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15Tarique_DSP_8ETC.PPT

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FFT is an efficient algorithm to compute the discrete Fourier transform (DFT) and convert a time domain signal to its frequency domain representation. Radix-2 FFT is the most common algorithm, in which the input is divided into groups of 2 samples at each stage. FFT algorithms generally have a number of samples that is a power of 2, like 2N, to efficiently compute the DFT. The radix-2 FFT breaks the computation into "butterflies" or decimation in time (DIT) and decimation in frequency (DIF) structures to recursively compute the DFT. Twiddle factors representing complex roots of unity are used to compute the outputs of each butterfly operation.

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DFT and FFT • FFT is an algorithm to convert a time domain signal to DFT efficiently. • FFT is not unique. Many algorithms are available. • Each algorithm has merits and demerits. • In each algorithm, depending on the sequence needed at the output, the input is regrouped. • The groups are decided by the number of samples.
FFTs • The number of points can be nine too. • It can be 15 as well. • Any number in multiples of two integers. • It can not be any prime number. • It can be in the multiples of two prime numbers.
FFTs • The purpose of this series of lectures is to learn the basics of FFT algorithms. • Algorithms having number of samples 2N, where N is an integer is most preferred. • 8 point radix-2 FFT by decimation is used from learning point of view. • Radix-x: here ‘x’ represents number of samples in each group made at the first stage. They are generally equal. • We shall study radix-2 and radix-3.
Radix-2: DIT or, DIF • Radix-2 is the first FFT algorithm. It was proposed by Cooley and Tukey in 1965. • Though it is not the efficient algorithm, it lays foundation for time-efficient DFT calculations. • The next slide shows the saving in time required for calculations with radix-2. • The algorithms appear either in (a) Decimation In Time (DIT), or, (b) Decimation In Frequency (DIF). • DIT and DIF, both yield same complexity and results. They are complementary. • We shall stress on 8 to radix 2 DIT FFT.
Other popular Algorithms Besides many, the popular algorithms are: • Goertzel algorithm • Chirp Z algorithm • Index mapping algorithm • Split radix in prime number algorithm. have modified approach over radix-2. Split radix in prime number does not use even the twiddles. We now pay attention to 8/radix-2 butterfly FIT FFT algorithm.
Relationship between exponential forms and twiddle factors (W) for Periodicity = N Sr. No. Exponential form Symbolic form 01 e-j2n/N = e-j2(n+N)/N WN n = WN n+N 02 e-j2(n+N/2)/N = - e-j2n/N WN n+N/2= - WN n 03 e-j2k = e-j2Nk/N = 1 WN N+K = 1 04 e-j2(2/N) = e-j2/(N/2) WN 2 = WN/2
Values of various twiddles WN n for length N=8 n WN n = e(-j2)(n/N) Remarks 0 1 = 10 WN 0 1 (1-j)/√2 = 1 -45 WN 1 2 -j = 1 -90 WN 2 3 - (1+j)/√2 = 1-135 WN 3 4 -1 = 1 -180 WN 4 = -WN 0 5 - (1-j)/√2 = 1 -225 WN 5 = -WN 1 6 j = 1 90 WN 6 = -WN 2 7 (1+j)/√2 = 1 45 WN 7 = -WN 3
DFT calculations • The forward DFT, frequency domain output in the range 0kN-1 is given by: • While the Inverse DFT, time domain output, again, in the range 0kN-1 is denoted by X k ( ) 0 n 1  n x n ( ) W N  nk        x n ( ) 1 N 0 n 1  k X k ( ) W N   nk                
Matrix Relations • The DFT samples defined by can be expressed in NxN matrix as where  T N X X X ] [ ..... ] [ ] [ 1 1 0   X  T N x x x ] [ ..... ] [ ] [ 1 1 0   x 1 0 , ] [ ] [ 1 0        N k W n x k X N n kn N     x(n) X(k) 1 0 n           N nk N W
Matrix Relations can be expanded as NXN DFT matrix                          2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 ) 1 ( 2 1 1 0 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N k nk N W W W W W W W W W W             1 0 N k nk N W
DFT: For N of length 4,range of n, k = [0 1 2 3] each. Hence X(n) = x(0)WN n.0+x(1)WN n.1+x(2)WN n.2 + x(3)WN n.3 X k ( ) 0 n 1  n x n ( ) W N  nk        x x(0) x(1) x(2) x(3) X(0) = W4 0x0 W4 0x1 W4 0x2 W4 0x3 X(1) = W4 1x0 W4 1x1 W4 1x2 W4 1x3 X(2) = W4 2x0 W4 2x1 W4 2x2 W4 2x3 X(3) = W4 3x0 W4 3x1 W4 3x2 W4 3x3
x(0) x(1) x(2) x(3) X(0) = W4 0x0 W4 0x1 W4 0x2 W4 0x3 X(1) = W4 1x0 W4 1x1 W4 1x2 W4 1x3 X(2) = W4 2x0 W4 2x1 W4 2x2 W4 2x3 X(3) = W4 3x0 W4 3x1 W4 3x2 W4 3x3 x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 W4 3 X(2) = W4 0 W4 2 W4 4 W4 6 X(3) = W4 0 W4 3 W4 6 W4 9 Can be rewritten as:
x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 W4 3 X(2) = W4 0 W4 2 W4 4 W4 6 X(3) = W4 0 W4 3 W4 6 W4 9 x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 -W4 1 X(2) = W4 0 W4 2 W4 0 W4 2 X(3) = W4 0 -W4 1 W4 2 W4 1
x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 -W4 1 X(2) = W4 0 W4 2 W4 0 W4 2 X(3) = W4 0 -W4 1 W4 2 W4 1 x(0) x(1) x(2) x(3) X(0) = 1 1 1 1 X(1) = 1 -j -1 j X(2) = 1 -1 1 -1 X(3) = 1 j -1 -j
Matrix Relations • Likewise, the IDFT is can be expressed in NxN matrix form as 1 0 , ] [ ] [ 1 0         N n W k X n x N k n k N     X(k) 1 0 n x 1            N n nk N W
Matrix Relations can also be expanded as NXN DFT matrix                                    2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 ) 1 ( 2 1 1 0 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N k nk N W W W W W W W W W W              1 0 N k nk N W Observe:     1 0 N k nk N W               1 0 1 * 1 0 N n nk N W W N N k nk N The inversion can be had by Hermitian conjugating j by –j and dividing by N.
x(0) x(1) x(2) x(3) X(0) = 1 1 1 1 X(1) = 1 -j -1 j X(2) = 1 -1 1 -1 X(3) = 1 j -1 -j X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               Characterisitc or system matrix
X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )             1 4 1 1 1 1 1 j 1  j  1 1  1 1  1 j  1  j             X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )                           x The effective determinant of above is 1/4. Conversion in system matrices of DFT and IDFT obtained by replacing by j by –j. Inversion of above matrix is: x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )             1 4 1 1 1 1 1 j 1  j  1 1  1 1  1 j  1  j             X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )                          
FOR 8 point DFT: N=8 • The system Matrix is: W8 0x0 W8 0x1 W8 0x2 W8 0x3 W8 0x4 W8 0x5 W8 0x6 W8 0x7 W8 1x0 W8 1x1 W8 1x2 W8 1x3 W8 1x4 W8 1x5 W8 1x6 W8 1x7 W8 2x0 W8 2x1 W8 2x2 W8 2x3 W8 2x4 W8 2x5 W8 2x6 W8 2x7 W8 3x0 W8 3x1 W8 3x2 W8 3x3 W8 3x4 W8 3x5 W8 3x6 W8 3x7 W8 4x0 W8 4x1 W8 4x2 W8 4x3 W8 4x4 W8 4x5 W8 4x6 W8 4x7 W8 5x0 W8 5x1 W8 5x2 W8 5x3 W8 5x4 W8 5x5 W8 5x6 W8 5x7 W8 6x0 W8 6x1 W8 6x2 W8 6x3 W8 6x4 W8 6x5 W8 6x6 W8 6x7 W8 7x0 W8 7x1 W8 7x2 W8 7x0 W8 7x4 W8 7x5 W8 7x6 W8 7x7
The characteristic matrix of 8-point DFT is: 1 1 1 1 1 1 1 1 1 1 j  2 j  1 j  ( )  2 1  1 j  ( )  2 j 1 j  2 1 j  1  j 1 j  1  j 1 1 j  ( )  2 j 1 j  2 1  1 j  2 j  1 j  ( )  2 1 1  1 1  1 1  1 1  1 1 j  ( )  2 j  1 j  2 1  1 j  2 j 1 j  ( )  2 1 j 1  j  1 j 1  j  1 1 j  2 j 1 j  ( )  2 1  1 j  ( )  2 j  1 j  ( ) 2                                  
DFT and its Inverse Conversion in system matrices of DFT and IDFT obtained by replacing by j by –j.. 1 1 1 1 1 1 1 1 1 1 j  2 j  1 j  ( )  2 1  1 j  ( )  2 j 1 j  2 1 j  1  j 1 j  1  j 1 1 j  ( )  2 j 1 j  2 1  1 j  2 j  1 j  ( )  2 1 1  1 1  1 1  1 1  1 1 j  ( )  2 j  1 j  2 1  1 j  2 j 1 j  ( )  2 1 j 1  j  1 j 1  j  1 1 j  2 j 1 j  ( )  2 1  1 j  ( )  2 j  1 j  ( ) 2                                   1 8 1 1 1 1 1 1 1 1 1 1 j  2 j 1 j  2  1  1 j  2  j  1 j  2 1 j 1  j  1 j 1  j  1 1 j  2  j  1 j  2 1  1 j  2 j 1 j  2  1 1  1 1  1 1  1 1  1 1 j  2  j 1 j  2 1  1 j  2 j  1 j  2  1 j  1  j 1 j  1  j 1 1 j  2 j  1 j  2  1  1 j  2  j 1 j  2                                                                      DFT IDFT Effective determinant
Calculation advantage in radix-2 algorithms for various values of N Improvement A/B N N^2 N^2-N A (N/2)ln(N) Nln(N) B 8 64 56 120 8.32 16.64 24.95 4.81 16 256 240 496 22.18 44.36 66.54 7.45 32 1024 992 2016 55.45 110.90 166.36 12.12 64 4096 4032 8128 133.08 266.17 399.25 20.36 128 16384 16256 32640 310.53 621.06 931.59 35.04 256 65536 65280 130816 709.78 1419.57 2129.35 61.43 512 262144 261632 523776 1597.01 3194.02 4791.03 109.32 1024 1048576 1047552 2096128 3548.91 7097.83 10646.74 196.88 2048 4194304 4192256 8386560 7807.61 15615.22 23422.83 358.05 4096 16777216 16773120 33550336 17034.79 34069.57 51104.36 656.51 8192 67108864 67100672 134209536 36908.70 73817.40 110726.10 1212.09 16384 268435456 268419072 536854528 79495.66 158991.33 238486.99 2251.09 No. of computation points DIRECT COMPUTATION FFT computaton complex multiplication Complex additions complex multiplication Complex additions Total Total
The Process of Decimation • First step of process of decimation is splitting a sequence in smaller sequences. • A sequence of 15 can be splitted in five sequences of threes or three sequences of fives. • 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;  There can be various combinations and varied complexities.
4-point DFT X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               Can be, by interchanging col. 2 and 3, seen as X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 1  1 1  1 j  1  j 1 j 1  j              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )              
4-point DFT X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 1  1 1  1 j  1  j 1 j 1  j              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               The system matrix is rewritten as X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1               x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )              
Matrix manipulation to get the desired input sequence from the actual x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )             1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1             x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )              
Process of decimation: example X[n] -1 -2 1 2 3 4 5 6 7 n X[1] X[3] X[5] X[7] X[n] -1 -2 1 2 3 4 5 6 7 n X[1] X[2] X[3] X[4] X[5] X[6] X[7] X[0] X[n] -1 -2 1 2 3 4 5 6 7 n X[2] X[4] X[6] X[0] Separating the above sequence for +ve ‘n’ in even and odd sequence numbers .
Process of decimation: example X[n] - - 1 2 3 4 5 6 7 n X[1] X[3] X[5] X[7] 2 3 4 6 X[n] - 1 5 7 n X[2] X[4] X[6] X[0] Compress the even sequence by two. Shift the sequence to left by one and compress by two X[n] - 1 2 3 n X[2] X[4] X[6] X[0] X[n] 2 4 6 n X[1] X[3] X[5] X[7] The compression is also called decimation
x 0 ( ) x 2 ( )  x 0 ( ) x 2 ( )  x 1 ( ) x 3 ( )  x 1 ( ) x 3 ( )              1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             x 0 ( ) x 2 ( )  x 0 ( ) x 2 ( )  x 1 ( ) x 3 ( )  x 1 ( ) x 3 ( )                X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1               x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               First stage of realization Second stage
Decimation of 4 point DFT into 2xradix-2 • The values of W4 0= 1; W4 2 = -1; W4 1= -j; and W4 3 = j X[0] x0 x2 x1 x3 x0+x2 xo -x2 x1+x3 x1-x3 X[1] X[2] X[3] Wo w1 W2 W3 -1 -1
Decimation of 4 point DFT into 2xradix-2 • The values of W4 0= 1; W4 2 = -1; W4 1= -j; and W4 3 = j X[0] N/4 point DFT even N/4 point DFT odd x0 x2 x1 x3 x0+x2 xo -x2 x1+x3 x1-x3 X[1] X[2] X[3] Wo w1 W2 W3
N=8-point radix-4 DIT-FFT: N/2 point DFT [EVEN] N/2 point DFT [ODD] X(0) X(4) X(2) X(6) X(1) X(3) X(5) X(7) X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] X0[0] X1 [0] X0[1] X1 [1] X0[2] X1[2] X0[3] X1[3] a -j  -b -1 -a  j b Note: -W4 = W0=1; -W5= W1 = a = (1-j)/2; -W2 = W6=j and -W3 = W7 = b = (1+j)/2 1
The matrix equation of the slide is: • Note the symmetry between 1st and 3rd partition and 2nd and 4th partition of system matrix. X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( ) X 4 ( ) X 5 ( ) X 6 ( ) X 7 ( )                        1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1  0 0 0 0 a 0 0 0 a  0 0 0 0 j  0 0 0 j 0 0 0 0 b  0 0 0 b                       X 0 0 ( ) X o 1 ( ) X o 2 ( ) X 0 3 ( ) X 1 0 ( ) X 1 1 ( ) X 1 2 ( ) X 1 3 ( )                                
N=8-point radix-2 DIT-FFT: N-point DFT N/2 point DFT N/4 point DFT X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] x[0] x[4] x[2] x[6] x[1] x[5] x[3] x[7] -1 -1 -1 -1 -1 -1 -1 -1 w2 w2 w2 w1 w3 -1 -1 -1 -1
System Matrix for N/2 point DFT Block Note that partitions 2 and 3 are null matrices while partitions 1 and 4 are identical. Also note that subpartitions 1st and 3rd are I matrices and 2nd and 4th sub-partitions have sign difference. Cross diagonals in either case are zero. X 0 0 ( ) X o 1 ( ) X o 2 ( ) X 0 3 ( ) X 1 0 ( ) X 1 1 ( ) X 1 2 ( ) X 1 3 ( )                               1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1  0 0 0 0 0 0 j  0 j 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1  0 0 0 0 0 0 j  0 j                       x 0 ( ) x 4 ( )  x 0 ( ) x 4 ( )  x 2 ( ) x 6 ( )  x 2 ( ) x 6 ( )  x 1 ( ) x 5 ( )  x 1 ( ) x 5 ( )  x 3 ( ) x 7 ( )  x 3 ( ) x 7 ( )                         
System Matrix for N/4 block Again see the simile between 1and 4 partitions. Note that 2 and 3 s are null matrices. Further note the sub-partition matrices. They also follow the same rule. All the elements are real. x 0 ( ) x 4 ( )  x 0 ( ) x 4 ( )  x 2 ( ) x 6 ( )  x 2 ( ) x 6 ( )  x 1 ( ) x 5 ( )  x 1 ( ) x 5 ( )  x 3 ( ) x 7 ( )  x 3 ( ) x 7 ( )                        1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1                        x 1 ( ) x 4 ( ) x 2 ( ) x 6 ( ) x 1 ( ) x 5 ( ) x 3 ( ) x 7 ( )                        
Signal flow graph for decimation of 8 point DFT N-point DFT N/2 point DFT N/4 point DFT X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] x[0] x[4] x[2] x[6] x[1] x[5] x[3] x[7] -1 -1 -1 -1 -1 -1 -1 -1 w2 w2 w2 w1 w3 -1 -1 -1 -1

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15Tarique_DSP_8ETC.PPT

  • 1. DFT and FFT • FFT is an algorithm to convert a time domain signal to DFT efficiently. • FFT is not unique. Many algorithms are available. • Each algorithm has merits and demerits. • In each algorithm, depending on the sequence needed at the output, the input is regrouped. • The groups are decided by the number of samples.
  • 2. FFTs • The number of points can be nine too. • It can be 15 as well. • Any number in multiples of two integers. • It can not be any prime number. • It can be in the multiples of two prime numbers.
  • 3. FFTs • The purpose of this series of lectures is to learn the basics of FFT algorithms. • Algorithms having number of samples 2N, where N is an integer is most preferred. • 8 point radix-2 FFT by decimation is used from learning point of view. • Radix-x: here ‘x’ represents number of samples in each group made at the first stage. They are generally equal. • We shall study radix-2 and radix-3.
  • 4. Radix-2: DIT or, DIF • Radix-2 is the first FFT algorithm. It was proposed by Cooley and Tukey in 1965. • Though it is not the efficient algorithm, it lays foundation for time-efficient DFT calculations. • The next slide shows the saving in time required for calculations with radix-2. • The algorithms appear either in (a) Decimation In Time (DIT), or, (b) Decimation In Frequency (DIF). • DIT and DIF, both yield same complexity and results. They are complementary. • We shall stress on 8 to radix 2 DIT FFT.
  • 5. Other popular Algorithms Besides many, the popular algorithms are: • Goertzel algorithm • Chirp Z algorithm • Index mapping algorithm • Split radix in prime number algorithm. have modified approach over radix-2. Split radix in prime number does not use even the twiddles. We now pay attention to 8/radix-2 butterfly FIT FFT algorithm.
  • 6. Relationship between exponential forms and twiddle factors (W) for Periodicity = N Sr. No. Exponential form Symbolic form 01 e-j2n/N = e-j2(n+N)/N WN n = WN n+N 02 e-j2(n+N/2)/N = - e-j2n/N WN n+N/2= - WN n 03 e-j2k = e-j2Nk/N = 1 WN N+K = 1 04 e-j2(2/N) = e-j2/(N/2) WN 2 = WN/2
  • 7. Values of various twiddles WN n for length N=8 n WN n = e(-j2)(n/N) Remarks 0 1 = 10 WN 0 1 (1-j)/√2 = 1 -45 WN 1 2 -j = 1 -90 WN 2 3 - (1+j)/√2 = 1-135 WN 3 4 -1 = 1 -180 WN 4 = -WN 0 5 - (1-j)/√2 = 1 -225 WN 5 = -WN 1 6 j = 1 90 WN 6 = -WN 2 7 (1+j)/√2 = 1 45 WN 7 = -WN 3
  • 8. DFT calculations • The forward DFT, frequency domain output in the range 0kN-1 is given by: • While the Inverse DFT, time domain output, again, in the range 0kN-1 is denoted by X k ( ) 0 n 1  n x n ( ) W N  nk        x n ( ) 1 N 0 n 1  k X k ( ) W N   nk                
  • 9. Matrix Relations • The DFT samples defined by can be expressed in NxN matrix as where  T N X X X ] [ ..... ] [ ] [ 1 1 0   X  T N x x x ] [ ..... ] [ ] [ 1 1 0   x 1 0 , ] [ ] [ 1 0        N k W n x k X N n kn N     x(n) X(k) 1 0 n           N nk N W
  • 10. Matrix Relations can be expanded as NXN DFT matrix                          2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 ) 1 ( 2 1 1 0 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N k nk N W W W W W W W W W W             1 0 N k nk N W
  • 11. DFT: For N of length 4,range of n, k = [0 1 2 3] each. Hence X(n) = x(0)WN n.0+x(1)WN n.1+x(2)WN n.2 + x(3)WN n.3 X k ( ) 0 n 1  n x n ( ) W N  nk        x x(0) x(1) x(2) x(3) X(0) = W4 0x0 W4 0x1 W4 0x2 W4 0x3 X(1) = W4 1x0 W4 1x1 W4 1x2 W4 1x3 X(2) = W4 2x0 W4 2x1 W4 2x2 W4 2x3 X(3) = W4 3x0 W4 3x1 W4 3x2 W4 3x3
  • 12. x(0) x(1) x(2) x(3) X(0) = W4 0x0 W4 0x1 W4 0x2 W4 0x3 X(1) = W4 1x0 W4 1x1 W4 1x2 W4 1x3 X(2) = W4 2x0 W4 2x1 W4 2x2 W4 2x3 X(3) = W4 3x0 W4 3x1 W4 3x2 W4 3x3 x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 W4 3 X(2) = W4 0 W4 2 W4 4 W4 6 X(3) = W4 0 W4 3 W4 6 W4 9 Can be rewritten as:
  • 13. x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 W4 3 X(2) = W4 0 W4 2 W4 4 W4 6 X(3) = W4 0 W4 3 W4 6 W4 9 x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 -W4 1 X(2) = W4 0 W4 2 W4 0 W4 2 X(3) = W4 0 -W4 1 W4 2 W4 1
  • 14. x(0) x(1) x(2) x(3) X(0) = W4 0 W4 0 W4 0 W4 0 X(1) = W4 0 W4 1 W4 2 -W4 1 X(2) = W4 0 W4 2 W4 0 W4 2 X(3) = W4 0 -W4 1 W4 2 W4 1 x(0) x(1) x(2) x(3) X(0) = 1 1 1 1 X(1) = 1 -j -1 j X(2) = 1 -1 1 -1 X(3) = 1 j -1 -j
  • 15. Matrix Relations • Likewise, the IDFT is can be expressed in NxN matrix form as 1 0 , ] [ ] [ 1 0         N n W k X n x N k n k N     X(k) 1 0 n x 1            N n nk N W
  • 16. Matrix Relations can also be expanded as NXN DFT matrix                                    2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 ) 1 ( 2 1 1 0 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N k nk N W W W W W W W W W W              1 0 N k nk N W Observe:     1 0 N k nk N W               1 0 1 * 1 0 N n nk N W W N N k nk N The inversion can be had by Hermitian conjugating j by –j and dividing by N.
  • 17. x(0) x(1) x(2) x(3) X(0) = 1 1 1 1 X(1) = 1 -j -1 j X(2) = 1 -1 1 -1 X(3) = 1 j -1 -j X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               Characterisitc or system matrix
  • 18. X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )             1 4 1 1 1 1 1 j 1  j  1 1  1 1  1 j  1  j             X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )                           x The effective determinant of above is 1/4. Conversion in system matrices of DFT and IDFT obtained by replacing by j by –j. Inversion of above matrix is: x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )             1 4 1 1 1 1 1 j 1  j  1 1  1 1  1 j  1  j             X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )                          
  • 19. FOR 8 point DFT: N=8 • The system Matrix is: W8 0x0 W8 0x1 W8 0x2 W8 0x3 W8 0x4 W8 0x5 W8 0x6 W8 0x7 W8 1x0 W8 1x1 W8 1x2 W8 1x3 W8 1x4 W8 1x5 W8 1x6 W8 1x7 W8 2x0 W8 2x1 W8 2x2 W8 2x3 W8 2x4 W8 2x5 W8 2x6 W8 2x7 W8 3x0 W8 3x1 W8 3x2 W8 3x3 W8 3x4 W8 3x5 W8 3x6 W8 3x7 W8 4x0 W8 4x1 W8 4x2 W8 4x3 W8 4x4 W8 4x5 W8 4x6 W8 4x7 W8 5x0 W8 5x1 W8 5x2 W8 5x3 W8 5x4 W8 5x5 W8 5x6 W8 5x7 W8 6x0 W8 6x1 W8 6x2 W8 6x3 W8 6x4 W8 6x5 W8 6x6 W8 6x7 W8 7x0 W8 7x1 W8 7x2 W8 7x0 W8 7x4 W8 7x5 W8 7x6 W8 7x7
  • 20. The characteristic matrix of 8-point DFT is: 1 1 1 1 1 1 1 1 1 1 j  2 j  1 j  ( )  2 1  1 j  ( )  2 j 1 j  2 1 j  1  j 1 j  1  j 1 1 j  ( )  2 j 1 j  2 1  1 j  2 j  1 j  ( )  2 1 1  1 1  1 1  1 1  1 1 j  ( )  2 j  1 j  2 1  1 j  2 j 1 j  ( )  2 1 j 1  j  1 j 1  j  1 1 j  2 j 1 j  ( )  2 1  1 j  ( )  2 j  1 j  ( ) 2                                  
  • 21. DFT and its Inverse Conversion in system matrices of DFT and IDFT obtained by replacing by j by –j.. 1 1 1 1 1 1 1 1 1 1 j  2 j  1 j  ( )  2 1  1 j  ( )  2 j 1 j  2 1 j  1  j 1 j  1  j 1 1 j  ( )  2 j 1 j  2 1  1 j  2 j  1 j  ( )  2 1 1  1 1  1 1  1 1  1 1 j  ( )  2 j  1 j  2 1  1 j  2 j 1 j  ( )  2 1 j 1  j  1 j 1  j  1 1 j  2 j 1 j  ( )  2 1  1 j  ( )  2 j  1 j  ( ) 2                                   1 8 1 1 1 1 1 1 1 1 1 1 j  2 j 1 j  2  1  1 j  2  j  1 j  2 1 j 1  j  1 j 1  j  1 1 j  2  j  1 j  2 1  1 j  2 j 1 j  2  1 1  1 1  1 1  1 1  1 1 j  2  j 1 j  2 1  1 j  2 j  1 j  2  1 j  1  j 1 j  1  j 1 1 j  2 j  1 j  2  1  1 j  2  j 1 j  2                                                                      DFT IDFT Effective determinant
  • 22. Calculation advantage in radix-2 algorithms for various values of N Improvement A/B N N^2 N^2-N A (N/2)ln(N) Nln(N) B 8 64 56 120 8.32 16.64 24.95 4.81 16 256 240 496 22.18 44.36 66.54 7.45 32 1024 992 2016 55.45 110.90 166.36 12.12 64 4096 4032 8128 133.08 266.17 399.25 20.36 128 16384 16256 32640 310.53 621.06 931.59 35.04 256 65536 65280 130816 709.78 1419.57 2129.35 61.43 512 262144 261632 523776 1597.01 3194.02 4791.03 109.32 1024 1048576 1047552 2096128 3548.91 7097.83 10646.74 196.88 2048 4194304 4192256 8386560 7807.61 15615.22 23422.83 358.05 4096 16777216 16773120 33550336 17034.79 34069.57 51104.36 656.51 8192 67108864 67100672 134209536 36908.70 73817.40 110726.10 1212.09 16384 268435456 268419072 536854528 79495.66 158991.33 238486.99 2251.09 No. of computation points DIRECT COMPUTATION FFT computaton complex multiplication Complex additions complex multiplication Complex additions Total Total
  • 23. The Process of Decimation • First step of process of decimation is splitting a sequence in smaller sequences. • A sequence of 15 can be splitted in five sequences of threes or three sequences of fives. • 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;  There can be various combinations and varied complexities.
  • 24. 4-point DFT X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 j  1  j 1 1  1 1  1 j 1  j              x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )               Can be, by interchanging col. 2 and 3, seen as X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 1  1 1  1 j  1  j 1 j 1  j              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )              
  • 25. 4-point DFT X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 1 1 1 1 1  1 1  1 j  1  j 1 j 1  j              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               The system matrix is rewritten as X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1               x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )              
  • 26. Matrix manipulation to get the desired input sequence from the actual x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )             1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1             x 0 ( ) x 1 ( ) x 2 ( ) x 3 ( )              
  • 27. Process of decimation: example X[n] -1 -2 1 2 3 4 5 6 7 n X[1] X[3] X[5] X[7] X[n] -1 -2 1 2 3 4 5 6 7 n X[1] X[2] X[3] X[4] X[5] X[6] X[7] X[0] X[n] -1 -2 1 2 3 4 5 6 7 n X[2] X[4] X[6] X[0] Separating the above sequence for +ve ‘n’ in even and odd sequence numbers .
  • 28. Process of decimation: example X[n] - - 1 2 3 4 5 6 7 n X[1] X[3] X[5] X[7] 2 3 4 6 X[n] - 1 5 7 n X[2] X[4] X[6] X[0] Compress the even sequence by two. Shift the sequence to left by one and compress by two X[n] - 1 2 3 n X[2] X[4] X[6] X[0] X[n] 2 4 6 n X[1] X[3] X[5] X[7] The compression is also called decimation
  • 29. x 0 ( ) x 2 ( )  x 0 ( ) x 2 ( )  x 1 ( ) x 3 ( )  x 1 ( ) x 3 ( )              1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1              x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             x 0 ( ) x 2 ( )  x 0 ( ) x 2 ( )  x 1 ( ) x 3 ( )  x 1 ( ) x 3 ( )                X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( )             1 0 1 0 0 1 0 1 1 0 1  0 0 j  0 j             1 1 0 0 1 1  0 0 0 0 1 1 0 0 1 1               x 0 ( ) x 2 ( ) x 1 ( ) x 3 ( )               First stage of realization Second stage
  • 30. Decimation of 4 point DFT into 2xradix-2 • The values of W4 0= 1; W4 2 = -1; W4 1= -j; and W4 3 = j X[0] x0 x2 x1 x3 x0+x2 xo -x2 x1+x3 x1-x3 X[1] X[2] X[3] Wo w1 W2 W3 -1 -1
  • 31. Decimation of 4 point DFT into 2xradix-2 • The values of W4 0= 1; W4 2 = -1; W4 1= -j; and W4 3 = j X[0] N/4 point DFT even N/4 point DFT odd x0 x2 x1 x3 x0+x2 xo -x2 x1+x3 x1-x3 X[1] X[2] X[3] Wo w1 W2 W3
  • 32. N=8-point radix-4 DIT-FFT: N/2 point DFT [EVEN] N/2 point DFT [ODD] X(0) X(4) X(2) X(6) X(1) X(3) X(5) X(7) X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] X0[0] X1 [0] X0[1] X1 [1] X0[2] X1[2] X0[3] X1[3] a -j  -b -1 -a  j b Note: -W4 = W0=1; -W5= W1 = a = (1-j)/2; -W2 = W6=j and -W3 = W7 = b = (1+j)/2 1
  • 33. The matrix equation of the slide is: • Note the symmetry between 1st and 3rd partition and 2nd and 4th partition of system matrix. X 0 ( ) X 1 ( ) X 2 ( ) X 3 ( ) X 4 ( ) X 5 ( ) X 6 ( ) X 7 ( )                        1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1  0 0 0 0 a 0 0 0 a  0 0 0 0 j  0 0 0 j 0 0 0 0 b  0 0 0 b                       X 0 0 ( ) X o 1 ( ) X o 2 ( ) X 0 3 ( ) X 1 0 ( ) X 1 1 ( ) X 1 2 ( ) X 1 3 ( )                                
  • 34. N=8-point radix-2 DIT-FFT: N-point DFT N/2 point DFT N/4 point DFT X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] x[0] x[4] x[2] x[6] x[1] x[5] x[3] x[7] -1 -1 -1 -1 -1 -1 -1 -1 w2 w2 w2 w1 w3 -1 -1 -1 -1
  • 35. System Matrix for N/2 point DFT Block Note that partitions 2 and 3 are null matrices while partitions 1 and 4 are identical. Also note that subpartitions 1st and 3rd are I matrices and 2nd and 4th sub-partitions have sign difference. Cross diagonals in either case are zero. X 0 0 ( ) X o 1 ( ) X o 2 ( ) X 0 3 ( ) X 1 0 ( ) X 1 1 ( ) X 1 2 ( ) X 1 3 ( )                               1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1  0 0 0 0 0 0 j  0 j 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1  0 0 0 0 0 0 j  0 j                       x 0 ( ) x 4 ( )  x 0 ( ) x 4 ( )  x 2 ( ) x 6 ( )  x 2 ( ) x 6 ( )  x 1 ( ) x 5 ( )  x 1 ( ) x 5 ( )  x 3 ( ) x 7 ( )  x 3 ( ) x 7 ( )                         
  • 36. System Matrix for N/4 block Again see the simile between 1and 4 partitions. Note that 2 and 3 s are null matrices. Further note the sub-partition matrices. They also follow the same rule. All the elements are real. x 0 ( ) x 4 ( )  x 0 ( ) x 4 ( )  x 2 ( ) x 6 ( )  x 2 ( ) x 6 ( )  x 1 ( ) x 5 ( )  x 1 ( ) x 5 ( )  x 3 ( ) x 7 ( )  x 3 ( ) x 7 ( )                        1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1  0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1                        x 1 ( ) x 4 ( ) x 2 ( ) x 6 ( ) x 1 ( ) x 5 ( ) x 3 ( ) x 7 ( )                        
  • 37. Signal flow graph for decimation of 8 point DFT N-point DFT N/2 point DFT N/4 point DFT X[0] X[1] X[2] X[3] X[4] X[5] X[6] X[7] x[0] x[4] x[2] x[6] x[1] x[5] x[3] x[7] -1 -1 -1 -1 -1 -1 -1 -1 w2 w2 w2 w1 w3 -1 -1 -1 -1