陆昱宽

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 3: Signal Representation by Fourier Series. Signal Processing and Linear Systems (2nd ed., pp. 297-300). Oxford University Press.

Sources:

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 193-195). Oxford University Press.

In this section I will illustrate that, for a system specified by the differential equation \[ \begin{equation} \label{eq_2_2} Q(D) y(t)=P(D) x(t) , \end{equation} \]

its transfer function is \(H(s)\), the bilateral Laplace transform of \(h(t)\), which is the unit impulse response of the system, and that \(H(s)\) also satisfies: \[ \color{teal} {H(s)=\frac{P(s)}{Q(s)}} . \]

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 168-195). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 151-162). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 168-195). Oxford University Press.

This section is devoted to the determination of the zero-state response(see past post) of an LTIC system. We shall assume that the systems discussed in this section are in the zero state unless mentioned otherwise.

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 168-195). Oxford University Press.

Stability is an important system property. Two types of system stability are generally considered: external (BIBO) stability and internal (asymptotic) stability. Let us consider both stability types in turn.

Sources:

1. Developing inside a Container from VS Code.
2. Custom Dev Container Features from VS Code.
3. See my github repo for my DevContainer confguration.

Sources:

1. Shiyu Zhao. Chapter 6: Stochastic Approximation. Mathematical Foundations of Reinforcement Learning
2. --> Youtube: Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) algorithm: minimize \(J(w)=\mathbb{E}[f(w, X)]\) using \(\left\{\nabla_w f\left(w_k, x_k\right)\right\}\) \[ w_{k+1}=w_k-\alpha_k \nabla_w f\left(w_k, x_k\right) \]

Sources:

1. Shiyu Zhao. Chapter 6: Stochastic Approximation. Mathematical Foundations of Reinforcement Learning
2. --> Youtube: Stochastic Gradient Descent

Robbins-Monro (RM) algorithm: solve \(g(w)=0\) using \(\left\{\tilde{g}\left(w_k, \eta_k\right)\right\}\) \[ w_{k+1}=w_k-a_k \tilde{g}\left(w_k, \eta_k\right) \]