Lecture 4 of Signal

convolution sum

Okay, let’s take a look at “Lecture 4 - LTI Systems” to give you a heads-up on the main topics covered.

Based on the document, this lecture focuses heavily on Linear Time-Invariant (LTI) Systems and a fundamental concept called Convolution. Here are the key ideas you’ll encounter:

1. Importance of LTI Systems: These systems have both linearity (superposition holds) and time-invariance (system behavior doesn’t change over time).

2. Impulse Response is Key: A central idea is that any LTI system can be completely characterized by its response to a single, simple signal: the unit impulse ($\delta [n]$ in discrete-time, $\delta (t)$ in continuous-time). This response is called the impulse response, usually denoted by $h[n]$ or $h(t)$.

3. Representing Signals with Impulses: The lecture will show how arbitrary signals can be broken down and represented as a sum (in discrete-time) or integral (in continuous-time) of scaled and shifted unit impulses (the sifting property).

4. Convolution Sum (Discrete-Time): By representing inputs as sums of impulses and using LTI properties, you can find the output $y[n]$ for any input $x[n]$ given the impulse response $h[n]$. This is the convolution sum:

   $$y[n]=\sum _{k=-\infty }^{\infty }x[k]h[n-k]$$

   often written as $y[n]=x[n]\ast h[n]$.

5. Convolution Integral (Continuous-Time): Similarly, for continuous-time LTI systems, the output is given by the convolution integral:

   $$y(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau$$

   often written as $y(t)=x(t)\ast h(t)$.

Essentially, this lecture introduces convolution as the fundamental operation describing how an LTI system transforms an input signal into an output signal, based on the system’s impulse response.