Linearity Property


Statement:

The DFT of the linear combination of two or more signals is the sum of the linear combination of DFT of individual signals.

Proof:

We will be proving the property

a1x1(n) + a2x2(n) ==> DFT[a1x1(n) + a2x2(n)] = a1X1[k] + a2X2[k]

We have the formula to calculate DFT:
X(k) = ∑n=0N-1 x(n)WNnk where k = 0, 1, 2, … N-1.
Here x(n) = a1x1(n) + a2x2(n)
Therefore,
X(k) = ∑n=0N-1 a1x1(n) + a2x2(n)WNnk
= ∑n=0N-1 a1x1(n)WNnk + ∑n=0N-1 a2x2(n)WNnk
a1 and a2 are constants and can be separated, therefore,
= a1 ∑n=0N-1 x1(n)WNnk + a2 ∑n=0N-1 x2(n)WNnk
= a1X1[k] + a2X2[k]
Hence, proved.

The step-by-step values for the provided input is shown at the bottom of the page:

light_mode dark_mode

Enter the constant a1:



Enter x1(n) sequence:



Enter the constant a2:



Enter x2(n) sequence :