Decimation-in-time (DIT)
Radix-2 FFT
Presented by
Asst.lecturer
Noora Hani
DIT FFT
Butterfly Diagram
Observing the basic computations performed at each stage, we can arrive
at the following conclusions:
(i) In each computation, two complex numbers a and b are considered.
(ii) The complex number b is multiplied by a phase factor𝑾𝒌𝑵 .
(iii) The product 𝒃𝑾𝒌𝑵 is added to the complex number a to form a new
complex number A.
(iv) The product 𝒃𝑾𝒌𝑵 is subtracted from complex number a to form new
complex number B.
The basic computations can be expressed by signal flow graph shown in the
next slide [ DIT FFT]
Table showing powers of the factor 𝑾𝒌𝑵 for 4-point and 8-point
Example: find the 4-point DFT of the sequence x(n)=[2,1,4,3] using DIT
FFT algorithm
Sol:
Example: given sequence x(n)=[0,1,2,3,4,5,6,7] determine X(k) using
DIT FFT
SOL:
Example: Given a sequence x(n) for 0 ≤ n ≤ 3, where x(0) = 1, x(1) = 2, x(2) = 3,
and x(3) = 4,
a. Evaluate its DFT X(k) using the decimation-in-frequency FFT method.
b. Evaluate its DFT X(k) using the decimation-in-time FFT method.
Sol: a.
4
x(0)=1
x(1)=2
6
x(2)=3
-2*𝑤40
4
10=X(0)
𝑤20
6
-2=X(2)
-1
-2
-2+2j=X(1)
-1
-2*𝑤41
x(3)=4
-1
𝑤20
2j
-1
-2-2j=X(3)
4
4
10=x(0)
-2
-2+2j=x(1)
6*𝑤40
-2=x(2)
X(0)=1
X(2)=3
𝑤20
-2
-1
6
X(1)=2
X(3)=4
-1
𝑤20
-2
-1
-2*𝑤41
-2-2j=x(3)
-1