#### Ambiguity function

The ambiguity function (AF) is a two-dimensional function
χ(τ, f_{D} ) with the signal delay time τ
in the filter and the Doppler frequency f_{D} of the echo signal.
It is an analytical tool for investigating the effects of target velocities on the output from a
matched filter (MF).
It is defined only by the characteristics of the echo pulse and the optimal filter and not by a specific target scenario.

The ambiguity function represents the response of the optimal filter both to the transmitted signal s(t), i.e., the known waveform for which the optimal filter was dimensioned, and to a received signal r(t) whose waveform was shifted by a Doppler frequency. The ambiguity function allows a statement about the suitability of a certain transmit waveform for the intended radar application..

The Optimum Filter is used to optimally determine the presence (detection) of the amplitude or location of a known waveform. It provides the best possible signal-to-noise ratio for the echo signals without a frequency shift due to the Doppler effect. The impulse response of an optimal filter is:

(1)

- s
^{*}= conjugate complex value of the transmitted signal s; - T = time at which the entire energy of the signal has entered the filter;
- t = current time

The response χ(τ) of an optimal filter to the non-doubler-shifted received signal r(t) (being as a copy of the transmitted signal in the baseband) at time t = τ is thus:

(2)

Here it was assumed in a simplified form that T = 0. If the received signal is now subject to a Doppler shift, then it has the form:

(3)

This received signal inserted into equation 2 gives the definition of the ambiguity function:

(4)

For f_{D} = 0,
the expression e^{0} = 1
and thus the ambiguity function corresponds exactly to the autocorrelation function of the signal s(t).

The optimal filter is no longer optimally tuned for this echo signal with a Doppler frequency unequal to zero. The output of the filter is now influenced by both the signal propagation time and the Doppler shift, it is distorted into a sinc-function. The Doppler shift causes a deterioration of the signal-to-noise ratio in the optimal filter. For this reason, non-optimal filters (mismatched filters, MMF) are also used in radar. The ambiguity function can be used to answer the question of which filter response is required.

Figure 1: Ambiguity function for a rectangular pulse with 2 s length (simulation with MatLab)

Figure 1: Ambiguity function for a rectangular pulse with 2 s length (simulation with MatLab)

##### Ambiguity function for a rectangular pulse

The transmit pulse shape of a classical pulse radar, i.e. a radar with a keyed on/off modulator is described by:

(5)

- T
_{PW}= pulse width of the transmitted pulse - t = actual time during the pass through the filter.

The echo signal with a Doppler frequency is therefore:

(6)

The ambiguity function for the square wave pulse is then:

(7)

Figure 2: Contours of the ambiguity function of an unmodulated rectangular pulse (simulation with MatLab)

Figure 2: Contours of the ambiguity function of an unmodulated rectangular pulse (simulation with MatLab)

Figure 3: Ambiguity-Function, cut on the line τ = 0

No numerical data can be taken from a three-dimensional representation as in Fig. 1.
Therefore, two-dimensional contour representations (Figure 2) or horizontal or vertical sections are also used.
A cut on the line f_{D} = 0 would result
in a triangular pulse waveform as filter response of the optimal filter.
A cut on the line τ = 0 is shown in Figure 3.

##### Basic properties of the ambiguity function

- The function has the maximum value at the origin (at τ = 0
and f
_{D}= 0) and has a smaller magnitude elsewhere.

- The volume (see Figure 1) under the surface of the ambiguity function and the
τ - f
_{D}plane is constant, and equal to one.

- The function is symmetrical with respect to the lines
τ = 0 and
f
_{D}= 0.

(Which is why used only plots with half of the respective values sometimes, a so called “partial ambiguity function” with |τ| < T_{PW})