1:21. Down sampling/under sampling. The spectrum has the same shape, but the periodicity is twice as dense.

Transcription

1 1:21 Down sampling/under sampling The spectrum has the same shape, but the periodicity is twice as dense.

2 2:21 SUMMARY 1) The DFT only gives a 100% correct result, if the input sequence is periodic. 2) Filtering (implemented as a multiplication in frequency domain) can only be carried out as a circular convolution. 3) Windows are inavoidable and they affect the analysis. The result is obtained by convolving the desired frequency response with the window's frequency response. 4) Apparent frequency resolution is not the same as Effective frequency resolution. Zero padding only improves the Apparent frequency resolution, which then becomes larger, i.e. smaller distance between the k's.

3 3:21 Periodicity in W 1 kn ormal DFT: X [ k]= x[ n]w, k=0, 1,... 1 n=0 =8 Complex conjugate symmetry: n kn kn * W k =W = W kn k n =W k n Periodicity in n og k: W =W

4 4:21 Fast Fourier Transform (FFT) 1) Periodicity can be utilised to create a number of fast algorithms for computing a DFT faster than the std. formula. 2) Such an algorithms is called Fast Fourier Transform. 3) An FFT is a DFT, usually with restrictions on lengths. 4) With 8 instructions per product/addition a straight DFT can be computed in 8 2 t instr. = ns=10,5 ms while an FFT only requires 8 log 2 t instr. =0,16 ms 2

5 5:21 From direct DFT to Goertzel W =e k j 2 / k =e j2 k =1 so we can always multiply with W k : 1 X [ k]=w k r=0 1 k r x[r ]W kr = x[ r]w r=0 Let us define the (special) sequence: y k [ n]= r= n r x[ r]w k u[n r ] yk [ n]= x [n] W u[ n] kn ote that: X [ k]= y k [n] n=

6 6:21 From direct DFT to Goertzel, contd (1) Flow graph for W kn u[ n] : x[n] y k [ n] z 1 kn W k instead of all W! The flow graph has the following transfer function: H k z = W k z but it can be made even simpler (and require less computation time) although it doesn't look so at the first glance...

7 7:21 From direct DFT to Goertzel, contd (2) Multiplying with 1 W k z 1 in nominator and denominator yields: k H k z = k 1 W z 1 k 1 W z 1 W z k 1 1 H k z = 1 W z cos 2 k / z z 1 2 with the flow graph (now a 2nd order recursive calculation of X [ k]): x [n] yk [ n] z 1 W k 2 cos 2 k/ Real multiplication z 1 Only after iterations not every iteration! 1 Simple sign change

8 8:21 What do we really need? Important! If a limited number of frequency points M must be found, it is more efficient to use a direct DFT (e.g. computed using Goertzel's algorithm) or even a DTFT, i.e. M log 2 FFTs are certainly faster if the above is not true, however, we get (as we shall see) also frequency points that we don't necessarily need. Besides, FFTs are normally only of lengths =2 p, p=1, 2, 3,...

9 9:21 FFT using decimation in time 1 X [ k]= x[ n]w nk, k=0,1,... 1 n=0 /2 1 X [ k]= r=0 x[2 r ] W 2 rk W k (even) / 2 1 r=0 x[2 r 1] W 2 rk, where n=2 r (odd)

10 10:21 FFT using decimation in time, contd (1) Since W 2 =e 2 j 2 / =e j2 / / 2 =W /2 the decimation can be considered as two DFTs with half length /2: / 2 1 X [ k]= r=0 rk k x[ 2 r]w /2 W / 2 1 r=0 rk x [ 2 r 1]W / 2 X [ k]=g[ k] W k H [ k] =8

11 11:21 FFT using decimation in time, contd (2) G [ k] and H [ k] can now be seperated in the same way: / 4 1 G [ k]= l=0 k g [2 l]w lk W /4 /2 /4 1 H [ k]= l=0 lk k h[2 l]w / 4 W / 2 37 / 4 1 l=0 /4 1 l=0 g [2 l 1]W lk and /4 lk h[2 l 1]W /4

12 12:21 Third seperation for an 8 points FFT Stage Computation time ormal DFT: 2 With full seperation we get: 1. seperation: 2 /2 2 log 2 2. seperation: 4 / 4 2

13 13:21 The Butterfly computation The original Butterfly: Since /2 j 2 / /2 W =e j =e = 1 /2 then W r can be written as r /2 W /2 r r =W W = W and the simplified Butterfly becomes:

14 14:21 The final 8 points FFT, now utilising the simplified Butterfly

15 15:21 In place computation of an FFT Rewriting to X m [ l], where m is the stage (column), and l is the row. In the 8 points case we then have: X 0 [0]= x[0] X 0 [1]= x[ 4] X 0 [2]= x[2] X 0 [7]= x[7] All calculations follow: X m [ p]= X m 1 [ p] W r X m 1 [ q] X m [ q]= X m 1 [ p] W r X m 1 [ q]

16 16:21 Scrambling of input Decimal Binary Binary reversed Scrambed Reversed: n0 n1 n2 n2 n1 n0 ormal: x[n2n1n0] ormal Reversed x[000] x[000] x[001] x[100] x[010] x[010] x[011] x[110] x[100] x[001] x[101] x[101] x[110] x[011] x[111] x[111]

17 17:21 Scrambling of input, contd Depending on the application it is not always necessary to scramble input. E.g. not needed for FFT convolution : x1 FFT X x2 Unscrambled FFT Scrambled IFFT x Unscrambled x= x 1 x 2 In this case we need FFTs, which don't have scambling as an integrated part of the algorithm. Always check that!

18 18:21 Alternative signal graphs, Singleton Simple change to sorted input and scrambled output:...and with sorted output (messy!):

19 19:21 Alternative signal graphs, Singleton, contd A minor re organisation yields: Given by Singleton in Benefit: Same procedure for each stage. Possibility for sequential data access.

20 20:21 Inverse FFT It can be shown that x[n]= 1 { 1 k=0 * X * [ k]w kn } Hence, the exact same algorithm can be applied for inverse FFT, if you: 1) Complex conjugate the input to X * [ k], i.e. change sign on the imaginary part of X [ k]. 2) Complex conjugate the output and divide this output by.

21 21:21 Practical FFT algorithms In practise you would rarely write your own FFT algorithm but use existing ones. Efficient FFT algorithms usually come with commercial DSP development cards etc. On higher level (e.g. C programming) for execution on a PC, free software exists, most notably the FFTW package which can handle FFTs of any size.