Expert: Josh - 11/16/2004

Question
Hi
thanx for ur reply but i would really appreciate if u can refer me to some site that has some detail with examples of it.
Thanx and regards

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Followup To
Question -
HI
Can u just explain to me in simple words and terms that what is a dirac delta function and step function.
How to detect that it is a diac delta function and step function and how to solve them.
I will really thankful to u.
Regards

Answer -
Hi Adeel,

1. A dirac delta function delta(x) has zero value everywhere, except that it has an impulse with one unit of amplitude at the origin. This unusual function has a simple definition,
delta(x) = {1, if x=0
         {0, otherwise [Note: x is a continuous variable]
but it is very tricky to define mathematically - with respect to limits and differentiability.

Translation property:
A more general definition is to shift the position of the critical value. For the dirac delta function,
delta(x-a)=1 if and only if x=a, otherwise it is zero.

2. A similar function is the "kronecker delta". This is a sampled version of the dirac delta function. i.e., it is only defined for sequences (in the discrete time domain).
delta[n-m] = {1, if n=m
         {0  if n not equals m
[Note: Here, "n" consists of the set of integers. For instance, n can be ...,-2,-1,0,1,2,3,...

3. The best way to think of the step function is to think of it as a switch. For all practical purpose, it has only two possible values. If x>a, s(x)=1. If x<a, s(x)=0.
It looks precisely like a step in a staircase.
An analogy is a transistor in an electronic circuit. If the input current exceeds a certain threshold, the device turns on. If the input is less than the threshold, it turns off.

There is more than one acceptable definition of the value at x=a. The point at x=a is sometimes inclusive, i.e., s(a) also equals 1. An alternative definition is that s(a)=0.5.

Okay, I guess you can now identify the dirac delta and step functions knowing what they look like.
I don't know what level you are at. So, I'll just briefly say a word about their applications, so you at least appreciate what they are used for. These two functions are often used as test signals to probe, measure and characterize the behavior of a complex system. The objective is to find out how a complex system responds to some stimulus and for engineers to control various processes (as wide ranging as telecommunication, automation in a chocolate manufacturing plant to the assessment of a person's loss of hearing.)

Cheers.

Answer
I don't know any good site off the top of my head.
Try typing "dirac delta" and "example" in your favorite search engine. You see, the dirac delta function has a host of formal properties. Unless your course requires you to study these in detail, a lot of the references will not be beneficial to you, because the mathematical properties are not easy to understand. I'm not sure what exactly it is that you need to know for your purpose.