## 2 Fundamental Concepts  
With present day digital computer systems, and in areas such as signal processing and control, the conversion of an analog signal into a digital version requires sampling and quantisation. This action is called the A/D conversion (analog to digital conversion) and its inverse is the D/A conversion. In general, there are infinite continuous signals that can generate a given discrete signal. However, when the signals are bandlimited, meaning that they are of bounded support in the frequency domain, we can recover the original signal. **Definition 1**. : _The process of associating a discrete signal to a continuous one such that_  
1. _the domain of the discrete signal is a discrete subset of the domain of the continuous signal,_
2. _the range of the discrete signal is also a discrete subset of the range of the continuous signal,_
3. _both signals, discrete and continuous, assume the same values at the intersection domains,_  
_is named sampling [6]._  
The inverse operation, that is, the calculation of intermediate continuous values based on the discrete samples, is called interpolation or reconstruction. Ideally, the sampling operation is performed by the delta comb that we introduce in the follow-up. **Definition 2**. : _Let \(F\) and \(T=\frac{1}{F}\) be two positive real constants. We define an almost linear time scale (ALTS) by [24]_  
\[t_{n}=n\cdot T+\tau_{n},\quad T=1/F,\quad n\in\mathbb{Z}, \tag{1}\]  
_where \(\tau_{n}\) a zero mean sequence such that_  
\[0\leq|\tau_{n}|<\frac{T}{4}. \tag{2}\]  
_The constant \(F\) is called sampling frequency, or density of the time scale, while its inverse \(T\) is the average graininess or sampling interval [25]._  
**Remark 1**. : _In some applications, we prefer a half sampling interval delayed sequence \(t_{n}=(n+1/2)\cdot T+\tau_{n}\)._  
**Definition 3**. : _On the above ALTS (1) we define a uniform (in amplitude) almost periodic delta comb by_  
\[p(t)=\sum_{n=-\infty}^{+\infty}\delta(t-t_{n}),\qquad t\in\mathbb{R}, \tag{3}\]  
_that is considered as the ideal sampler, since its multiplication by any signal, \(y(t)\), gives its sampled version_  
\[y_{p}(t)=y(t)p(t)=\sum_{n=-\infty}^{+\infty}y(t_{n})\delta(t-t_{n}),\qquad t\in \mathbb{R}. \tag{4}\]  
_This distribution has the Fourier transform (FT)_  
\[Y_{p}(\omega)=\sum_{n=-\infty}^{+\infty}y(t_{n})e^{-j\omega t_{n}},\qquad \omega\in\mathbb{R}, \tag{5}\]  
_where \(j=\sqrt{-1}\) and_  
\[Y_{p}(\omega)=\int_{-\infty}^{\infty}y_{p}(t)e^{-j\omega t}\mathrm{d}t.\]  
**Theorem 1**. : _Let \(\Omega\in\mathbb{R}^{+}\) be the density of an odd ALTS, that is, so that \(\omega_{n}=-\omega_{-n}\), given by \(\omega_{n}=n\Omega+\
u_{n}\), with \(|\
u_{n}|<\Omega/4\), \(n\in\mathbb{Z}\). The FT of the comb (3) in the time domain is another comb [24] in the frequency domain given by:_  
\[C(\omega)=2\pi\sum_{n=-\infty}^{+\infty}C_{n}\delta\left(\omega-\omega_{n} \right), \tag{6}\]  
_where the coefficients,_  
\[C_{k}=\frac{1}{T}\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\mathrm{e}^{- \mathrm{j}\omega_{k}t_{n}},\]  
_form a bounded sequence and_  
\[\frac{\Omega T}{2\pi}=1. \tag{7}\]  
This has as consequence that  
\[Y_{p}(\omega)=\sum_{n=-\infty}^{+\infty}C_{n}Y(\omega-\omega_{n}), \tag{8}\]  
showing that \(Y_{p}(\omega)\) is an almost periodic function that results from the almost periodic repetition of \(Y(\omega)\). Therefore, if \(y(t)\) is not bandlimited, or if \(\Omega\) is not high enough, then two consecutive repeated images of \(Y(\omega)\) may overlap (phenomenon called aliasing) and it is not possible to recover the original signal by low-pass filtering. Unless something else is specified, we will assume that our signals are bandlimited in the follow-up. If the bandwidth is \(W\), then we conclude that \(2W<\Omega-\Omega/4<\omega_{1}\). Otherwise, if we allow a spectral superposition, \(2W>\omega_{1}\), then we cannot recover the original signal by low-pass filtering.