# 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*(*ω*) = *e ^{−i}*

*+*

^{ω}*e*

^{−i}^{2}

^{ω}and you are asked to find *x*. Because the DTFT is linear, you can find
the inverse DTFT for each component,

*X*_{1}(*ω*)
= *e ^{−i}*

^{ω}*X*_{2}(*ω*)
= *e ^{−i}*

^{2}

^{ω}which we recognize as

*x*_{1}(*n*)
= *δ *(*n *−
1)

*x*_{2}(*n*)
= *δ *(*n *−
2)

Since

*X*(*ω*) = *X*_{1}(*ω*)
+ *X*_{2}(*ω*)

we get the result

*x*(*n*) = *x*_{1}(*n*)
+ *x*_{2}(*n*)
= *δ *(*n *−
1)* + δ *(*n *−
2).