Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Reading the signal literature I often come across the expression $k \delta (t)$ where $k$ is a constant. I presume this is a notation to suggest we are referring to the area or strength of the Dirac function since the multiplication of a constant, $k$, by $\delta (t)$ seems to make no sense.

My question is what does $k \delta(t)$ mean? Am I correct in assuming that on its own it doesn't represent $k$ multiplied by $\delta(t)$?

share|improve this question
up vote 5 down vote accepted

You are correct. In your example, $k$ would refer to the "area underneath" the impulse. Mathematically speaking, the Dirac delta isn't a function in the typical sense of the term. Instead, it is more of a distribution, characterized by the fact that when integrated across any interval that contains $t=0$, the result is unity. That is:

$$ \int_{-\epsilon}^{\epsilon} \delta(t) dt = 1 \ \forall\ \epsilon > 0 $$

The distribution is typically defined as follows:

$$ \delta(t) = \begin{cases} \infty,\ t = 0 \\ 0, \text{ otherwise}\end{cases} $$

Multiplication by a constant $k$ obviously wouldn't change this definition at all. So instead, the scaling merely changes the area underneath the distribution:

$$ \int_{-\epsilon}^{\epsilon} k\delta(t) dt = k \ \forall\ \epsilon > 0 $$

Extending this concept of multiplication by a constant $k$ to multiplication by a function $f(t)$ yields the following:

$$ \int_{-\epsilon}^{\epsilon} f(t) \delta(t) dt = f(0) \ \forall\ \epsilon > 0 $$

or more generally:

$$ \int_{T-\epsilon}^{T+\epsilon} f(t) \delta(t-T) dt = f(T) \ \forall\ \epsilon > 0 $$

which is known as the sifting property of the Dirac delta and is used extensively in linear systems theory.

share|improve this answer
Remark that the Dirac delta is typically defined as the limit of a sequence of functions with unit area and support that approaches to 0. – thang Jan 22 '13 at 19:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.