Proof of the scaling property of an impulse function

In summary, the conversation discusses proving the scaling property of the impulse function, delta(a(t-to)) = 1/abs(a)*delta(t-to), in a signals and systems course. One method suggested is to substitute k(t-to) for x in the definition of the delta function, which results in a pulling in of the factor a by 1/k and ultimately proving the desired property. However, the rigor of this method may be questionable according to mathematical standards.
  • #1
amolraf
2
0
I am presently taking my first course in signals and systems and I have been charged with proving the scaling property of the impulse function; that:

delta(a(t-to)) = 1/abs(a)*delta(t-to)

I am seriously miffed and need some help.
 
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  • #2
Well I messed around with it for a bit and found one way to do it- take the definition of the delta function before the limit- like this:

0 for x < -a
delta(a, x) = 1/2a for -a < x < a
0 for x > a

and substitude k(t-to) for x...you'll find after messing around with it that a gets pulled in by a factor of 1/k. The area becomes 1/k and but the function is not translated, and hence your property is proved, although how rigorous this would be considered by mathematical standards I don't know.
 
  • #3


The scaling property of an impulse function is an important concept in signals and systems, and it is essential to understand it in order to fully grasp the behavior of signals in different systems. In order to prove this property, we will need to use the definition of the impulse function and some basic properties of scaling.

First, let's start with the definition of the impulse function. The impulse function, also known as the Dirac delta function, is a mathematical function that is defined as zero everywhere except at t=to, where it is infinity. This can be represented mathematically as:

delta(t-to) = 0, for t ≠ to
delta(t-to) = ∞, for t = to

Now, let's consider the scaled impulse function, delta(a(t-to)). This means that the impulse function is being multiplied by a scaling factor, a. We can represent this mathematically as:

delta(a(t-to)) = 0, for t ≠ to
delta(a(t-to)) = ∞, for t = to

Now, we need to use the property of scaling, which states that if we multiply a function by a constant, the function is scaled by the same constant. In other words, if we have a function f(t), and we multiply it by a constant k, then the resulting function is kf(t).

Applying this property to our scaled impulse function, we get:

delta(a(t-to)) = a*0, for t ≠ to
delta(a(t-to)) = a*∞, for t = to

Since 0 multiplied by any constant is still 0, we can simplify the above equations to:

delta(a(t-to)) = 0, for t ≠ to
delta(a(t-to)) = ∞, for t = to

Now, we can compare this with the definition of the impulse function. We can see that the only difference between the two is the scaling factor, a. This means that we can rewrite our scaled impulse function as:

delta(a(t-to)) = a*delta(t-to)

But, we know that the impulse function is only defined at t=to, where it is infinity. So, we can rewrite the above equation as:

delta(a(t-to)) = a*delta(t-to), for t = to

Now, we can apply the definition of the impulse function to the right side of the equation, and we get:

delta(a(t-to)) = a*
 

1. What is the scaling property of an impulse function?

The scaling property of an impulse function, also known as the sifting property, states that an impulse function can be scaled by a constant factor without affecting its integral. This means that the area under the impulse function remains the same regardless of the scaling factor.

2. How is the scaling property of an impulse function proved?

The scaling property of an impulse function can be proved using mathematical techniques such as integration and substitution. By substituting a scaled impulse function into the integral and using the properties of the Dirac delta function, it can be shown that the area under the scaled impulse function remains the same.

3. Why is the scaling property of an impulse function important?

The scaling property of an impulse function is important because it allows us to manipulate and analyze signals and systems using impulse functions. This simplifies calculations and makes it easier to understand the behavior of a system or signal.

4. Can the scaling property of an impulse function be applied to any function?

No, the scaling property of an impulse function can only be applied to the Dirac delta function, which is a special type of function. Other functions may have different scaling properties or may not have a scaling property at all.

5. How is the scaling property of an impulse function used in real-world applications?

The scaling property of an impulse function is used in various applications, such as signal processing, control systems, and image processing. It allows us to analyze and design systems by representing signals as a combination of impulse functions, making calculations more efficient and accurate.

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