WEEK10 Fast Fourier Transform (FFT) (Theory and Implementation)
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1 WEEK10 Fast Fourier Transform (FFT) (Theory and Implementation)
2 Learning Objectives DFT algorithm. Conversion of DFT to FFT algorithm. Implementation of the FFT algorithm. Chapter 19, Slide 2
3 DFT Algorithm The Fourier transform of an analogue signal x(t) is given by: The Discrete Fourier Transform (DFT) of a discrete-time signal x(nt) is given by: Where: Chapter 19, Slide 3
4 DFT Algorithm If we let: then: Chapter 19, Slide 4
5 DFT Algorithm x[n] = input X[k] = frequency bins W = twiddle factors X(0) X(1) X(k) X(N) = x[0]w N0 + x[1]w 0*1 N + + x[n]w 0*(N) N = x[0]w N0 + x[1]w 1*1 N + + x[n]w 1*(N) N : = x[0]w N0 + x[1]w k*1 N + + x[n]w k*(n) N : = x[0]w N0 + x[1]w (N)*1 N + + x[n]w (N)(N) N Note: For N samples of x we have N frequencies representing the signal. Chapter 19, Slide 5
6 Performance of the DFT Algorithm The DFT requires N 2 (NxN) complex multiplications: Each X(k) requires N complex multiplications. Therefore to evaluate all the values of the DFT ( X(0) to X(N) ) N 2 multiplications are required. The DFT also requires (N)*N complex additions: Each X(k) requires N additions. Therefore to evaluate all the values of the DFT (N)*N additions are required. Chapter 19, Slide 6
7 Performance of the DFT Algorithm Can the number of computations required be reduced? Chapter 19, Slide 7
8 DFT FFT A large amount of work has been devoted to reducing the computation time of a DFT. This has led to efficient algorithms which are known as the Fast Fourier Transform (FFT) algorithms. Chapter 19, Slide 8
9 DFT FFT [1] x[n] = x[0], x[1],, x[n] Lets divide the sequence x[n] into even and odd sequences: x[2n] = x[0], x[2],, x[n-2] x[2n+1] = x[1], x[3],, x[n] Chapter 19, Slide 9
10 DFT FFT Equation 1 can be rewritten as: [2] Since: Then: Chapter 19, Slide 10
11 DFT FFT The result is that an N-point DFT can be divided into two N/2 point DFT s: N-point DFT Where Y(k) and Z(k) are the two N/2 point DFTs operating on even and odd samples respectively: Two N/2- point DFTs Chapter 19, Slide 11
12 DFT FFT Periodicity and symmetry of W can be exploited to simplify the DFT further: [3] Or: : Symmetry And: Chapter 19, Slide 12 : Periodicity
13 DFT FFT Symmetry and periodicity: W 8 4 W 8 5 W 8 3 W 8 6 W 8 2 W 8 7 W 8 0 =W 8 8 W 8 1 = W 8 9 W k+n/2 N = - W k N W k+n/2 N/2 =W k N/2 W k+4 8 = - W k 8 W k+8 8 =W k 8 Chapter 19, Slide 13
14 DFT FFT Finally by exploiting the symmetry and periodicity, Equation 3 can be written as: [4] Chapter 19, Slide 14
15 DFT FFT Y(k) and W N k Z(k) only need to be calculated once and used for both equations. Note: the calculation is reduced from 0 to N to 0 to (N/2-1). Chapter 19, Slide 15
16 DFT FFT Y(k) and Z(k) can also be divided into N/4 point DFTs using the same process shown above: Chapter 19, Slide 16 The process continues until we reach 2 point DFTs.
17 DFT FFT x[0] x[2] x[4] N/2 point DFT y[0] y[1] y[2] X[0] = y[0]+ z[0] X[1] = y[1]+w 1 z[1] x[n-2] y[n-2] x[1] x[3] x[5] N/2 point DFT z[0] z[1] z[2] W 1 X[N/2] = y[0]- z[0] X[N/2+1] = y[1]-w 1 z[1] x[n] z[n/2] Illustration of the first decimation in time FFT. Chapter 19, Slide 17
18 FFT Implementation Calculation of the output of a butterfly : Y(k) = U r +ju i U =U r +ju i = X(k) W Nk Z(k) = (L r +jl i )(W r +jw i ) L =L r +jl i = X(k+N/2) Key: U = Upper r = real L = Lower i = imaginary Different methods are available for calculating the outputs U and L. The best method is the one with the least number of multiplications and additions. Chapter 19, Slide 18
19 FFT Implementation Calculation of the output of a butterfly : U r +ju i U =U r +ju i (L r +jl i )(W r +jw i ) L =L r +jl i Chapter 19, Slide 19
20 FFT Implementation Calculation of the output of a butterfly : U r +ju i U =U r +ju i = (L r W r -L i W i + U r )+j(l r W i + L i W r + U i ) (L r +jl i )(W r +jw i ) To further minimise the number of operations (* and +), the following are calculated only once: temp1 = L r W r temp2 = L i W i temp3 = L r W i temp4 = L i W r temp1_2 = temp1 - temp2 L =L r +jl i = (U r -L r W r + L i W i )+j(u i -L r W i -L i W r ) temp3_4 = temp3 + temp4 U r = temp1 - temp2 + U r = temp1_2 + U r U i = temp3 + temp4 + U i = temp3_4 + U i L r = U r - temp1 + temp2 = U r - temp1_2 L i = U i - temp3 - temp4 = U i - temp3_4 Chapter 19, Slide 20
21 FFT Implementation (Butterfly Calculation) Converting the butterfly calculation into C code: temp1 = (y[lower].real * WR); temp2 = (y[lower].imag * WI); temp3 = (y[lower].real * WI); temp4 = (y[lower].imag * WR); temp1_2 = temp1 - temp2; temp3_4 = temp 3 + temp4; y[upper].real = temp1_2 + y[upper].real; y[upper].imag = temp3_4 + y[upper].imag; y[lower].imag = y[upper].imag - temp3_4; y[lower].real = y[upper].real - temp1_2; Chapter 19, Slide 21
22 FFT Implementation To efficiently implement the FFT algorithm a few observations are made: Each stage has the same number of butterflies (number of butterflies = N/2, N is number of points). The number of DFT groups per stage is equal to (N/2 stage ). The difference between the upper and lower leg is equal to 2 stage. The number of butterflies in the group is equal to 2 stage. Chapter 19, Slide 22
23 FFT Implementation Example: 8 point FFT W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 23 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
24 FFT Implementation Example: 8 point FFT (1) Number of stages: W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 24 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
25 Stage 1 FFT Implementation Example: 8 point FFT (1) Number of stages: N stages = 1 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Chapter 19, Slide 25 Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
26 Stage 1 FFT Implementation Stage 2 Example: 8 point FFT (1) Number of stages: N stages = 2 W 1 W 3 Chapter 19, Slide 26 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
27 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 W 1 W 3 Chapter 19, Slide 27 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
28 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: W 1 W 3 Chapter 19, Slide 28 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
29 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 1 W 1 W 3 Chapter 19, Slide 29 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
30 Block 1 Block 2 Stage 1 FFT Implementation Stage 2 Stage 3 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 2 W 1 W 3 Chapter 19, Slide 30 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
31 Block 1 Block 2 Block 3 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 3 W 3 Chapter 19, Slide 31 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
32 Block 1 Block 2 Block 3 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Block 4 W 3 Chapter 19, Slide 32 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
33 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 1 W 3 Chapter 19, Slide 33 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
34 Block 1 Block 2 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 W 3 Chapter 19, Slide 34 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
35 Block 1 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 W 3 Chapter 19, Slide 35 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage
36 Chapter 19, Slide 36 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1:
37 Chapter 19, Slide 37 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1
38 Chapter 19, Slide 38 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 1
39 Chapter 19, Slide 39 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2
40 Chapter 19, Slide 40 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 1
41 Chapter 19, Slide 41 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 2
42 Chapter 19, Slide 42 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 3
43 Chapter 19, Slide 43 Stage 1 FFT Implementation Stage 2 Stage 3 W 1 W 3 Decimation in time FFT: Number of stages = log 2 N Number of blocks/stage = N/2 stage Number of butterflies/block = 2 stage Example: 8 point FFT (1) Number of stages: N stages = 3 (2) Blocks/stage: Stage 1: N blocks = 4 Stage 2: N blocks = 2 Stage 3: N blocks = 1 (3) B flies/block: Stage 1: N btf = 1 Stage 2: N btf = 2 Stage 3: N btf = 4
44 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 Chapter 19, Slide 44
45 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 4/2 = 2 Chapter 19, Slide 45
46 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index Input Index Twiddle Factor Index N/2 = 4 4/2 = 2 2/2 = 1 Chapter 19, Slide 46
47 FFT Implementation Stage 1 Stage 2 Stage 3 W 1 W 3 Start Index 0 Input Index 1 Twiddle Factor Index N/2 = 4 4/2 = 2 2/2 = 1 Indicies Used W Chapter 19, Slide 47 W 3
48 FFT Implementation The most important aspect of converting the FFT diagram to C code is to calculate the upper and lower indicies of each butterfly: GS = N/4; /* Group step initial value */ step = 1; /* Initial value */ NB = N/2; /* NB is a constant */ for (k=n; k>1; k>>1) /* Repeat this loop for each stage */ { for (j=0; j<n; j+=gs) /* Repeat this loop for each block */ { for (n=j; n<(j+gs); n+=step) /* Repeat this loop for each butterfly */ { upperindex = n; lowerindex = n+step; } } /* Change the GS and step for the next stage */ } GS = GS << 1; step = step << 1; Chapter 19, Slide 48
49 FFT Implementation How to declare and access twiddle factors: (1) How to declare the twiddle factors: struct { short real; // * cos (2*pi*n) -> Q15 format short imag; // * sin (2*pi*n) -> Q15 format } w[] = { 32767,0, 32767,-201,... }; (2) How to access the twiddle factors: short temp_real, temp_imag; temp_real = w[i].real; temp_imag = w[i].imag; Chapter 19, Slide 49
50 Links: FFT Implementation Project location: \Code\Chapter 19 - Fast Fourier Transform\ Projects: FFT Decimation in Time: \FFT_C_Fixed_DIT\ FFT Decimation in Frequency: \FFT_C_Fixed_DIF\ Decimation in Time Code: FFTMain.c FFT code (fft1024.c) Laboratory sheet: FFTLabSheet.pdf Chapter 19, Slide 50
51 WEEK10 Fast Fourier Transform (FFT) (Theory and Implementation) - End -
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