Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs

We discuss determining a finite sample linear time-invariant (FS-LTI) system’s impulse response function, h[n], in the frequency domain when the input testing function is a uniform window function with a width of L and the output is limited to a finite number of effective samples, M. Assuming that the samples beyond M are all zeros, the corresponding infinite sample LTI (IS-LTI) system is a marginally stable system. The ratio of the discrete Fourier transforms (DFT) of the output to input of such an FS-LTI system, H0[k], cannot be directly used to find h[n] via inverse DFT (IDFT). Nevertheless, H0[k] contains sufficient information to determine the system’s impulse response function (IRF). In the frequency-domain approach, we zero-pad the output array to a length of N. We present methods to recover h[n] from H0[k] for two scenarios: (1) N≥max(L,M+1) and N is a coprime of L, and (2) N≥L+M+1. The marginal stable system discussed here is an artifact due to the zero-value assumption on unavailable samples. The IRF obtained applies to any LTI system up to the number of effective data samples, M. In demonstrating the equivalence of H0[k] and h[n], we derive two interesting DFT pairs. These DFT pairs can be used to find trigonometric sums that are otherwise difficult to prove. The frequency-domain approach makes mitigating the effects of interferences and random noise easier. In an example application in radar remote sensing, we show that the frequency-domain processing method can be used to obtain finer details than the range resolution provided by the radar system’s transmitter.