I find myself confused how two properties of the ambiguity function relate to one another. The properties in question are (presented here from wikipedia, but similar formulas can be found in books by Levanon, Mahafza, etc...)
The maximum value of the ambiguity function is in point $(0, 0)$ $$ \tag{1} |\chi(\tau, f)|^2 \le |\chi(0, 0)|^2 $$
The volume under the ambiguity function is invariant $$ \tag{2} \intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}|\chi(\tau, f)|^2 d\tau df = |\chi(0, 0)|^2 = E^2 $$
From these 2 properties doesn't it naturally follow that all the volume of the ambiguity function is completely concentrated at point $(0, 0)$? And the only valid form for the ambiguity function to satisfy $(2)$ would be $$ \tag{3} |\chi(\tau, f)|^2 = E^2 \cdot \delta(f) \cdot \delta(\tau) $$ Obviously, this can't be the case. But I can't seem to grasp how $(2)$ could be satisfied otherwise.
UPDATE:
My logic for this reasoning is this following. If someone could walk me through where it should fall apart, that would be helpful. From my perspective:
- The first property states that the ambiguity function at point $|\chi(0, 0)|^2$ will equal $E^2$ (or 1, if you normalize it), and at all other points it will be smaller.
- The second property states that its entire volume, integrated over $f$ and $\tau$ will equal $E^2$ (or 1, if you normalize it).
- Since $|\chi(\tau, f)|^2$ is non-negative, if at any point other than $(0, 0)$ the ambiguity function is non-zero, its volume will be greater than $E^2$, which contradicts the second property.
- Hence, the only possible form for the ambiguity function from these two properties is a delta-function.
