3.4 Energy and Power Spectral Density

Energy and power spectral density functions describe how a signal's energy or power is distributed across frequencies. They bridge the time-domain and frequency-domain views of a signal, making them essential for tasks like filter design, bandwidth estimation, and noise analysis.

This topic builds directly on the Fourier Transform. Once you can represent a signal in the frequency domain, spectral density tells you where the energy or power actually lives along the frequency axis.

Energy and Power Spectral Density Functions

Definitions and Applications

Energy Spectral Density (ESD) is the magnitude squared of a signal's Fourier transform. It tells you how the signal's energy is spread across different frequencies.

• Used for energy signals: signals with finite total energy, like transient pulses or decaying exponentials.
• Units: energy per unit frequency (joules/Hz).

Power Spectral Density (PSD) describes how the power of a signal is distributed over frequency. It's defined as the Fourier transform of the signal's autocorrelation function.

• Used for power signals: signals with finite average power but infinite total energy, like periodic signals or random processes. These signals go on forever, so total energy is infinite, but average power per unit time is finite.
• Units: power per unit frequency (watts/Hz).

The key distinction: if a signal eventually dies out, you use ESD. If it persists indefinitely, you use PSD.

Computation and Properties

The total energy of a signal equals the integral of its ESD over all frequencies:

$E_{total} = \int_{-\infty}^{\infty} E(\omega)\, d\omega$

Similarly, the average power equals the integral of the PSD:

$P_{avg} = \int_{-\infty}^{\infty} P(\omega)\, d\omega$

Both ESD and PSD are always non-negative:

$E(\omega) \geq 0 \quad \text{and} \quad P(\omega) \geq 0 \quad \text{for all } \omega$

This follows directly from the fact that they involve squared magnitudes.

Connection to the autocorrelation function: Both spectral densities relate to the autocorrelation function $R_x(\tau)$ through the Fourier transform. This relationship is known as the Wiener-Khinchin theorem. For energy signals, the ESD is the Fourier transform of the energy autocorrelation. For power signals, the PSD is the Fourier transform of the time-averaged autocorrelation:

$S_x(\omega) = \int_{-\infty}^{\infty} R_x(\tau)\, e^{-j\omega\tau}\, d\tau$

Fourier Transform and Spectral Density

Relationship between Fourier Transform and Spectral Density

For an energy signal $x(t)$ with Fourier transform $X(\omega)$, the ESD is simply:

$E(\omega) = |X(\omega)|^2$

You just take the Fourier transform and square its magnitude. That's it.

For a power signal, you can't directly take the Fourier transform (it may not converge). Instead, you truncate the signal to a window of duration $T$, compute the transform $X_T(\omega)$, and take a limit:

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

where $X_T(\omega)$ is the Fourier transform of $x(t)$ restricted to $[-T/2,\; T/2]$. Dividing by $T$ normalizes the result to give power rather than energy.

For any real-valued signal, both ESD and PSD are even functions of frequency:

$E(\omega) = E(-\omega) \quad \text{and} \quad P(\omega) = P(-\omega)$

This is because $|X(\omega)|^2 = X(\omega) \cdot X^*(\omega)$, and for real signals $X(-\omega) = X^*(\omega)$.

Parseval's Theorem

Parseval's theorem connects time-domain energy to frequency-domain energy:

$\int_{-\infty}^{\infty} |x(t)|^2\, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2\, d\omega = \frac{1}{2\pi} \int_{-\infty}^{\infty} E(\omega)\, d\omega$

The left side computes energy in the time domain; the right side computes it in the frequency domain. Both give the same answer. This is useful because sometimes one integral is much easier to evaluate than the other. It also confirms that the Fourier transform preserves energy (up to the $1/2\pi$ scaling factor that depends on which convention you use).

Signal Energy and Power Calculation

Continuous-Time Signals

To find total energy or average power from spectral density:

1. Compute the Fourier transform $X(\omega)$ of the signal.
2. Find the spectral density: $|X(\omega)|^2$ for ESD, or use the PSD limit formula for power signals.
3. Integrate over all frequencies:

$E_{total} = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(\omega)|^2\, d\omega \qquad \text{or} \qquad P_{avg} = \frac{1}{2\pi}\int_{-\infty}^{\infty} P(\omega)\, d\omega$

In practice, you'll often integrate over a specific frequency band rather than all frequencies to find the energy or power within that band.

Discrete-Time Signals

For a discrete-time signal $x[n]$ with discrete-time Fourier transform (DTFT) $X(e^{j\omega})$, the formulas are analogous but the integration is over $[-\pi, \pi]$ instead of $(-\infty, \infty)$:

• ESD: $E(\omega) = |X(e^{j\omega})|^2$
• Total energy: $E_{total} = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2\, d\omega$

For power signals, truncate to $N$ samples and take the limit:

• PSD: $P(\omega) = \lim_{N\to\infty} \frac{1}{N} |X_N(e^{j\omega})|^2$
• Average power: $P_{avg} = \frac{1}{2\pi} \int_{-\pi}^{\pi} P(\omega)\, d\omega$

The finite integration limits $[-\pi, \pi]$ reflect the fact that discrete-time signals have periodic spectra with period $2\pi$.

Properties of Spectral Density Functions

Non-Negativity and Bandwidth

Non-negativity ($E(\omega) \geq 0$, $P(\omega) \geq 0$) is guaranteed because spectral densities are defined through squared magnitudes. You can never have "negative energy" at a frequency.

Bandwidth is determined from the spectral density by identifying the range of frequencies where the density is significant. There are several common definitions:

• 3 dB bandwidth: the range of frequencies where the spectral density is at least half its peak value.
• Null-to-null bandwidth: the width of the main lobe (between the first zeros on either side of the peak).
• Fractional energy bandwidth: the narrowest band containing some percentage (e.g., 99%) of the total energy.

Signals with wider bandwidth contain more significant frequency components and generally require higher sampling rates and more storage or transmission capacity.

Applications in Signal Analysis and Processing

Spectral density functions show up throughout signal processing:

• Frequency analysis: Identify dominant frequency components, harmonics, and noise floors by examining peaks and shapes in the PSD.
• Sampling rate selection: The bandwidth visible in the spectral density tells you the minimum sampling rate needed (via the Nyquist criterion) for discrete-time processing.
• Filter design: If you know which frequency bands carry useful signal versus noise, you can design lowpass, highpass, or bandpass filters to keep what you want and reject the rest.
• Signal compression: Frequency components with very low spectral density contribute little to the signal. Discarding or coarsely quantizing them reduces data size with minimal perceptual loss.
• System identification: For a linear time-invariant (LTI) system, the output PSD relates to the input PSD through the system's transfer function: $S_y(\omega) = |H(\omega)|^2 S_x(\omega)$. This lets you estimate $|H(\omega)|^2$ by comparing input and output spectral densities.