12. If two finite duration sequence x1(n) & x2(n) are finite duration
Sequence both of length N with DFTs X1(K)& X2(K)
Now we find x3(n) for which the DFT is X3(K) Where
Similarly
N
17. DFT used to perform linear filtering
FFT Set of Algorithms, proposed by
Cooly and Tukey 1965
FFT: computing DFT with less number of
computation than direct evaluation of DFT.
FFT used in digital spectral analysis,
filter simulation, auto correlation & pattern
recognition.
FFT reduce the computation time
Performance improved compared to DFT.
18. Evaluate one value of K No.of complex Multiplication N.
All the N value of X(k)
No.of Complex Multiplication is N*N=N2
No.of Addition
one value of k: (N-1) Addition
For N value of k: N(N-1)
19. =
No.of multiplication
Computation of X(k) for each K Requires
4N (each Summation N multiplication) Real Multiplication
X(k) for all k from 0 to N-1 N(4N)=4 N2 Real Multiplication
No.of Addition:
Each of 4 summation of N terms
Requires N-1 Real two input addition4(N-1)
Combine the Sum to Get the real part & imaginary part 2
For one k value 4(N-1)+ 2=4N-2
For all the value of K N*(4N-2)
20. Fast Fourier Transform
FFT applicable for two basic properties for twiddle
factor
It reduces the no.of complex multiplication
Basic Principle of FFT
Decomposing the computation of Discrete Fourier
transform of sequence of length N into successively smaller
discrete FTs
Types of FFT
1. Decimation in Time [DIT]
2. Decimation Frequency [DIF]
21. The sequence for which we need the DFT is
successively divided into smaller sequence &
DFTs of these subsequence are combined in a
certain pattern to obtain the required DFT of the
entire sequence.
It also known as Radix-2 DIT FFT Algorithm.
The No.of output points N can be expressed as a
power 2 that is N= 2M, M Integer
22.
23. 0 1 2 3 4 5 6 7
0 2 4 6 1 3 5 7
0 4 2 6 1 5 3 7
N Point
DFT
N/2 Point
DFT
N/4 Point
DFT
Even Odd
Even-Even Even Odd Odd -Even Odd Odd