The document explains the Discrete Fourier Transform (DFT), its mathematical formulation, and its significance in converting discrete signals from the time domain to the frequency domain. It outlines the properties and applications of the DFT and inverse DFT, including linearity, periodicity, and the effects of sequence manipulation. Additionally, the document provides examples of DFT computations and a homework assignment related to the DFT.

1. THE DISCRETE
FOURIER
TRANSFORM
Prepared By:
Bongoyan, Rom Carlvren P.
Anabeso, Jose
Jorge, Cedric John

2. 1
2
3
4
5
Students will be able to
understand the DFT.
Students will be able to
solve the DFT and IDFT.
Students will be able to
solve the DFT as a
linear transformation.
Examine its properties
Bring to the student's
attention the importance
and the advantages of
the DFT
OBJECTIVES:

3. INTRODUCTION
The discrete Fourier transform (DFT) operates on a finite set of
numbers or a finite segment of a discrete signal. Physically, the DFT
of a discrete-time function x(n) may be viewed as its frequency-
domain representation and used as a tool to approximate the DTFT,
FT, or the spectrum of the continuous-time function x(r) [from
which x(n) is produced). Mathematically, the DFT may be
formulated by extending the classical Fourier transform of
continuous-time signals to a finite-length discrete-time sequence
through sampling and windowing. Alternatively, the DFT can be
introduced as an operation specified by a transformation matrix
that converts a sequence x(n) to another sequence X (k) of the
same length.

4. Discrete Fourier Transform
The DFT transforms a finite-length sequence X(n), 0 ≤ n ≤ (N-1), into another sequence X(k), 0 ≤ k ≤
(N-1), of the same length, where
X(k) k= 0, 1, 2, … N-1
X(k) is called the N-point DFT of x(n) and x(n) is called the inverse discrete fourier transform (IDFT) of
X(k). The latter may be recovered from X(k) by
X(n) , n= 0, 1, 2, … N-1
Customarily, x(n) represents a function in the time domain and X (k) is referred to as its
representation in the frequency domain. The sequences x(n) and X (k) always begin with the zeroth
term. They may be real-valued or complex numbers. In this chapter we will focus mostly on a real-
valued x(n).

5. Example of DFT
Compute the 4-point DFT of the sequence
x(n)= { 1, 2, 1, 2 }

6. Example of IDFT
Compute the 4-point IDFT of the sequence
X(k)= { 6, 0, -2, 0 }

7. The DFT as a linear transformation
• Now we will define the new term 'W' as
• This is called as “ Twiddle Factor”.
• Twiddle factor makes the computation of DFT a bit easy and fast.
• Using twiddle factor we can write equation of DFT and IDFT as follows:
X(k) k= 0, 1, 2, … N-1
And
X(n) , n= 0, 1, 2, … N-1

8. The DFT as a linear transformation
• The twiddle factor is denoted by W, and is given by,
• Now the discrete time sequence x(n) can be denoted by .Here 'N' stands for 'N' point DFT.
• Range of 'n' is from 0 to 'N-1'.
• can be represented in the matrix form as follows:
=
‘n’ varies from ‘0’ to ‘N-1’
x(0)
x(1)
x(2)
.
.
x(N-1)

9. The DFT as a linear transformation
• Now the DFT of x(n) is denoted by X(k). In the matrix form X(k) can be represented as follows:
=
‘k’ varies from ‘0’ to ‘N-1’
X(0)
X(1)
X(2)
.
.
X(N-1)

10. The DFT as a linear transformation
• We can also represent in the matrix form. Remember that 'k' varies from 0 to N-1 and 'n' also
varies from 0 to N-1.
• Note that each value is obtained by taking multiplication of ‘k’ and ‘n’.
• For example, if k=2, n=2 then we get,
= =
n=0 n=1 n=2 n=3
k=0
= k=1
k=2
k=3

11. The DFT as a linear transformation
• Thus, DFT can be represented in the matrix form as
=
• Similarly IDFT can be represented in the matrix form as
=
• Here is the complex conjugate of .

12. Example of DFT as Linear Transformation
Calculate the 4-point DFT of the 4-point sequence
x(n)= { 1, 2, 1, 2 }

13. PROPERTIES
OF THE
DISCRETE
FOURIER
TRANSFORM

14. Properties of DFT:
1. Linearity
- The N-point DFT of ax(n)+by(n) is aX(k)+bY(k) for any x(n) and y(n) of the same length N and
any constants a and b.
x(n) X(k)
y(n) Y(k)
ax(n)+by(n) aX(k)+bY(k)
2. Periodicity
- If x(n) X(k) then,
x(n) = x(n+N), for all n
X(k) = X(k+N), for all k.

15. Properties of DFT:
3. Circular Symmetries of a sequence
- As we have seen, the N-point DFT of a finite duration sequence x(n), of length L ≤ N, is
equivalent to the N-point DFT of a periodic sequence (n), of period N, which is obtained by
periodically extending x(n), that is,
(n)=
Now suppose that we shift the periodic sequence x,,(n) by k units to the right. Thus we obtain
another periodic sequence
(n)=(n-k)=
The finite-duration sequence
X’(n)= 0 ≤ n ≤ N-1

16. Multiplication of two DFTs
Suppose that we have two finite-duration sequences of length N, (n) and (n).
Their respective N-point DFT's are
(k) k= 0, 1, … N-1
(k) k= 0, 1, … N-1
If we multiply the two DFTS together, the result is a DFT, say (k), of a sequence (n) of length N. Let us
determine the relationship between (n) and the sequences (n) and (n).
We have
(k) = (k) (k) , k=0,1, …, N-1
The IDFT of {(k)} is
(m)
(m)

17. Circular Convolution
If,
DFT
(n) (k)
N
and
DFT
(n) (k)
N
then
DFT
(n) (n) (k) (k)
N
where (n) (n) denotes the circular convolution of the sequence (n) and (n).

18. ADDITIONAL
DFT
PROPERTIES

19. Time reversal of a sequence
If
DFT
x(n) X(k)
N
Then
DFT
N
Hence reversing the N-point sequence in time is equivalent to reversing the DFT values.
Circular time shift of a sequence
If
DFT
x(n) X(k) then
N
DFT
X
N

20. Circular Frequency Shift
If
DFT
x(n) X(k)
N
Then
DFT
x
N
Complex-conjugate properties
If
DFT
x(n) X(k) then
N
DFT
X*(n) X*
N

21. Multiplication of two sequences
If
DFT
(n) (k)
N
And
DFT
(n) (k)
N
Then
DFT
(n) (n) (k) (k)
N

22. Parseval’s Theorem
If
DFT
x(n) X(k)
N
And
DFT
y Y(k)
N
Then
=

23. ASSIGNMENT:
Compute the 5-point DFT of
a sequence x(n)= {1, 2, 3, 4}.
Then, find the IDFT of the
obtained DFT.

24. THANK YOU
E V E R Y O N E