EECS20N: Signals and Systems

Using Linearity

We can often use linearity to avoid calculating integrals.


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

X(ω) = e−iω + e−i2ω

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

X1(ω) = e−iω

X2(ω) = e−i2ω

which we recognize as

x1(n) = δ (n − 1)

x2(n) = δ (n − 2)


X(ω) = X1(ω) + X2(ω)

we get the result

x(n) = x1(n) + x2(n) = δ (n − 1) + δ (n − 2).