[Advanced] MATLAB Stochastic Process Analysis

# 1. Foundations of Stochastic Processes**

Stochastic processes are mathematical models used to describe random phenomena that evolve over time or space. It is a collection of random variables, where each variable corresponds to a specific moment or location in time or space. Stochastic processes are widely applied in engineering, physics, finance, and biology, among other fields, to describe phenomena such as noise, signals, traffic flow, and stock prices.

This section will introduce the basic concepts of stochastic processes, including definitions, classifications, and fundamental properties. We will discuss the types of stochastic processes, such as stationary and non-stationary processes, as well as their time and frequency domain characteristics. In addition, we will explore the statistical properties of stochastic processes, such as mean, variance, and autocorrelation function.

# 2.1 Time Domain Modeling

### 2.1.1 Autocorrelation Function and Power Spectral Density

**Autocorrelation Function**

The autocorrelation function measures the correlation between different time points of a stochastic process. For a real-valued stochastic process $X(t)$, its autocorrelation function is defined as:

R_X(\tau) = E[X(t)X(t+\tau)]

where $\tau$ is the time shift. The autocorrelation function represents the correlation between the stochastic process at times $t$ and $t+\tau$ when the time shift is $\tau$.

**Power Spectral Density**

The power spectral density is the Fourier transform of the autocorrelation function, representing the power distribution of the stochastic process across different frequencies. The power spectral density is defined as:

S_X(f) = \frac{1}{2\pi}\int_{-\infty}^{\infty} R_X(\tau) e^{-i2\pi f\tau} d\tau

where $f$ is the frequency. The power spectral density provides the power information of each frequency component in the stochastic process.

### 2.1.2 Response of Linear Time-Invariant Systems

**Linear Time-Invariant Systems**

Linear time-invariant systems are those that have a linear relationship between input and output and are invariant to time shifts. For a linear time-invariant system, the output $Y(t)$ can be expressed as the convolution of the input $X(t)$:

Y(t) = h(t) * X(t)

where $h(t)$ is the system's impulse response.

**Response of Stochastic Processes**

If the input is a stochastic process $X(t)$, then the output is also a stochastic process $Y(t)$. The autocorrelation function of $Y(t)$ can be expressed as:

R_Y(\tau) = h(\tau) * R_X(\tau)

where $*$ denotes the convolution operation.

**Power Spectral Density**

The power spectral density of $Y(t)$ can be expressed as:

S_Y(f) = |H(f)|^2 S_X(f)

where $H(f)$ is the system's frequency response.

**Code Block:**

import numpy as np
import matplotlib.pyplot as plt

# Define a stochastic process
x = np.random.randn(1000)

# Calculate the autocorrelation function
tau = np.arange(-100, 100)
r_x = np.correlate(x, x, mode='full')

# Calculate the power spectral density
f = np.fft.fftfreq(len(x))
s_x = np.abs(np.fft.fft(r_x))**2 / len(x)

# Plot the autocorrelation function and power s