1) Here is some example code in Python that demonstrates how to simulate the Kalman filter:

Convert this python code to matlab code and provide output:

import numpy as np

import matplotlib.pyplot as plt

# Initialize the state of the system

x = np.array([[0], [0]])

# Initialize the state transition matrix

A = np.array([[1, 1], [0, 1]])

# Initialize the input matrix

B = np.array([[0], [0]])

# Initialize the output matrix

C = np.array([[1, 0]])

# Initialize the process noise covariance

Q = np.array([[0, 0], [0, 0]])

# Initialize the measurement noise covariance

R = np.array([[1]])

# Initialize the state estimate error covariance

P = np.array([[1000, 0], [0, 1000]])

# Initialize the control input

u = np.array([[0]])

# Initialize the measurement data

y = np.array([[0]])

# Initialize the Kalman gain

K = np.array([[0], [0]])

# Initialize the time steps

T = 100

# Initialize the arrays to store the true state and the estimated state

x_true = np.zeros((2, T))

x_est = np.zeros((2, T))

# Iterate over the time steps

for t in range(T):

# Generate the true state of the system

x = A.dot(x) + B.dot(u) + np.random.multivariate_normal([0, 0], Q).reshape((2, 1))

x_true[:, t] = x.reshape((2))

# Generate the measurement data

y = C.dot(x) + np.random.normal(0, np.sqrt(R)).reshape((1, 1))

# Estimate the state of the system using the Kalman filter

x_est[:, t] = A.dot(x_

# Compute the Kalman gain

S = C.dot(P).dot(C.T) + R

K = P.dot(C.T).dot(np.linalg.inv(S))

# Update the state estimate

x = x_est[:, t].reshape((2, 1)) + K.dot(y - C.dot(x_est[:, t].reshape((2, 1))))

x_est[:, t] = x.reshape((2))

# Update the state estimate error covariance

P = (np.eye(2) - K.dot(C)).dot(P)

# Plot the true state and the estimated state

plt.plot(x_true[0, :], label='True state')

plt.plot(x_est[0, :], label='Estimated state')

plt.legend()

plt.show()