Energy and 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 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 (, 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
Etotal=∫−∞∞E(ω)dω
Express the average power of a signal as the integral of the power spectral density function over all frequencies
Pavg=∫−∞∞P(ω)dω
State that energy and power spectral density functions are always non-negative
E(ω)≥0 and P(ω)≥0 for all ω
Relate the energy and power spectral density functions to the autocorrelation function of a signal via the Fourier transform
E(ω)=∫−∞∞Rx(τ)e−jωτdτ and P(ω)=∫−∞∞Rx(τ)e−jωτdτ, where Rx(τ) 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(ω) in terms of the Fourier transform X(ω) of a signal x(t)
E(ω)=∣X(ω)∣2
Express the power spectral density function P(ω) in terms of the Fourier transform X(ω) of a signal x(t)
P(ω)=limT→∞T1∣XT(ω)∣2, where XT(ω) 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(ω)=E(−ω) and P(ω)=P(−ω)
Parseval's Theorem
Relate the energy of a signal in the time domain to its energy spectral density in the frequency domain using
∫−∞∞∣x(t)∣2dt=2π1∫−∞∞E(ω)dω
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
Etotal=∫−∞∞E(ω)dω
Calculate the average power of a continuous-time signal by integrating the power spectral density function over all frequencies
Pavg=∫−∞∞P(ω)dω
Discrete-Time Signals
Express the energy spectral density E(ω) for a discrete-time signal x[n] in terms of its discrete-time Fourier transform X(ejω)
E(ω)=∣X(ejω)∣2
Calculate the total energy of a discrete-time signal using the energy spectral density function
Etotal=2π1∫−ππE(ω)dω
Express the power spectral density P(ω) for a discrete-time signal x[n] in terms of its discrete-time Fourier transform XN(ejω) of the truncated signal over the interval [0,N−1]
P(ω)=limN→∞N1∣XN(ejω)∣2
Calculate the average power of a discrete-time signal using the power spectral density function
Pavg=2π1∫−ππP(ω)dω
Properties of Spectral Density Functions
Non-Negativity and Bandwidth
State that energy and power spectral density functions are always non-negative
E(ω)≥0 and P(ω)≥0 for all ω
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