How can I call the area under an impulse/spike?

I am looking for an appropriate term to express the area under a spike or impulse for time series signals. The term "Briefness" is probably not appropriate since the amplitude should be taken into account

What you're describing "literally" is just the area under a curve, the integral. I don't think this helps you.

What you want, functionally, is not quite clear, but I'd bet it's either

• the energy of the pulse or
• the energy-bandwidth product.

The energy of the signal is the integral over the (magnitude) squared signal in the continuous time case, or the sum over the (magnitude) squared signal in continuous time.

The bandwidth is a frequency domain concept, and will tend to be larger the "shorter and sharper" the pulse is. The product of bandwidth and energy has important estimation quality effects for example in radar applications.

• Thank you for your answer Commented Aug 18, 2020 at 11:45
• You're welcome, but you don't have to say "thank you"; an upvote to the answers that you found helpful is the right thing to do. Commented Aug 18, 2020 at 11:46

The integral of a function can be called it's zeroth moment, with the $$n^{th}$$ moment being defined as

$$m_n=\int_{-\infty}^{\infty}t^nf(t)dt\tag{1}$$

That value also corresponds to the Fourier transform of $$f(t)$$ evaluated at $$\omega=0$$ (i.e., at DC):

$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\tag{2}$$

Hence,

$$\int_{-\infty}^{\infty}f(t)dt=m_0=F(0)\tag{3}$$

• Thank you for your answer Commented Aug 18, 2020 at 11:45
• @afousafous: Sure. Please accept the answer that was most helpful by clicking on the green check mark next to it, thanks! Commented Aug 18, 2020 at 11:49