Notes › ENGR 2341: Signals and Systems Lecture 3

Continuous Time Systems

calendar_today January 20, 2025 article 280 words access_time 2 min replay Modified January 20, 2025

Table of Contents

System

Manipulates (changes, records, transmits) signals
Receives an input signal, operates on it, and produces an output signal
Common representations:
- Differential equations
- Impulse response/convolution integral
- Transfer/system function
- Frequency response

Impulse Response

Impulse responses, $h(t)$, is the system’s response to a unit impulse input

$$ \begin{align} y(t) &= f(y(t), x(t)) \\ h(t) &= y(t) \ \vert_{x(t)=\delta(t)} \end{align} $$

The impulse response of an integrator is the unit step

$$ \begin{align} y(t) &= \int_{-\infty}^{t}x(\tau) \ \mathrm{d}\tau \\ h(t) &= \int_{-\infty}^{t}\delta(\tau) \ \mathrm{d}\tau \\ &= u(t) \end{align} $$

System Properties

Causality

Causal (non-predictive): the output only depends on the current and past inputs
Causal iff the output at $t_1$ solely depends on the input from $t \in (-\infty, t_1]$
Memoryless iff the output at $t_1$ solely depends on the input at $t_1$
Non-causal systems cannot be physically implemented because that implies future values are contributing to the current output (reverse time)
$h(t) = 0, \ t < 0$ for a causal LTI system

Linearity

Linear: the output is a linear mapping where additivity and homogeneity hold
The superposition of additivity and homogeneity is possible:

$$ f(\alpha x_{1} + \beta x_{2}) = \alpha f(x_{1}) + \beta f(x_{2}) $$

Time Invariance

Time-invariant: a system that does not change over time
- An input applied at some time or another time is the same, just delayed (shifted in time)
Linear time-invariant (LTI) systems are desirable:
- Can be characterized by their impulse response
- Cannot produce frequency components not in the input

BIBO Stability

Bounded-Input-Bounded-Output (BIBO) stable: a system with an absolutely integrable impulse response
Typically have finite duration impulse responses or one-sided and decay to zero

$$ \int_{-\infty}^{\infty} \vert h(t) \vert \ \mathrm{d}t \lt \infty $$

References

Continuous-time systems (course handout)

Sources

Continuous-time systems

Graph

mindmap ↗