Laplace Transform
Analyze LTI systems in the s-domain.
Definition
$$ X(s) = \mathcal{L}{x(t)} = \int_0^{\infty} x(t) e^{-st} dt $$
Where $s = \sigma + j\omega$ is a complex frequency.
Common Transforms
| $x(t)$ | $X(s)$ | ROC |
|---|---|---|
| $\delta(t)$ | $1$ | All $s$ |
| $u(t)$ | $\frac{1}{s}$ | $\text{Re}(s) > 0$ |
| $e^{-at}u(t)$ | $\frac{1}{s+a}$ | $\text{Re}(s) > -a$ |
| $t^n u(t)$ | $\frac{n!}{s^{n+1}}$ | $\text{Re}(s) > 0$ |
| $\sin(\omega t)u(t)$ | $\frac{\omega}{s^2 + \omega^2}$ | $\text{Re}(s) > 0$ |
| $\cos(\omega t)u(t)$ | $\frac{s}{s^2 + \omega^2}$ | $\text{Re}(s) > 0$ |
Transfer Function
$$ H(s) = \frac{Y(s)}{X(s)} $$
Properties
- Linearity: $\mathcal{L}{ax_1 + bx_2} = aX_1(s) + bX_2(s)$
- Time shift: $\mathcal{L}{x(t-t_0)u(t-t_0)} = e^{-st_0}X(s)$
- Differentiation: $\mathcal{L}{\frac{dx}{dt}} = sX(s) - x(0^-)$
- Integration: $\mathcal{L}{\int_0^t x(\tau)d\tau} = \frac{X(s)}{s}$
Python (Symbolic)
1from scipy import signal
2import numpy as np
3
4# Define transfer function H(s) = 1/(s^2 + 2s + 1)
5num = [1]
6den = [1, 2, 1]
7sys = signal.TransferFunction(num, den)
8
9# Frequency response
10w, H = signal.freqs(num, den)
11
12# Step response
13t, y = signal.step(sys)
14
15# Impulse response
16t, y = signal.impulse(sys)
Further Reading
Related Snippets
- Convolution
Linear systems and filtering operations - Correlation
Signal similarity and pattern matching - Fourier Transform
DFT, FFT, and frequency analysis - Sampling Theory
Nyquist theorem, reconstruction, and interpolation - Signal Theory Basics
Fundamental concepts in signal processing - Window Functions
Spectral leakage reduction for FFT analysis