EE123 Digital Signal Processing

Transcription

1 EE23 Digital Signal Processing Lecture 8 FFT II Lab based on slides by JM Kahn M Lustig, EECS UC Berkeley

2 Announcements Last time: Started FFT Today Lab Finish FFT Read Ch 002 Midterm : Feb 22nd M Lustig, EECS UC Berkeley

3 Lab Generate a chirp M Lustig, EECS UC Berkeley

4 Lab Play and record chirp M Lustig, EECS UC Berkeley

5 Lab Auto-correlation of a chirp - pulse compression M Lustig, EECS UC Berkeley

6 Lab I part II - Sonar Generate a pulse - analytic Use real part for pulse train Transmit and record Sent and recorded: M Lustig, EECS UC Berkeley

7 Lab I part II - Sonar Extract a pulse sent: received: M Lustig, EECS UC Berkeley

8 Lab I part II - Sonar Matched Filtering received: Filter: Envelope Matched Filtered M Lustig, EECS UC Berkeley

9 Lab I part II - Sonar Display echos vs distance Matched Filter: samples t=samp /fs d=samp /fs *v_s M Lustig, EECS UC Berkeley

10 Lab I part II - Sonar Real time demo M Lustig, EECS UC Berkeley

11 Decimation-in-Time Fast Fourier Transform Combining all these stages, the diagram for the 8 sample DFT is: x[0] X[0] x[4] x[2] x[6] x[] x[5] x[3] x[7] 0 /2 /2 0 /2 / X[] X[2] X[3] X[4] X[5] X[6] X[7] This the decimation-in-time FFT algorithm Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

12 Decimation-in-Time Fast Fourier Transform In general, there are log 2 stages of decimation-in-time Each stage requires /2 complex multiplications, some of which are trivial The total number of complex multiplications is (/2) log 2 The order of the input to the decimation-in-time FFT algorithm must be permuted First stage: split into odd and even Zero low-order bit first ext stage repeats with next zero-lower bit first et e ect is reversing the bit order of indexes Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

13 Decimation-in-Time Fast Fourier Transform This is illustrated in the following table for = 8 Decimal Binary Bit-Reversed Binary Bit-Reversed Decimal Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

14 Decimation-in-Frequency Fast Fourier Transform The DFT is X [k] = X n=0 x[n] nk If we only look at the even samples of X [k], we can write k =2r, X [2r] = X n=0 x[n] n(2r) e split this into two sums, one over the first /2 samples, and the second of the last /2 samples X [2r] = (/2) X n=0 (/2) x[n] 2rn + X n=0 x[n + /2] 2r(n+/2) Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

15 Decimation-in-Frequency Fast Fourier Transform But 2r(n+/2) = 2rn = 2rn e can then write = rn /2 X [2r] = = = (/2) X n=0 (/2) X n=0 (/2) X n=0 (/2) x[n] 2rn + X n=0 (/2) x[n] 2rn + X n=0 (x[n]+x[n + /2]) rn /2 x[n + /2] 2r(n+/2) x[n + /2] 2rn This is the /2-length DFT of first and second half of x[n] summed Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

16 Decimation-in-Frequency Fast Fourier Transform X [2r] = DFT 2 X [2r + ] = DFT 2 {(x[n]+x[n + /2])} {(x[n] x[n + /2]) n } (By a similar argument that gives the odd samples) Continue the same approach is applied for the /2 DFTs, and the /4 DFT s until we reach simple butterflies Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

17 Decimation-in-Frequency Fast Fourier Transform The diagram for and 8-point decimation-in-frequency DFT is as follows x[0] X[0] x[] x[2] x[3] x[4] x[5] x[6] x[7] /2 /2 0 /2 /2 X[4] X[2] X[6] X[] X[5] X[3] X[7] This is just the decimation-in-time algorithm reversed! The inputs are in normal order, and the outputs are bit reversed Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

18 on-power-of-2 FFT s A similar argument applies for any length DFT, where the length is a composite number For example, if = 6, a decimation-in-time FFT could compute three 2-point DFT s followed by two 3-point DFT s x[0] x[3] 2-Point DFT Point DFT X[0] X[2] x[] x[4] 2-Point DFT 6 X[4] X[] x[2] x[5] 2-Point DFT Point DFT X[3] X[5] Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

19 on-power-of-2 FFT s Good component DFT s are available for lengths up to 20 or so Many of these exploit the structure for that specific length For example, a factor of /4 = e j 2 (/4) = e j 2 = j hy? just swaps the real and imaginary components of a complex number, and doesn t actually require any multiplies Hence a DFT of length 4 doesn t require any complex multiplies Half of the multiplies of an 8-point DFT also don t require multiplication Composite length FFT s can be very e cient for any length that factors into terms of this order Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

20 For example = 693 factors into = (7)(9)() each of which can be implemented e 9 DFT s of length 7 7 DFT s of length 9, and 7 9 DFT s of length ciently e would perform Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

21 Historically, the power-of-two FFTs were much faster (better written and implemented) For non-power-of-two length, it was faster to zero pad to power of two Recently this has changed The free FFT package implements very e cient algorithms for almost any filter length Matlab has used FFT since version 6 Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

22 Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

23 FFT as Matrix Operation 0 X [0] X [k] X [ ] C A = n k0 kn ( )0 ( )n 0( ) k( ) ( )( ) 0 B A x[0] x[n] x[ ] C A is fully populated ) 2 entries Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

24 FFT as Matrix Operation 0 X [0] X [k] X [ ] C A = n k0 kn ( )0 ( )n 0( ) k( ) ( )( ) 0 B A x[0] x[n] x[ ] C A is fully populated ) 2 entries FFT is a decomposition of into a more sparse form: F = apple I/2 D /2 I /2 D /2 apple /2 0 0 /2 apple Even-Odd Perm Matrix I /2 is an identity matrix D /2 is a diagonal with entries,,, /2 Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

25 FFT as Matrix Operation Example: =4 F 4 = Miki Lustig UCB Based on Course otes by JM Kahn SP Fall , EE23 Digital Signal Processing

26 Beyond log hat if the signal x[n] has a k sparse frequency A Gilbert et al, ear-optimal sparse Fourier representations via sampling H Hassanieh et al, early Optimal Sparse Fourier Transform Others O(K Log ) instead of O( Log ) From: M Lustig, EECS UC Berkeley