Sumudu Transform - Properties and Theorems

Properties and Theorems

• The transform of a Heaviside unit step function is a Heaviside unit step function in the transformed domain.
• The transform of a Heaviside unit ramp function is a Heaviside unit ramp function in the transformed domain.
• The transform of a monomial tn is the scaled monomial S{tn} = n!·un.
• If f(t) is a monotonically increasing function, so is F(u) and the converse is true for decreasing functions.
• The Sumudu transform can be defined for functions which are discontinuous at the origin. In that case the two branches of the function should be transformed separately. If f(t) is Cn continuous at the origin, so is the transformation F(u).
• The limit of f(t) as t tends to zero is equal to the limit of F(u) as u tends to zero provided both limits exist.
• The limit of f(t) as t tends to infinity is equal to the limit of F(u) as u tends to infinity provided both limits exist.
• Scaling of the function by a factor c > 0 to form the function f(ct) gives a transform F(cu) which is the result of scaling by the same factor.
• By taking the Sumudu transform of the output signal of a dynamic system when the input is a unit step, the transfer function of the dynamic system in the u–domain can be defined. This is an easily comprehensible concept for the transfer function of a system.

All of these properties may be deduced from the corresponding properties of the Laplace transform using no more than simple high school algebra.