frac{N_0\omega_c}{4}$$

Interpretation: The output power equals the noise PSD level ($N_0/2$) multiplied by the equivalent noise bandwidth of the filter ($\omega_c/2$ rad/s or $f_c\pi/2$ Hz for a first-order system). Wider bandwidth filters pass more noise power.

PSD of Periodic Signals

For a periodic signal with Fourier Series coefficients $c_n$: $$S_{xx}(\omega) = 2\pi\sum_{n=-\infty}^{\infty}|c_n|^2\delta(\omega - n\omega_0)$$

The PSD consists of impulses at the harmonic frequencies, with areas proportional to the power in each harmonic. This is consistent with Parseval's theorem for Fourier Series: $P = \sum|c_n|^2$.

Example: For $x(t) = A\cos(\omega_0 t)$, the PSD is: $$S_{xx}(\omega) = \frac{\pi A^2}{2}[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$$

Total power: $\frac{1}{2\pi}\cdot 2\cdot\frac{\pi A^2}{2} = \frac{A^2}{2}$ ✓

PSD Estimation Methods

Periodogram

The simplest estimator — compute the DFT of the data and square the magnitude: $$\hat{S}_{xx}(\omega_k) = \frac{1}{N}|X[k]|^2$$

Problem: high variance (noisy estimate) even for large $N$. The periodogram is an inconsistent estimator — its variance does not decrease as $N$ increases. Each frequency bin fluctuates wildly.

Welch's Method (Averaged Periodogram)

Divide data into overlapping segments, window each segment, compute periodograms, and average:

1. Split $N$ samples into $K$ segments of length $L$ with overlap $D$
2. Apply a window function (Hanning, Hamming) to each segment
3. Compute periodogram of each windowed segment
4. Average the $K$ periodograms

• Reduces variance by approximately factor $K$
• Trades frequency resolution for reduced variance (shorter segments = less resolution)
• The standard method in practice (used by MATLAB's pwelch and SciPy's signal.welch)

Correlogram Method

Estimate the autocorrelation $\hat{R}_{xx}(\tau)$, window it, then take the Fourier Transform. Based directly on the Wiener-Khinchin theorem. The window applied to the autocorrelation estimate controls the bias-variance tradeoff of the spectral estimate.

Multitaper Method

Uses multiple orthogonal tapers (DPSS windows) to produce independent spectral estimates that are then averaged. Provides excellent bias and variance properties but requires more computation. Preferred for short data records where every sample matters.

White Noise

White noise has a flat (constant) PSD across all frequencies: $$S_{nn}(\omega) = N_0/2 \quad \text{for all } \omega$$

The corresponding autocorrelation is an impulse: $R_{nn}(\tau) = (N_0/2)\delta(\tau)$, meaning different time samples are completely uncorrelated. White noise is the reference noise model in communications and filter theory.

Total power of ideal white noise: Integrating flat PSD over all frequencies gives infinite power — a mathematical idealization. Real noise always has finite bandwidth, but the white noise model is accurate whenever the noise bandwidth far exceeds the system bandwidth.

Colored Noise

Real noise is often "colored" (non-flat PSD):

• Pink noise ($1/f$): Power decreases with frequency. Common in electronic devices, financial data, and biological signals. Each octave carries equal power.
• Brown noise ($1/f^2$): Even steeper rolloff. Models Brownian motion and random walks. Generated by integrating white noise.
• Bandlimited white noise: Flat within a band, zero outside. The realistic version of white noise.
• Blue noise ($f$): Power increases with frequency. Used in dithering algorithms.

Applications

Noise characterization: Determining the noise floor and spectral shape in measurement systems allows engineers to identify noise sources and design appropriate mitigation.

Filter design: Knowing the signal and noise PSDs allows designing optimal Wiener filters that minimize mean-square error between estimated and true signals.

Communication system design: Allocating power across frequency bands (water-filling algorithm) based on channel noise PSD maximizes channel capacity.

Vibration analysis: Identifying resonant frequencies in mechanical systems from PSD peaks enables predictive maintenance and structural health monitoring.

Biomedical signal processing: EEG signals are characterized by their PSD, with clinically significant frequency bands (delta, theta, alpha, beta) defined by power concentration regions.

Key Takeaways

• PSD describes power distribution across frequency for power signals (infinite energy, finite power)
• Wiener-Khinchin theorem: PSD = Fourier Transform of autocorrelation
• PSD is always real, non-negative, and even (for real signals)
• Filtered signal PSD: $S_{yy} = |H|^2 S_{xx}$ — the system shapes spectral power
• White noise has flat PSD; colored noise has frequency-dependent PSD
• Welch's method (averaged periodogram) is the standard practical estimation technique
• Power in a band equals the integral of PSD over that band (divided by $2\pi$)
• PSD units are watts/Hz (or V²/Hz), representing power density per unit bandwidth