Properties of LTI system and Convolution

Signal and System (IERG 2051)

This is a very interesting course. We use the mobile phone, TV , radio every day. However, most people don’t know the principle of these types of things working on. In this course, we can have a simple concept on how to transformation of signals, properties of LTI system, how to represent signals on frequency domain and so on.

In the previous week, we have learnt how to identify the properties of LTI system. To fulfill LTI system, we should consider the following properties. They are Memory, Invertibility, Causality, Stability, Time-invariance and Linearity.

Let’s see the example as below:

Determine if the system is: linear, causal, time-invariant, memoryless, invertible, or stable.

y(t) = e^x(t)

 Not linear: The output of x(t) = ax₁(t) + bx₂(t) is e^{ax1(t)+bx2(t)} =6 ae^x1(t) + be^x2(t)

Causal: The output just depends on input at time t.

Time invariant: The output of x(t) = x(t -t0) is e^x(t-t0) = y(t -t0).

 Memoryless: Output just depends on input at time t.

 Invertible: The inverse system is y(t) = log x(t).

 Stable: Let x(t) < M. Then y(t) < e^M.

The other thing I have learnt is convolution. The convolution is very mathematical.

Here is the definition and properties of convotion.

One of the application of convolution is act as matched filter for signal processing.

If the maximum of output is sufficiently large, the signal is friend. Otherwise it is foe.

http://www.fourier-series.com/Convolution/conv-integral.html

This website has some tutorial of convolution. It may help you to understand the concept of convolution.