Dirac Delta Function: Scaling and Shifting

[math&science]

Just have a question about the dirac delta function. I understand how you would write it if you want to shift it but how would you scale it assuming we are using discrete time. Would you write 2*diracdelta[n] or diracdelta[2n]. Also, would that increase it or reduce it by 2 meaning that instead of it being 1 at n=0, it would be 2 instead or would it be 1/2. Does it work the same way as scaling other functions in other words? Thank you!

 

[Cexy]

Remember the defining property of the Dirac delta:

$$\int_{-\infty}^{\infty} \delta(x)dx = 1$$

Thus

$$\int_{-\infty}^{\infty} \delta(ax)dx = \frac{1}{a} \int_{-\infty}^{\infty} \delta(y)dy = \frac{1}{a}$$

So we can think of multipling the argument by a as being the same thing as dividing the function by a:

$$\delta(ax) = \frac{1}{a}\delta(x)$$

 

[tacman]

Hello,

Thank you for your question about the Dirac delta function and how it can be scaled and shifted in discrete time.

To answer your question, let's first review the definition of the Dirac delta function. The Dirac delta function, denoted by δ(t), is a mathematical function that is defined to be 0 for all values of t except for t=0, where it is infinity (or undefined). In discrete time, the Dirac delta function is often denoted as δ[n] and is defined to be 0 for all values of n except for n=0, where it is equal to 1.

Now, to scale the Dirac delta function, you would use the same principles as scaling any other function. In other words, if you want to scale the function by a factor of 2, you would write 2*δ[n] or δ[2n]. This means that the function will now have a value of 2 at n=0 instead of 1. Similarly, if you want to scale the function by a factor of 1/2, you would write 1/2*δ[n] or δ[n/2]. This means that the function will now have a value of 1/2 at n=0 instead of 1.

It is important to note that scaling the Dirac delta function does not change its essential properties. It will still be 0 for all values of n except for n=0, where it will have a value of 1 (or any scaled value). This is because the Dirac delta function is a distribution, not a regular function, and thus cannot be scaled in the same way as other functions.

I hope this answers your question. Please let me know if you have any further inquiries.

Best regards,