3.4 Energy and Power Spectral Density

Energy and power spectral density functions are key tools in signal analysis. They show how a signal's energy or power is spread across frequencies, helping us understand its composition and behavior.

These concepts are crucial in the Fourier Transform chapter. They link time-domain signals to their frequency-domain representations, allowing us to analyze and process signals more effectively in various applications.

Energy and Power Spectral Density Functions

Definitions and Applications

• Define the energy spectral density function as the magnitude squared of the Fourier transform of a signal
  • Describes how the energy of a signal is distributed over different frequencies
  • Used for signals with finite energy (transient signals, pulses)
• Define the power spectral density function as the Fourier transform of the autocorrelation function of a signal
  • Describes how the power of a signal is distributed over different frequencies
  • Used for signals with finite average power but infinite energy (periodic signals, random processes)
• Specify the units of energy spectral density as energy per unit frequency (joules/Hz) and power spectral density as power per unit frequency (watts/Hz)

Computation and Properties

• Express the total energy of a signal as the integral of the energy spectral density function over all frequencies
  • $E_{total} = \int_{-\infty}^{\infty} E(\omega) d\omega$
• Express the average power of a signal as the integral of the power spectral density function over all frequencies
  • $P_{avg} = \int_{-\infty}^{\infty} P(\omega) d\omega$
• State that energy and power spectral density functions are always non-negative
  • $E(\omega) \geq 0$ and $P(\omega) \geq 0$ for all $\omega$
• Relate the energy and power spectral density functions to the autocorrelation function of a signal via the Fourier transform
  • $E(\omega) = \int_{-\infty}^{\infty} R_x(\tau) e^{-j\omega\tau} d\tau$ and $P(\omega) = \int_{-\infty}^{\infty} R_x(\tau) e^{-j\omega\tau} d\tau$, where $R_x(\tau)$ is the autocorrelation function of $x(t)$

Fourier Transform and Spectral Density

Relationship between Fourier Transform and Spectral Density

• Express the energy spectral density function $E(\omega)$ in terms of the Fourier transform $X(\omega)$ of a signal $x(t)$
  • $E(\omega) = |X(\omega)|^2$
• Express the power spectral density function $P(\omega)$ in terms of the Fourier transform $X(\omega)$ of a signal $x(t)$
  • $P(\omega) = \lim_{T\to\infty} \frac{1}{T} |X_T(\omega)|^2$, where $X_T(\omega)$ is the Fourier transform of the truncated signal $x(t)$ over the interval $[-T/2, T/2]$
• State that for a real-valued signal $x(t)$, the energy and power spectral density functions are even functions
  • $E(\omega) = E(-\omega)$ and $P(\omega) = P(-\omega)$

Parseval's Theorem

• Relate the energy of a signal in the time domain to its energy spectral density in the frequency domain using Parseval's theorem
  • $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} E(\omega) d\omega$
• Explain that Parseval's theorem allows for the computation of signal energy in either the time or frequency domain
  • Useful for analyzing the energy distribution of a signal across different frequencies

Signal Energy and Power Calculation

Continuous-Time Signals

• Calculate the total energy of a continuous-time signal by integrating the energy spectral density function over all frequencies
  • $E_{total} = \int_{-\infty}^{\infty} E(\omega) d\omega$
• Calculate the average power of a continuous-time signal by integrating the power spectral density function over all frequencies
  • $P_{avg} = \int_{-\infty}^{\infty} P(\omega) d\omega$

Discrete-Time Signals

• Express the energy spectral density $E(\omega)$ for a discrete-time signal $x[n]$ in terms of its discrete-time Fourier transform $X(e^{j\omega})$
  • $E(\omega) = |X(e^{j\omega})|^2$
• Calculate the total energy of a discrete-time signal using the energy spectral density function
  • $E_{total} = \frac{1}{2\pi} \int_{-\pi}^{\pi} E(\omega) d\omega$
• Express the power spectral density $P(\omega)$ for a discrete-time signal $x[n]$ in terms of its discrete-time Fourier transform $X_N(e^{j\omega})$ of the truncated signal over the interval $[0, N-1]$
  • $P(\omega) = \lim_{N\to\infty} \frac{1}{N} |X_N(e^{j\omega})|^2$
• Calculate the average power of a discrete-time signal using the power spectral density function
  • $P_{avg} = \frac{1}{2\pi} \int_{-\pi}^{\pi} P(\omega) d\omega$

Properties of Spectral Density Functions

Non-Negativity and Bandwidth

• State that energy and power spectral density functions are always non-negative
  • $E(\omega) \geq 0$ and $P(\omega) \geq 0$ for all $\omega$
  • Follows from the definition of spectral density functions as the magnitude squared of the Fourier transform or the Fourier transform of the autocorrelation function
• Determine the bandwidth of a signal from its energy or power spectral density function
  • Bandwidth is the range of frequencies over which the spectral density is significant (above a certain threshold)
  • Signals with wider bandwidth have more significant frequency components and require more resources (sampling rate, storage, transmission) to process

Applications in Signal Analysis and Processing

• Use the energy and power spectral density functions to analyze the frequency content of a signal
  • Identify dominant frequency components, harmonics, and noise
  • Determine the required sampling rate for discrete-time processing based on the signal bandwidth
• Apply spectral density functions in signal filtering and compression
  • Design filters (lowpass, highpass, bandpass) based on the desired frequency response
  • Compress signals by discarding or quantizing frequency components with low spectral density
• Utilize spectral density functions for system identification and characterization
  • Estimate the transfer function of a linear time-invariant system from its input and output spectral densities
  • Analyze the effects of noise and distortion on signal quality using spectral density techniques