Linear Systems – Lecture C

Applications of Convolution

In the last two lectures we saw that if we know the impulse response of an LTI system, then we can calculate its response to any input signal. Now we shall look at some specific instances where the convolution calculation is useful.

First and Second Order Systems

A system whose input-output relation can be expressed a a first order differential equation is called a first order system, and a system whose input-output relation can be expressed as a second order differential equation is called a second order system. And so on for higher order system. We shall see later how the impulse response of such systems can be obtained.

Here we shall look briefly at first order and second order system behaviour. The impulse response of a first order system comprises a single exponential. Such a first order system will respond well to slowly changing inputs but will respond poorly to quickly changing inputs. Therefore such systems are called low-pass systems.

In addition to an exponential a first order system’s impulse response may contain a single delta function. In such a case, the system responds well to quickly changing inputs but responds poorly to slowly changing inputs. This is a high-pass filter.

The expressions for the impulse responses are given below, Low-pass h₁(t) and High-pass h₂(t) :

Using the following interactive example, study the behaviour of the two types of first order systems on various signals. See how changing the time constant 1/a changes the behaviour of the system.

Mechanical systems with mass, viscosity and elasticity have second order behaviour. Unlike first order systems, second order systems can exhibit oscillatory behaviour even with non-oscillatory input. The vibration in mechanical systems is one such example.  For a second order system with low-pass behaviour, the system parameters determine the frequency of oscillations (natural frequency) and also the amplitude of oscillations. If the system is highly damped then oscillations may be very small. Again use the interactive example to study the behaviour of such second order systems. Only second order low pass filters are shown in this example.

See how the output of these systems is obtained from the impulse response.

Use the program to:

• Study the difference between low-pass and high-pass filtering systems
• Study the relationship between time constant of a first order system and the filter behaviour.
• Does the shape of the input signal determine how much it is affected by the low-pass and high pass filters?
• Compare the behaviour of first order and second order systems.
• Study the influence of natural frequency and damping factor on second order system behaviour.

© Suresh Devasahayam