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Anna university
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Fatima Michael College of Engineering & Technology
UNIT - I
DISCRETE FOURIER TRANSFORMS (DFT)
1. DIRECT COMPUTATION
2. RADIX-2 FFT
3. DECIMATION-IN-TIME FFT
4. FLOWGRAPHS
5. BIT REVERSAL PERMUTATION
6. COMPLEXITY
7. DECIMATION-IN-FREQUENCY FFT
Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology
THE FFT
A fast Fourier transform (FFT) is any fast algorithm for computing
the DFT. The development of FFT algorithms had a tremendous
impact on computational aspects of signal processing and applied
science. The DFT of an N -point signal
{x[n], 0 n N
1}
is defined as
X[k] =
N
X1
x[n] WN kn ,
0kN
n=0
where
✓
2⇡
WN = e = cos
N
is the principal N -th root of unity.
j 2⇡
N
◆
✓
2⇡
+ j sin
N
1
◆
DIRECT DFT COMPUTATION
Direct computation of X[k] for 0 k N
(N
1 requires
1)2 complex multiplications
N (N
1) complex additions
Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology
RADIX-2 FFT
The radix-2 FFT algorithms are used for data vectors of lengths
N = 2K . They proceed by dividing the DFT into two DFTs of
length N/2 each, and iterating. There are several types of radix2 FFT algorithms, the most common being the decimation-in-time
(DIT) and the decimation-in-frequency (DIF). This terminology will
become clear in the next sections.
Preliminaries
The development of the FFT will call on two properties of WN .
The first property is:
WN2 = WN/2
which is derived as
WN2 = e
=e
j 2⇡
N ·2
2⇡
j N/2
= WN/2 .
More generally, we have
nk
WN2nk = WN/2
.
The second property is:
k+ N2
WN
=
WNk
Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology
which is derived as
k+ N2
WN
2⇡
N
= ej N (k+ 2 )
2⇡
2⇡ N
= e j N k · ej N ( 2 )
2⇡
= ej N k · ej⇡
2⇡
=
ej N k
=
WNk
Fatima Michael College of Engineering & Technology

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