Expressing Unit Step Function (u(t)) in Terms of Signum Function (sgn(t))

The unit step function (u(t)) is a fundamental function in various fields, including control theory, signal processing, and mathematics. It is defined as:

(u(t) 0), for (t (u(t) 1), for (t geq 0)

Another useful function in this context is the signum function, commonly denoted as (sgn(t)), which is defined as:

(sgn(t) -1), for (t (sgn(t) 0), for (t 0) (sgn(t) 1), for (t > 0)

This article will delve into the relationship between these two functions and provide a detailed derivation of the expression (u(t) frac{1}{2} , sgn(t) - frac{1}{2}).

Derivation of the Relationship

We can express the unit step function (u(t)) in terms of the signum function (sgn(t)) using the following relationship:

1. Case (t

For (t [ u(t) frac{1}{2} , sgn(t) - frac{1}{2} frac{1}{2}(-1) - frac{1}{2} -frac{1}{2} - frac{1}{2} -1 ]

However, we know that (u(t) 0) for (t

2. Case (t 0)

For (t 0), the signum function (sgn(0) 0). Substituting this into the expression:

[ u(t) frac{1}{2} , sgn(t) - frac{1}{2} frac{1}{2}(0) - frac{1}{2} -frac{1}{2} ]

In practice, we define (u(0) frac{1}{2}) to ensure the function is continuous and well-defined at (t 0).

3. Case (t > 0)

For (t > 0), the signum function (sgn(t) 1). Substituting this into the expression:

[ u(t) frac{1}{2} , sgn(t) - frac{1}{2} frac{1}{2}(1) - frac{1}{2} frac{1}{2} - frac{1}{2} 1 ]

This matches the definition of the unit step function (u(t) 1) for (t > 0).

Conclusion

Based on the above derivations, we can conclude the expression:

[ u(t) frac{1}{2} , sgn(t) - frac{1}{2} ]

This accurately represents the unit step function (u(t)) for all values of (t).

Example Values of (t)

Let's verify the expression with some example values of (t):

For (t -2):

(u(-2) frac{1}{2} , sgn(-2) - frac{1}{2} frac{1}{2}(-1) - frac{1}{2} -1)

For (t 0):

(u(0) frac{1}{2} , sgn(0) - frac{1}{2} frac{1}{2}(0) - frac{1}{2} -frac{1}{2})

For (t 2):

(u(2) frac{1}{2} , sgn(2) - frac{1}{2} frac{1}{2}(1) - frac{1}{2} 0.5)

Related Keywords

unit step function signum function mathematical representation

By understanding this relationship, you can apply it in various mathematical and engineering contexts, enhancing your analytical and problem-solving skills.