# Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs
## 5 Discussion
\(h_{1f}[n]\) can be a better approximation of \(q_{0}[n]\) than \(h_{3f}[n]\) when the missing samples contain negligible signal power. This situation resembles an FIR system, where \(h_{1}[n]\) is \(h[n]\) if \(M\) is sufficiently long. Additional operations in obtaining \(h[n]\) add to the noise variance. In practical applications, one should make the transmitter pulse wide (i.e., large \(L\)) to increase the signal-to-noise ratio and sample long enough so that all the desired signals are inside the sampling window. \(h_{3f}[n]\) is most useful when external signals are truncated as in the case of continuous clutter. Another application of the outlined method is when the radar inter-pulse period needs to be short, such as in measuring the ionosphere D-region incoherent scatter spectra [17]. The minimum value of \(N\) is the coprime of \(L\) larger than \(M\). We further note that filtering \(h[n]\) does not work well. To reduce the statistical fluctuation of \(h[n]\), it is necessary to smooth in the time domain or low-pass filter in the frequency domain. These operations will reduce the time resolution, filling the gap at \(n=26\) and \(27\) so that the first two signals cannot be well resolved. Figure 7: Example of frequency filtering in the presence of noise without averaging. The black line (\(q_{0}\)) represents the ideal input signal without noise. The recovered signal without filtering (\(h\)) is shown as black stems. Symbols corresponding to \(h_{1f}\), \(h_{2f}\), and \(h_{3f}\) are filtered results of \(h_{1}[n]\), \(h_{2}[n]\), and \(h_{3}[n]\), respectively. The green line is the received signal (\(y[n]\)) divided by 8. The simplicity of the frequency domain expressions in Lemmas 1 and 2 makes them suitable for finding trigonometry identities. Applying \(L=2\), odd \(N\), \(0\leq i_{L}\leq N_{\#2}=\frac{N-1}{2}\), and \(L_{i}=N-1-i_{L}L\) to Lemma 1, we have  
\[\frac{-j}{N}\sum_{k=0}^{N-1}\frac{e^{2\pi_{N}(i_{L}+n+1)k}}{\cos(\pi_{N}k)}\sin \biggl{(}2\pi_{N}\biggl{(}\frac{1}{2}+i_{L}\biggr{)}k\biggr{)}=(\delta[n,\;2]- \delta[n-1,\;2])(u[n]-u[n-L_{i}]). \tag{60}\]  
The real and imaginary parts of the equation lead to  
\[\frac{1}{N}\sum_{k=0}^{N-1}\frac{sin(2\pi_{N}(i_{L}+n+1)k)}{\cos(\pi_{N}k)}\sin \left(2\pi_{N}\biggl{(}\frac{1}{2}+i_{L}\biggr{)}k\right)\]  
\[=(\delta[n,\;2]-\delta[n-1,\;2])(u[n]-u[n-L_{i}]) \tag{61}\]  
and  
\[\sum_{k=0}^{N-1}\frac{\cos(2\pi_{N}(i_{L}+n+1)k)}{\cos(\pi_{N}k)}\sin\left(2 \pi_{N}\biggl{(}\frac{1}{2}+i_{L}\biggr{)}k\right)=0. \tag{62}\]  
The second equation can be directly proved by applying the symmetric properties for odd N: \(\sin\Bigl{(}2\pi_{N}\biggl{(}\frac{1}{2}+i_{L}\biggr{)}k\Bigr{)}=\sin\Bigl{(}2 \pi_{N}\Bigl{(}\frac{1}{2}+i_{L}\Bigr{)}(N-k)\Bigr{)}\), \(cos(\pi_{N}k)=-cos(\pi_{N}(N-k))\), and \(\cos(2\pi_{N}(i_{L}+n+1)k)=cos(2\pi_{N}(i_{L}+n+1)(N-k))\). A specific case for \(i_{L}=0\) and odd \(N\) leads to  
Figure 7.0  
Lemma 2 can also be used to derive trigonometric sums. These sums, without a trigonometric function in the denominator, can be more readily proved by converting trigonometric functions to their complex exponential representations. Figure 7.1 Example of frequency filtering in the presence of noise with 1000 averages. The ideal input signal without noise is represented by the blue asterisks (\(q_{0}\)). The black squares and red circles are the recovered target signals averaged over 1000 independent realizations without and with filtering, respectively, The black and red lines represent the standard deviations multiplied by 10 for the recovered signals without and with filtering, respectively.