10.2 Signal reconstruction

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This module describes the reconstruction, also known as interpolation, of a continuous time signal from a discrete time signal, including a discussion of cardinal spline filters.

Introduction

The sampling process produces a discrete time signal from a continuous time signal by examining the value of the continuous time signal at equally spaced points in time. Reconstruction, also known as interpolation, attempts to perform an opposite process that produces a continuous time signal coinciding with the points of the discrete time signal. Because the sampling process for general sets of signals is not invertible, there are numerous possible reconstructions from a given discrete time signal, each of which would sample to that signal at the appropriate sampling rate. This module will introduce some of these reconstruction schemes.

Reconstruction process

The process of reconstruction, also commonly known as interpolation, produces a continuous time signal that would sample to a given discrete time signal at a specific sampling rate. Reconstruction can be mathematically understood by first generating a continuous time impulse train

${x}_{imp}\left(t\right)=\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)\delta \left(t-n{T}_{s}\right)$

from the sampled signal ${x}_{s}$ with sampling period ${T}_{s}$ and then applying a lowpass filter $G$ that satisfies certain conditions to produce an output signal $\stackrel{˜}{x}$ . If $G$ has impulse response $g$ , then the result of the reconstruction process, illustrated in [link] , is given by the following computation, the final equation of which is used to perform reconstruction in practice.

$\begin{array}{cc}\hfill \stackrel{˜}{x}\left(t\right)& =\left({x}_{imp}*g\right)\left(t\right)\hfill \\ & ={\int }_{-\infty }^{\infty }{x}_{imp}\left(\tau \right)g\left(t-\tau \right)d\tau \hfill \\ & ={\int }_{-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)\delta \left(\tau -n{T}_{s}\right)g\left(t-\tau \right)d\tau \hfill \\ & =\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right){\int }_{-\infty }^{\infty }\delta \left(\tau -n{T}_{s}\right)g\left(t-\tau \right)d\tau \hfill \\ & =\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)g\left(t-n{T}_{s}\right)\hfill \end{array}$

Reconstruction filters

In order to guarantee that the reconstructed signal $\stackrel{˜}{x}$ samples to the discrete time signal ${x}_{s}$ from which it was reconstructed using the sampling period ${T}_{s}$ , the lowpass filter $G$ must satisfy certain conditions. These can be expressed well in the time domain in terms of a condition on the impulse response $g$ of the lowpass filter $G$ . The sufficient condition to be a reconstruction filters that we will require is that, for all $n\in \mathbb{Z}$ ,

$g\left(n{T}_{s}\right)=\left\{\begin{array}{cc}1& n=0\\ 0& n\ne 0\end{array}\right)=\delta \left(n\right).$

This means that $g$ sampled at a rate ${T}_{s}$ produces a discrete time unit impulse signal. Therefore, it follows that sampling $\stackrel{˜}{x}$ with sampling period ${T}_{s}$ results in

$\begin{array}{cc}\hfill \stackrel{˜}{x}\left(n{T}_{s}\right)& =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)g\left(n{T}_{s}-m{T}_{s}\right)\hfill \\ & =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)g\left(\left(n-m\right){T}_{s}\right)\hfill \\ & =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)\delta \left(n-m\right)\hfill \\ & ={x}_{s}\left(n\right),\hfill \end{array}$

which is the desired result for reconstruction filters.

Cardinal basis splines

Since there are many continuous time signals that sample to a given discrete time signal, additional constraints are required in order to identify a particular one of these. For instance, we might require our reconstruction to yield a spline of a certain degree, which is a signal described in piecewise parts by polynomials not exceeding that degree. Additionally, we might want to guarantee that the function and a certain number of its derivatives are continuous.

This may be accomplished by restricting the result to the span of sets of certain splines, called basis splines or B-splines. Specifically, if a $n\mathrm{th}$ degree spline with continuous derivatives up to at least order $n-1$ is required, then the desired function for a given ${T}_{s}$ belongs to the span of $\left\{{B}_{n}\left(t/{T}_{s}-k\right)|k\in \mathbb{Z}\right\}$ where

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