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 Bsplines. Specifically, if a n th n th degree spline with continuous derivatives up to at least order n1n1 is required, then the desired function for a given TsTs belongs to the span of {Bn(t/Tsk)k∈Z}{Bn(t/Tsk)k∈Z} where
B
n
=
B
0
*
B
n

1
B
n
=
B
0
*
B
n

1
(5)for n≥1n≥1 and
B
0
(
t
)
=
1

1
/
2
<
t
<
1
/
2
0
otherwise
.
B
0
(
t
)
=
1

1
/
2
<
t
<
1
/
2
0
otherwise
.
(6)However, the basis splines BnBn do not satisfy the conditions to be a reconstruction filter for n≥2n≥2 as is shown in Figure 2. Still, the BnBn are useful in defining the cardinal basis splines, which do satisfy the conditions to be reconstruction filters. If we let bnbn be the samples of BnBn on the integers, it turns out that bnbn has an inverse bn1bn1 with respect to the operation of convolution for each nn. This is to say that bn1*bn=δbn1*bn=δ. The cardinal basis spline of order nn for reconstruction with sampling period TsTs is defined as
η
n
(
t
)
=
∑
k
=

∞
∞
b
n

1
(
k
)
B
n
(
t
/
T
s

k
)
.
η
n
(
t
)
=
∑
k
=

∞
∞
b
n

1
(
k
)
B
n
(
t
/
T
s

k
)
.
(7)In order to confirm that this satisfies the condition to be a reconstruction filter, note that
η
n
(
m
T
s
)
=
∑
k
=

∞
∞
b
n

1
(
k
)
B
n
(
m

k
)
=
(
b
n

1
*
b
n
)
(
m
)
=
δ
(
m
)
.
η
n
(
m
T
s
)
=
∑
k
=

∞
∞
b
n

1
(
k
)
B
n
(
m

k
)
=
(
b
n

1
*
b
n
)
(
m
)
=
δ
(
m
)
.
(8)Thus, ηnηn is a valid reconstruction filter. Since ηnηn is an n th n th degree spline with continuous derivatives up to order n1n1, the result of the reconstruction will be a n th n th degree spline with continuous derivatives up to order n1n1.
The lowpass filter with impulse response equal to the cardinal basis spline η0η0 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 η1η1 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 th n th order cardinal basis spline ηnηn, the result is a piecewise degree nn polynomial. Furthermore, it has continuous derivatives up to at least order n1n1. However, unless all samples are points on a polynomial of degree at most nn, the derivative of order nn will be discontinuous.
Reconstructions of the discrete time signal given in Figure 4 using several of these filters are shown in Figure 5. As the order of the cardinal basis spline increases, notice that the reconstruction approaches that of the infinite order cardinal spline η∞η∞, 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.
"My introduction to signal processing course at Rice University."