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
digital signal
digital signal
A/D
DSP
D/A
analog signal
analog 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
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 10
-10 20 40 60 80
100
t (ms) 10
-10 10 20 30 40 50
n (samples)

13. Basic Sequences and Operations
Delaying (Shifting) a sequence
y[n]  x[n  no]
Unit sample (impulse) sequence
1.5 1
n  0
1 n  0 0
[n]  
0.5
-10 -5 5 10
Unit step sequence
1.5 n 1
u[n]  0 
0.5
-10 -5 5 10
Exponential sequences
x[n]  An 1
0.5
-10 -5 5 10

14. Sinusoidal Sequences  A  nejon  n    j A 
Important class of sequences
xn  coson  
An exponential sequence with complex
   ejo and A  A ej
xn  An  A ej  nejon
 A  nejon
xn  A  cos n
 n    j A  n
sin n  
o o
x[n] is a sum of weighted sinusoids
Different from continuous-time, discrete-time sinusoids
– Have ambiguity of 2k in frequency
coso  2kn    coson  
– Are not necessary periodic with 2/o
coson    coson  oN   only if N 
2k
is an integer  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]
T{.}
y[n]  T{x[n]}
x[n]
y[n]
Example Discrete-Time Systems
– Moving (Running) Average
y[n]  x[n]  x[n  1]  x[n  2]  x[n  3]
Maximum
y[n]  maxx[n], x[n  1], x[n  2]
Ideal Delay System
y[n]  x[n  no]

17. Memoryless System y[n]  x[n]2 y[n]  signx[n] y[n]  x[n  no]
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
y[n]  x[n]2
Sign
y[n]  signx[n]
Counter Example
Ideal Delay System
y[n]  x[n  no]

18. Linear Systems T{x1[n]  x2[n]}  Tx1[n]  Tx2[n] and (add
Linear System: A system is linear if and only if
T{x1[n]  x2[n]}  Tx1[n]  Tx2[n] and
Tax[n] 
(add
Examples
– Ideal Delay System
y[n]  x[n  no]
 x1[n  no]  x2[n  no]
 ax1[n  no]
T{x1[n]  x2[n]}
T{x2[n]}  Tx1[n] Tax[n] aTx[n]

19. Time-Invariant Systems
Time-Invariant (shift-invariant) Systems
A time shift at the input causes corresponding time-shift at output
y[n]  T{x[n]}  y[n  no]  Tx[n  no ]
Example
Square
Delay the input the output is y   
Delay the output gives
n  x[n  n ] 2
y[n]  x[n]2 1 o
   x[n  no]
y n - no 2
Counter Example
– Compressor System
Delay the input the output is
y1n  x[Mn  no] yn - no   xMn  no 
y[n]  x[Mn]
Delay the output gives

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

21. Stable System x[n] x[n]  Bx    y[n]  By   y[n]  x[n]2
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
x[n]  Bx    y[n]  By  
Example
– Square
y[n]  x[n]2
if input is bounded by x[n]  B
output is b
Counter Example
– Log
x[n]
y[n]  log 10
even if input is bounded by x[n]  Bx  
output not bounded for xn  0  y0  log10 xn  

22. DFS,DTFT & DFT

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

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

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

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

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

28. The Four Fourier Methods28

29. Relations Among Fourier Methods29

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

31.  X ( jk )  1 x(t)e j T x(t)     2F  2 T
The Family of Fourier Transform
Periodic-Continuous－Fourier Series
T0 2
X ( jk )  1 
x(t)e j T
T 2 
x(t)  
  2F  2 T

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

33. 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.
 Introduction

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

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

36. DFT and IDFT

37. IDFT

38. The Discrete Fourier Transform (DFT)
The definition of DFT
N 1 
X (k )  DFT[x(n)] 
x(n)W
nk N
, 0  k  N 1
n0 1
N 1 
n0
x(n)  IDFT[ X (k )] 
X (k )W
nk N
, 0  n  N 1 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
Polar plot for
Time shift Property of DFT

47. Example of DFT

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

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

50. Example of IDFT
Remember:

51. Example of IDFT
Remember:

52. Problems on DFT

53. Problems on DFT

56. Properties of DFT

57. Circular shift of a sequence

61. DFT[x((n  m)) R DFT[W x(n)]  X ((k  l)) R (k) The Properties of DFT
Linearity
DFT[ax1 (n)  bx (n)] 
N3-point DFT, N3=max(N1,N2)
Circular shift of a sequence
DFT[x((n  m)) R
Circular shift in the frequency domain
DFT[W
nl N
x(n)]  X ((k  l)) R (k)
N N

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

63. 1 DFT[x1 (n)  x2 (n)]  X (k) X (k) N N 1 N 1 DFT[x1 (n) N x (n)]
The Properties of DFT
Circular convolution
N 1
x2 (n)   x1 (m)x2 ((n  m)) N RN (n) 
x1 (n)N
m0 
N 1
  x2 (m)x1 ((n  m)) N RN (n)  x2 (n) 
N x1 (n)
m0
DFT[x1 (n) N x (n)] 
Multiplication 1
DFT[x1 (n)  x2 (n)] 
X (k)
X (k) N N 1 2

64.   DFT[x1 (n) x2 (n)]  X1 (k) X 2 (k) DFT[x1 (n)  x2 (n)]  1
The Properties of DFT
Circular convolution 
N 1  
x (n) x (n) 
x1 (m)x2 ((n  m))
R (n)
N  N N 
1 2
m0  
N 1   
x2 (m)x1 ((n  m))
R (n)  x (n)
x (n)
N  N N 
m0  2 1
DFT[x1 (n) x2 (n)]  X1 (k) X 2 (k) N
Multiplication
DFT[x1 (n)  x2 (n)]  1
X1 (k)N
X 2 (k) N

65. The Properties of DFT rxy (m)   x(n) y *((n  m))N RN (m)
Circular correlation
Linear correlation
 
rxy (m)   x(n) y *(n  m)   x(n  m) y *(n)
n n
Circular correlation
rxy (m)   x(n) y *((n  m))N RN (m)
n0
N 1
N 1
  x((n  m))N y *(n)RN (m)
n0

66. then rxy (m)  IDFT[Rxy (k )]
The Properties of DFT
if Rxy (k )  X (k ) Y (k ) *
then rxy (m)  IDFT[Rxy (k )]
  x(n) y *((n  m))N RN (m)
n0
N 1
  x((n  m))N y *(n)RN (m)
n0
N 1

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

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

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

70. The Properties of DFT x (n)  x* ((N  n)) R xop (n)  x(n)  x (n)
ep ep
xop (n) 

71. The Properties of DFT X (k)  X * ((N  k)) R (k)
X ( and X op (k )
X (k)  X (k)
X (k)  X * ((N  k)) R (k)
ep ep N N
X (k)  X * ((N  k)) R (k)
op op N N

72. The Properties of DFT Re[Xep (k)]  Re[Xep ((N  k))N RN (k)]
Im[Xep (k)]   Im[Xep ((N  k))N RN (k)]
Re[Xop (k)]   Re[Xop ((N  k))N RN (k)]
Im[Xop (k)]  Im[Xop ((N  k))N RN (k)]

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

74. Circular convolution in Time domain

77. Circular convolution

83. Circular convolution- Example

84. 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 1 * * 6 4 + + + +
= =28
= =31
* 4
5 *
* 5
6 * 3
1st term 2 3
2st term 2 84 + +

85.  y(n)   x ( ˜ ˜ y(n)  ˜ ˜ ˜ x2(k)x (n  k) * 6 x1(n)*
We have the same symmetry as before
3rd term 1 * 5
y(n)   x (
N 1
˜ ˜ + +
=5+8+18=31
N 1 
k  0
y(n)  ˜
˜ ˜
x2(k)x (n  k) 1
* 6
4 * 3 2 + 85

86. Cirular convolution-Matrix method

87. Circular convolution property

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

93. Linear convolution using DFT
( a ) z e r o  p a d d in g
x [ n ]
a n d h [ n ] to
le n g th of N  L1  L 2  1

94. LINEAR & CIRCULAR CONVOLUTION
calculate N point circular convolution by linear convolution
( a ) y [ n ]  x [ n ] * h [ n ]
(b) x[n]( N )h[n]   y[n  rN ]RN [n]
calculate linear convolution by circular convolution 
r  
( a ) z e r o  p a d d in g x [ n ] a n d h [ n ]
( b ) x [ n ]  h [ n ] 
(3) calculate linear convolution by DFT
( a ) z e r o  p a d d in g x [ n ] a n d h [ n ]
( b ) N po in t s D F T of x [ n ] and
h [ n ]
(c ) x[n ]( N )h[n ]  ID FT { X [k ]H [k ]}
x[n ] * h[n ]  x[n ]( N )h[n ]

95. Example:Cir.conv

96. Circular conv

97. FAST FOURIER TRANSFORM

98. Xk   x[n]e j2 / Nkn
Discrete Fourier Transform
The DFT pair was given as
N1
Xk   x[n]e j2 / Nkn
n0
Baseline for computational complexity:
x[n]  1 N1 Xkej2 / Nkn
N k 0
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
e j 2 / N k N n   e j 2 / N kNe j 2 / N k  n   e j 2 / N kn
e j 2 / N kn  e j 2 / N k n N   e j 2 / N k  N n

99. Properties- Example

100. 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
N 1 N 1 N 1
X k   x[n]e j 2 / N kn   x[n]e j 2 / N kn   x[n]e j 2 / N kn
n0 n even n odd
Substitute variables n=2r for n even and n=2r+1 for odd
 2rk  N
X k 
N / 21 N / 21
2r 1k N
x[2r]W
 x[2r 1]W
r0 r0
N / 21 N / 21 
N / 2 N  rk
r0 
x[2r]W
 W k x[2r 1]W rk
N / 2
r0
 Gk  W H 
k  k N
G[k] and H[k] are the N/2-point DFT’s of each subsequence

101. 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

103. DIT- FFT

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

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

108. 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

109. Decimation-in-time flow graphs require two sets of registers
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
X0 0  x0  X0 000  x000
X0 1  x4  X0 001  x100 X0 2  x2  X0 010  x010 X0 3  x6  X0 011  x110 X0 4  x1  X0 100  x001 X0 5  x5  X0 101  x101 X0 6  x3  X0 110  x011 X0 7  x7  X0 111  x111

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

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

112.      X [k]  x[n]W x[n]W X [k]  x[2r]W  x[2r 1]W 
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].
 
X [k] 
x[n]W
nk nk
x[n]W N N
n even n odd
or, with the substitution of variable n=2r for n even and n=2r+1 for n odd
( N / 2)1 ( N / 2)1  
X [k] 
x[2r]W
2rk 
x[2r 1]W
(2r 1)k N N
r 0
r 0
( N / 2)1
( N / 2)1 
r 0
N N 
r 0 
x[2r](W ) W k
2 rk
x[2r 1](W )
2 rk N

113.  N N  W 2  e2 j (2 / N )  e j 2 /( N / 2 X [k]  x[2r](W ) W k
Since N
That is, WN is the root of the equation W =1
2 N/2
Consequently,
( N / 2)1 ( N / 2)1
 N N 
X [k] 
x[2r](W ) W k
2 rk
x[2r 1](W )
2 rk N
r 0
k  0,1,..., N 1
r 0
 G[k ]  W k H[k ], N
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.

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

119. We can further decompose the (N/2)-point DFT into two (N/4)-point DFTs
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

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

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

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

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

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

125. 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.

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

128. Split the DFT equation into even and odd frequency indexes
Decimation-In-Frequency FFT Algorithm
The DFT equation
Xk 
N1 
x[n]W nk N
n0
Split the DFT equation into even and odd frequency indexes
N 1 N / 21
N 1
X 2r
  
x[n]W
n2r
 x[n]W 
n2r
x[n]W
n2r
N N N
n0 n0
n N / 2
Substitute variables to get
N / 21 N / 21
N / 21
X 2r
 
n N / 22r 
x[n]  x[n  N / 2]W
x[n]W
n2r
 x[n  N / 2]W  nr
N N
N / 2
Similarly for odd-numbered frequencies
n0 n0 n0
N / 21
X   
n0
x[n]  x[n  N / 2]W
2r 1
n2r 1
N / 2

130. Decimation in Frequency algorithm

131. Decimation in Frequency

133. DIF-FFT ALGORITHM

134. DIF-FFT ALGORITHM

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

138. Solution and signal flow graph of the example

142. IDFT Using DIF-FFT