Waveform design course Chapters 7

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Waveform design courseChapters 7 8 from
Waveform Design for Active Sensing SystemsA
computational approach
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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).
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Discrete-CAF synthesis
Under the assumptions that
It can be proved that
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Design problem
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Cyclic algorithm (CA) for discrete-CAF synthesis
Using the following notations
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CA contd..
C2 can be re-written as
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CA steps
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Discrete CAF with weights
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Numerical examples
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Numerical examples
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Numerical examples
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Numerical examples
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Numerical examples
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Continuous time CAF synthesis
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Continuous time CAF synthesis
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CA for CAF synthesis
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Numerical example
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Numerical example
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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)

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Data model and problem formulation
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MSE of the mis-matched filter
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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.

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A frequency domain approach
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Contd..
Using the circulant parameterization
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Contd..
Using the DFT matrices to diagonalize the
circulant matrices
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CREW (fre)
The design problem can be re-written as
Minimizer over hp
Minimization over ep
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CREW (fre)
Minimization over zp is convex, it can be
solved using the Lagrangian methods
Using Lagrangian multipliers
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CREW (fre)

Once ep is obtained, x can be obtained via
which can be solved by a CA, unimodular and PAR
constraints can be imposed.
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Lower bound on MSE
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CREW (mat)
MSE for the matched filter
Minimization over ep
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Numerical examples
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Jamming scenarios
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Numerical example
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Barrage jamming
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Robust design
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Robust design