ENT 315
Medical Signal Processing
CHAPTER 3
FAST FOURIER TRANSFORM
Dr. Lim Chee Chin
Outline
• Definition and Introduction FFT
– Properties of FFT
– Algorithm of FFT
§ Decimate in Time (DIT) FFT
§ Steps for radix 2 for 4point and 8point
§ Decimate in Frequency (DIF) FFT
§ Steps for radix 2 for 4point and 8point
• Definition and Introduction IFFT
– IDFT using FFT algorithm
FFT
• DFT can be used to perform linear filtering,
correlation analysis and spectral analysis.
• Direct computation of DFT involves large number
of computations.
• In, 1965, Cooley and Turkey proposed fast Fourier
transform, which is a highly efficient procedure
for computing the DFT of a finite series with less
number of computations.
• FFT is used in digital spectral analysis, filter
simulation, auto-correlation and pattern
recognition.
FFT
• Definition FFT:
• Direction computation of DFT involves (N-1) complex
additions and N complex multiplications
• For all N value of X(k), the number of complex
multiplications required is
and total number of
complex additions required is N(N-1)
Properties of FFT
• FFT exploit the two basic properties of the
twiddle factor as shown above and reduces
the number of complex multiplications
required to perform DFT from
to
FFT Algorithms
FFT Algorithm
Decimation in
Time
Decimation in
Frequency
Decimation In Time
Fast Frequency Transform
-
DIT FFT Definition
Steps of radix 2-DIT FFT Algorithm
4 point DIT FFT (Flow Graph)
4 point DIT FFT (Reduced Flow Graph)
8 point DIT FFT (Flow Graph)
8 point DIT FFT (Reduced Flow Graph)
Decimation-In-Time (DIT) Algorithm
Even,
Odd,
4-point DIT FFT
X(n) = { x(0), x(1), x(2), x(3)}
Recall
x(0)
Even
Odd
x(2)
-1
-1
x(1)
x(3)
-1
-1
Recall
4-point DIT FFT - Final Graph
-1
-1
-1
In order decimation
-1
In order
Example: 4-point DIT FFT
Question:
Given
algorithm.
Solution:
N=4
, find X(k) using DIT FFT
8-point DIT FFT
First Stage
Second Stage 8-point DIT FFT
Second Stage computation
Second Stage of DIT-FFT
• DFT computation into two 4-point DFT
computations.
• Branches entering a node are added to
produce the node variable.
• If no coefficient is indicated, it means that the
branch transmittance is equal to one.
• Transmittance is an integer power of WN.
Three Stages of Computation
Combined butterfly diagram
8-point DFT
• In DIT algorithm, we can find that for the output sequence
to be in a natural order the input sequence has to be stored
in a shuffled order.
• Computation of 8-point DFT using radix - 2 FFT, involves 3
stages of computation.
• The given 8-point sequence is decimated to 2-point
sequences.
• For each 2-point sequence, the 2-point DFT is computed.
• From the result of 2-point of DFT, the 4-point DFT can be
computed.
• From the result of 4-point DFT, the 8-point DFT can be
computed.
Example
Given
using DIT FFT algorithm.
Solution:
N=8
x(n)={-1,0,2,0,-4,0,2,0}
n= 012 3 45 67
, find X(k)
Decimation In Frequency
Fast Frequency Transform
-
DIF FFT Definition
Steps of radix 2-DIT FFT Algorithm
4 point DIF FFT (Flow Graph)
4 point DIF FFT (Reduced Flow Graph)
8 point DIF FFT (Flow Graph)
8 point DIF FFT (Reduced Flow Graph)
• Odd indexed samples of X(k) can be obtained
from the 4-point DFT of the sequence g(n)
Reduced Flow Graph DIF FFT for N = 8
DIT vs. DIF
Differences
• For DIT, the input is bitreversed, while the output is
in natural order.
• Whereas, for DIF, the input is
in natural order while the
output is in bit reversed.
• The DIF butterfly is slightly
different from the DIT wherein
DIF the complex multiplication
takes place after the addsubtract operation.
Similarities
• Both algorithms requires
Nlog2N operations to
compute the DFT.
• Both algorithms can be
done in-place and both
need to perform bit reversal
at some place during
computation.
First stage of 8-point DIF FFT
Second stage of 8-point DIF FFT
Reduced Flow Graph DIF FFT for N = 8
DIF FFT
Question
• Given x(n)={1,2,-1,2,4,2,-1,2} find X(k) using
DIF FFT.
Inverse Fast Fourier Transform
(IFFT)
Inverse Fast Fourier Transform (IFFT)
• An FFT algorithm for calculating the DFT samples can also
be used to evaluate efficiently the inverse DFT (IDFT).
• The inverse DFT is given by
• Taking this complex conjugate of above equation.
• RHS is DFT of sequence X*(k). Therefore,
• Taking the complex conjugate of both side, the
desired output sequence x(n) which is given by
• Use to compute IDFT
– if the output is divided by N
– The “twiddle factors” are negative power of
• IFFT flow graph obtained from an FFT flow graph
by
–
–
–
–
replacing all the x(n) by X(k),
dividing the input data by N, or
dividing each stage by 2 when N is power of 2
Changing the exponents of
to negative values.
• DIT IFFT and DIF IFFT are same.
4-point IFFT Flow Graph
X(n) = { x(0), x(1), x(2), x(3)}
x(0)
Even
x(1)
x(2)
x(3)
-1
-1
Odd
-1
-1
Example: 4-point IFFT
Question:
Given
using DIT IFFT algorithm.
Solution:
N=4
Answer:
, find x(n)
8 point IFFT
Example
Given X(k)={-1,3,-9,3,-1,3,-9,3}, find x(n) using
DIT IFFT algorithm.
Solution:
N=8
Answer:
Example (from DIF)
Given X(k)={11,-3,7,-3,-5,-3,7,-3}, find x(n)
using DIF IFFT algorithm.
Solution:
N=8
Answer: