Unit Impulse Signal in Control System

In this article, we are going to learn about the unit impulse signal in control systems. We know that a signal is a source of information that varies with time, space, temperature, and other independent variables. We can also define a signal as a function that conveys information about a phenomenon or a process. There are two types of signals continuous time domain signals that are defined for every instant of time and discrete time domain signals that are defined for discrete time instants. Standard signals are the signals that are used to know the performance of the control systems using the time response of the output. These signals have the known form of the mathematical expression. Standard signals are impulse, step, ramp, parabolic, and sinusoidal. Unit impulse signal is also one of the standard signals. Let us now discuss about the impulse and unit impulse signal in detail.

What is a Unit Impulse Signal?

In control systems, an impulse signal is a fundamental concept used to analyze and understand the behavior of LTI systems. Impulse signal serves as a building block and crucial tool for understanding system behavior and analyzing system responses.An impulse signal is often denoted by δ(t), is an idealized mathematical function that is zero everywhere except at time=0, at time t=0 it is infinity (∞). Mathematically it is represented as follows:

[Tex]\int_{-\infty }^{\infty }\delta (t) dt=\infty[/Tex] for t = 0

[Tex]\int_{-\infty }^{\infty }\delta (t) dt= 0[/Tex] for t ≠ 0

The below figure represents the impulse signal graph with the magnitude infinity .

Impulse signal

An unit impulse impulse signal is fundamental concept used to analyze and underastand the behaviour of LTI systems.Unir impulse signal serves as building block and crucial tool for understanding system behaviour and analyzing system responses .An unit impulse signal is often denoted by δ(t) in continuous time domain and δ[n] in discrete time domain , is an idealized mathematical function that is zero everywhere except at time t=0,at time t=0 it is unity (1).It is represented in the form of Dirac delta function . Mathematically it is represented as follows:

Continuous Time Domain Equation:

[Tex]\int_{-\infty }^{\infty }\delta (t) dt= 1[/Tex] for t = 0

[Tex]\int_{-\infty }^{\infty }\delta (t) dt= 0[/Tex] for t ≠ 0

We can also represents the above equation as follows :

[Tex]\delta (t) = 1[/Tex] for t=0

[Tex]\delta (t) =0[/Tex] for t≠0

The below figure represents the unit impulse signal in continuous time domain

Continuous time domain unit impulse signal

Discrete Time Domain Equation

[Tex]\delta [n]= 1 [/Tex]for n=0

[Tex]\delta [n]=0[/Tex] for n≠0

The below figure represents the unit impulse in discrete time domain

Discrete time domain unit impulse signal

Types of unit impulse signal

Continuous time unit impulse signal

The continuous time unit impulse signal is denoted as δ(t) .The mathematical equation and graph of continuous time unit impulse signal is shown below.

[Tex]\delta (t) = 1[/Tex] for t=0

[Tex]\delta (t) =0[/Tex] for t≠0

The below figure represents the unit impulse signal in continuous time domain

Unit impulse in continuous time domain

Discrete time unit impulse signal

The discrete time unit impulse signal is denoted as δ[n].The mathematical equation and graph of discrete time unit impulse signal is shown below.

[Tex]\delta [n]= 1 [/Tex]for n=0

[Tex]\delta [n]=0[/Tex] for n≠0

The below figure represents the unit impulse in discrete time domain

Unit impulse in discrete time domain

Doublet unit impulse signal

A doublet unit impulse signal is the signal that can be formed by the derivative of the unit impulse signal . It is oftenly denoted by [Tex]\delta {}'[n][/Tex] in discrete time domain and [Tex]\delta {}'(t)[/Tex] in continuous time domain .

Properties of unit impulse signal

• The unit impulse signal is an even signal . The mathematical expression to determine the unit impulse signal is even signal is shown as :

[Tex]\delta(t)=\delta(-t)[/Tex] in continuous time domain

[Tex]\delta[n]=\delta[-n][/Tex] in discrete time domain

• Area under the unit impulse signal is equal to one , this property represent the energy or amplitude of the unit impulse signal to normalized to 1 .

[Tex]\int_{-\infty }^{\infty } \delta (t) dt = 1[/Tex] in continuous time domain

[Tex]\sum_{n=-\infty }^{\infty } \delta [n] = 1 [/Tex]in discrete time domain

• The scaling property can be represented in many ways . Scaling property is the property that will compress or decompress the signal by a constant . Let us look at the each way representation of scaling property :

1. [Tex]\int_{-\infty }^{\infty } C . \delta (t) dt = C [/Tex] in continuous time domain where C is a constant .

[Tex]\sum_{n=-\infty }^{\infty } C . \delta [n] = C[/Tex] in discrete time domain where C is a constant .

2. [Tex]\int_{-\infty }^{\infty } \delta (Ct) dt = \frac{1}{C} \delta (t)[/Tex] in continuous time domain where C is a constant .

[Tex] \sum_{n=-\infty }^{\infty } \delta [C.n] = \frac{1}{C} \delta[n][/Tex] in discrete time domain where C is a constant

3. [Tex]\int_{-\infty }^{\infty } \delta (\frac{t}{C}) dt = C .\delta (t)[/Tex] in continuous time domain where C is a constant .

[Tex]\sum_{n=-\infty }^{\infty } \delta [n/C] = C. \delta[n] [/Tex] in discrete time domain where C is a constant .

• Shifting property is the property of the signal that will shift the signal either right or left of the origin by a constant unit . Let us look at the mathematical representation of the shifting property in continuous time domain and discrete time domain.

[Tex]\int_{-\infty }^{\infty } \delta (t-to) dt = \delta (to)[/Tex] in continuous time domain where to is the constant that the signal shifted.

The value of to will be either positive or negative based on the value of to the signal shifted to left or right of the origin.

[Tex] \sum_{n=-\infty }^{\infty } \delta [n-K] = \delta[K][/Tex] in discrete time domain where K is the constant that the signal shifted.

The value of to will be either positive or negative based on the value of to the signal shifted to left or right of the origin.

• The mathematical expressions for the sifting property is shown below .Sifting property is the property that sifts out the signal.

[Tex]\int_{-\infty }^{\infty } y(t).\delta (t-to) dt = y (to)[/Tex] in continuous time domain where to is a constant value .

[Tex]\sum_{n=-\infty }^{\infty } y[n] \delta [n-K] = y[K][/Tex] in discrete time domain where K is a constant.

• Unit impulse exhibits the convolution identity property . Convolution is combining of two or more signals to get a third signal . The mathematical expressions for the convolution property of unit impulse signal in both continuous time domain and discrete time domain are represented below:

[Tex]y(t)*\delta (t) = y(t)[/Tex] in continous time domain representation .

[Tex]y[n]*\delta [n] = y[n][/Tex] in discrete time domain representation .

• Unit impulse signal has the property of linearity in both continuous time and discrete time domains. Linearity is the property of the system that follows the prniciple of superposition.

Characteristics of unit impulse signal

• Unit impulse can be represented in both continuous time domain and discrete time domain.
• Area under the unit impulse signal is equal to one , this characteristic represent the energy or amplitude of the unit impulse signal to normalized to 1.
• By the derivation of unit step signal in continuous time domain we get the unit impulse signal. ([Tex]\frac{\mathrm{d} u(t)}{\mathrm{d} t} =\delta (t) [/Tex]).
• By the integration of the unit impulse signal will gives the unit step signal in continuous time domain. ( [Tex]\int_{-\infty }^{\infty }\delta (t) dt = u(t)[/Tex]).
• This unit impulse signal is orthogonal to all other signals , but it is not orthogonal to itself .
• At the point of concentration unit impulse signal has the amplitude that is equal to one.
• Arthimetic operations like addition in discrete time domain , multiplication in continuous time domiain , scaling , shifting can performed on unit impulse signal .

Applications of unit impulse signal

• Unit impulse signal can be used for modeling and analysing the linear time invariant system. We can find the information of the behaviour of the system with the help of response of the unit impulse response.
• Unit impulse signal can be used in the signal reconstruction and can be used as a sampling function.
• Unit impulse response is used in the control systems to understand and analyze the dynamic behaviour of the system and gives the information regrading the stability and performance of the system.
• It can be used for the edge detection ,enhance the spatial characteristics of the image in the image processing .
• This unit impulse signal can be used for the generation of many controlled input signals .
• The frequency response of the unit impulse signal provides the information which is helpful for the designing of the filters.
• Unit impulse signal is used in the network analysis to determine the output of the networks and circuits.
• Unit impulse response is used in solving the diffential equations , it makes the computation easy to get the output of the system.

Advantages of unit impulse signal

• Unit impulse response provides an easy ways for the mathematical computations.
• For the system analysing and modeling , we almost use the unit impulse signal.
• This can be helpful for the calculation of the transfer function.
• It can be served as an analytical tool for the differential equations.
• Shifting property of the unit impulse signal makes the impulse signal as a sampling function . Sampling function can be used for the analyze of the system easily.
• Unit impulse response be an important tool for the determination of spectrum and frequency compenents of the continuous and discrete signals.

Disadvantages of unit impulse signal

• Graphical visualization of unit impulse signal is difficult and can we cannot understand its proprties with the help of visualization.
• Unit impulse signal has the discontinuities in its derivates .
• In some case unit impulse may leads to the divergent while using in integrals.
• Practical implementation of the unit impulse signal is difficult.
• Unit impulse signal can be used as a theoritical tool but not as a practical tool.

Conclusion

In this article we have learnt about the impulse signal , unit impulse signal in both continuous time domain and discrete time domain . We have learnt about the different types of the unit impulse signals , its applications , advantages and disadvantages . We also learned about the properties of the unit impulse signal and at the same time we also learned about the characteristics of the unit impulse signal. We have discussed about the applications of the unit impulse signal in different fields and domain . From the above discussion we can conclude that the unit impulse function has the versatile nature . It is one of the most important tool for the system analysis and modelling . To know the performance , stability and to understand the system behaviour unit impulse response is the best component . It plays a key role in the mathematical caluculations , convolution and analyzing system stability .

Frequently Asked Questions

What is unit impulse signal ?

An unit impulse signal is often denoted by δ(t) in continuous time domain and δ[n] in discrete time domain , is an idealized mathematical function that is zero everywhere except at time t=0,at time t=0 it is unity (1).It is represented in the form of Dirac delta function .

Name the types of unit impulse signal ?

Some of the types of unit impulse signals are :

• Discrete time unit impulse signal
• Continuous time unit impulse signal
• Doublet unit impulse signal

Give the equation of continuous time unit impulse signal ?

The mathematical equation and graph of continuous time unit impulse signal is shown below.

[Tex]\delta (t) = 1[/Tex] for t=0

[Tex]\delta (t) =0[/Tex] for t≠0

Give the equation of discrete time unit impulse signal ?

The mathematical equation and graph of discrete time unit impulse signal is shown below.

[Tex]\delta [n]= 1[/Tex] for n=0

[Tex]\delta [n]=0 [/Tex] for n≠0

Determine the scaling property of unit impulse signal ?

The scaling property can be represented in many ways . Scaling property is the property that will compress or decompress the signal by a constant . Let us look at the each way representation of scaling property :

1.[Tex] \int_{-\infty }^{\infty } C . \delta (t) dt = C[/Tex] in continuous time domain where C is a constant .

[Tex]\sum_{n=-\infty }^{\infty } C . \delta [n] = C[/Tex] in discrete time domain where C is a constant .

2. [Tex]\int_{-\infty }^{\infty } \delta (Ct) dt = \frac{1}{C} \delta (t) [/Tex] in continuous time domain where C is a constant .

[Tex]\sum_{n=-\infty }^{\infty } \delta [C.n] = \frac{1}{C} \delta[n][/Tex] in discrete time domain where C is a constant

3. [Tex]\int_{-\infty }^{\infty } \delta (\frac{t}{C}) dt = C .\delta (t)[/Tex] in continuous time domain where C is a constant .

[Tex]\sum_{n=-\infty }^{\infty } \delta [n/C] = C. \delta[n] [/Tex] in discrete time domain where C is a constant .