# The Ambiguity Function and the Cyclic Autocorrelation Function: Are They the Same Thing?

Let’s talk about ambiguity and correlation. The ambiguity function is a core component of radar signal processing practice and theory. The autocorrelation function and the cyclic autocorrelation function, are key elements of generic signal processing and cyclostationary signal processing, respectively. Ambiguity and correlation both apply a quadratic functional to the data or signal of interest, and they both weight that quadratic functional by a complex exponential (sine wave) prior to integration or summation.

Are they the same thing? Well, my answer is both yes and no.

### The Ambiguity Function

The common definition of the ambiguity function is

$\displaystyle A_x(\tau, \nu) = \int_{-\infty}^\infty x(t) x^*(t-\tau) e^{-i2\pi\nu t}\, dt, \hfill (1)$

where $\tau$ is the delay or lag parameter and $\nu$ is a frequency parameter that we will interpret below. Notice that the integral is over all time. This implies that $x(t)$ should be an energy signal. An energy signal has finite energy, so that the integral

$\displaystyle E_x = \int_{-\infty}^\infty \left| x(t) \right|^2 \, dt \hfill (2)$

exists and is greater than zero. Because it exists, it is also finite. Any bounded integrable (or summable in discrete time) signal that has finite temporal support is an energy signal; think of a rectangle, triangle, rapidly decaying exponential pulse, etc. An energy signal doesn’t have to have finite support, but that is a common property of energy signals, especially those used in radar.

The cross ambiguity function replaces one of the $x(t)$ terms in (1) with another energy signal $y(t)$:

$\displaystyle A_{xy} (\tau, \nu) = \int_{-\infty}^\infty x(t) y^*(t-\tau) e^{-i2\pi\nu t}\, dt, \hfill (3)$

Consider a radar-like signal consisting of a pulse $s(t)$ with finite temporal support, such as a burst of a sine wave or a burst of a chirp waveform. The signal is transmitted, strikes a reflective target, experiences the Doppler effect, and a portion of its energy propagates back to a receiver (often co-located with the transmitter). So the received reflected pulse is

$\displaystyle y(t) = C s(t-D) e^{-i2\pi f_d t} + n(t) \hfill (4)$

where $C$ is a complex-valued constant, $D$ is the round-trip delay from transmitter to target to receiver, and $f_d$ is the Doppler shift frequency, which is a good approximation to the Doppler effect in many cases of practical interest.

If you substitute $y(t)$ from (4) and $x(t) = s(t)$ into (3), and ignore the additive noise $n(t)$, you can see that the ambiguity magnitude will peak when $\nu = f_d$ and $D = -\tau$. For this reason, the $\nu$ parameter is called the Doppler frequency or Doppler parameter.

The “auto” ambiguity in (1) is useful for studying the radar-processing properties of candidate radar waveforms, whereas the cross version in (3) is more directly related to radar signal processing itself.

When using the ambiguity functions, the range of the Doppler frequency is limited to those Doppler shifts that are physically possible given the possible relative motions between the target and the transmitter/receiver. This is a (thankfully, for computational cost reasons) small range. For example, for a stationary transmitter/receiver, a target moving at $1000$ mph relative to the transmitter/receiver, and an operating frequency of $1.0$ GHz, the Doppler shift is about $1.5$ kHz.

I think the ambiguity function is most often called just that: ambiguity or ‘the ambiguity function.’ But the cross version is often called the CAF, and some people call the auto ambiguity function the ‘complex ambiguity function,’ or CAF. So that’s two CAFs.

### The Cyclic Autocorrelation Function

Back to our wheelhouse. The cyclic autocorrelation function is defined by an infinite-time average:

$\displaystyle R_x(\tau, \alpha) = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t+\tau/2) x^*(t-\tau/2) e^{-i2\pi\alpha t}\, dt \hfill (5)$

Notice here that if $x(t)$ is an energy signal, the limit will be zero. On the other hand, if $x(t)$ is a power signal, the limit can be finite and non-zero, depending on the nature of the signal(s) in $x(t)$, the value of the delay $\tau$, and crucially on the value of the cycle frequency $\alpha$. The power of a signal is the temporal average of the energy:

$\displaystyle P_x = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} \left| x(t) \right|^2 \, dt \hfill (6)$

The cyclic cross correlation function is defined by

$\displaystyle R_{xy}(\tau, \alpha) = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t+\tau/2) y^*(t-\tau/2) e^{-i2\pi\alpha t}\, dt \hfill (7)$

In the correlation functions, the frequency parameter–the cycle frequency–that gives rise to non-zero results is a function of the modulation types and parameters of the signals that are present in $x(t)$ and $y(t)$. Cycle frequencies can be quite small, such as those for narrowband digital signals operating at $2.4$ kHz, or quite large, such as those for wideband digital signals such as CDMA ($1.2288$ MHz), WCDMA ($3.84$ MHz), WiFi DSSS BPSK ($1, 2, 3, ...$ MHz), etc. The range of $\alpha$ is typically much larger than the range of $\nu$.

The cyclic autocorrelation function is also called the CAF. So that’s three CAFs, increasing the potential for confusion. Which CAF are we talking about? If they are all the same, there is no problem. If they aren’t, then we need to know how they differ.

### Estimators

To answer the post-title question, we need to introduce estimators of the ambiguity and cyclic autocorrelation functions. The two functions involve an integral over all time (a sum for discrete-time versions), so that natural estimators are simply the infinite-time definitions with the infinite intervals replaced by finte intervals. For the cyclic autocorrelation function, this is the estimator that follows simply from the definition in the fraction-of-time probability framework (The Literature [R67, R8, R131]), and is also the familiar estimator that arises in the context of a cycloergodic cyclostationary stochastic process.

The cyclic correlation estimators:

$\displaystyle \hat{R}_x(\tau, \alpha) = \frac{1}{T} \int_{-T/2}^{T/2} x(t+\tau/2) x^*(t-\tau/2) e^{-i2\pi\alpha t}\, dt, \hfill (8)$

$\displaystyle \hat{R}_{xy}(\tau, \alpha) = \frac{1}{T} \int_{-T/2}^{T/2} x(t+\tau/2) y^*(t-\tau/2) e^{-i2\pi\alpha t} \, dt. \hfill (9)$

For the ambiguity, we have the estimators:

$\displaystyle \hat{A}_x(\tau, \nu) = \int_{-T/2}^{T/2} x(t) x^*(t-\tau)e^{-i2\pi\nu t} \, dt \hfill (10)$

$\displaystyle \hat{A}_{xy}(\tau, \nu) = \int_{-T/2}^{T/2} x(t) y^*(t-\tau)e^{-i2\pi\nu t} \, dt \hfill (11)$

### The Question of Equivalence for Infinite-Time Mathematical Objects

When we think of ambiguity and cyclic correlation in terms of infinite-time mathematical operations ((1), (3), (5), and (7)), they are not equivalent since they produce very different answers for identical inputs. Ambiguity in this context applies to energy signals (think: signals of finite duration) and cyclic correlation applies to power signals (think: persistent random processes or signals). Ambiguity will blow up if applied to a power signal, and cyclic correlation will be zero for an energy signal.

Conclusion: They are not the same thing.

### The Question of Equivalence for Finite-Time Estimators

When we think of ambiguity and cyclic correlation in terms of finite-time estimators, there is an equivalence in that one function is a scaled version of the other. Let’s look at the ambiguity for $\tau \ll T$. First, impose symmetry with respect to $\tau$:

$\displaystyle \hat{A}_x(\tau, \nu) = \int_{-T/2-\tau/2}^{T/2-\tau/2} x(t+\tau/2)x^*(t-\tau/2) e^{i2\pi\nu(t + \tau/2)} \, dt \hfill (12)$

or

$\displaystyle \hat{A}_x(\tau, \nu) \approx e^{-i\pi\nu \tau} \int_{-T/2}^{T/2} x(t+\tau/2) x^*(t-\tau/2) e^{-i2\pi\nu t} \, dt. \hfill (13)$

We recognize the integral in (13) as the estimate of the cyclic autocorrelation multiplied by $T$,

$\displaystyle \hat{A}_x(\tau,\nu) \approx e^{-i\pi\nu\tau} T \hat{R}_x(\tau, \nu). \hfill (14)$

So the estimates differ by a scaling factor $T$ as well as a phase shift of $-i\pi\nu\tau$. Since the user of the estimator selects $\nu$ and $\tau$, this phase factor is easily compensated. Therefore, the estimates of ambiguity and cyclic correlation contain the same information.

Conclusion: They are the same thing.

### Related Topics

In the theory of complex-valued cyclostationary signals, the involved random variables and random signals are not proper. This means that their conjugate correlation functions can be non-zero. The normal correlation between random variables $X$ and $Y$ is

$\displaystyle R_{XY} = E\left[ XY^* \right]. \hfill (15)$

The conjugate correlation is defined as the correlation between $X$ and the conjugate of $Y$, or $Y^*$:

$\displaystyle r_{XY} = E\left[X(Y^*)^*\right] = E[XY]. \hfill (16)$

As we’ve explained elsewhere on the CSP Blog, this impropriety leads to the conjugate cyclic autocorrelation and conjugate cross correlation functions

$\displaystyle r_x(\tau, \beta) = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t + \tau/2) x(t - \tau/2) e^{-i2\pi\beta t}\, dt \hfill (17)$

and

$\displaystyle r_{xy}(\tau, \beta) = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t +\tau/2) y(t - \tau/2) e^{-i2\pi\beta t} \, dt, \hfill (18)$

It is unclear whether there is any useful analog of these conjugate correlation functions in the world of ambiguity processing. So in this sense, the conclusion is that the cyclic autocorrelation functions are not the same as the ambiguity function.

Let me know what you think in the Comments section below.

I'm a signal processing researcher specializing in cyclostationary signal processing (CSP) for communication signals. I hope to use this blog to help others with their cyclo-projects and to learn more about how CSP is being used and extended worldwide.

## 6 thoughts on “The Ambiguity Function and the Cyclic Autocorrelation Function: Are They the Same Thing?”

1. So in fourier transformed space do the peaks of lagged correlations of frequency show up the doppler shift, and of phase show up the delay?

Sounds similar to camera autofocus and automatic sub-pixel image registration.

The convolution version would be the correlation after reversing one time-series; which leaves frequency unaffected but flips its phase.

1. So in fourier transformed space do the peaks of lagged correlations of frequency show up the doppler shift, and of phase show up the delay?

I confess I do not grasp the meaning of this question. Can you rephrase and use mathematical notation? In your comments, you can use LaTeX to format equations and symbols. Start a mathematical object (equation, symbol, etc.) with a dollar sign, type the word ‘latex’, include the latex commands, and end with a dollar sign.

“* latex \alpha *” produces $\alpha$, where * should be replaced by \$

Up to constants and scaling the fourier transform A of a is defined

$A(f)=\int_{-\infty}^\infty a(t)e^{ift}\,dt$

and the Convolution operator in time domain is equivalent to pointwise multiplication in the frequency domain

$w(t) = u(t) \otimes v(t) = u^*(-t) \star v(t) \Leftrightarrow W(f) = U^*(f) V(f)$

If signal $u( t ) = v( t/d + c )$ then $U(f)=e^{(-idcf)}.V(df)$,

a scaling of frequencies.

1. I would write

$A(f) = \int_{-\infty}^\infty a(t) e^{-i2\pi f t} \, dt$

and the convolution theorem for Fourier transforms says

$w(t) = u(t) \otimes v(t) \Leftrightarrow W(f) = U(f) V(f)$

But, yes, I agree that convolution implies that each frequency component in the signal $u(t)$ is scaled by the corresponding frequency component in the signal $v(t)$. In the context of filtering, identify $u(t)$ as the “signal” and $v(t)$ as the “impulse response of the filter” (linear time-invariant transformation).

I think by “additive frequency shift” you mean a simple frequency shift like those implied by the multiplication of a signal by a complex sine wave, $y(t) = x(t) e^{i2\pi f_0 t}$. This kind of simple frequency shift is a good approximation to the full Doppler effect (which affects different frequency components of the signal differently in general) under mild assumptions on the bandwidth, center frequency, and involved speeds.

2. Jonas says: