In this paper we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth B, and the received signal is observed over a finite time interval of length T only. This gives rise to a delay and Doppler resolution of 1/B and 1/T. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay-Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay-Doppler pairs is at least 2.37/B in time or 2.37/T in frequency. Furthermore, this result allows the total number of delay-Doppler pairs to be linear up to a log-factor in BT, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time-frequency components of a signal that is S-sparse in the continuous dictionary of time-frequency shifts of a random window function, from a number of measurements, that is linear up to a log-factor in S.
翻译:在本文中,我们研究从对已知输入信号的反应中找出一个有时间变化的线性系统。 更具体地说, 我们考虑那些对输入信号的反应是通过延迟和多普勒转换版本的加权叠加而得到的系统。 这个问题出现在无线通信和雷达成像等多种应用中。 由于实际限制, 输入信号的带宽B 有限, 接收的信号只在T 的有限时间间隔内观察到。 这导致延迟- 多普勒配对的延迟和多普勒解析 1/B 和 1/T 的延迟和多普勒解析。 我们表明, 能够克服这一分辨率限制, 也就是说, 我们完全可以恢复持续延迟- 多普勒配对和相应的减速因子, 解决一个螺旋优化问题。 这个结果显示, 延迟- 多普勒配对之间的距离至少是2. 37/B 或 2. 37/ T 频率。 此外, 这个结果使得延迟- 多普勒配对的总数可以直线性到 BT 的对数, 系统的反应的尺寸, 也就是系统的尺寸和相应的减速度调数度度度度度值的频率的值值, 状态的精确度函数是不同的, 我们可以显示, 方向的频率的逻辑的值的值的值的值的值的值值的值值的值的值值值值的值的值的值值值值值值的值的值的值值。 。