Purpose of radar: measure round trip time delay

Purpose of radar: measure round trip time delay

1. 1
Radar Signals
Tutorial II: The Ambiguity Function

2. 2
oPurpose of radar: measure round trip time delay.
Brief Review

3. 3
oRadar equation:
oMatched filter:
•Maximizes the SNR in the received signal.
•Response is described by the autocorrelation
function of the signal.
Brief Review

4. 4
oAutocorrelation of a signal:
Brief Review

5. 5
oDefinition: The ambiguity function is the time
response of a filter matched to a given finite energy
signal when the signal is received with a delay
and a Doppler shift relative to the nominal values
expected by the filter.
The Ambiguity Function

6. 6
oComplex envelope of a constant frequency pulse:
Example(1)

7. 7
oPartial AF:
Example(1)

8. Contour 0.707
8
oContour plot of the AF:
Example(1)
Contour 0.1

9. 9
Why is the AF important?

10. 10
oWhy is the AF important?
•Chirp waveform
Example(2)
Ambiguity Function SISO range-Doppler image

11. 11
oWhy is the AF important?
•Unmodulated pulse
Example(2)
Ambiguity Function SISO range-Doppler image

12. 12
oProperty 1: Maximum at (0,0).
AF Properties (1)

13. Apply CS
13
oProof of property 1:
AF Properties (1)

14. 14
oProperty 2: Constant volume.
AF Properties (2)

15. 15
oProof of property 2:
•Rewrite , replacing with .
AF Properties (2)

16. 16
oProof of property 2:
•Apply Parseval’s theorem –the energy in the
time domain is equal to the energy in the
frequency domain.
AF Properties (2)

17. 17
oProof of property 2:
•Integrate both sides with respect to to yield
volume .
AF Properties (2)

18. 18
oProof of property 2:
•Change variables and solve.
AF Properties (2)

19. 19
oImplications of property 2.
•Additional volume constraints:
•No matter how we design our waveform, the
volume of the AF remains constant.
AF Properties (2)

20. 20
oProperty 3: Symmetry with respect to the origin.
AF Properties (3)

21. 21
oProperty 4: Linear FM effect.
If
,
then adding linear frequency modulation (LFM)
implies that:
.
AF Properties (4)

22. 22
oProof of property 4:
AF Properties (4)

23. 23
oImplications of property 4:
AF Properties (4)

24. 24
oImplications of property 4:
AF Properties (4)

25. 25
oLinear frequency-modulated (LFM) pulse (Chirp).
•The most popular pulse compression method.
•Conceived during WWII.
•Basic idea: sweep the frequency band linearly
during the pulse duration .
Chirp Waveform

26. Chirp rate
26
oLinear frequency-modulated (LFM) pulse (Chirp).
•Complex envelope:
Chirp Waveform

27. 27
oLinear frequency-modulated (LFM) pulse (Chirp).
•Complex envelope:
Chirp Waveform

28. 28
oLinear frequency-modulated (LFM) pulse (Chirp).
•Ambiguity Function:
Chirp Waveform

29. 29
oLinear frequency-modulated (LFM) pulse (Chirp).
•Ambiguity Function:
Chirp Waveform

30. 30
oAdvantageof chirp: improved range resolution.
•Zero-Doppler cut:
•For a large time-bandwidth product
( ), the first null occurs at:
Chirp Waveform

31. 31
oAdvantageof chirp: improved range resolution.
•Zero-Doppler cut:
Chirp Waveform

32. 32
oAdvantageof chirp: improved range resolution.
•Spectrum of unmodulated pulse:
Chirp Waveform

33. 33
oAdvantageof chirp: improved range resolution.
•Spectrum of LFM pulse:
Chirp Waveform
LFM improves range resolution according to
the time-bandwidth product!

34. 34
oDisadvantageof chirp: delay-Doppler coupling.
•For small Doppler shift , the delay location of
the peak response is shifted from true delay by:
•Preferred in situations with ambiguous Doppler
shifts.
Chirp Waveform

35. 35
oDisadvantageof chirp: delay-Doppler coupling.
Chirp Waveform
Contour 0.707
Contour 0.1
A target with positive Doppler appears closer
than its true range!

36. 36
oSISO range-Doppler imaging example
•Bandwidth , duration , chirp-rate .
Example(3)
40 dB target

37. 37
oSISO range-Doppler imaging example
•, fix
Example(3)

38. 38
oOther forms of frequency modulation:
•LFM amplitude weighting.
•Costas coding.
•Nonlinear FM.
oPhased-coded waveforms:
•Barker code.
•Chirp-like sequences.
Future Talks