1 Radar Signals Tutorial II: The Ambiguity Function.

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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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