Content deleted Content added
Line 140:
(2) If <math>x(t) = exp[-\alpha_{1}\pi (t-t_{1})^{2}+j2\pi f_{1}t] + exp[-\alpha_{2}\pi (t-t_{2})^{2}+j2\pi f_{2}t]</math>
:<math>A_{x}(\tau,n)</math>
:<math>= \int^\infty_{-\infty}x_{1}(t+\tau/2)x_{1}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +<br>
:<math>\int^\infty_{-\infty}x_{2}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> + <br>
:<math>\int^\infty_{-\infty}x_{1}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +<br>
:<math> \int^\infty_{-\infty}x_{2}(t+\tau/2)x_{1}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math><br>
<br>
:<math>A_{x}(\tau,n) = A_{x1}(\tau,n) + A_{x2}(\tau,n) + A_{x1x2}(\tau,n) + A_{x2x1}(\tau,n)</math><br>
<br>
:<math>A_{x}(\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)]</math>
:<math>A_{x}(\tau,n) = \sqrt\frac{1}{2\alpha_{2}}exp[-\pi (\frac{\alpha_{2}\tau^{2}}{2}+\frac{n^{2}}{2\alpha_{1}})]exp[j2\pi (f_{2}\tau-t_{2}n)]</math><br>
<br>
When <math>\alpha_{1} = \alpha_{2}</math>
:<math>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})]</math><br>
where<br>
*t_{u} = (t_{1}+t_{2}/2), <br>
*f_{u} = (f_{1}+f_{2})/2, <br>
*\alpha_{u} = (\alpha_{1}+\alpha_{2})/2,<br>
*t_{d} = t_{1}+t_{2}, <br>
*f_{d} = f_{1}-f_{2}, <br>
*\alph_{d} = \alpha_{1}-\alpha_{2}</math><br>
*:<math>A_{x2x1}(\tau,n) = A_{x1x2}^{*}(-\tau,-n)</math>
When <math>\alpha_{1} not equal \alpha_{2}</math>
:<math>A_{x1x2}(\tau,n) = \sqrt\frac{1}{2\alpha_{u}}exp[-\pi \frac{[(n-f_{d})+j(\alpha_{1}t_{1}+\alpha_{2}t_{2})-j\alpha_{d}\tau /2]^{2}}{2\alpha_{u}}exp[-\pi(\alpha_{1}(t_{1}-\frac{\tau}{2})^{2})+\alpha_{2}(t_{2}-\frac{\tau}{2})^{2})]exp[j2\pi
f_{u}\tau]</math><br>
[[File:WDF AF 2.png|thumb|WDF and AF for the signal with 2 terms]]
*:<math>A_{x2x1}(\tau,n) = A_{x1x2}^{*}(-\tau,-n)</math><br>
WDF and AF for the signal with 2 terms<br>
<br>
For the ambiguity function:<br>
*The auto term is always near to the origin
*The auto term is always near to the origin
== See also ==
| |||