Summary of POWER SPECTRAL DENSITY OF RANDOM PROCESS
Key Concepts:
- Power Spectral Density (PSD): A measure of the power of a signal as a function of frequency, which characterizes the distribution of power across different frequency components of a signal.
- Random Processes: Stochastic processes that are used to model signals that vary over time.
- Wide Sense Stationary (WSS) Processes: A type of random process where the mean and variance are constant over time, and the covariance depends only on the time difference.
- Convolution: A mathematical operation used to determine the output of a linear time-invariant (LTI) system when a random process is applied as input. The Convolution integral is a key formula discussed.
- Impulse Response: The output of a system when presented with a brief input signal (impulse). The Impulse Response characterizes the behavior of the system.
- Autocorrelation Function: A function that describes how the values of a random process are correlated with themselves over time.
Methodology:
- Convolution Integral:
- For continuous signals, the output is calculated using the Convolution integral.
- For discrete signals, linear Convolution is applied.
- Expectation Calculation: The expected value of the output of a random process is obtained by multiplying the expected value of the input signal by the DC value of the Impulse Response.
- Fourier Transform: The PSD can be obtained as the Fourier Transform of the Autocorrelation Function of the random process.
Properties of Power Spectral Density:
- Non-negativity: PSD is always greater than or equal to zero.
- Even Function: PSD is an even function of frequency.
- Area under Curve: The area under the PSD curve at zero frequency corresponds to the total power of the signal.
- Mean Square Value: The mean square value of a random process is related to the PSD.
- Gaussian Processes: When a Gaussian random process is passed through a linear filter, the output remains a Gaussian process.
Researchers/Sources Featured:
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Science and Nature