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Amity University
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DISCRETE FOURIER TRANSFORM
1. Introduction
The sampled discrete-time fourier transform (DTFT) of a finite length, discrete-time signal is
known as the discrete Fourier transform (DFT). The DFT contains a finite number of samples
equal to the number of samples N in the given signal. Computationally efficient algorithms for
implementing the DFT go by the generic name of fast Fourier transforms (FFTs). This chapter
describes the DFT and its properties, and its relationship to DTFT.
2. Definition of DFT and its Inverse
Lest us consider a discrete time signal x (n) having a finite duration, say in the range 0 ≤ n ≤N-1.
The DTFT of the signal is
N-1
X ( ω) =
Σ
-jwn
x (n)e
(1)
n-0
Let us sample X using a total of N equally spaced samples in the range : ω ∈(0,2π),
sampling interval is 2π That is, we sample X(ω) using the frequencies.
N
ω= ωk = 2πk , 0 ≤ k ≤N-1.
N-1
-jwn
Thus
X (k) =
x (n)e
n-0
Σ
N-1
X (k) =
Σ
n-0
x (n)e
- j2πkn
so
the
(2)
(2)
N
The result is, by definition the DFT.
That is ,
Equation (0.2) is known as N-point DFT analysis equation. Fig 0.1 shows the Fourier transform
of a discrete – time signal and its DFT samples.

x(w)
π
0
→w
2π
Fig.1 Sampling of X(w) to get x(k)
While working with DFT, it is customary to introduce a complex quantity
WN = e-j2π /N
Also, it is very common to represent the DFT operation
N=1
X(k) = DFT ( x(n)) =
Σ
x(n) WNkn, 0≤n≤N-1
n=0
The complex quantity Wn is periodic with a period equal to N. That is,
WNa+N = e-j+2π/N(a+N) = e-j2π /N n = WNa where a is any integer.
Figs. 0.2(a) and (b) shows the sequence
odd respectively.
6
5
7
4
for 0≤n≤N-1 in the z-plane for N being even and
5
4
6
0
3
1
2
(a)
0
3
1
2
(b)
Fig.2 The Sequence for even N (b) The sequence for odd N.
The sequence WkNN for 0 ≤ n ≤ N-1 lies on a circle of unit radius in the complex plane
and the phases are equally spaced, beginning at zero.
The formula given in the lemma to follow is a useful tool in deriving and analyzing
various DFT oriented results.

2.1. Lemma
N-1
Σ Wkn = N δ (k) = { N, k = 0
n-0 N
(3)
0, k ≠ 0
Proof :
N-1
Σ an = 1 - aN : a≠ 1
n-0
1-a
We know that
Applying the above result to the left side of equation (3.3), we get
N-1
Σ
(WkN) n = 1- WkNN
n= 0
1- WkNN
=
1- e-j2π kNN
1- e-j2π kNN
=
1- 1
1- e- j2π kNN
=
0, k ≠ 0
: k ≠0
when k = 0, the left side of equation (3.3) becomes
N-1
N-1
Σ WN0xn =
n= 0
Σ
1=N
n =0
Hence, we may write
N-1
N, k = 0
Σ WN0xn =
0, k ≠ 0
n=0
= N δ (k), 0≤ k≤ N-1
2.2 Inverse DFT
The DFT values (X(k), 0≤ k≤ N-1), uniquely define the sequence x(n) through the inverse DFT
formula (IDFT) :
N-1
x (n) = IDFT (X(k) = 1 Σ X(k) WN-kn , 0≤ k≤ N-1
N k=0
The above equation is known as the synthesis equation.

N-1
N-1 N-1
1 Σ X(k) WN-kn = 1 Σ Σ x(m) WNkm
N k=0
N k=0 m=0
N-1
N-1
N-1
[
Proof :
=
] = WN-kn
1 Σ x(m) Σ WN-(n-m)k
N k=0
m=0
[
]
It can be shown that
N-1
Σ WN(n-m)k
=
N, n=m
0, n ≠ m
Hence,
N-1
1
N
Σ
x(m) Nδ (n-m)
= 1 x Nx (m)
N
m=n
( sifting property)
= x(n)
2.3 Periodicity of X (k) and x (n)
The N-point DFT and N-point IDFT are implicit period N. Even though x (n) and X (k) are
sequences of length – N each, they can be shown to be periodic with a period N because the
exponentials WN±kn in the defining equations of DFT and IDFT are periodic with a period N. For
this reason, x (n) and X (k) are called implicit periodic sequences. We reiterate the fact that for
finite length sequences in DFT and IDFT analysis periodicity means implicit periodicity. This
can be proved as follows :
N-1
X (k) = Σ x(n) WNkn
N-p=0
⇒
X (k+N) = Σ x(n) WN(k+N)n
Since, WNNn = e-j2πnNn = 1, we get
N-1
X (k+N) =
Σ
x(n) WN-kn
n=0
= X (k)

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