z transform and laplace transform table, Study notes of Linear Control Systems

Download z transform and laplace transform table and more Study notes Linear Control Systems in PDF only on Docsity!

Fourier Series (FS)

𝑘

𝑗 2 𝜋𝑘𝑡/𝑇

𝑘

−𝑗 2 𝜋𝑘𝑡/𝑇

𝑇

0

0

∞

𝑘=−∞

Fourier Transform (FT)

𝑗𝜔𝑡

−𝑗𝜔𝑡

∞

−∞

∞

−∞

Convolution:

1

2

1

2

∞

−∞

1

2

Discrete-Time Fourier Transform (DTFT)

𝑥[𝑛] =

𝑗𝜔

𝑗𝜔𝑛

𝑗𝜔

) = ∑ 𝑥[𝑛]𝑒

−𝑗𝜔𝑛

∞

𝑛=−∞

𝜋

−𝜋

Convolution:

1

[𝑛] ∗ 𝑥

2

[𝑛] = ∑ 𝑥

1

[𝑘]𝑥

2

[𝑛 − 𝑘] ⇔ 𝑋

1

𝑗𝜔

2

𝑗𝜔

∞

𝑘=−∞

Laplace Transform

−𝑠𝑡

∞

−∞

Convolution:

1

2

1

2

∞

−∞

1

2

z-Transform

𝑋(𝑧) = ∑ 𝑥[𝑛]𝑧

−𝑛

∞

𝑛=−∞

Convolution:

1

[𝑛] ∗ 𝑥

2

[𝑛] = ∑ 𝑥

1

[𝑘]𝑥

2

[𝑛 − 𝑘] ⇔ 𝑋

1

2

∞

𝑘=−∞