1. Stopband constraint case and the ambiguity function Daniel Jansson

2. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Stopband constraint case Goal Generate discrete, unimodular sequences with frequency notches and good correlation properties Why? Avoiding reserved frequency bands is important in many applications (communications, navigation..) Avoiding other interference How? SCAN (Stopband CAN) / WeSCAN (Weighted Stopband CAN)

3. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Stopband CAN (SCAN) Let {x(n)}, n = 1...N be the sought sequence Express the bands to be avoided as Define the DFT matrix with elements Form matrix S from the columns of F Ñ corresponding to the frequencies in Ω We suppress the spectral power of {x(n)} in Ω by minimizing where

4. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Stopband CAN (SCAN) The problem on the previous slide is equivalent to where G are the remaining columns of F Ñ. Suppressing the correlation sidelobes is done using the CAN formulation

5. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Stopband CAN (SCAN) Combining the frequency band suppression and the correlation sidelobe suppression problems we get where 0 ≤ λ ≤ 1 controls the relative weight on the two penalty functions. The problem is solved by using the algorithm on the next slide

6. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

7. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Stopband CAN (SCAN) If a constrained PAR is preferable to unimodularity the problem can be solved in the same way except x for each iteration is given by the solution to

8. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Weighted SCAN (WeSCAN) Minimization of J 2 is a way of minimizing the ISL The more general WISL (weighted ISL) is given by where are weights

9. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Weighted SCAN (WeSCAN) Let and D be the square root of Γ. Then the WISL can be minimized by solving where and Replace in the SCAN problem with and perform the SCAN algorithm, but do necessary changes that are straightforward.

10. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples The spectral power of a SCAN sequence generated with parameters N = 100, Ñ = 1000, λ = 0.7 and Ω = [0.2,0.3] Hz. P stop = -8.3 dB (peak stopband power)

11. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples The autocorrelation of a SCAN sequence generated with parameters N = 100, Ñ = 1000, λ = 0.7 and Ω = [0.2,0.3] Hz, P corr = -19.2 dB (peak sidelobe level)

12. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples P stop and P corr vs λ

13. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples The spectral power of a WeSCAN sequence generated with γ 1 =0, γ 2 =0 and γ k =1 for larger k. P stop = -34.9 dB (peak stopband power)

14. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples The autocorrelation of the WeSCAN sequence

15. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se Numerical examples The spectral power of a SCAN sequence generated with PAR ≤ 2

16. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function The response of a matched filter to a signal with various time delays and Doppler frequency shifts (extension of the correlation concept). The (narrowband) ambiguity function is where u(t) is a probing signal which is assumed to be zero outside [0,T], τ is the time delay and f is the Doppler frequency shift.

17. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Three properties worth noting 1. The maximum value of |χ(τ,f)| is achieved at | χ(0,0)| and is the energy of the signal, E 2. d |χ(τ,f)|= |χ(-τ,-f)| 3. D

18. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Proofs 1. Cauchy-Schwartz gives and since | χ(0,0)| = E, property 1 follows. 2. Use the variable change t -> t+ τ which implies property 2.

19. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Proofs 3. The volume of |χ(τ,f)| 2 is given by Let W τ (f) be the Fourier transform of u(t)u*(t- τ). Parseval gives therefore

20. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Ambiguity function of a chirp

21. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Ambiguity function of a Golomb sequence

22. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Ambiguity function of CAN generated sequences

23. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Ambiguity Function Why is there a vertical stripe at the zero delay cut? The ZDC is nothing but the Fourier transform of u(t)u*(t). Since u(t) is unimodular we get and the sinc-function decreases quickly as f increases. No universal method that can synthesize an arbirtrary ambiguity function.

24. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF Assume u(t) is on the form where p n (t) is an ideal rectangular pulse of length t p The ambiguity function can be written as Inserting τ = kt p and f = p/(Nt p ) gives where is called the discrete AF. If |p|<<N then

25. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF Minimizing the sidelobes of the discrete AF in a certain region where and are the index sets specifying the region. Define the set of sequences as

26. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF Denote the correlation between {x m (n)} and {x l (n)} by All values of are contained in the set Minimizing the correlations is thus equivalent to minimizing the discrete AF sidelobes.

27. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF Define where All elements of appear in We can thus minimize which as we saw before is almost equivalent to Minimize by using the cyclic algorithm on the next slide

28. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF

29. Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se The Discrete AF