GVPW DIGITAL SIGNAL PROCESSING Page 1
BHARATI VIDYAPEETH COLLEGE OF ENGINEERING, PUNE
DEPARTMENT OF ELECTRONICS ENGINEERING
SUB: DIGITAL SIGNAL PROCESSING
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DSP notes Unit-2 with examples, Lecture notes of Digital Signal Processing
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2023/2024
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BHARATI VIDYAPEETH COLLEGE OF ENGINEERING, PUNE
DEPARTMENT OF ELECTRONICS ENGINEERING
SUB: DIGITAL SIGNAL PROCESSING
𝐀
𝐀
BHARATI VIDYAPEETH COLLEGE OF ENGINEERING, PUNE
DEPARTMENT OF ELECTRONICS ENGINEERING
SUB: DIGITAL SIGNAL PROCESSING
EXPERIMENT No. - 02
Aim : - To study the DITFFT using MATLAB.
Apparatus : - Language software MATLAB
THEORY;
Decimation – In – Time (DIT) FFT algorithm
In this algorithm, the time – domain sequence x[n] is decimated into two
𝑁
- point sequences, one composed
2
of even – indexed values of x[n], and other composed of odd – indexed values of x[n].i.e.,
[
]
[
]
𝑎𝑛𝑑, ℎ[𝑛] = 𝑥[2𝑛 + 1 ].....𝑒𝑞𝑛. 2
The N – point DFT of x[n] is given by
𝑁−
𝑋(𝑘) = ∑ 𝑥[𝑛]𝑊
𝑛𝑘
This can be rewritten as
𝑛=
𝑁−1 𝑁−
𝑋
( 𝑘
) = ∑ 𝑥[𝑛]𝑊
𝑛𝑘
+ ∑ 𝑥[𝑛]𝑊
𝑛𝑘
𝑁
−
1 2
𝑛=0,𝑒𝑣
𝑒𝑛
𝑁
𝑁
−
1 2
𝑁
𝑛=0,𝑜𝑑𝑑
= ∑ 𝑥[2𝑛]𝑊
2𝑛𝑘
+ ∑ 𝑥[2𝑛 + 1]𝑊
(2𝑛+1)𝑘
𝑛=
0
𝑁
−
1 2
𝐀
𝐀
𝑛=
0
𝑁
𝑁
−
1 2
= ∑ 𝑥[2𝑛]𝑊
2𝑛𝑘
𝑘
∑ 𝑥[2𝑛 + 1]𝑊
2𝑛𝑘
𝑛=
𝑁
𝑁
𝑁
𝑛=
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
𝐀
And using the symmetry property of the twiddle factor, W N
, and equations4 & 5
𝑘
Equations 3 and 6 result in the following butterfly diagram
For example, for N = 8, the DFT points in terms of G and H are
0
1
2
3
For the remaining 4 points X(4) to X(7), we use equations 4, 5, 6 and 7 to get
0
1
2
3
The butterfly diagram for the above set of equations is
2
2
The above process is repeated for calculating the N/2 point DFTs of g[n] and h[n], and this is continued till
we get two point DFTs. Once we reach a two – point sequence, say p[n]={p[0], p[1]}, its 2 – point DFT
would be
1
𝑃(𝑘) = ∑ 𝑝[𝑛]𝑊
𝑛𝑘
𝑛=
𝑜𝑟, 𝑃(𝑘) = 𝑝[ 0 ] + 𝑝[ 1 ]𝑊
𝑘
[
]
[
]
[
]
2
[
]
[
]
− 𝑝[1]
The overall butterfly diagram for DIT FFT algorithm for N =8 is