What is the value of the impulse function?

In summary, the value of a delta function is infinity due to its arbitrarily high amplitude, but its area is what matters. When taking the inverse Fourier transform, the resulting expression reflects the amplitude and frequency of the original function.
  • #1
clw
2
0

Homework Statement


I am confused with the idea that the value of a delta function is infinity but when given a graph why is there an amplitude/magnitude value for the delta function. I have attached a graph. Assuming the phase is zero I'm trying to write a Fourier transform for it.

Homework Equations



G(f) = |G(f)| [itex]e^{j\vartheta}[/itex]

The Attempt at a Solution


I know that it is a shifted impulse function so would it just be 3δ(f-5)+3δ(f+5)? Any help would be very much appreciated I'm just having a hard time grasping the idea of impulse function having an infinite value but there's an amplitude of 3?
 

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  • #2
clw said:

Homework Statement


I am confused with the idea that the value of a delta function is infinity but when given a graph why is there an amplitude/magnitude value for the delta function. I have attached a graph. Assuming the phase is zero I'm trying to write a Fourier transform for it.

Homework Equations



G(f) = |G(f)| [itex]e^{j\vartheta}[/itex]

The Attempt at a Solution


I know that it is a shifted impulse function so would it just be 3δ(f-5)+3δ(f+5)? 3?

Your expression is correct.

The delta function has arbitrarily high amplitude but also arbitrarily small spread. The area is what matters.

If you take the inverse Fourier transform of your expression you would get 6cos[2π(5)t]. So what you are displaying is the Fourier transform of a cosine wave with frequency = 5 Hz and amplitute = 6.
 

1. What is an impulse function?

The impulse function, also known as the Dirac delta function, is a mathematical function that is zero everywhere except at a single point, where it is infinitely large. It is used in physics and engineering to represent a sudden and instantaneous change in a system.

2. What is the value of an impulse function?

The value of an impulse function at its single point of existence is considered to be infinite. However, in practical applications, it is often approximated as a very large but finite value.

3. How is the impulse function used in signal processing?

In signal processing, the impulse function is used to represent a signal that is instantaneous and has a very short duration. It is also used to model the response of a system to an instantaneous input.

4. What is the significance of the impulse function in Fourier analysis?

The impulse function is used as a basis function in Fourier analysis, where it represents a single frequency component at time zero. It is also used to transform time-domain signals into frequency-domain signals.

5. How is the impulse function related to the concept of derivatives?

The impulse function is closely related to the concept of derivatives. In fact, the impulse function can be thought of as the derivative of the Heaviside step function, which represents a sudden change in a system. This relationship is used in the mathematical formulation of differential equations.

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