DSP_FOEHU - Lec 09 - Fast Fourier Transform
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The document discusses the Fast Fourier Transform (FFT) algorithm. It explains that the FFT decomposes an N-point discrete Fourier transform (DFT) into smaller DFTs of size N/2, taking advantage of the periodicity and symmetry of complex numbers. For N that is a power of 2, it separates the input sequence into even and odd indexed parts, then recursively applies this decomposition until 2-point DFTs are reached. This decimation-in-time approach reduces the computational complexity from O(N^2) for the direct DFT to O(NlogN) for the FFT.
![FAST FOURIER TRANSFORM ALGORITHMS
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
Substitute variables n=2r for n even and n=2r+1 for odd
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DSP_FOEHU - Lec 09 - Fast Fourier Transform
- 1. رَـدْقِـن،،،لمااننا نصدقْْقِنرَد LECTURE (09) Fast Fourier Transform Assist. Prof. Amr E. Mohamed
- 2. FAST FOURIER TRANSFORM ALGORITHMS The basic idea behind the fast Fourier transform (FFT) Decompose the N-point DFT computation into computations of smaller-size DFTs Take advantage of the periodicity and symmetry of the complex number . Assuming that N is an even number. Divide the data sample into two groups by decimation 2 1 2 ,...1,0' )1'2()'( )'2()'( N nfor nxnv nxnu )...7()3();5()2();3()1();1()0( )...6()3();4()2();2()1();0()0( xvxvxvxv xuxuxuxu
- 3. FAST FOURIER TRANSFORM ALGORITHMS 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 Substitute variables n=2r for n even and n=2r+1 for odd 3 1 oddn /2 1 evenn /2 1 0 /2 ][][][ N knNj N knNj N n knNj enxenxenxkX 1 2 ]12[]2[ 1 2 0]12[]2[ 12/ 0r )2/( 2/ 12/ 0r )2/( 2/ 12/ 0r 2/ 12/ 0r 2/ Nk N WrxWWrx N kWrxWWrx kX N Nkr N k N N Nkr N N rk N k N N rk N
- 4. Computation Complexity To calculate the DFT of N-Points discrete time signal, we need: (N-1)2 Complex Multiplications N(N-1) Complex Additions. To calculate the FFT of N-Points discrete time signal, we need: Complex Multiplications Complex Additions. 4 NlogN 2 Nlog 2 N 2
- 5. 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 5
- 6. 6 After two steps of decimation in time On the whiteboard Repeat until we’re left with two-point DFT’s On the whiteboard Decimation In Time
- 7. 7 Decimation-In-Time FFT Algorithm Final flow graph for 8-point decimation in time On the whiteboard
- 8. 8