1. Discrete Fourier Transform
(DFT)
Presented by: SHAHRYAR ALI

2. Discrete Time FT (DTFT)
DTFT defined as: Note: continuous frequency domain!
(frequency density function)
s
si +∞
ly
na
a S(f) = ∑ s[n] ⋅ e − j 2 π f n
n = −∞ Holds for Aperiodic
signals
is
hes 2π
nt 1
sy s[n] = ⋅ ∫ S(f)e j 2 π f ndf
2π
0

3. Problem With DTFT
– Defined for infinite-length sequences.
– From numerical computation viewpoint:
“It is troublesome as one has to evaluate infinite sums at
uncountable infinite frequencies”
– To use Matlab, we have to truncate sequences and then
evaluate the expression at many finite points.

4. Therefore:
• We turn our attention to a numerically computable
transform.
• It is obtained by sampling the DTFT transform in the
frequency domain (or the z-transform on the unit circle).
But..
• We know that a periodic function can always be
represented by:
“A linear combination of harmonically related complex
exponentials”

5. The Discrete Fourier Series
• So we have Discrete Fourier Series representation.
• Definition: Periodic sequence.
~ (n) = ~ (n + kN ), ∀n, k
x x
N: the fundamental period of the sequences

6. Discrete Fourier Series
Analysis equation:
~ N −1
~[n]e − j( 2 π / N)kn
X[k ] = ∑x
n=0
Synthesis equation:
~[n] = 1 N −1 ~
x ∑ X[k ]e j( 2 π / N)kn
N k =0

7. • For convenience we sometimes use:
− j( 2 π / N )
WN = e
So..
~ N −1
~[n]Wkn
X[k ] = ∑x N
n=0
~
{ X ( K ), k = 0,±1, } called the discrete Fourier series
are
coefficients.
~[n] = 1
N −1
~
x ∑ X[k ]WN kn
−
N k =0

8. Properties of DFS
• Linearity
~ [n] ~
x1 ← DFS →
  X1 [k ]
~ [n] ~
x 2 ← DFS →
  X2 [k ]
~ ~
a~1 [n] + b~2 [n]
x x ← DFS → aX1 [k ] + bX2 [k ]
 
• Shift of a Sequence
~[n] ~
x ← DFS →
 X[k ]
~
X[n] ← DFS → N~[ − k ]
 x
• Duality
~[n] ~
x DFS
←  →
 X[k ]
~[n − m] ~
x ←  → e − j2 πkm / NX[k ]
DFS

e j2 πnm / N~
x [n] ← → ~[k − m]
DFS
 X

9. The Fourier Transform of Periodic Signals
• Periodic sequences are not absolute or square summable
– Hence they don’t have a Fourier Transform.
• We can represent them as sums of complex exponentials:
DFS.
• We can define a periodic signal whose primary shape is
that of the finite duration signal .
• We then use the DFS on this periodic signal.
• So we define a new transform called the Discrete Fourier
Transform (DFT), which is the primary Period of the DFS.

10. Discrete Fourier Transform (DFT)

11. Discrete Fourier Transform (DFT)
• Discrete Fourier transform (or DFT) takes a finite number
of samples of a signal.
• It then transforms them into a finite number of
frequency samples .
• The discrete Fourier transform does not act on signals
that exist at all time.
• The DFT can be used in practice using a fast Fourier
transform (FFT) algorithm.

12. Fourier analysis
Input Time Signal Frequency spectrum
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8
Periodic FS Discrete
time, t
Continuous (period T)
2.5
2
1.5 Aperiodic FT Continuous
1
0.5
0
0 2 4 6 8 10 12
time, t
2.5
2
DFS Discrete
1.5
Periodic
1
0.5
0
(period T)
0 1 2 3 4 5 6 7 8
time, tk
Discrete
DTFT Continuous
2.5
2
Aperiodic
1.5
1
0.5
0
time, tk
DFT
0 2 4 6 8 10 12
Discrete

13. Discrete Fourier Transform (DFT)
Definition: The Discrete Fourier Transform (DFT) is defined by:
Where n = 0, 1, 2, …., N-1
The Inverse Discrete Fourier Transform (IDFT) is defined by:
where k = 0, 1, 2, …., N-1.
Same form of DFS but for aperiodic signals.
Signal treated as periodic for computational purpose only.

14. Sample X at N points
O<w<2π
x(2)
x(1)
x(o) w
x(N-1)

15. DFT at work
• To see how DFT equation actually works in practice,
let’s do a simple example - calculate DFT of 4
element sequence, x(n)={1,1,0,0}
for k=0
4−1
X ( 0 ) = ∑ x ( n ) e− j 2π ×0×n 4
n= 0
= x ( 0 ) e − j 2π ×0×0 4 + x ( 1) e − j 2π ×0×1 4 + x ( 2 ) e− j 2π ×0×2 4 + x ( 3) e − j 2π ×0×3 4
= 1×e− j 2π ×0×0 4 + 1×e− j 2π ×0×1 4 + 0 ×e − j 2π ×0×2 4 + 0 ×e− j 2π ×0×3 4
=2

16. DFT at work
for k=1
X ( 1) = x ( 0 ) e − j 2π ××0 4 + x ( 1) e − j 2π ×× 4 + x ( 2 ) e − j 2π ××2 4 + x ( 3) e − j 2π ××3 4
1 11 1 1
= 1×e − j 2π ××0 4 + 1 ×e − j 2π ×× 4 + 0 ×e − j 2π ××2 4 + 0 ×e − j 2π ××3 4
1 11 1 1
 π   π 
= 1 +  cos  ÷− j sin  ÷÷
 2  2 
= 1− j
• Following the same procedure we also get:
X ( 2) = 0 X ( 3) = 1 + j
• The result: DFT({1,1,0,0})={2,1+j,0,1-j}

17. DFT Properties
Time Frequency
Linearity a·s[n] + b·u[n] a·S(k)+b·U(k)
1 N−1
Multiplication s[n] ·u[n] ⋅ ∑S(h)U(k - h)
N h =0
N− 1
Convolution S(k)·U(k)
∑ s[m] ⋅ u[n − m]
m= 0
Time shifting s[n - m]
2π k ⋅m
−j
e T ⋅ S(k)
Frequency shifting S(k - h)
2π h t
+j
e T ⋅ s[n]

18. s[n]
S(f)
(a) (b)
0 T/2 T 2T f
s”[n] IDFT
DFT
(c) (d)
(e)
(f) cK
(a) Aperiodic discrete signal. (b) DTFT transform magnitude.
(c) Periodic version of (a). (d) DFS coefficients = samples of (b).
(e) Inverse DFT estimates a single period of s[n]
(f) DFT estimates a single period of (d).

19. Why DFT is important?
 To find the frequency content of a signal.
• To design an audio format (e.g., CD audio).
• To design a communications system (what bandwidth is
required?).
 To determine the frequency response of a structure.
• A musical instrument.

20. The Fast Fourier Transform
• The fast Fourier transform (FFT) is simply a class of
special algorithms which implement the discrete Fourier
transform .
• It calculates with considerable savings in computational
time.
• Maximum efficiency of computation is obtained by
constraining the points to be an integer power of two,
e.g. 1024 or 2048.

21. QUESTIONS???