LFM信号模型-01

LFM信号由于其具有较大的带宽时宽积，广泛应用于现代雷达系统中。频率线性地向上或向下扫过脉冲宽带，以向上线性调频为例，LFM 信号可以表示为

\[x\left(t\right)= exp(j2\pi(f_0t+\frac{\mu}{2}t^2))\text{Rect}\left(\frac{t}{\tau_0}\right) \tag{1} \]

其中，\(\text{Rect}\left(\frac{t}{\tau_0}\right)\)表示宽带为\(\tau_0\)的矩形脉冲，该式为线性调频信号的解析信号。其频谱主要由\(exp(j\pi\mu t^2)\text{Rect}\left(\frac{t}{\tau_0}\right)\)决定,而\(exp(j2\pi f_0t)\)相当于对频谱引入了关于中心频率为\(f_0\)的频移。

​考虑基带LFM信号，调频信号的频率范围从\(f_0\to f_0+\mu\tau_0\),故\(f_0\)可设置为\(-\frac{\mu\tau_0}{2}\).设LFM信号的带宽为\(B\),求得下列信号的傅里叶变换，即

\[\begin{aligned} \hat{X}(f)&=\int_{-\infty}^{+\infty}exp(j\pi\mu t^2)\text{Rect}\left(\frac{t}{\tau_0}\right)e^{-j2\pi ft}dt\\ &=\int_{-\frac{\tau_0}{2}}^{\frac{\tau_0}{2}}e^{j\pi\mu t^2}e^{-j2\pi ft}df \end{aligned} \tag{2} \]

考虑菲涅耳积分的形式，进行变量代换，设\(\mu'=\pi\mu\),令\(z=\sqrt{\frac{2}{\pi}}\left(\sqrt{\mu'}t-\frac{\pi f}{\sqrt{\mu'}}\right)\),并且有\(dt=\sqrt{\frac{\pi}{2\mu'}}dz\),式(2)可以改写为

\[\begin{aligned} \hat{X}(f)&=\sqrt{\frac{\pi}{2\mu'}}e^{-j(\pi f)^2/\mu'}\int_{-z_1}^{z_2}e^{j\pi z^2/2}dz\\ &=\sqrt{\frac{\pi}{2\mu'}}e^{-j(\pi f)^2/\mu'}\left\{\int_{0}^{z_1}e^{j\pi z^2/2}+\int_0^{z_2}e^{j\pi z^2/2}\right\}\\ z_1 &=-\sqrt{\frac{2\mu'}{\pi}}\left(\frac{\tau_0}{2}+\frac{\pi f}{\mu'}\right)=\sqrt{\frac{B\tau_0}{2}}\left(1+\frac{f}{B/2}\right)\\ z_2 &=\sqrt{\frac{2\mu'}{\pi}}\left(\frac{\tau_0}{2}-\frac{f}{\mu}\right)=\sqrt{\frac{B\tau_0}{2}}\left(1-\frac{f}{B/2}\right)\\ \end{aligned} \tag{3} \]

利用\(C(z),S(z)\)表示菲涅尔(Fresnel)积分，定义为

\[C(z)=\int_0^zcos(\frac{\pi v^2}{2})dv\\ S(z)=\int_0^zsin(\frac{\pi v^2}{2})dv \tag{4} \]

有\(C(-z)=-C(z),S(-z)=-S(z)\),其中，若\(z >> 1\),菲涅尔积分可近似表示为

\[C(z)\approx\frac{1}{2}+\frac{1}{\pi z}sin(\frac{\pi}{2}z^2)\\ S(z)\approx\frac{1}{2}-\frac{1}{\pi z}cos(\frac{\pi}{2}z^2) \tag{5} \]

进而可得到\(\hat{X}(f)\)的频谱为

\[\hat{X}(f)=\sqrt{\frac{\pi}{2\mu'}}e^{-j(\pi f)^2/(\mu')}\left\{[C(z_1)+C(z_2)]+j[S(z_1)+S(z_2)]\right\}\\ =\sqrt{\frac{1}{2\mu}}e^{-j(\pi f)^2/(\mu')}\left\{[C(z_1)+C(z_2)]+j[S(z_1)+S(z_2)]\right\} \tag{6} \]

令时宽带宽积为\(Q=B\tau_0\),若该值较大，从式(6)中可以推导，当频段处于调频信号的频段，即\(-B/2\sim B/2\),此时\(z_1\)和\(z_2\)近似相等，且此时\(C(z_1),C(z_2),S(z_1),S(z_2)\)均近似等于1/2, LFM信号的频谱幅值近似等于 \(\sqrt{\frac{1}{\mu}}\),当频段远大于B/2或远小于-B/2时，\(z_1，z_2\)可近似认为互为相反数，于是其频谱幅值近似为0.后续仿真可验证此结论。