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Ambiguity Function

In the intricate dance of radar and sonar signal processing, where echoes are coaxed from the ether and silence, the ambiguity function emerges as a crucial, if somewhat vexing, mathematical construct. It’s a two-dimensional landscape, charting the interplay between propagation delay, denoted by the Greek letter tau ($\tau$), and Doppler frequency, represented by f. This function, $\chi(\tau, f)$, is essentially a measure of how a returned pulse gets distorted, a consequence of the receiver's matched filter grappling with the signal from a moving target. Think of it as the system's inherent uncertainty, its blind spot, a consequence of its own design rather than the target itself. This isn't some abstract theory; it's baked into the very definition of the pulse and the filter used, particularly in the sophisticated world of pulse compression radar.

Now, the definitions proliferate like weeds. Some are tailored for the relatively simple realm of narrowband signals, while others, more robust, can grapple with the complexities of wideband signals. It's worth noting that often, what's presented as the ambiguity function is the magnitude squared of a more fundamental definition, as seen in the work of Weiss.

For a given complex baseband pulse, $s(t)$, the narrowband ambiguity function is formally expressed as:

$\chi(\tau, f) = \int_{-\infty}^{\infty} s(t) s^{*}(t-\tau) e^{i2\pi ft} dt$

Here, the asterisk ($^{*}$) signifies the complex conjugate, and $i$ is the ubiquitous imaginary unit. A quick observation: when the Doppler shift is zero ($f=0$), this expression simplifies beautifully, reducing to the autocorrelation of the original pulse, $s(t)$. To get a more intuitive grasp, we often examine specific "cuts" through this function: the zero-delay cut, $\chi(0, f)$, and the zero-Doppler cut, $\chi(\tau, 0)$. The output of the matched filter itself, the signal you'd actually observe in a radar system, corresponds to a Doppler cut, where the frequency is fixed by the target's Doppler shift: $\chi(\tau, f_D)$.

Background and Motivation

The genesis of the ambiguity function lies in the fundamental operation of pulse-Doppler radar systems. These systems emit a series of precisely shaped radio frequency pulses. The characteristics of these pulses—their duration, frequency, and any internal modulations—are critical. When these pulses strike a target and reflect back, the receiver is tasked with deciphering the echo. In an ideal scenario, the received signal is a faithful, albeit delayed, replica of the transmitted pulse. The delay, $\tau$, directly correlates with the target's distance, while a frequency shift, $f$, reveals its velocity through the Doppler shift. If the original pulse is $s(t)$, the received signal, stripped of real-world complications like noise and attenuation, would be:

$s_{\tau, f}(t) \equiv s(t-\tau) e^{i2\pi ft}$

The reality, of course, is messier. Noise is an unwelcome constant. Still, a strong correlation between the received signal and a specific $s_{\tau, f}$ for a given delay and Doppler shift $(\tau, f)$ strongly suggests the presence of a target at that range and velocity. The problem, however, is the specter of false positives. Sometimes, the received signal might exhibit a high correlation with a different $(\tau', f')$ pair, leading to an ambiguous measurement. The signal is, in essence, ambiguous.

This ambiguity arises precisely when $s_{\tau, f}$ and $s_{\tau', f'}$ are highly correlated for distinct $(\tau, f)$ and $(\tau', f')$ pairs. This is where the ambiguity function, $\chi$, steps in. Its defining characteristic is that the correlation between $s_{\tau, f}$ and $s_{\tau', f'}$ is precisely $\chi(\tau - \tau', f - f')$. Different pulse shapes, or waveforms, $s(t)$, yield different ambiguity functions. This makes the choice of waveform paramount, a strategic decision impacting the system's clarity.

The ambiguity function itself is a complex entity. The degree of ambiguity, the measure of our uncertainty, is tied to its magnitude squared, $|\chi(\tau, f)|^2$.

Relationship to Time-Frequency Distributions

The ambiguity function is not just an isolated concept; it occupies a central position in the sophisticated field of time–frequency signal processing. It shares a profound connection with the Wigner–Ville distribution, linked by a two-dimensional Fourier transform. This relationship is the bedrock upon which other time–frequency distributions are built. Specifically, the class of bilinear time–frequency distributions are derived by performing a two-dimensional filtering operation directly in the ambiguity domain, essentially manipulating the ambiguity function of the signal. This approach allows for distributions that are better tailored to the specific characteristics of the signals being analyzed.

Furthermore, the ambiguity distribution can be re-imagined as the short-time Fourier transform of a signal, with the signal itself acting as the window function. This insight has paved the way for defining ambiguity distributions not just in the time-frequency domain, but also in the time-scale domain, offering a different perspective on signal analysis.

Wideband Ambiguity Function

When signals exhibit significant bandwidth, the simpler narrowband model falls short. The wideband ambiguity function steps in to handle these more complex scenarios. For a signal $s \in L^2(\mathbb{R})$, it is defined as:

$WB_{ss}(\tau, \alpha) = \sqrt{|\alpha|} \int_{-\infty}^{\infty} s(t) s^{*}(\alpha(t-\tau)) dt$

Here, $\alpha$ is a time-scale factor, intrinsically linked to the relative speed of the target. Specifically, for a target moving with radial velocity $v$, $\alpha = \frac{c+v}{c-v}$, where $c$ is the wave speed. The essence of this function is that the received signal isn't just time-shifted and frequency-shifted; it's also time-compressed or expanded by the factor $\alpha$. This time scaling is equivalent to a frequency scaling. For radar applications, where the wave speed $c$ is typically much greater than the target speed $v$, this frequency compression is closely approximated by a frequency shift $\Delta f = f_c \frac{v}{c}$, the familiar Doppler shift. This approximation is what allows us to revert to the narrowband ambiguity function, which can be computed efficiently using the FFT algorithm.

Ideal Ambiguity Function

The theoretical ideal, a sort of holy grail in signal processing, is the ambiguity function that behaves like a two-dimensional Dirac delta function, often called a "thumbtack" function. It would be infinitely sharp at $(0,0)$ and precisely zero everywhere else:

$\chi(\tau, f) = \delta(\tau)\delta(f)$

Such a function would, in theory, eliminate all ambiguities. Both the zero-delay and zero-Doppler cuts would be perfect impulses. While this offers perfect range and velocity resolution, it’s not always desirable. A target with even a slight Doppler shift would vanish from the radar picture. However, if Doppler processing is performed independently, knowing the exact Doppler frequency allows for precise ranging without interference from targets moving at different velocities.

This idealized function is theoretically produced by infinite white noise – a signal with infinite duration and infinite bandwidth. The catch? It requires infinite power, rendering it physically impossible. No finite pulse $s(t)$ can ever achieve this perfect $\delta(\tau)\delta(f)$ ambiguity function. Nevertheless, approximations exist. Noise-like signals, such as those generated by maximal-length sequences in binary phase-shift keyed waveforms, come remarkably close, offering excellent performance in practice.

Properties

The ambiguity function, $\chi(\tau, f)$, possesses several key properties that govern its behavior and utility:

1. Maximum Value: The magnitude squared of the ambiguity function is bounded by its value at the origin: $|\chi(\tau, f)|^2 \leq |\chi(0, 0)|^2$ This means the peak correlation occurs when there is no delay and no Doppler shift.

2. Symmetry about the Origin: The function exhibits a specific symmetry: $\chi(\tau, f) = \exp[j2\pi \tau f] \chi^{*}(-\tau, -f)$ This relationship links the function's values at $(\tau, f)$ to its complex conjugate at $(-\tau, -f)$, with a phase factor dependent on the product of delay and frequency.

3. Volume Invariance: The total energy, or volume under the magnitude squared surface, is invariant and equal to the squared energy of the original signal: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |\chi(\tau, f)|^2 \,d\tau \,df = |\chi(0, 0)|^2 = E^2$ This property, known as Woodward's ambiguity theorem, is fundamental. $E$ represents the energy of the signal $s(t)$.

4. Modulation by a Linear FM Signal: Applying a linear frequency modulation (FM) to the signal $s(t)$ results in a predictable shift in the ambiguity function. If $s(t)$ produces $|\chi(\tau, f)|$, then $s(t)\exp[j\pi kt^2]$ (a signal with linear FM modulation) produces $|\chi(\tau, f+k\tau)|$. This highlights how frequency modulation affects the ambiguity function's shape in the Doppler domain.

5. Frequency Energy Spectrum: The zero-Doppler cut of the ambiguity function, $\chi(\tau, 0)$, is directly related to the signal's energy spectrum through a Fourier transform: $S(f)S^{*}(f) = \int_{-\infty}^{\infty} \chi(\tau, 0) e^{-j2\pi \tau f} \,d\tau$ This connects the time-domain correlation properties (at zero Doppler) to the frequency-domain energy distribution.

6. Bounds for Power Integrals: For $p>2$, upper bounds and for $p<2$, lower bounds exist for the $p^{th}$ power integrals of the ambiguity function: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |\chi(\tau, f)|^p \,d\tau \,df$ These bounds are sharpest, achieved precisely when the signal $s(t)$ is a Gaussian function.

Square Pulse

Consider the simplest of waveforms: a rectangular pulse of duration $\tau$ and amplitude $A$, represented as $A(u(t) - u(t-\tau))$, where $u(t)$ is the Heaviside step function. The zero-Doppler cut, which is the autocorrelation of this pulse, results in a triangular pulse with a peak amplitude of $\tau^2 A^2$ and a duration of $2\tau$. However, if the received pulse is subject to a Doppler shift, the output of the matched filter is distorted into a sinc function. The larger the Doppler shift, the smaller the peak of this sinc function becomes, making target detection increasingly difficult. [citation needed]

In general, the square pulse is far from ideal for pulse compression. Its autocorrelation function is relatively short in amplitude, hindering target detection in noisy environments, and too wide in time, making it challenging to distinguish between closely spaced targets.

LFM Pulse

A more practical and widely adopted waveform in radar and sonar applications is the linear frequency modulated (LFM) pulse, often referred to as a "chirp." Its primary advantage lies in its ability to achieve a large bandwidth while maintaining a constant pulse duration and a constant envelope. An LFM pulse with a constant envelope exhibits an ambiguity function that, while similar to that of a square pulse, is distinctly skewed in the delay-Doppler plane. Crucially, minor Doppler shifts do not significantly alter the fundamental shape of the pulse or drastically reduce its amplitude. Instead, they manifest as a temporal shift. This phenomenon, known as range-Doppler coupling, means that an uncompensated Doppler shift can lead to an inaccurate estimate of the target's range.

Multistatic Ambiguity Functions

The concept of the ambiguity function extends beyond the familiar monostatic radar configuration to encompass multistatic radars. These are systems characterized by multiple, non-colocated transmitters and/or receivers, with bistatic radar being a specific instance.

In multistatic scenarios, the straightforward linear relationship between time delay and range, so characteristic of monostatic systems, dissolves. The range is no longer solely a function of delay but becomes intricately dependent on the specific geometric arrangement of the transmitters, receivers, and the target. Consequently, the multistatic ambiguity function is most effectively defined not as a simple function of delay and Doppler, but rather as a function of two- or three-dimensional position and velocity vectors, tailored to a particular multistatic geometry and transmitted waveform.

Just as the monostatic ambiguity function arises from the matched filter, its multistatic counterpart is derived from the optimal multistatic detector. This detector is designed to maximize the probability of detection while maintaining a fixed probability of false alarm, typically through the joint processing of signals received at all receivers. The precise nature of this optimal detection algorithm hinges on whether the target fluctuations observed by different bistatic pairs within the multistatic system are correlated. If they are correlated, the optimal detector employs phase-coherent summation of the received signals, potentially yielding exceptionally high target location accuracy. If the fluctuations are uncorrelated, the optimal detector resorts to incoherent summation, which provides a diversity gain. These types of systems are sometimes referred to as MIMO radars, drawing parallels with the information-theoretic principles found in MIMO communication systems.

Ambiguity Function Plane

The "ambiguity function plane" is a conceptual visualization tool. It can be thought of as an infinite arrangement of radial lines, each representing a specific relationship between delay and Doppler. Within this framework, each radial line can be interpreted as the fractional Fourier transform of a stationary random process, offering a deeper mathematical insight into the signal's behavior.

Example

The ambiguity function (AF) is intrinsically linked to the WDF, or Wigner-Ville Distribution. A common mathematical representation for the ambiguity function $A_x(\tau, n)$ of a signal $x(t)$ is given by:

$A_{x}(\tau, n) = \int_{-\infty}^{\infty} x(t + \frac{\tau}{2}) x^{*}(t - \frac{\tau}{2}) e^{-j2\pi tn} dt$

Let's consider a specific case:

(1) Gaussian Pulse: If the signal is a complex Gaussian function: $x(t) = \exp[-\alpha \pi (t-t_0)^2 + j2\pi f_0 t]$ Then its ambiguity function becomes: $A_{x}(\tau, n) = \sqrt{\frac{1}{2\alpha}} \exp[-\pi (\frac{\alpha \tau^2}{2} + \frac{n^2}{2\alpha})] \exp[j2\pi (f_0 \tau - t_0 n)]$ This shows a Gaussian-shaped distribution in the $\tau-n$ plane, centered at $(0,0)$ for the delay and Doppler terms, but with a phase term dependent on the original signal's center frequency $f_0$ and time shift $t_0$.

(2) Sum of Two Gaussian Pulses: When the signal is a sum of two such Gaussian pulses, say $x(t) = x_1(t) + x_2(t)$, where: $x_1(t) = \exp[-\alpha_1 \pi (t-t_1)^2 + j2\pi f_1 t]$ $x_2(t) = \exp[-\alpha_2 \pi (t-t_2)^2 + j2\pi f_2 t]$ The total ambiguity function $A_x(\tau, n)$ is the sum of the auto-ambiguity functions of each component ($A_{x1}$ and $A_{x2}$) plus their cross-ambiguity functions ($A_{x1x2}$ and $A_{x2x1}$): $A_{x}(\tau, n) = A_{x1}(\tau, n) + A_{x2}(\tau, n) + A_{x1x2}(\tau, n) + A_{x2x1}(\tau, n)$

The individual auto-terms are: $A_{x1}(\tau, n) = \sqrt{\frac{1}{2\alpha_1}} \exp[-\pi (\frac{\alpha_1 \tau^2}{2} + \frac{n^2}{2\alpha_1})] \exp[j2\pi (f_1 \tau - t_1 n)]$ $A_{x2}(\tau, n) = \sqrt{\frac{1}{2\alpha_2}} \exp[-\pi (\frac{\alpha_2 \tau^2}{2} + \frac{n^2}{2\alpha_2})] \exp[j2\pi (f_2 \tau - t_2 n)]$

The cross-terms, $A_{x1x2}(\tau, n)$ and $A_{x2x1}(\tau, n)$, represent the interaction between the two pulses. Their form depends on whether the decay constants $\alpha_1$ and $\alpha_2$ are equal.

If $\alpha_1 = \alpha_2 = \alpha_u$: $A_{x1x2}(\tau, n) = \sqrt{\frac{1}{2\alpha_u}} \exp[-\pi (\alpha_u \frac{(\tau - t_d)^2}{2} + \frac{(n - f_d)^2}{2\alpha_u})] \exp[j2\pi (f_u \tau - t_u n + f_d t_u)]$ where $t_u = (t_1 + t_2)/2$, $f_u = (f_1 + f_2)/2$, $t_d = t_1 + t_2$, and $f_d = f_1 - f_2$.

The term $A_{x2x1}(\tau, n)$ is related to the complex conjugate of $A_{x1x2}(\tau, n)$ evaluated at $(-\tau, -n)$.

If $\alpha_1 \neq \alpha_2$: The cross-term $A_{x1x2}(\tau, n)$ becomes more complex, involving coupled quadratic phase terms.

A crucial observation for the ambiguity function is that the auto-terms (representing the contribution of each individual pulse) tend to be localized near the origin of the $\tau-n$ plane, while the cross-terms, which arise from the interaction between different parts of the signal, can be spread out and may introduce significant ambiguities.

See Also

• Matched filter
• Pulse compression
• Pulse-Doppler radar
• Digital signal processing
• Philip Woodward