Define impulse signal

An impulse signal, also known as a delta function or Dirac delta function in continuous time, is a fundamental concept in signal processing and systems analysis. It is used to model an idealized point source or an instantaneous event.

Impulse Signal in Continuous Time

Definition: The continuous-time impulse signal, denoted as δ(t)\delta(t)δ(t), is a mathematical function that is zero everywhere except at t=0t = 0t=0, where it is infinitely large in such a way that its integral over all time is equal to one.

Properties:

1. Sifting Property:

   ∫−∞∞δ(t−t0)f(t) dt=f(t0)\int_{-\infty}^{\infty} \delta(t - t_0) f(t) \, dt = f(t_0)∫−∞∞?δ(t−t0?)f(t)dt=f(t0?)

   This property implies that the impulse signal can "sample" a function at a specific point.

2. Unit Area:

   ∫−∞∞δ(t) dt=1\int_{-\infty}^{\infty} \delta(t) \, dt = 1∫−∞∞?δ(t)dt=1

   This ensures that the impulse has a finite area under its curve.

3. Scaling Property:

   δ(at)=1?a?δ(t)\delta(a t) = \frac{1}{|a|} \delta(t)δ(at)=?a?1?δ(t)

   This property indicates how the impulse signal behaves under scaling transformations.

4. Shifting Property:

   δ(t−t0)\delta(t - t_0)δ(t−t0?)

   This represents an impulse occurring at time t=t0t = t_0t=t0?.

Impulse Signal in Discrete Time

Definition: The discrete-time impulse signal, denoted as δ[n]\delta[n]δ[n], is defined as a sequence that is zero for all nnn except at n=0n = 0n=0, where it is equal to one.

Mathematical Representation:

δ[n]={1,n=00,n≠0\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}δ[n]={1,0,?n=0n?=0?

Properties:

1. Sifting Property:

   ∑n=−∞∞δ[n−n0]x[n]=x[n0]\sum_{n=-\infty}^{\infty} \delta[n - n_0] x[n] = x[n_0]n=−∞∑∞?δ[n−n0?]x[n]=x[n0?]

   Similar to the continuous case, the discrete impulse can "sample" a sequence at a specific point.

2. Unit Sample:

   ∑n=−∞∞δ[n]=1\sum_{n=-\infty}^{\infty} \delta[n] = 1n=−∞∑∞?δ[n]=1

   This indicates that the total sum of the impulse is one.

3. Shifting Property:

   δ[n−n0]\delta[n - n_0]δ[n−n0?]

   This represents an impulse occurring at the discrete time index n=n0n = n_0n=n0?.

Applications

• Impulse Response: The response of a system to an impulse signal is known as the impulse response, which characterizes the system completely in both continuous and discrete domains.
• Convolution: In both continuous and discrete systems, the impulse signal is used in the convolution process to determine the output of a system given its input and impulse response.
• Sampling: The impulse signal is used in sampling theory to represent the sampling of continuous signals.
• Testing and Analysis: Impulse signals are used to test and analyze the behavior of circuits and systems, as their effects can be easily isolated and studied.

Summary

The impulse signal is a crucial tool in signal processing and systems analysis due to its unique properties and the ability to represent instantaneous events or point sources. It serves as a foundation for understanding system responses, convolution, and various other applications in engineering and applied mathematics.