Impulse Signal

An Impulse signal is one of the standard test signal. It is also known as Dirac delta function. An Impulse signal exists only at zero. The magnitude of an Impulse signal is infinity. It is denoted by δ(t), mathematically δ(t) will be infinity at t = 0 and zero for t ≠ 0. An impulse signal is an infinitesimally narrow pulse with unit area and infinite amplitude. It is used to analyze system response to Impulse – like inputs. It gives overall system response. Continuous Impulse signal is defined by its magnitude whereas Discrete Impulse signal is defined by its Amplitude.

There are two types of Impulse signal, Continuous Impulse signal and Discrete Impulse signal.

Continuous Ramp signal can be defined as the following expression given below-

δ(t) = ∞ ; for t = 0

0 ; for t ≠ 0

Area definition of Impulse function :

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

Thus the above expression implies that it is a continuous Unit Impulse signal of Area ‘ 1 ‘

How we generate Continuous impulse signal?

First of all we take a Rectangular ( Gate function ) of area ‘ 1 ‘ unit. Then decrease its width in such a way that its area remain ‘ 1 ‘ unit. When width of Rectangular function tends to zero then height of rectangular function will be of infinite magnitude. It can also be generated by Triangular function with same concept.

Properties of Unit Impulse Signal

Given Below are Some of the properties of Unit Impulse Signals

• Area under Unit Impulse function is Unity.

Expression :

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

• Impulse function is an Even function ( As its graph is symmetric about vertical axis )

Expression :

[Tex] \delta(t) \space = \space \delta(-t)[/Tex]

• Scaling Property of Impulse signal.

Expression :

[Tex] \delta(at) \space = \space \frac{1}{|a|}\delta(t)[/Tex]

• Product property

If X(t) is Continuous at t = 0. Then

1. [Tex] x(t)\delta(t) \space = \space x(0)\delta(t) [/Tex]
2. [Tex]x(t)\delta(t-t_{0}) \space = \space x(t_{0})\delta(t-t_{0}) [/Tex]
3. [Tex] x(t-t_{1})\delta(t-t_{0}) \space = \space x(t_{0}-t_{1})\delta(t-t_{0}) [/Tex]

• Sampling/ Integration property of Impulse function.

1. [Tex]\displaystyle \int_{-\infty}^{\infty}x(t)\delta(t)dt \space = \space \displaystyle\int_{-\infty}^{\infty}x(0)\delta(t)dt \space = \space x(0)\displaystyle\int_{-\infty}^{\infty}\delta(t)dt \space = x(0) [/Tex]
2. [Tex] \displaystyle \int_{-\infty}^{\infty}x(t)\delta(t-t_{0})dt = \displaystyle\int_{-\infty}^{\infty}x(t_{0})\delta(t-t_{0})dt = x(t_{0})\displaystyle\int_{-\infty}^{\infty}\delta(t-t_{0})dt = x(t_{0}) [/Tex]

Discrete-Time Impulse signal can be defined as the following expression given below

δ[n] = 1 ; for n = 0

0 ; for n ≠ 0

Where n is an integer.

Discrete Impulse signal is defined by its magnitude whereas Continuous Impulse signal is defined by area. Discrete Unit Impulse signal is also known as Karnecker’s delta or Unit Sample. δ[n] is not affected by scaling.

Properties of Discrete-Time Unit Impulse signal

• Symmetric Property: Impulse function is an Even function ( As its graph is symmetric about vertical axis ).

Expression :

[Tex] \delta[n] \space = \space \delta[-n][/Tex]

• Scaling Property : Scaling does not affect δ[n].

[Tex] \delta[mn] \space = \space \delta[n] [/Tex]

Proof :

[Tex] \delta[n] \space = \space \begin{cases}\\ 1 , \text{for n = 0}\\ 0, \text{otherwise} \end{cases}[/Tex]

[Tex] \delta[mn] \space = \space \begin{cases}\\ 1 , \text{for mn = 0 \space or n = 0}\\ 0, \text{otherwise} \end{cases}[/Tex]

• Product Property
  

[Tex]x[n]\delta[n] \space = \space x[0]\delta[n] [/Tex]

[Tex] x[n]\delta[n-n_{0}] \space = \space x[n_{0}]\delta[n-n_{0}] [/Tex]

[Tex]x[n-n_{1}]\delta[n-n_{0}] \space = \space x[n_{0} -n_{1}]\delta[n-n_{0}] [/Tex]

• Summation Property

[Tex] \displaystyle\sum_{n=-\infty}^{\infty}x[n]\delta[n] \space = \space \displaystyle\sum_{n=-\infty}^{\infty}x[0]\delta[n] \space = \space x[0] [/Tex]

[Tex]\displaystyle\sum_{n=-\infty}^{\infty}x[n]\delta[n-n_{0}] \space = \space \displaystyle\sum_{n=-\infty}^{\infty}x[n_{0}]\delta[n-n_{0}] \space = \space x[n_{0}] [/Tex]

[Tex] \displaystyle\sum_{n=n_{1}}^{n_{2}}x[n]\delta[n-n_{0}] \space = \space \displaystyle\sum_{n=n_{1}}^{n_{2}}x[n_{0}]\delta[n-n_{0}] \space = \space x[n_{0}], \\ provided \space n_{1} \leq \space n \leq \space n_{2}[/Tex]

Applications of Impulse Signals

• System Analysis : Impulse signals are widely used to analyze the behavior of linear time-invariant (LTI) systems. By applying an impulse signal as input to a system, We can obtain its impulse response, which characterizes the system’s behavior.
• Control System Design: Impulse signals are valuable tools for designing and analyzing control systems. They help to determine the system stability and performance of control systems by evaluating their Impulse response.
• Signal Processing: Impulse signals play a crucial role in signal processing applications such as filtering, convolution, and system identification. For example, in digital signal processing, convolving a signal with an impulse response helps to characterize the system’s behavior and filter out undesired components from the input signal.
• Communication Systems: Impulse signals used in communication systems for channel characterization and modulation schemes. By transmitting impulse-like signals through communication channels, can analyze channel characteristics such as impulse response and frequency response which is crucial for optimizing data transmission and reception.

Advantages of Impulse Signals

• Simplified System Analysis: Impulse signals simplify the analysis of Linear Time-Invariant (LTI) systems. By convolving the impulse response of a system with any input signal, we can obtain the system’s response for that input.
• Universal Response Representation: The impulse response of a system contain its behavior to a unit impulse input, which serves as a basis for understanding its response to any input signal. This universality allows us to analyze and predict the system’s behavior for various input signals without needing to test each input individually.
• Mathematical Simplicity: Impulse signals are easy to represent mathematically. The impulse function, typically denoted as δ(t) or δ[n] in continuous and discrete-time domains, respectively. Its mathematical simplicity facilitates analytical solutions and computations in system analysis and signal processing.
• Time-Invariance Property: Impulse signals exhibit the property of time-invariance, meaning that their response remains consistent over time.

Disadvantages of Impulse Signals

• Idealization vs. Practical Realization: The concept of an impulse signal is mathematically useful for system analysis, practical realization of an ideal impulse signal is not feasible. Ideal impulse signals have zero duration and infinite amplitude, which are physically impossible to generate or measure.
• Physical Constraints: Practical systems and instruments have limitations in terms of bandwidth, resolution, and response time, making it challenging to generate or detect signals with infinitesimally short durations.
• Signal Distortion: Due to limitations in bandwidth and signal processing capabilities, real-world impulse signals are subject to distortion and attenuation.

The standard signals are often used in control systems, signal processing, communication and various engineering applications. These are predefined signals with known characteristics. To clarify standard test signals, their uses and application in the control systems.

In this article, we will be going through standard signals. First of all, we will discuss the basic concepts of signals, and then we will go through 4 standard test signals that are mainly used in control systems followed by definitions, mathematical expressions, representation, properties, real-life examples, and relationships between all 4 standard test signals. At last, we will conclude our article with some interesting FAQs.