Contents

Introduction

In

the previous section
we considered digital resamplers on the basis of Lagrange piecewise and polynomial interpolation.
The equation for polynomial coefficients was received in the form of:

where

is a vector of polynomial coefficients,

is an inverse system matrix,

is a vector
of delayed input signal samples.

As a result of multiplier minimization equations for polynomial coefficients were received:

The functional chart of the optimized signal digital resampling filter on
the basis of Lagrange piecewise and polynomial interpolation corresponding to (2)
is given in Figure 1.

Figure 1. Functional Chart of the Optimized Digital Resampling Filter

Thus, we received the filter structure which calculates coefficients of
Lagrange polynomial interpolation using only one multiplier by

and two trivial multipliers by

.

Upon considering examples we mentioned that the pulse characteristic

of Farrow filter under signal digital interpolation has no continuous
derivative in interpolation knots.
As a result, the rejection level within the scope of the received filter
blocking is only 28 dB as it is shown in Figure 2.

Figure 2. Pulse Characteristic and AFR of Digital Interpolation Farrow Filter

Besides Lagrange interpolation there are also other methods of piecewise
and polynomial interpolation, for example, spline interpolation [1] which
provides continuous derivatives in interpolation knots in contrast to
Lagrange polynomial interpolation.

In this section we will consider creating Farrow filter on the basis of Hermite splines [1].

Creating Cubic Hermite Spline

Upon creating cubic Lagrange polynomial four signal samples

are used and the received polynomial interpolation

passes through these
knots as it is shown in Figure 3a.

Figure 3. Creating Polynomial Interpolation: a - Cubic Lagrange Polynomial b - Cubic Hermite Spline

However such creating polynomial does not impose restrictions upon values of
derivatives in extreme points

and

.
As a result there is discontinuity of the pulse characteristic derivative
of the interpolator filter as it is shown in Figure 2.

To provide derivative continuity upon “conjugating” cubic polynomial we will
use cubic Hermite spline which is created in the interval

as it is shown in Figure 3b.

We will calculate Hermite spline coefficients by solving the linear equations set.
Two set equations result from Figure 3b:

It is necessary to add two more equations into the set (3).
For this purpose we demand for Hermite spline derivatives

in the knots

and

to be equal to derivatives of the input
signal

in these points, i.e.

Uniting Equations (3) and (4) in the unified equations set to calculate
cubic Hermite spline coefficients, we get:

Or in the matrix form:

Then the system solution (6) can be as follows:

After multiplying the return matrix

by the vector

we get the equations for cubic Hermite spline coefficients in the form:

Thus, we received the equations for cubic Hermite spline coefficients.
At the same time the coefficients depend on values of signal derivatives

and

which we have to value taking
into account input signal samples.

Values of a Discrete Signal Derivative

The numerical differentiation problem of discretely set signals
is solved when using approximation of a signal derivative by means of final differences.
The simplest derivative approximation is the finite difference of the form:

Values of derivatives (9) are realized upon using the input signal linear interpolation.
This method requires only one subtraction, however, it has the greatest inaccuracy as
the residual member in case of derivative value (9) is equal to

[2], i.e.
it decreases linearly with reducing the sampling step.

The more exact derivative value method is the central difference method:

The central difference results from the derivative value with using the input
signal parabolic interpolation, and the residual member in case of the derivative
value (10) is equal to

[2], i.e. it decreases quadratically in case of
reducing the sampling step

.

At the same time the central difference (10) requires additional multiplication
by

which can be realized in integer arithmetics as the single
position shifting to the right.

Thus, we can use values of derivative (10) in Equation (8) to calculate Hermite
spline coefficients and provide derivative continuity at the digital resampling
filter output.

Optimized Structure of Farrow Filter on the Basis of Hermite Splines

Plug (10) into (8) and get equations for Hermite spline coefficients in the form:

Equation (11) allows to note that:

then

and finally it is possible to write down:

Calculation of Hermite spline coefficients requires only multiplication by

and

which can be considered as trivial.
Besides we can get rid of multiplication by

when calculating

if we consider that

is nothing but the delayed single position value

.

The functional chart of the optimized digital resampling filter on the basis
of cubic Hermite splines is shown in Figure 4.

Figure 4. Optimized Digital Resampling Filter on the Basis of Cubic Hermite Splines

Comparing Figures 1 and 4, it is possible to note that the optimized digital resampling
filter on the basis of cubic Hermite splines requires only one trivial multiplication
by

(multiplication by 2 can be replaced with one adder)
while the filter on the basis of Lagrange polynomial interpolation requires
two trivial multiplication by

and one multiplier by

.
The total quantity of adders required to calculate coefficients is also less
when using cubic Hermite splines.

Conclusions

In this section the Farrow filter structure of signal digital resampling on the basis
of cubic Hermite splines is considered. The question of input signal derivative values
to provide derivative continuity when using cubic Hermite splines is also considered.

The received digital resampling filter requires only one multiplication by

which can be considered as trivial. The total quantity of adders required to calculate
polynomial coefficients is also less when using cubic Hermite splines in comparison
with Lagrange polynomial interpolation.

In the following section we will consider examples of using the Farrow filter of signal
digital resampling on the basis of cubic Hermite splines and compare results of this using
with the signal resampling filter on the basis of Lagrange polynomial interpolation.

Reference

[1]
Kahaner D., Moler C., Nash S.
Numerical Methods and Software.
Prentice Hall, 1988.

[2]
Farrow C.W.
A Continuously Variable Digital Delay Element.
Circuits and Systems, IEEE International Symposium. 1988, p. 2641–2645. vol. 3

[3]
Gardner Floyd M.
Interpolation in Digital Modems-Part I: Fundamentals:
IEEE Transactions on Communications, Vol. 41, No. 3, March 1993, P. 501-507.

[4]
Erup L., Gardner Floyd M., Harris Robert A.
Interpolation in Digital Modems-Part II: Implementation and Performance:
IEEE Transactions on Communications, Vol. 41, No. 6, June 1993, p.998-1008.

[5]
Franck A.
Efficient Algorithms for Arbitrary Sample Rate
Conversion with Application to Wave Field
Synthesis.
PhD thesis. Universitätsverlag Ilmenau, Ilmenau, 2012. [PDF]
[6]
McConnell J.
Analysis of Algorithms: An Active Learning Approach.
Jones and Bartlett Publishers, 2001.