# Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs
## 5 Discussion
In the case of Theorem 2, the poles overlap with zeros. No modifications are needed to the process of finding \(g_{4}[n]\) or \(h_{3}[n]\) in numerical implementation. Our primary motivation in this study is to bridge the gap of equivalence between the time-domain and frequency-domain methods in characterizing an FS-LTI(\(L,\ M,\ N\)) system. Possible applications of the frequency-domain method discussed here lie in mitigating the effects of interferences and random noise and improving the computing speed. These are the same reasons for choosing the frequency-domain over the time-domain approach in many signal processing applications. One example application involves using incoherent scatter radars to detect meteor head echoes [16]. In radar remote sensing, the voltage samples scattered off targets as a function of time (or range) in response to an impulse is the IRF. A typical incoherent scatter radar transmitter has a bandwidth of 1 MHz, which limits the pulse width to 1 \(\upmu\)s or longer, corresponding to a 150 m best range resolution. Meteor echoes have much finer structures than 150 m. In such a scenario, one can over-sample the return echoes at a finer resolution (say 15 m) to study the details of the meteoric ablation. In Figure 7, we simulate a situation when a radar's range samples contain multiple targets from a uniform transmitter pulse with a length of 20 units. The IRF without noise, i.e., the ideal signal describing the targets, is assumed to be  
\[q_{0}[n]=0.5(u[n-22]-u[n-26])+2\times 0.95^{n-28}u[n-28]+0.3u[n-90] \tag{58}\]  
The three terms represent three targets and their values are shown in the black line in Figure 7. The return samples, \(y[n]\), are the convolution of \(q_{0}[n]\) with \(x_{L}[n]\) (\(L=20\)) superposed with a random noise component and an undesirable sinusoid interference via the following equation:  
\[y[n]=q_{0}[n]*x_{w}[n]+0.15\cos[\frac{2\pi n}{2.23}]+0.05r_{n}[n] \tag{59}\]  
where \(r_{n}[n]\) is the standard normal random variable. A realization of \(y[n]\) is shown in Figure 7 as the green line. The first two targets cannot be resolved because the width of the radar pulse exceeds the gap between them. In Figure 7, the black stems are the results of \(h[n]\) without filtering. \(h[n]\) differs from \(q_{0}[n]\) because of the additive noise and interference. To reduce noise, we low-pass filter \(H_{1}[k]\) at a normalized cut-off frequency of 0.4375 and perform an IDFT to generate \(h_{1f}[n]\), which is the filtered form of \(h_{1}[n]\). Following the shift equations in Theorem 1, we obtain the filtered form of \(h_{2}[n]\) and \(h_{3}[n]\) from \(h_{1f}[n]\), which are denoted as \(h_{2f}[n]\) and \(h_{3f}[n]\) and shown as blue \"x\" and red \"o\", respectively, in Figure 7. \(h_{3f}[n]\) matches with \(q_{0}[n]\) better than \(h[n]\). \(h_{3f}[n]\) resolves the two targets (with a gap at \(n=26\), 27) while large error in \(h[n]\) diminishes its ability to do so. To see the statistical characteristics, the average and standard deviations of 1000 simulations are shown in Figure 8. The mean of the unfiltered (h avg) and filtered (h\({}_{3f}\) avg) results are represented by black squares and red circles, respectively. We see that the filtered results follow the ideal signal (\(q_{0}\)) closely while the unfiltered results deviate from the ideal signal significantly. The unfiltered and filtered standard deviations of \(h_{1}[n]\) and \(h_{3f}[n]\) multiplied by 10 are shown as a black and red line, respectively. The standard deviations for \(h[n]\) are about 40% larger than that for \(h_{3f}[n]\). The standard deviation of both \(h[n]\) and \(h_{3f}[n]\) increases with \(n\) piece-wise with a step width of 20 (L). This is because \(h[n]\) is an iterative sum of \(y[n]\) as seen from (1). The deviation of the mean \(h[n]\) from \(q_{0}[n]\) in Figure 8 is due to the sinusoidal interference item, instead of random noise. One can increase the width of the transmitter pulse (i.e., \(L\)) to decrease the random fluctuation.