EEO 401 Digital Signal Processing Prof. Mark Fowler

Transcription

1 EEO 401 Digital Signal Processing Prof. Mark Fowler Note Set #26 FFT Algorithm: Divide & Conquer Viewpoint Reading: Sect & of Proakis & Manolakis

2 Divide & Conquer Approach The previous note set s FFT development was somewhat ad hoc. Here we develop a more formalized & generalized approach that can be used to develop other FFT approaches. To illustrate the basic ideas consider the case of an N-pt DFT where N is not prime can be factored into two integer factors: N = LM Can always zero-pad out to appropriate number We now can use either of two mappings from 2-D indices l,m to the actual time index n: n = Ml + m n = l + ml Row-Wise Input Mapping Column-Wise Input Mapping We use one or the other of these mappings to convert the input to a matrix and then take DFTs along either the rows or columns. We now can use either of two mappings from 2-D indices p,q to the actual DFT result index k: k = Mp + q k = p + ql Row-Wise Output Mapping Column-Wise Output Mapping 2/15

3 n = Ml + m Row-Wise Input Mapping DFTs Down Columns n = l + ml Column-Wise Input Mapping DFTs Across Rows 3/15

4 Now to illustrate how to use this machinery Use Column-wise input mapping = + Row-wise output mapping n l ml k = Mp + q N 1 X[ k] = xnw [ ] n= 0 kn N M L 1 X[ p, q] xl [, mw ] = m= 0 l= 0 ( Mp+ q)( l+ ml) N X[ p, q] = W xl [, mw ] W L M 1 lq mq pl N N L N M l= 0 m= 0 F[, lq] Glq [, ] = X[ pq, ] = W W W W MLmp mlq Mpl lq N N N N Nmp N = W = 1 W mq N L = W pl N M Compute M-pt DFTs of Rows Apply Twiddle Factors Compute L-pt DFTs of Colns 4/15

5 Illustration of this Approach Compute M-pt DFTs of Rows Apply Twiddle Factors Compute L-pt DFTs of Colns Although this LOOKS more complicated it is actually more efficient! 5/15

6 Application to Develop Dec-In-Time Radix-2 FFT We apply the divide-and-conquer approach with M = N/2 & L = 2 n = l + ml Column-Wise Input Mapping Let N = 2 v x[0] x[2] x[4] x[ N 2] x[1] x[3] x[5] x[ N 1] DFTs Across Rows Compute N/2-pt DFTs of Rows Compute 2-pt DFTs of Colns Apply Twiddle Factors Now repeat this for each N/2-pt DFT Etc., Etc., Etc. 6/15

7 Application to Develop Dec-In-Frequency Radix-2 FFT N = 2 v We apply the divide-and-conquer approach with M = 2 & L = N/2 n = l + ml Column-Wise Input Mapping x[0] x[1] x[n/2] x[n/2 + 1] Compute N/2-pt DFTs of Columns DFTs Across Rows X[1] X[3] X[0] X[2] x[2] x[n/2 1] x[n/2 + 2] x[n 1] Apply Twiddle Factors g 2 [n] k W N N/2-pt DFT X[N 2] Compute 2-pt DFTs of Rows x[0] x[1] 2-pt DFT x[n/2] x[n/2+1] Now repeat this for each N/2-pt DFT Etc., Etc., Etc. g 1 [n] x[n/2-1] x[n 1] 7/15

8 N = 8 First Stage of Dec-in-Freq FFT 8/15

9 Complete Dec-in-Freq FFT for N = 8 DFT Values are in Bit Reversed Order! 9/15

10 Butterfly Structure: DiT vs DiF Butterfly Structure: Dec-in-Time Butterfly Structure: Dec-in-Freq 10/15

11 3 Different Configurations of D-in-Time FFT N=8 D-in-Time FFT w/ BR Inputs Can be done in-place N=8 D-in-Time FFT w/ BR Outputs Can be done in-place N=8 D-in-Time FFT w/ both sides Normal Order Can NOT be done in-place Diagrams from R. Lyon, Understanding Digital Signal Processing, 3 rd Ed., Prentice- Hall, 2011 Writes over Data Needed Later!! 11/15

12 3 Different Configurations of D-in-Freq FFT N=8 D-in-Freq FFT w/ BR Inputs Can be done in-place N=8 D-in-Freq FFT w/ BR Outputs Can be done in-place Writes over Data Needed Later!! Diagrams from R. Lyon, Understanding Digital Signal Processing, 3 rd Ed., Prentice- Hall, 2011 N=8 D-in-Freq FFT w/ both sides Normal Order Can NOT be done in-place 12/15

13 Implementation Issues We ve looked at two radix-two methods. - Other radices: 4 & 8 - split radix (2 and 4) In-Place computation requires only 2N memory locations - But complicates the indexing & control operations - Doubling the memory to 4N locations can be advantageous o Reduces complexity of indexing & control o Allows natural ordering for both input and output In general, many factors come into play when determining best method - Parallelism, HW vs SW, fixed-point vs floating-point, etc. Also no need to develop distinct IFFT algorithm N 1 N 1 1 j2 πkn/ N 1 * j2 πkn/ N xn [ ] = Xke [ ] = X[ ke ] N n= 0 N n= 0 * { } { } { * IDFT X [ k] = conj DFT X [ k] } 1 N 13/15

14 Two Tricks for Real-Valued Signals 1. Efficient DFT of two Real-Valued Signals Let x 1 [n] and x 2 [n] be real-valued signals, each length N Form: x[ n] x [ n] + jx [ n] 1 2 Then we have Thus + 2 2j * * [ ] [ ] [ ] [ ] 1[ ] = xn x n & x2[ n] = xn x n x n { [ ]} + { * [ ]} { [ ]} { * [ ]} DFT x n DFT x n DFT x n DFT x n X1[ k] = & X2[ k] = 2 2j But DFT{ x * [n]} = X * [N k] 1 * 1 * X1[ k] = Xk [ ] X[ N k] & X2[ k] Xk [ ] X[ N k] 2 + = 2j 14/15

15 2. Efficient DFT of 2N-pt Real-Valued Signal Let g[n] be a real-valued signal of length 2N Then define: x [ n] g[2 n] & x [ n] g[2n+ 1] 1 2 And: xn [ ] = x1[ n] + j x2[ n] From Trick #1 we have 1 * 1 * X1[ k] = Xk [ ] X[ N k] & X2[ k] Xk [ ] X[ N k] 2 + = 2j But using ideas from Dec-in-Time FFT we know N 1 N 1 2 nk (2n+ 1) k 2N 2N n= 0 n= 0 G[ k] = g[2 n] W + g[2n+ 1] W N 1 N 1 2 nk (2n+ 1) k x1[ nw ] 2N x2[ nw ] 2N n= 0 n= 0 = + So then we get Gk [ ] = X[ k] + W X[ k], k= 0,1,, N 1 k 1 2N 2 Gk [ + N] = X[ k] W X[ k], k= 0,1,, N 1 k 1 2N 2 So computing one N-pt DFT of x[n] gets us the 2N-pt DFT of g[n]! 15/15