Inverse Transform An inverse transform is the operation that undoes the effect of the original transform. It allows us to recover the original signal or fun...
Inverse Transform
An inverse transform is the operation that undoes the effect of the original transform. It allows us to recover the original signal or function from its transformed version.
Formal Definition:
Let x(t) be a function of time, and let X(ω) be its Fourier transform. The inverse transform X^-1(ω) of X(ω) tells us the following:
X^-1(ω) = ∫ X(τ) * δ(ω - τ) dτ
where δ(ω) is the Dirac delta function.
Interpretation:
The inverse transform represents a convolution operation between the original signal and a Dirac delta function. This means that the inverse transform shifts the original signal to the frequency domain, where it is multiplied by the Dirac delta function.
Examples:
The inverse transform of δ(ω) is itself, indicating that the original signal is the same as the input signal.
The inverse transform of sin(ωt) is cos(ωt).
The inverse transform of a heavy (or rectangular) function may not exist in all cases.
Key Points:
Inverse transforms are unique.
They are obtained by reversing the order of the Fourier transform.
Inverse transforms can be used to obtain the original signal from its transformed version