Decimation of time,a type of fast fourier transform in digital signal processing.

1. Decimation-In-Time
Suraj Kumar Saini
ID: 2015KUEC2015
Department of Electronics and Communication Engineering
Indian Institute of Information Technology Kota
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2. Overview
1 Discrete Fourier Transform
2 Decimation-In-Time
3 Radix-2 DFT Algorithm
4 Butterfly Structure
5 Complexity in DIT
6 Inverse Discrete Fourier Transform
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3. Discrete Fourier Transform
The DFT pair was given as :
X[k] =
N−1
n=0
x[n]e−j(2π/N)kn
x[n] =
1
N
N−1
k=0
X[k]ej(2π/N)kn
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
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4. Continue...
Most fast methods are based on symmetry properties
Symmetry:
W k
N [N − n] = W −kn
N = (W kn
N )∗
Periodicity in n and k:
W kn
N = W
k[n+N]
N = W
[k+N]n
N
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5. Decimation-In-Time
DIT algorithm is used to calculate the DFT of a N-point sequence.
First step of process of decimation is splitting a sequence in smaller
sequences.
A sequence of 16 numbers can be splitted in 2 sequences of 8.
Further,
Each sequence of 8 can be be splitted in two sequences of 4.
Subsequently each sequence of 4 can be splitted in two sequences of
two.
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6. Decimation-In-Time
Separate x[n] into two sequence of length N/2 sequence.
Even indexed samples in the first sequence
Odd indexed samples in the other sequence
X[k] =
N−1
n=0
x[n]e−j(2π/N)kn
=
N−1
n,even
x[n]e−j(2π/N)kn
+
N−1
n,odd
x[n]e−j(2π/N)kn
Substitute variables n=2r for n even and n=2r+1 for odd
X[k] =
N/2−1
r=0
x[2r]W 2rk
N +
N/2−1
r=0
x[2r + 1]W
(2r+1)k
N
= G[k] + W k
NH[k]
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7. 8-point Radix-2 DFT Algorithm
Repeat until were left with two-point DFTs
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8. Continue...
Final flow graph for 8-point decimation in time
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9. Butterfly Structure
Cross feed of G[k] and H[k] in flow diagram is called a butterfly, due
to shape
We can implement each butterfly with one multiplication
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10. Complexity in DIT
8-point DFT example using Decimation-in-time
Total complexity
N2
/2 + N complex multiplications
N2
/2 + N complex additions
More efficient than direct DFT
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11. Example
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12. Inverse Discrete Fourier Transform
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13. Thank you!
Questions and Suggestions
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