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In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of time delay and Doppler frequency
χ(τ,f){displaystyle chi (tau ,f)} showing the distortion of a returned pulse due to the receiver matched filter^[1] (commonly, but not exclusively, used in pulse compression radar) due to the Doppler shift of the return from a moving target. The ambiguity
function is determined by the properties of the pulse and the matched filter, and not any particular target scenario. Many definitions of the ambiguity function exist; Some are restricted to narrowband signals and others are suitable to describe the propagation delay and Doppler relationship of wideband signals. Often the definition of the ambiguity function is given as the magnitude squared of other definitions (Weiss^[2]).
For a given complex baseband pulse s(t){displaystyle s(t)}, the narrowband ambiguity function is given by

χ(τ,f)=∫−∞∞s(t)s∗(t−τ)ei2πftdt{displaystyle chi (tau ,f)=int _{-infty }^{infty }s(t)s^{*}(t-tau )e^{i2pi ft},dt}

where ∗{displaystyle ^{*}} denotes the complex conjugate and i{displaystyle i} is the imaginary unit. Note that for zero Doppler shift (f=0{displaystyle f=0}) this reduces to the autocorrelation of s(t){displaystyle s(t)}. A more concise way of representing the
ambiguity function consists of examining the one-dimensional
zero-delay and zero-Doppler "cuts"; that is, χ(0,f){displaystyle chi (0,f)} and
χ(τ,0){displaystyle chi (tau ,0)}, respectively. The matched filter output as a function of a time (the signal one would observe in a radar system) is a Doppler cut, with constant frequency given by the target's Doppler shift: χ(τ,fD){displaystyle chi (tau ,f_{D})}.

Background and motivation

Pulse-Doppler radar equipment sends out a series of radio frequency pulses. Each pulse has a certain shape (waveform)—how long the pulse is, what its frequency is, whether the frequency changes during the pulse, and so on. If the waves reflect off a single object, the detector will see a signal which, in the simplest case, is a copy of the original pulse but delayed by a certain time τ{displaystyle tau }—related to the object's distance—and shifted by a certain frequency f{displaystyle f}—related to the object's velocity (Doppler shift). If the original emitted pulse waveform is s(t){displaystyle s(t)}, then the detected signal (neglecting noise, attenuation, and distortion, and wideband corrections) will be:

sτ,f(t)≡s(t−τ)ei2πft.{displaystyle s_{tau ,f}(t)equiv s(t-tau )e^{i2pi ft}.}

The detected signal will never be exactly equal to any sτ,f{displaystyle s_{tau ,f}} because of noise. Nevertheless, if the detected signal has a high correlation with sτ,f{displaystyle s_{tau ,f}}, for a certain delay and Doppler shift (τ,f){displaystyle (tau ,f)}, then that suggests that there is an object with (τ,f){displaystyle (tau ,f)}. Unfortunately, this procedure may yield false positives, i.e. wrong values (τ′,f′){displaystyle (tau ',f')} which are nevertheless highly correlated with the detected signal. In this sense, the detected signal may be ambiguous.

The ambiguity occurs specifically when there is a high correlation between sτ,f{displaystyle s_{tau ,f}} and sτ′,f′{displaystyle s_{tau ',f'}} for (τ,f)≠(τ′,f′){displaystyle (tau ,f)neq (tau ',f')}. This motivates the ambiguity function χ{displaystyle chi }. The defining property of χ{displaystyle chi } is that the correlation between sτ,f{displaystyle s_{tau ,f}} and sτ′,f′{displaystyle s_{tau ',f'}} is equal to χ(τ−τ′,f−f′){displaystyle chi (tau -tau ',f-f')}.

Different pulse shapes (waveforms) s(t){displaystyle s(t)} have different ambiguity functions, and the ambiguity function is relevant when choosing what pulse to use.

The function χ{displaystyle chi } is complex-valued; the degree of "ambiguity" is related to its magnitude |χ(τ,f)|2{displaystyle |chi (tau ,f)|^{2}}.

Relationship to time–frequency distributions

The ambiguity function plays a key role in the field of time–frequency signal processing,^[3] as it is related to the Wigner–Ville distribution by a 2-dimensional Fourier transform. This relationship is fundamental to the formulation of other time–frequency distributions: the bilinear time–frequency distributions are obtained by a 2-dimensional filtering in the ambiguity domain (that is, the ambiguity function of the signal). This class of distribution may be better adapted to the signals considered.^[4]

Moreover, the ambiguity distribution can be seen as the short-time Fourier transform of a signal using the signal itself as the window function. This remark has been used to define an ambiguity distribution over the time-scale domain instead of the time-frequency domain.^[5]

Wideband ambiguity function

The wideband ambiguity function of s∈L2(R){displaystyle sin L^{2}(R)} is:^[2][6]

WBss(τ,α)=|α|∫−∞∞s(t)s∗(α(t−τ))dt{displaystyle WB_{ss}(tau ,alpha )={sqrt {|{alpha }|}}int _{-infty }^{infty }s(t)s^{*}(alpha (t-tau )),dt}

where α{displaystyle {alpha }} is a time scale factor of the received signal relative to the transmitted signal given by:

α=c+vc−v{displaystyle alpha ={frac {c+v}{c-v}}}

for a target moving with constant radial velocity v. The reflection of the signal is represented with compression (or expansion) in time by the factor α{displaystyle alpha }, which is equivalent to a compression by the factor α−1{displaystyle alpha ^{-1}} in the frequency domain (with an amplitude scaling). When the wave speed in the medium is sufficiently faster than the target speed, as is common with radar, this compression in frequency is closely approximated by a shift in frequency Δf = f_c*v/c (known as the doppler shift). For a narrow band signal, this approximation results in the narrowband ambiguity function given above, which can be computed efficiently by making use of the FFT algorithm.

Ideal ambiguity function

An ambiguity function of interest is a 2-dimensional Dirac delta function or "thumbtack" function; that is, a function which is infinite at (0,0) and zero elsewhere.

χ(τ,f)=δ(τ)δ(f){displaystyle chi (tau ,f)=delta (tau )delta (f),}

An ambiguity function of this kind would be somewhat of a misnomer; it would have no ambiguities at all, and both the zero-delay and zero-Doppler cuts would be an impulse. This is not usually desirable (if a target has any Doppler shift from an unknown velocity it will disappear from the radar picture), but if Doppler processing is independently performed, knowledge of the precise Doppler frequency allows ranging without interference from any other targets which are not also moving at exactly the same velocity.

This type of ambiguity function is produced by ideal white noise (infinite in duration and infinite in bandwidth).^[7] However, this would require infinite power and is not physically realizable. There is no pulse s(t){displaystyle s(t)} that will produce δ(τ)δ(f){displaystyle delta (tau )delta (f)} from the definition of the ambiguity function. Approximations exist, however, and noise-like signals such as binary phase-shift keyed waveforms using maximal-length sequences are the best known performers in this regard.^[8]

Properties of the ambiguity function

(1) Maximum value

|χ(τ,f)|2≤|χ(0,0)|2{displaystyle |chi (tau ,f)|^{2}leq |chi (0,0)|^{2}}

(2) Symmetry about the origin

χ(τ,f)=exp⁡[j2πτf]χ∗(−τ,−f){displaystyle chi (tau ,f)=exp[j2pi tau f]chi ^{*}(-tau ,-f),}

(3) Volume invariance

∫−∞∞∫−∞∞|χ(τ,f)|2dτdf=|χ(0,0)|2=E2{displaystyle int _{-infty }^{infty }int _{-infty }^{infty }|chi (tau ,f)|^{2},dtau ,df=|chi (0,0)|^{2}=E^{2}}

(4) Modulation by a linear FM signal

If s(t)→|χ(τ,f)| then s(t)exp⁡[jπkt2]→|χ(τ,f+kτ)|{displaystyle {text{If }}s(t)rightarrow |chi (tau ,f)|{text{ then }}s(t)exp[jpi kt^{2}]{rightarrow }|chi (tau ,f+ktau )|,}

(5) Frequency energy spectrum

S(f)S∗(f)=∫−∞∞χ(τ,0)e−j2πτfdτ{displaystyle S(f)S^{*}(f)=int _{-infty }^{infty }chi (tau ,0)e^{-j2pi tau f},dtau }

(6) Upper bounds for p>2{displaystyle p>2} and lower bounds for p<2{displaystyle p<2} exist ^[9]for the pth{displaystyle p^{th}} power integrals

∫−∞∞∫−∞∞|χ(τ,f)|pdτdf{displaystyle int _{-infty }^{infty }int _{-infty }^{infty }|chi (tau ,f)|^{p},dtau ,df}.

These bounds are sharp and are achieved if and only if s(t){displaystyle s(t)} is a Gaussian function.

Square pulse

Ambiguity function for a square pulse

Consider a simple square pulse of duration τ{displaystyle tau } and
amplitude A{displaystyle A}:

A(u(t)−u(t−τ)){displaystyle A(u(t)-u(t-tau )),}

where u(t){displaystyle u(t)} is the Heaviside step function. The
matched filter output is given by the autocorrelation of the pulse, which is a triangular pulse of height τ2A2{displaystyle tau ^{2}A^{2}} and
duration 2τ{displaystyle 2tau } (the zero-Doppler cut). However, if the
measured pulse has a frequency offset due to Doppler shift, the
matched filter output is distorted into a sinc function. The
greater the Doppler shift, the smaller the peak of the resulting sinc,
and the more difficult it is to detect the target.

In general, the square pulse is not a desirable waveform from a pulse compression standpoint, because the autocorrelation function is too short in amplitude, making it difficult to detect targets in noise, and too wide in time, making it difficult to discern multiple overlapping targets.

LFM pulse

Ambiguity function for an LFM pulse

A commonly used radar or sonar pulse is the linear frequency modulated (LFM) pulse (or "chirp"). It has the advantage of greater bandwidth while keeping the pulse duration short and envelope constant. A constant envelope LFM pulse has an ambiguity function similar to that of the square pulse, except that it is skewed in the delay-Doppler plane. Slight Doppler mismatches for the LFM pulse do not change the general shape of the pulse and reduce the amplitude very little, but they do appear to shift the pulse
in time. Thus, an uncompensated Doppler shift changes the target's apparent range; this phenomenon is called range-Doppler coupling.

Multistatic ambiguity functions

The ambiguity function can be extended to multistatic radars, which comprise multiple non-colocated transmitters and/or receivers (and can include bistatic radar as a special case).

For these types of radar, the simple linear relationship between time and range that exists in the monostatic case no longer applies, and is instead dependent on the specific geometry – i.e. the relative location of transmitter(s), receiver(s) and target. Therefore, the multistatic ambiguity function is mostly usefully defined as a function of two- or three-dimensional position and velocity vectors for a given multistatic geometry and transmitted waveform.

Just as the monostatic ambiguity function is naturally derived from the matched filter, the multistatic ambiguity function is derived from the corresponding optimal multistatic detector – i.e. that which maximizes the probability of detection given a fixed probability of false alarm through joint processing of the signals at all receivers. The nature of this detection algorithm depends on whether or not the target fluctuations observed by each bistatic pair within the multistatic system are mutually correlated. If so, the optimal detector performs phase coherent summation of received signals which can result in very high target location accuracy.^[10] If not, the optimal detector performs incoherent summation of received signals which gives diversity gain. Such systems are sometimes described as MIMO radars due to the information theoretic similarities to MIMO communication systems.^[11]

See also

• Matched filter


• Pulse compression


• Pulse-Doppler radar


• Digital signal processing


• Philip Woodward

References

Further reading

• Richards, Mark A. Fundamentals of Radar Signal Processing. McGraw–Hill Inc., 2005.
  ISBN 0-07-144474-2.


• Ipatov, Valery P. Spread Spectrum and CDMA. Wiley & Sons, 2005.
  ISBN 0-470-09178-9
  


• Chernyak V.S. Fundamentals of Multisite Radar Systems, CRC Press, 1998.


• Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.


• M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.


• Nadav Levanon, and Eli Mozeson. Radar signals. Wiley. com, 2004.


• Augusto Aubry, Antonio De Maio, Bo Jiang, and Shuzhong Zhang. "Ambiguity function shaping for cognitive radar via complex quartic optimization." IEEE Transactions on Signal Processing 61 (2013): 5603-5619.


• Mojtaba Soltanalian, and Petre Stoica. "Computational design of sequences with good correlation properties." IEEE Transactions on Signal Processing, 60.5 (2012): 2180-2193.


• G. Krötzsch, M. A. Gómez-Méndez, Transformada Discreta de Ambigüedad, Revista Mexicana de Física, Vol. 63, pp. 505--515 (2017). "Transformada Discreta de Ambigüedad".