1. Waveform design course Chapters 7 & 8 from Waveform Design for Active Sensing Systems A computational approach

2. Cross ambiguity function (CAF)
CAF has more degrees of freedom compared to that of the conventional ambiguity function, a case where v(t) equals u(t).

3. Discrete-CAF synthesis
Under the assumptions that
It can be proved that

4. Design problem

5. Cyclic algorithm (CA) for discrete-CAF synthesis
Using the following notations

6. CA contd..
C2 can be re-written as

7. CA steps

8. Discrete CAF with weights

9. Numerical examples

10. Numerical examples

11. Numerical examples

12. Numerical examples

13. Numerical examples

14. Continuous time CAF synthesis

15. Continuous time CAF synthesis

16. CA for CAF synthesis

17. Numerical example

18. Numerical example

20. Joint design of transmit sequence and receive filter
In Radars/Sonars.
Conventional receiver : Matched filter (MF) (in the case of Doppler shifts, a bank of filters).
MF maximizes the signal-to-noise ratio (SNR).
Apart from noise here one can also have clutters.
Signal to clutter-plus interference ratio (SCIR)

21. Data model and problem formulation

22. MSE of the mis-matched filter

23. CREW (gra) Minimization of MSE wrt to w Concentrated MSE :
Minimization problem :
which can be tackled via gradient methods like BFGS (Broyden-Fletcher-Goldfarb-Shanno) method – requires only gradient.

24. A frequency domain approach

25. Contd..
Using the circulant parameterization

26. Contd..
Using the DFT matrices to diagonalize the circulant matrices

27. CREW (fre) The design problem can be re-written as Minimizer over {hp}
Minimization over {εp}

28. CREW (fre)
Minimization over {zp} is convex, it can be solved using the Lagrangian methods
Using Lagrangian multipliers

29. CREW (fre) Once {|εp|} is obtained, x can be obtained via
which can be solved by a CA, unimodular and PAR constraints can be imposed.

30. Lower bound on MSE

31. CREW (mat)
MSE for the matched filter
Minimization over {εp}

32. Numerical examples

33. Jamming scenarios

34. Numerical example

38. Barrage jamming

41. Robust design

42. Robust design