10.2 Signal reconstruction  (Page 2/2)

for $n\ge 1$ and

However, the basis splines $B_n$ do not satisfy the conditions to be a reconstruction filter for $n\ge 2$ as is shown in [link] . Still, the $B_n$ are useful in defining the cardinal basis splines, which do satisfy the conditions to be reconstruction filters. If we let $b_n$ be the samples of $B_n$ on the integers, it turns out that $b_n$ has an inverse $b_n^{-1}$ with respect to the operation of convolution for each $n$ . This is to say that $b_n^{-1}*b_n=\delta$ . The cardinal basis spline of order $n$ for reconstruction with sampling period $T_s$ is defined as

In order to confirm that this satisfies the condition to be a reconstruction filter, note that

Thus, ${\eta }_n$ is a valid reconstruction filter. Since ${\eta }_n$ is an $n\mathrm{th}$ degree spline with continuous derivatives up to order $n-1$ , the result of the reconstruction will be a $n\mathrm{th}$ degree spline with continuous derivatives up to order $n-1$ .

The lowpass filter with impulse response equal to the cardinal basis spline ${\eta }_0$ of order 0 is one of the simplest examples of a reconstruction filter. It simply extends the value of the discrete time signal for half the sampling period to each side of every sample, producing a piecewise constant reconstruction. Thus, the result is discontinuous for all nonconstant discrete time signals.

Likewise, the lowpass filter with impulse response equal to the cardinal basis spline ${\eta }_1$ of order 1 is another of the simplest examples of a reconstruction filter. It simply joins the adjacent samples with a straight line, producing a piecewise linear reconstruction. In this way, the reconstruction is continuous for all possible discrete time signals. However, unless the samples are collinear, the result has discontinuous first derivatives.

In general, similar statements can be made for lowpass filters with impulse responses equal to cardinal basis splines of any order. Using the $n\mathrm{th}$ order cardinal basis spline ${\eta }_n$ , the result is a piecewise degree $n$ polynomial. Furthermore, it has continuous derivatives up to at least order $n-1$ . However, unless all samples are points on a polynomial of degree at most $n$ , the derivative of order $n$ will be discontinuous.

Reconstructions of the discrete time signal given in [link] using several of these filters are shown in [link] . As the order of the cardinal basis spline increases, notice that the reconstruction approaches that of the infinite order cardinal spline ${\eta }_{\infty }$ , the sinc function. As will be shown in the subsequent section on perfect reconstruction, the filters with impulse response equal to the sinc function play an especially important role in signal processing.

Reconstruction summary

Reconstruction of a continuous time signal from a discrete time signal can be accomplished through several schemes. However, it is important to note that reconstruction is not the inverse of sampling and only produces one possible continuous time signal that samples to a given discrete time signal. As is covered in the subsequent module, perfect reconstruction of a bandlimited continuous time signal from its sampled version is possible using the Whittaker-Shannon reconstruction formula, which makes use of the ideal lowpass filter and its sinc function impulse response, if the sampling rate is sufficiently high.