The document discusses discrete Fourier transforms (DFT). It begins with an overview of the syllabus which includes discrete signals and systems, introduction to DFT, properties of DFT, and applications of DFT such as filtering and fast Fourier transform (FFT) algorithms. It then provides background on digital signal processing, discussing how analog signals are converted to digital signals using sampling and then processed digitally before being reconstructed back to analog signals. The benefits and limitations of digital versus analog signal processing are also summarized.

1. UNIT-I
DISCRETE FOURIER TRANSFORM

2. Syllabus
• Discrete Signals and Systems- A Review
• Introduction to DFT
• Properties of DFT
• Circular Convolution
• Filtering methods based on DFT
• FFT Algorithms
• Decimation in time Algorithms,
• Decimation in frequency Algorithms
• Use of FFT in Linear Filtering.

3. Signal Processing
• Humans are the most advanced signal processors
– speech and pattern recognition, speech synthesis,…
• We encounter many types of signals in various applications
– Electrical signals: voltage, current, magnetic and electric fields,…
– Mechanical signals: velocity, force, displacement,…
– Acoustic signals: sound, vibration,…
– Other signals: pressure, temperature,…
• Most real-world signals are analog
– They are continuous in time and amplitude
– Convert to voltage or currents using sensors and transducers
• Analog circuits process these signals using
– Resistors, Capacitors, Inductors, Amplifiers,…
• Analog signal processing examples
– Audio processing in FM radios
– Video processing in traditional TV sets

4. Limitations of Analog Signal Processing
• Accuracy limitations due to
– Component tolerances
– Undesired nonlinearities
• Limited repeatability due to
– Tolerances
– Changes in environmental conditions
• Temperature
• Vibration
• Sensitivity to electrical noise
• Limited dynamic range for voltage and currents
• Inflexibility to changes
• Difficulty of implementing certain operations
– Nonlinear operations
– Time-varying operations
• Difficulty of storing information

5. Digital Signal Processing
• Represent signals by a sequence of numbers
– Sampling or analog-to-digital conversions
• Perform processing on these numbers with a digital processor
– Digital signal processing
• Reconstruct analog signal from processed numbers
– Reconstruction or digital-to-analog conversion
A/D DSP D/A
analog
signal
analog
signal
digital
signal
digital
signal
• Analog input – analog output
– Digital recording of music
• Analog input – digital output
– Touch tone phone dialing
• Digital input – analog output
– Text to speech
• Digital input – digital output
– Compression of a file on computer

7. Pros and Cons of Digital Signal Processing
• Pros
– Accuracy can be controlled by choosing word length
– Repeatable
– Sensitivity to electrical noise is minimal
– Dynamic range can be controlled using floating point numbers
– Flexibility can be achieved with software implementations
– Non-linear and time-varying operations are easier to implement
– Digital storage is cheap
– Digital information can be encrypted for security
– Price/performance and reduced time-to-market
• Cons
– Sampling causes loss of information
– A/D and D/A requires mixed-signal hardware
– Limited speed of processors
– Quantization and round-off errors

8. Discrete Signals and
Systems

12. Discrete-Time Signals: Sequences
• Discrete-time signals are represented by sequence of numbers
– The nth number in the sequence is represented with x[n]
• Often times sequences are obtained by sampling of continuous-time signals
– In this case x[n] is value of the analog signal at xc(nT)
– Where T is the sampling period
0 20 40 60 80 100
-10
0
10
t (ms)
0 10 20 30 40 50
-10
0
10
n (samples)

13. Basic Sequences and Operations
• Delaying (Shifting) a sequence
• Unit sample (impulse) sequence
• Unit step sequence
• Exponential sequences
]
n
n
[
x
]
n
[
y o









0
n
1
0
n
0
]
n
[






0
n
1
0
n
0
]
n
[
u
n
A
]
n
[
x 

-10 -5 0 5 10
0
0.5
1
1.5
-10 -5 0 5 10
0
0.5
1
1.5
-10 -5 0 5 10
0
0.5
1

14. Sinusoidal Sequences
• Important class of sequences
• An exponential sequence with complex
• x[n] is a sum of weighted sinusoids
• Different from continuous-time, discrete-time sinusoids
– Have ambiguity of 2k in frequency
– Are not necessary periodic with 2/o
   



 n
cos
n
x o





 j
j
e
A
A
and
e o
   
     















 




n
sin
A
j
n
cos
A
n
x
e
A
e
e
A
A
n
x
o
n
o
n
n
j
n
n
j
n
j
n o
o
 
   








 n
cos
n
k
2
cos o
o
    integer
an
is
k
2
N
if
only
N
n
cos
n
cos
o
o
o
o













16. Discrete-Time Systems
• Discrete-Time Sequence is a mathematical operation that maps a given
input sequence x[n] into an output sequence y[n]
• Example Discrete-Time Systems
– Moving (Running) Average
– Maximum
– Ideal Delay System
]}
n
[
x
{
T
]
n
[
y  T{.}
x[n] y[n]
]
3
n
[
x
]
2
n
[
x
]
1
n
[
x
]
n
[
x
]
n
[
y 






 
]
2
n
[
x
],
1
n
[
x
],
n
[
x
max
]
n
[
y 


]
n
n
[
x
]
n
[
y o



17. Memoryless System
• Memoryless System
– A system is memoryless if the output y[n] at every value of n
depends only on the input x[n] at the same value of n
• Example Memoryless Systems
– Square
– Sign
• Counter Example
– Ideal Delay System
 2
]
n
[
x
]
n
[
y 
 
]
n
[
x
sign
]
n
[
y 
]
n
n
[
x
]
n
[
y o



18. Linear Systems
• Linear System: A system is linear if and only if
• Examples
– Ideal Delay System
   
    (scaling)
]
n
[
x
aT
]
n
[
ax
T
and
y)
(additivit
]
n
[
x
T
]
n
[
x
T
]}
n
[
x
]
n
[
x
{
T 2
1
2
1




]
n
n
[
x
]
n
[
y o


 
 
  ]
n
n
[
ax
]
n
[
x
aT
]
n
n
[
ax
]
n
[
ax
T
]
n
n
[
x
]
n
n
[
x
]
n
[
x
T
]}
n
[
x
{
T
]
n
n
[
x
]
n
n
[
x
]}
n
[
x
]
n
[
x
{
T
o
1
o
1
o
2
o
1
1
2
o
2
o
1
2
1















19. Time-Invariant Systems
• Time-Invariant (shift-invariant) Systems
– A time shift at the input causes corresponding time-shift at output
• Example
– Square
• Counter Example
– Compressor System
 
]
n
n
[
x
T
]
n
n
[
y
]}
n
[
x
{
T
]
n
[
y o
o 




 2
]
n
[
x
]
n
[
y 
   
   2
o
o
2
o
1
]
n
n
[
x
n
-
n
y
gives
output
the
Delay
]
n
n
[
x
n
y
is
output
the
input
the
Delay




]
Mn
[
x
]
n
[
y 
 
   
 
o
o
o
1
n
n
M
x
n
-
n
y
gives
output
the
Delay
]
n
Mn
[
x
n
y
is
output
the
input
the
Delay





20. Causal System
• Causality
– A system is causal it’s output is a function of
only the current and previous samples
• Examples
– Backward Difference
• Counter Example
– Forward Difference
]
n
[
x
]
1
n
[
x
]
n
[
y 


]
1
n
[
x
]
n
[
x
]
n
[
y 



21. Stable System
• Stability (in the sense of bounded-input bounded-output BIBO)
– A system is stable if and only if every bounded input produces a
bounded output
• Example
– Square
• Counter Example
– Log






 y
x B
]
n
[
y
B
]
n
[
x
 2
]
n
[
x
]
n
[
y 






2
x
x
B
]
n
[
y
by
bounded
is
output
B
]
n
[
x
by
bounded
is
input
if
 
]
n
[
x
log
]
n
[
y 10

     
  







n
x
log
0
y
0
n
x
for
bounded
not
output
B
]
n
[
x
by
bounded
is
input
if
even
10
x

22. DFS,DTFT & DFT

23. Fourier Transform of continuous time
signals
  





 dt
e
t
x
t
x
FT
f
X ft
j 
2
)
(
)
(
)
(
  




 df
e
f
X
f
X
IFT
t
x ft
j 
2
)
(
)
(
)
(

24. Linear Time Invariant (LTI) Systems and z-Transform
]
[n
x ]
[n
y
]
[n
h
If the system is LTI we compute the output with the convolution:







m
m
n
x
m
h
n
x
n
h
n
y ]
[
]
[
]
[
*
]
[
]
[
If the impulse response has a finite duration, the system is called FIR (Finite Impulse
Response):
]
[
]
[
...
]
1
[
]
1
[
]
[
]
0
[
]
[ N
n
x
N
h
n
x
h
n
x
h
n
y 






25.   






n
n
z
n
x
n
x
Z
z
X ]
[
]
[
)
(
Z-Transform
Facts:
)
(
)
(
)
( z
X
z
H
z
Y 
]
[n
x ]
[n
y
)
(z
H
Frequency Response of a filter:
f
j
e
z
z
H
f
H 
2
)
(
)
( 


26. Frequency-Domain Properties
Time
Continuous
Discrete
Periodicity
Periodic Aperiodic
Fourier Series
Continuous-Time
Fourier Transform
DFT
Duration
Finite Infinite
Discrete-Time
Fourier Transform
and z-Transform
Discrete
&
Aperiodic
Continuous
&
Aperiodic
Continuous
&
Periodic (2)
Discrete
&
Periodic

27. Signal Processing Methods
Time
Continuous
Discrete
Periodicity
Periodic Aperiodic
Fourier Series
Continuous-Time
Fourier Transform
DFS
Duration
Finite Infinite
Discrete-Time
Fourier Transform
and z-Transform

28. The Four Fourier Methods
28

29. Relations Among Fourier Methods
29

30. The Family of Fourier Transform
 Aperiodic-Continuous－Fourier Transform
















d
e
j
X
t
x
dt
e
t
x
j
X
t
j
t
j
)
(
2
1
)
(
)
(
)
(


31. The Family of Fourier Transform
 Periodic-Continuous－Fourier Series













k
t
jk
T
T
t
jk
e
jk
X
t
x
dt
e
t
x
T
jk
X
0
0
0
0
)
(
)
(
)
(
1
)
(
0
2
2
0
0
0
0
2
2
T
F

 



32. The Family of Fourier Transform
 Periodic-Discrete－DFS (DFT)









1
0
2
1
0
2
)
(
1
)
(
)
(
)
(
N
k
nk
N
j
N
n
nk
N
j
e
k
X
N
n
x
e
n
x
k
X



33.  Introduction
The Discrete Fourier Transform (DFT)
 The DFS provided us a mechanism for numerically
computing the discrete-time Fourier transform. But most
of the signals in practice are not periodic. They are likely
to be of finite length.
 Theoretically, we can take care of this problem by
defining a periodic signal whose primary shape is that of
the finite length signal and then using the DFS on this
periodic signal.
 Practically, we define a new transform called the
Discrete Fourier Transform, which is the primary period
of the DFS.
 This DFT is the ultimate numerically computable
Fourier transform for arbitrary finite length sequences.

34.  Finite-length sequence & periodic sequence
The Discrete Fourier Transform (DFT)
)
(n
x Finite-length sequence that has N samples
)
(
~ n
x periodic sequence with the period of N
)
(
)
(
~
)
(
,
0
1
0
),
(
~
)
(
n
R
n
x
n
x
elsewhere
N
n
n
x
n
x
N



 



))
((
)
(
~
)
(
)
(
~
N
r
n
x
n
x
rN
n
x
n
x


 



Window operation
Periodic extension

35. The Family of Fourier Transform
 Aperiodic-Discrete－DTFT

















d
e
e
X
n
x
e
n
x
e
X
n
j
j
n
n
j
j
)
(
2
1
)
(
)
(
)
(

36. DFT and IDFT

37. IDFT

38.  The definition of DFT
The Discrete Fourier Transform (DFT)
1
0
,
)
(
1
)]
(
[
IDFT
)
(
1
0
,
)
(
)]
(
[
DFT
)
(
1
0
1
0

















N
n
W
k
X
N
k
X
n
x
N
k
W
n
x
n
x
k
X
N
n
nk
N
N
n
nk
N
)
(
)
(
~
)
(
)
(
1
)
(
)
(
)
(
~
)
(
)
(
)
(
1
0
1
0
n
R
n
x
n
R
W
k
X
N
n
x
k
R
k
X
k
R
W
n
x
k
X
N
N
n
N
nk
N
N
N
n
N
nk
N












39. Twiddle factor

40. Relationship of DFT to other
transforms

42. Problems on DFT

44. Example of DFT
• Find X[k]
– We know k=1,.., 7; N=8

45. Example of DFT

46. Example of DFT
Time shift Property of DFT
Polar plot for

47. Example of DFT

48. Example of DFT

49. Example of DFT
Summation for X[k]
Using the shift property!

50. Example of DFT
Summation for X[k]
Using the shift property!

51. Example of IDFT
Remember:

52. Example of IDFT
Remember:

53. Problems on DFT

54. Problems on DFT

58. Circular shift of a sequence

62. The Properties of DFT
 Linearity
)
(
)
(
)]
(
)
(
[
DFT 2
1
2
1 k
bX
k
aX
n
bx
n
ax 


N3-point DFT, N3=max(N1,N2)
 Circular shift of a sequence
)
(
)]
(
))
((
[
DFT k
X
W
n
R
m
n
x km
N
N
N 

)
(
))
((
)]
(
[
DFT k
R
l
k
X
n
x
W N
N
nl
N 


 Circular shift in the frequency domain

63. The Properties of DFT
 The sum of a sequence










1
0
0
1
0
0
)
(
)
(
)
(
N
n
k
N
n
nk
N
k
n
x
W
n
x
k
X
 The first sample of sequence




1
0
)
(
1
)
0
(
N
k
k
X
N
x

)
(
))
((
)]
(
[
)
(
)]
(
[
k
R
k
N
Nx
n
X
DFT
k
X
n
x
DFT
N
N




64. The Properties of DFT
 Circular convolution
)
(
)
(
)
(
))
((
)
(
)
(
))
((
)
(
)
(
)
(
1
2
1
0
1
2
1
0
2
1
2
1
n
x
n
x
n
R
m
n
x
m
x
n
R
m
n
x
m
x
n
x
n
x
N
N
m
N
N
N
m
N























N
N
)
(
)
(
)]
(
)
(
[
DFT 2
1
2
1 k
X
k
X
n
x
n
x 
N
)
(
)
(
1
)]
(
)
(
[
DFT 2
1
2
1 k
X
k
X
N
n
x
n
x 
 N
 Multiplication

65. The Properties of DFT
 Circular convolution
)
(
)
(
)
(
))
((
)
(
)
(
))
((
)
(
)
(
)
(
1
2
1
0
1
2
1
0
2
1
2
1
n
x
n
x
n
R
m
n
x
m
x
n
R
m
n
x
m
x
n
x
n
x
N
N
m
N
N
N
m
N























N
N
)
(
)
(
)]
(
)
(
[
DFT 2
1
2
1 k
X
k
X
n
x
n
x 
N
)
(
)
(
1
)]
(
)
(
[
DFT 2
1
2
1 k
X
k
X
N
n
x
n
x 
 N
 Multiplication

66. The Properties of DFT
 Circular correlation












n
n
xy n
y
m
n
x
m
n
y
n
x
m
r )
(
*
)
(
)
(
*
)
(
)
(
Circular correlation
)
(
)
(
*
))
((
)
(
))
((
*
)
(
)
(
1
0
1
0
m
R
n
y
m
n
x
m
R
m
n
y
n
x
m
r
N
N
n
N
N
N
n
N
xy










Linear correlation

67. The Properties of DFT
)
(
)
(
*
))
((
)
(
))
((
*
)
(
)]
(
[
IDFT
)
(
)
(
)
(
)
(
1
0
1
0
*
m
R
n
y
m
n
x
m
R
m
n
y
n
x
k
R
m
r
then
k
Y
k
X
k
R
if
N
N
n
N
N
N
n
N
xy
xy
xy














68. The Properties of DFT
 Parseval’s theorem







1
0
*
1
0
*
)
(
)
(
1
)
(
)
(
N
k
N
n
k
Y
k
X
N
n
y
n
x















1
0
2
1
0
2
1
0
*
1
0
*
)
(
1
)
(
)
(
)
(
1
)
(
)
(
then
)
(
)
(
let
N
k
N
n
N
k
N
n
k
X
N
n
x
k
X
k
X
N
n
x
n
x
n
y
n
x

69. The Properties of DFT
 Conjugate symmetry properties of DFT
 and
)
(n
xep )
(n
xop
Let be a N-point sequence
)
(n
x N
n
x
n
x ))
((
)
(
~ 
]
))
((
))
((
[
2
1
)]
(
~
)
(
~
[
2
1
)
(
~
]
))
((
))
((
[
2
1
)]
(
~
)
(
~
[
2
1
)
(
~
N
N
o
N
N
e
n
N
x
n
x
n
x
n
x
n
x
n
N
x
n
x
n
x
n
x
n
x
















It can be proved that
)
(
~
)
(
~
)
(
~
)
(
~
*
*
n
x
n
x
n
x
n
x
o
o
e
e






70. The Properties of DFT
 
  )
(
))
((
))
((
2
1
)
(
)
(
~
)
(
)
(
))
((
))
((
2
1
)
(
)
(
~
)
(
n
R
n
N
x
n
x
n
R
n
x
n
x
n
R
n
N
x
n
x
n
R
n
x
n
x
N
N
N
N
o
op
N
N
N
N
e
ep










Circular
conjugate
symmetric
component
Circular
conjugate
antisymmetric
component

71. The Properties of DFT
)
(
)
(
)
( n
x
n
x
n
x op
ep 

)
(
))
((
)
(
)
(
))
((
)
(
*
*
n
R
n
N
x
n
x
n
R
n
N
x
n
x
N
N
op
op
N
N
ep
ep






72. The Properties of DFT
)
(
)
(
)
( k
X
k
X
k
X op
ep 

)
(
))
((
)
(
)
(
))
((
)
(
*
*
k
R
k
N
X
k
X
k
R
k
N
X
k
X
N
N
op
op
N
N
ep
ep





 and
)
(k
Xep )
(k
Xop

73. The Properties of DFT
)]
(
))
((
Im[
)]
(
Im[
)]
(
))
((
Re[
)]
(
Re[
k
R
k
N
X
k
X
k
R
k
N
X
k
X
N
N
ep
ep
N
N
ep
ep





)]
(
))
((
Im[
)]
(
Im[
)]
(
))
((
Re[
)]
(
Re[
k
R
k
N
X
k
X
k
R
k
N
X
k
X
N
N
op
op
N
N
op
op






74. The Properties of DFT
)
(
))
((
)
(
then
)
(
))
((
)
(
if
k
R
k
N
X
k
X
n
R
n
N
x
n
x
N
N
N
N




 Circular even sequences
 Circular odd sequences
)
(
))
((
)
(
then
)
(
))
((
)
(
if
k
R
k
N
X
k
X
n
R
n
N
x
n
x
N
N
N
N







75. Circular convolution in Time
domain

78. Circular convolution

84. Circular convolution-Example

85. 85
x1(n)=(1,2,3) and x2(n)=(4,5,6).
One sequence is distributed clockwise and
the other counterclockwise and the shift of the inner circle is
clockwise.
1
2
3
6
5
4
*
+
* *
+
+
=6+10+12=28
1st term
1
2
3
4
6
5
*
+
* *
+
+
=4+12+15=31
2st term

86. 86
1
2
3
5
4
6
*
+
* *
+
+
=5+8+18=31
3rd term
x1(n)*
x2(n)=(1,2,3)*(4,5,6)=(18,31,31)
.
We have the same symmetry
as before
˜
y(n)  ˜
x1 (k) ˜
x2 (n  k)
k 0
N 1

˜
y(n)  ˜
x2(k) ˜
x1(n  k)
k 0
N 1


87. Cirular convolution-Matrix method

88. Circular convolution property

93. Use of DFT in linear filtering





1
0
]
[
]
[
]
[
L
k
k
n
h
k
x
n
y
DFT- based filtering always performs circular convolution.
But by zero padding adequetly,Circular convoltion can
yield the same result.
where n=0.....(L+M-1)-1

94. Linear convolution using DFT
1
]
[
]
[
)
( 2
1 


 L
L
N
of
length
to
n
h
and
n
x
padding
zero
a

95. ]
[
*
]
[
]
[
)
( n
h
n
x
n
y
a 
(2) calculate linear convolution by circular convolution
1
]
[
]
[
)
( 2
1 


 L
L
N
of
length
to
n
h
and
n
x
padding
zero
a
1
]
[
]
[
)
( 2
1 


 L
L
N
of
length
to
n
h
and
n
x
padding
zero
a
(1)calculate N point circular convolution by linear convolution
(3) calculate linear convolution by DFT
]
[
]
[
]
[
)
](
[
)
( n
R
rN
n
y
n
h
N
n
x
b N
r






]
[
)
](
[
]
[
]
[
)
( n
h
N
n
x
n
h
n
x
b 

]
[
]
[
int
)
( n
h
and
n
x
of
D FT
s
po
N
b
]
[
)
](
[
]
[
*
]
[
]}
[
]
[
{
]
[
)
](
[
)
(
n
h
N
n
x
n
h
n
x
k
H
k
X
IDFT
n
h
N
n
x
c


LINEAR & CIRCULAR CONVOLUTION

96. Example:Cir.conv

97. Circular conv

98. FAST FOURIER TRANSFORM

99. Discrete Fourier Transform
• The DFT pair was given as
• Baseline for computational complexity:
– Each DFT coefficient requires
• N complex multiplications
• N-1 complex additions
– All N DFT coefficients require
• N2 complex multiplications
• N(N-1) complex additions
• Complexity in terms of real operations
• 4N2 real multiplications
• 2N(N-1) real additions
• Most fast methods are based on symmetry properties
– Conjugate symmetry
– Periodicity in n and k
   





1
N
0
k
kn
N
/
2
j
e
k
X
N
1
]
n
[
x
   






1
N
0
n
kn
N
/
2
j
e
]
n
[
x
k
X
           kn
N
j
n
k
N
j
kN
N
j
n
N
k
N
j
e
e
e
e /
2
/
2
/
2
/
2 




 




        n
N
k
N
j
N
n
k
N
j
kn
N
j
e
e
e 




 /
2
/
2
/
2 



100. Properties-Example

101. Decimation-In-Time FFT Algorithms
• Makes use of both symmetry and periodicity
• Consider special case of N an integer power of 2
• Separate x[n] into two sequence of length N/2
– Even indexed samples in the first sequence
– Odd indexed samples in the other sequence
• Substitute variables n=2r for n even and n=2r+1 for odd
• G[k] and H[k] are the N/2-point DFT’s of each subsequence
       













1
odd
n
/
2
1
even
n
/
2
1
0
/
2
]
[
]
[
]
[
N
kn
N
j
N
kn
N
j
N
n
kn
N
j
e
n
x
e
n
x
e
n
x
k
X 


   
   
k
H
W
k
G
W
r
x
W
W
r
x
W
r
x
W
r
x
k
X
k
N
N
rk
N
k
N
N
rk
N
N
k
r
N
N
rk
N





















]
1
2
[
]
2
[
]
1
2
[
]
2
[
1
2
/
0
r
2
/
1
2
/
0
r
2
/
1
2
/
0
r
1
2
1
2
/
0
r
2

102. Decimation In Time
• 8-point DFT example using
decimation-in-time
• Two N/2-point DFTs
– 2(N/2)2 complex multiplications
– 2(N/2)2 complex additions
• Combining the DFT outputs
– N complex multiplications
– N complex additions
• Total complexity
– N2/2+N complex multiplications
– N2/2+N complex additions
– More efficient than direct DFT
• Repeat same process
– Divide N/2-point DFTs into
– Two N/4-point DFTs
– Combine outputs

104. DIT-FFT

107. Decimation In Time Cont’d
• After two steps of decimation in time
• Repeat until we’re left with two-point DFT’s

108. Decimation-In-Time FFT Algorithm
• Final flow graph for 8-point decimation in time
• Complexity:
– Nlog2N complex multiplications and additions

109. Butterfly Computation
• Flow graph constitutes of butterflies
• We can implement each butterfly with one multiplication
• Final complexity for decimation-in-time FFT
– (N/2)log2N complex multiplications and additions

110. In-Place Computation
• Decimation-in-time flow graphs require two
sets of registers
– Input and output for each stage
• Note the arrangement of the input indices
– Bit reversed indexing
       
       
       
       
       
       
       
       
111
x
111
X
7
x
7
X
011
x
110
X
3
x
6
X
101
x
101
X
5
x
5
X
001
x
100
X
1
x
4
X
110
x
011
X
6
x
3
X
010
x
010
X
2
x
2
X
100
x
001
X
4
x
1
X
000
x
000
X
0
x
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

























111. Example: 8 point FFT
(1)Number of stages:
– Nstages = 3
(2)Blocks/stage:
– Stage 1: Nblocks = 4
– Stage 2: Nblocks = 2
– Stage 3: Nblocks = 1
(3)B’flies/block:
– Stage 1: Nbtf = 1
– Stage 2: Nbtf = 2
– Stage 3: Nbtf = 2
FFT Implementation
W0 -1
W0 -1
W0 -1
W0 -1
W2 -1
W0
-1
W0
W2 -1
-1
W0
W1 -1
W0
W3 -1
-1
W2
-1
Stage 2 Stage 3
Stage 1
• Decimation in time FFT:
– Number of stages = log2N
– Number of blocks/stage = N/2stage
– Number of butterflies/block = 2stage-1

112. FFT Implementation
Start Index 0
Input Index 1
Twiddle Factor Index N/2 = 4 4/2 = 2 2/2 = 1
Indicies Used W0 W0
W2
W0
W1
W2
W3
0
2
0
4
W0 -1
W0 -1
W0 -1
W0 -1
W2 -1
W0
-1
W0
W2 -1
-1
W0
W1 -1
W0
W3 -1
-1
W2
-1
Stage 2 Stage 3
Stage 1

113. Decimation-in-time FFT algorithm
Most conveniently illustrated by considering the special case of N
an integer power of 2, i.e, N=2v.
Since N is an even integer, we can consider computing X[k] by
separating x[n] into two (N/2)-point sequence consisting of the
even numbered point in x[n] and the odd-numbered points in x[n].
or, with the substitution of variable n=2r for n even and n=2r+1
for n odd

 

dd
]
[
]
[
]
[
o
n
nk
N
even
n
nk
N W
n
x
W
n
x
k
X










1
)
2
/
(
0
)
1
2
(
1
)
2
/
(
0
2
]
1
2
[
]
2
[
]
[
N
r
k
r
N
N
r
rk
N W
r
x
W
r
x
k
X









1
)
2
/
(
0
2
1
)
2
/
(
0
2
)
](
1
2
[
)
](
2
[
N
r
rk
N
k
N
N
r
rk
N W
r
x
W
W
r
x

114. Since
That is, WN
2 is the root of the equation WN/2=1
Consequently,
Both G[k] and H[k] can be computed by (N/2)-point DFT, where
G[k] is the (N/2)-point DFT of the even numbered points of the
original sequence and the second being the (N/2)-point DFT of
the odd-numbered point of the original sequence.
Although the index ranges over N values, k = 0, 1, …, N-1, each of
the sums must be computed only for k between 0 and (N/2)-1,
since G[k] and H[k] are each periodic in k with period N/2.
2
/
)
2
/
/(
2
)
/
2
(
2
2
N
N
j
N
j
N W
e
e
W 

 

1
,...,
1
,
0
],
[
]
[ 


 N
k
k
H
W
k
G k
N









1
)
2
/
(
0
2
1
)
2
/
(
0
2
)
](
1
2
[
)
](
2
[
]
[
N
r
rk
N
k
N
N
r
rk
N W
r
x
W
W
r
x
k
X

119. Decomposing N-point DFT into two (N/2)-point DFT for the
case of N=8

120. We can further decompose the (N/2)-point DFT into two
(N/4)-point DFTs. For example, the upper half of the previous
diagram can be decomposed as

121. Hence, the 8-point DFT can be obtained by the following
diagram with four 2-point DFTs.

122. Flow graph of a 2-point DFT
Finally, each 2-point DFT can be implemented by the following
signal-flow graph, where no multiplications are needed.

123. Flow graph of complete decimation-in-time decomposition of an 8-point DFT.

124. In each stage of the decimation-in-time FFT algorithm, there
are a basic structure called the butterfly computation:
The butterfly computation can be simplified as follows:
Flow graph of a basic butterfly
computation in FFT.
Simplified butterfly computation.
]
[
]
[
]
[
]
[
]
[
]
[
1
1
1
1
q
X
W
p
X
q
X
q
X
W
p
X
p
X
m
r
N
m
m
m
r
N
m
m









125. Flow graph of 8-point FFT using the simplified butterfly computation

126. In the above, we have introduced the decimation-in-time
algorithm of FFT.
Here, we assume that N is the power of 2. For N=2v, it requires
v=log2N stages of computation.
The number of complex multiplications and additions required was
N+N+…N = Nv = N log2N.
When N is not the power of 2, we can apply the same principle
that were applied in the power-of-2 case when N is a composite
integer. For example, if N=RQ, it is possible to express an N-
point DFT as either the sum of R Q-point DFTs or as the sum of
Q R-point DFTs.
In practice, by zero-padding a sequence into an N-point sequence
with N=2v, we can choose the nearest power-of-two FFT
algorithm for implementing a DFT.
The FFT algorithm of power-of-two is also called the Cooley-
Tukey algorithm since it was first proposed by them.
For short-length sequence, Goertzel algorithm might be more
efficient.

128. Decimation-In-Frequency FFT Algorithm
• Final flow graph for 8-point decimation in
frequency

129. Decimation-In-Frequency FFT Algorithm
• The DFT equation
• Split the DFT equation into even and odd
frequency indexes
• Substitute variables to get
• Similarly for odd-numbered frequencies
  



1
N
0
n
nk
N
W
]
n
[
x
k
X
  











1
2
/
2
1
2
/
0
2
1
0
2
]
[
]
[
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[
2
N
N
n
r
n
N
N
n
r
n
N
N
n
r
n
N W
n
x
W
n
x
W
n
x
r
X
   
 
















1
2
/
0
2
/
1
2
/
0
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/
1
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0
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]
2
/
[
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/
[
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[
2
N
n
nr
N
N
n
r
N
n
N
N
n
r
n
N W
N
n
x
n
x
W
N
n
x
W
n
x
r
X
     








1
2
/
0
1
2
2
/
]
2
/
[
]
[
1
2
N
n
r
n
N
W
N
n
x
n
x
r
X

131. Decimation in Frequency algorithm

132. Decimation in Frequency

134. DIF-FFT ALGORITHM

135. DIF-FFT ALGORITHM

138. Example
• Using decimation-in-time FFT algorithm compute
DFT of the sequence
{-1 –1 –1 –1 1 1 1 1}
• Solution: Twiddle factors are

139. Solution and signal flow graph of the example

143. IDFT Using DIF-FFT