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Radar SignalsTutorial II: The Ambiguity Function
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o Purpose of radar: measure round trip time delay.
Brief Review
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o Radar equation:
o Matched filter:• Maximizes the SNR in the received signal.• Response is described by the autocorrelation function of the signal.
Brief Review
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o Autocorrelation of a signal:
Brief Review
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o Definition: 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
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o Complex envelope of a constant frequency pulse:
Example(1)
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o Partial AF:
Example(1)
Contour 0.707
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o Contour plot of the AF:
Example(1)
Contour 0.1
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Why is the AF important?
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o Why is the AF important?• Chirp waveform
Example(2)
Ambiguity Function SISO range-Doppler image
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o Why is the AF important?• Unmodulated pulse
Example(2)
Ambiguity Function SISO range-Doppler image
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o Property 1: Maximum at (0,0).
AF Properties (1)
Apply CS
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o Proof of property 1:
AF Properties (1)
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o Property 2: Constant volume.
AF Properties (2)
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o Proof of property 2:
• Rewrite , replacing with .
AF Properties (2)
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o Proof 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)
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o Proof of property 2:
• Integrate both sides with respect to to yield volume .
AF Properties (2)
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o Proof of property 2:
• Change variables and solve.
AF Properties (2)
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o Implications of property 2.
• Additional volume constraints:
• No matter how we design our waveform, the volume of the AF remains constant.
AF Properties (2)
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o Property 3: Symmetry with respect to the origin.
AF Properties (3)
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o Property 4: Linear FM effect.
If,
then adding linear frequency modulation (LFM) implies that:
.
AF Properties (4)
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o Proof of property 4:
AF Properties (4)
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o Implications of property 4:
AF Properties (4)
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o Implications of property 4:
AF Properties (4)
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o Linear 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
Chirp rate
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o Linear frequency-modulated (LFM) pulse (Chirp).
• Complex envelope:
Chirp Waveform
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o Linear frequency-modulated (LFM) pulse (Chirp).
• Complex envelope:
Chirp Waveform
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o Linear frequency-modulated (LFM) pulse (Chirp).
• Ambiguity Function:
Chirp Waveform
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o Linear frequency-modulated (LFM) pulse (Chirp).
• Ambiguity Function:
Chirp Waveform
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o Advantage of chirp: improved range resolution.
• Zero-Doppler cut:
• For a large time-bandwidth product ( ), the first null occurs at:
Chirp Waveform
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o Advantage of chirp: improved range resolution.
• Zero-Doppler cut:
Chirp Waveform
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o Advantage of chirp: improved range resolution.
• Spectrum of unmodulated pulse:
Chirp Waveform
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o Advantage of chirp: improved range resolution.
• Spectrum of LFM pulse:
Chirp Waveform
LFM improves range resolution according to the time-bandwidth product!
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o Disadvantage of 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
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o Disadvantage of chirp: delay-Doppler coupling.
Chirp Waveform
Contour 0.707
Contour 0.1
A target with positive Doppler appears closer than its true range!
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o SISO range-Doppler imaging example• Bandwidth , duration , chirp-rate .
Example(3)
40 dB target
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o SISO range-Doppler imaging example• , fix
Example(3)
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o Other forms of frequency modulation:• LFM amplitude weighting.• Costas coding.• Nonlinear FM.
o Phased-coded waveforms:• Barker code.• Chirp-like sequences.
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