How to Calculate Discrete Fourier Transform - Tutorial

How to Calculate DFT - Definition, Formula, Example

Definition:

Discrete Fourier transform (DFT ) is the transform used in fourier analysis, which works with a finite discrete-time signal and discrete number of frequencies. This tutorial explains how to calculate the discrete fourier transform.

Formula:

N-1 X(k) = ∑ x(n) e -j2πnk / N n=0 Where n - nth value series k - iterative value N - number of period

Example:

Generalization of derivation in a four-point DFT x={1,2,3,4}

Solution:

N-1 X(k) = ∑ x(n) e -j2πnk / N n=0

X(0) = x(0)e-j2π(0)(0)/4 + x(1)e-j2π(1)(0)/4 + x(2)e-j2π(2)(0)/4 + x(3)e-j2π(3)(0)/4

= 1(1) + 2(1) + 3(1) + 4(1)

= 10 + 0j

X(1) = x(0)e-j2π(0)(1)/4 + x(1)e-j2π(1)(1)/4 + x(2)e-j2π(2)(1)/4 + x(3)e-j2π(3)(1)/4

= x(0)e0 + x(1)e-j2π/4 + x(2)e-j4π/4 + x(3)e-j6π/4

= 1+x(1)(cos(-π/2)+jsin(-π/2)) + x(2)(cos(-π)+jsin(-π)) + x(3)(cos(-3π/2)+jsin(-3π/2))

= 1 + 2(-0.4480-0.8939j) + 3 (-0.5984+0.80114j) + 4 (0.9843+0.1760j)

= 1 - 0.8961 -1.7879j-1.7953 + 2.40345j + 3.9375 + 0.7041j

= 2.2460 + 1.3196j

X(2) = x(0)e-j2π(0)(2)/4 + x(1)e-j2π(1)(2)/4 + x(2)e-j2π(2)(2)/4 + x(3)e-j2π(3)(2)/4

= x(0)e0 + x(1)e-jπ + x(2)e-j2π + x(3)e-j3π

= 1+x(1)(cos(-π)+jsin(-π)) + x(2)(cos(-2π)+jsin(-2π)) + x(3)(cos(-3π)+jsin(-3π))

= 1 - 1.1969 + 1.6023j - 0.8510 - 2.8767j + 3.7520 + 1.3863j

= 2.7040 + 0.1119j

X(3) = x(0)e-j2π(0)(3)/4 + x(1)e-j2π(1)(3)/4 + x(2)e-j2π(2)(3)/4 + x(3)e-j2π(3)(3)/4

= x(0)e0 + x(1)e-j3π/2 + x(2)e-j4π + x(3)e-j9π/2

= 1+x(1)(cos(-3π/2)+jsin(-3π/2)) + x(2)(cos(-3π)     +jsin(-3π)) + x(3)(cos(-9π/2)+jsin(-9π/2))

= 1 + 1.9687 + 0.3520j + 2.8140 - 1.0397j + 3.4493 + 2.0252j

= 9.2322 + 3.4171j

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