# Frequency-Domain Characterization of Finite Sample Linear Systems with Uniform Window Inputs
## 1 Introduction  
A continuous linear time-invariant (LTI) system can be characterized by its frequency transfer function (FTF) or impulse response function (IRF) (e.g., ref. [1, 2]). In the time domain, an LTI system's output is the convolution of the input and the IRF. In the frequency domain, the output is the product of the input and the FTF. The corresponding expressions in the time domain and frequency domain are Fourier transform pairs. For theoretical convenience, one can determine a continuous system's characteristics using the Dirac delta function with an infinitely narrow and infinitely high-power pulse. In practice, the test pulse must have a finite width and amplitude. The input and output of continuous time-invariant systems are sampled to produce the sequences of the input, \(x[n]\), and the output, \(y[n]\), of discrete-time LTI systems. The foundational theory of discrete-time LTI systems is based on an infinite number of samples (e.g., [2]). However, realistic systems allow only for a finite number of samples. The finite-sample LTI (FS-LTI) system complicates the applications of LTI theories, especially in the frequency domain. The purpose of this study is to bridge the gap between time-domain and frequency-domain operations for FS-LTI systems. A system with finite samples cannot be a time-invariant system. To preserve the characteristics of an LTI system, explicit or implicit assumptions about the remaining data samples need to be made. Typical assumptions are that the unavailable samples have zero values or are periodic repetitions of the available samples. For an FS-LTI system, one can use a unit sample function, \(\delta[n]\), to determine the finite sample IRF, \(h[n]\), up to the length of the number of available samples, \(M\). The IRF of an FS-LTI system can only match the IRF of the corresponding infinite-sample LTI (IS-LTI) system for \(M\) samples. For a continuous LTI system, we can obtain the FT via the ratio of the Fourier transform of the output and input functions. For an FS-LTI discrete system with \(M\) samples, we can also obtain \(H_{r}[k]=Y[k]/X[k]\), where \(X[k]\) and \(Y[k]\), with a length of \(N\geq M\), are the discrete Fourier transforms (DFTs) of \(x[n]\) and \(y[n]\), respectively. In this study, \(n\) and \(k\) are used for time and frequency domain indices, respectively. When data samples are unavailable, zero values will be assumed to ensure proper operations or theoretical completeness. \(H_{r}[k]\) and \(h[n]\) are interchangeable in describing the FS-LTI system for finite impulse response (FIR) systems if \(M\) is sufficiently long. In general, however, the inverse DFT (IDFT) of \(H_{r}[k]\) does not directly recover any sample of \(h[n]\) for infinite impulse response (IIR) systems because the output is the sum of circular convolution in frequency-domain and time-domain exchanges. When \(x[n]\) is a finite uniform window sequence, we use \(H_{0}[k]\) to denote \(Y[k]/X[k]\). In such a case, \(H_{0}[k]\) may contain singularities. A fundamental question is whether \(H_{0}[k]\) or \(H_{r}[k]\) contains sufficient information to derive \(h[n]\) for an FS-LTI system. In this study, we are primarily concerned with deriving \(h[n]\) from \(H_{0}[k]\) for FS-LTI systems. When the input testing function is a uniform window function having a width of \(L\) and the output is or is assumed to be zeros after \(M\) points, the LTI system is marginally stable since the zeros of the z-domain transfer function are all on the unit circle. The IRF of such marginally stable systems is a periodic function after \(M-L+1\) points [3]. For a marginally stable system, as shown in [3], the non-redundant or effective part of \(h[n]\) can indeed be obtained from \(H_{0}[k]\) if the zero-padded length, \(N\), of \(x[n]\) and \(y[n]\) is chosen to be a sufficiently large multiple of \(L\). The current work extends that of [3]. We show here that \(N\) does not need to be a multiple of \(L\) in order to use \(H_{0}[k]\) to describe the FS-LTI system.