Using Linearity

We can often use linearity to avoid calculating integrals.

Example

Suppose you are given the DTFT X for a discrete-time signal x,

X(w) = e^-iw + e^-i2w

and you are asked to find x. Because the DTFT is linear, you can find the inverse DTFT for each component,

X₁(w) = e^-iw

X₂(w) = e^-i2w

which we recognize as

x₁(n) = d (n - 1)

x₂(n) = d (n - 2)

Since

X(w) = X₁(w) + X₂(w)

we get the result

x(n) = x₁(n) + x₂(n) = d (n - 1) + d (n - 2).