Lu, Yukuan

秦观

水边沙外。城郭春寒退。花影乱，莺声碎。飘零疏酒盏，离别宽衣带。人不见，碧云暮合空相对。

忆昔西池会。鹓鹭同飞盖。携手处，今谁在。日边清梦断，镜里朱颜改。春去也，飞红万点愁如海。

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 455-466). Oxford University Press.
2. B. P. Lathi & Roger Green. (2021). Chapter 7: Filter Design by Placement of Poles and Zeros. Signal Processing and Linear Systems (2nd ed., pp. 666-700). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 445-455). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 436-445). Oxford University Press.

In this section we explore the strong dependence of frequency response on the location of poles and zeros of \(H(s)\). This dependence points to a simple intuitive procedure to filter design.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 412-419). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 8: Sampling: The Bridge From Continuous To Discrete. Signal Processing and Linear Systems (3rd ed., pp. 775-784). Oxford University Press.

In this article, we show that a real signal \(x(t)\) whose spectrum is bandlimited to \(B\) Hz, i.e., \(X(\omega)=0\) for \(|\omega|>2 \pi B)\), can be reconstructed exactly (without any error) from its samples taken uniformly at a rate \(f_s>2 B\) samples per second. In other words, the minimum sampling frequency is \[ f_s>2 B \] Hz.

NOTE: In some literature, \(f_s=2 B\) is be used, which is not correct.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 386-401). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 373-386). Oxford University Press.

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 360-370). Oxford University Press.

The time-differentiation property of the Laplace transform has set the stage for solving linear differential (or integro-differential) equations with constant coefficients.

Because \(d^k y / d t^k \Longleftrightarrow\) \(s^k Y(s)\), the Laplace transform of a differential equation is an algebraic equation that can be readily solved for \(Y(s)\).

Next we take the inverse Laplace transform of \(Y(s)\) to find the desired solution \(y(t)\).

Sources:

1. B. P. Lathi & Roger Green. (2018). Chapter 4: The Laplace Transform. Signal Processing and Linear Systems (3rd ed., pp. 350-360). Oxford University Press.

For a quick reference table, see Wikipedia page on the Laplace transform.