## Windowed Sinc Interpolation

Bandlimited interpolation of discrete-time signals is a basic tool
having extensive application in digital signal
processing.^{5.8}In general, the problem is to correctly compute signal values at
arbitrary continuous times from a set of discrete-time samples of the
signal amplitude. In other words, we must be able to interpolate the
signal between samples. Since the original signal is always assumed
to be *bandlimited* to half the sampling rate, (otherwise
aliasing distortion would occur upon sampling), Shannon's sampling
theorem tells us the signal can be exactly and uniquely reconstructed
for all time from its samples by bandlimited interpolation.

Considerable research has been devoted to the problem of interpolating
discrete points. A comprehensive survey of ``fractional delay filter
design'' is provided in [267]. A comparison between
classical (*e.g.*, Lagrange) and bandlimited interpolation is given in
[407]. The book **Multirate Digital Signal
Processing **[97] provides a comprehensive summary and
review of classical signal processing techniques for sampling-rate
conversion. In these techniques, the signal is first interpolated by
an integer factor and then decimated by an integer factor
. This provides sampling-rate conversion by any rational factor
. The conversion requires a digital lowpass filter whose cutoff
frequency depends on
. While sufficiently general, this
formulation is less convenient when it is desired to resample the
signal at arbitrary times or change the sampling-rate conversion
factor smoothly over time.

In this section, a public-domain resampling algorithm is described which will evaluate a signal at any time specifiable by a fixed-point number. In addition, one lowpass filter is used regardless of the sampling-rate conversion factor. The algorithm effectively implements the ``analog interpretation'' of rate conversion, as discussed in [97], in which a certain lowpass-filter impulse response must be available as a continuous function. Continuity of the impulse response is simulated by linearly interpolating between samples of the impulse response stored in a table. Due to the relatively low cost of memory, the method is quite practical for hardware implementation.

In the next section, the basic theory is presented, followed by sections on implementation details and practical considerations.

###

Theory of Ideal Bandlimited Interpolation

We review briefly the ``analog interpretation'' of sampling rate conversion
[97] on which the present method is based. Suppose we have
samples of a continuous absolutely integrable signal ,
where is time in seconds (real), ranges over the integers, and
is the sampling period. We assume is bandlimited to
, where is the sampling rate. If
denotes the Fourier transform of , *i.e.*,
, then we assume
for
. Consequently, Shannon's sampling
theorem gives us that can be uniquely reconstructed from the
samples via

where

To resample at a new sampling rate , we need only evaluate Eq.(4.13) at integer multiples of .When the new sampling rate is less than the original rate , the lowpass cutoff must be placed below half the new lower sampling rate. Thus, in the case of an ideal lowpass, sinc, where the scale factor maintains unity gain in the passband.

A plot of the sinc function
sinc
to the left and right of the origin is shown in Fig.4.21.
Note that peak is at amplitude , and zero-crossings occur at all
nonzero integers. The sinc function can be seen as a hyperbolically
weighted sine function with its zero at the origin canceled out. The
name *sinc function* derives from its classical name as the
*sine cardinal* (or *cardinal sine*) function.

If ``'' denotes the convolution operation for digital signals, then the summation in Eq.(4.13) can be written as .

Equation Eq.(4.13) can be interpreted as a superpositon of
shifted and scaled sinc functions . A sinc function instance is
translated to each signal sample and scaled by that sample, and the
instances are all added together. Note that zero-crossings of
sinc occur at all integers except . That means at time
, (*i.e.*, on a sample instant), the only contribution to the
sum is the single sample . All other samples contribute sinc
functions which have a zero-crossing at time . Thus, the
interpolation goes precisely through the existing samples, as it
should.

A plot indicating how sinc functions sum together to reconstruct bandlimited signals is shown in Fig.4.22. The figure shows a superposition of five sinc functions, each at unit amplitude, and displaced by one-sample intervals. These sinc functions would be used to reconstruct the bandlimited interpolation of the discrete-time signal . Note that at each sampling instant , the solid line passes exactly through the tip of the sinc function for that sample; this is just a restatement of the fact that the interpolation passes through the existing samples. Since the nonzero samples of the digital signal are all , we might expect the interpolated signal to be very close to over the nonzero interval; however, this is far from being the case. The deviation from unity between samples can be thought of as ``overshoot'' or ``ringing'' of the lowpass filter which cuts off at half the sampling rate, or it can be considered a ``Gibbs phenomenon'' associated with bandlimiting.

A second interpretation of Eq.(4.13) is as follows: to obtain the
interpolation at time , shift the signal samples under *one* sinc
function so that time in the signal is translated under the peak of the
sinc function, then create the output as a linear combination of signal
samples where the coefficient of each signal sample is given by the value
of the sinc function at the location of each sample. That this
interpretation is equivalent to the first can be seen as a result of the
fact that convolution is commutative; in the first interpretation, all
signal samples are used to form a linear combination of shifted sinc
functions, while in the second interpretation, samples from one sinc
function are used to form a linear combination of samples of the shifted
input signal. The practical bandlimited interpolation algorithm presented
below is based on the second interpretation.

### From Theory to Practice

The summation in Eq.(4.13) cannot be implemented in practice because
the ``ideal lowpass filter'' impulse response actually extends
from minus infinity to infinity. It is necessary in practice to *window* the ideal impulse response so as to make it finite. This is the basis
of the *window method* for digital filter design
[115,362]. While many other filter design techniques
exist, the window method is simple and robust, especially for very
long impulse responses. In the case of the algorithm presented below,
the filter impulse response is very long because it is heavily
oversampled. Another approach is to design optimal decimated
``sub-phases'' of the filter impulse response, which are then
interpolated to provide the ``continuous'' impulse response needed for
the algorithm [358].

Figure 4.23 shows the frequency response of the ideal lowpass filter. This is just the Fourier transform of .

If we truncate at the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig.4.24. Note that the stopband exhibits only slightly more than 20 dB rejection.

If we instead use the Kaiser window [221,438] to taper to zero by the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig.4.25. Note that now the stopband starts out close to dB. The Kaiser window has a single parameter which can be used to modify the stop-band attenuation, trading it against the transition width from pass-band to stop-band.

### Implementation

The implementation below provides signal evaluation at an arbitrary time, where time is specified as an unsigned binary fixed-point number in units of the input sampling period (assumed constant).

Figure 4.26 shows the time register , and Figure 4.27 shows an example configuration of the input signal and lowpass filter at a given time. The time register is divided into three fields: The leftmost field gives the number of samples into the input signal buffer, the middle field is an initial index into the filter coefficient table , and the rightmost field is interpreted as a number between 0 and for doing linear interpolation between samples and (initially) of the filter table. The concatenation of and are called which is interpreted as the position of the current time between samples and of the input signal.

Let the three fields have , , and bits, respectively. Then the input signal buffer contains samples, and the filter table contains ``samples per zero-crossing.'' (The term ``zero-crossing'' is precise only for the case of the ideal lowpass; to cover practical cases we generalize ``zero-crossing'' to mean a multiple of time , where is the lowpass cutoff frequency.) For example, to use the ideal lowpass filter, the table would contain sinc.

Our implementation stores only the ``right wing'' of a symmetric
finite-impulse-response (FIR) filter (designed by the window method
based on a Kaiser window [362]). Specifically, if
,
, denotes a length symmetric
finite impulse response, then the
*right wing*
of is defined
as the set of samples for
. By symmetry, the
*left wing* can be reconstructed as
,
.

Our implementation also stores a table of *differences*
between successive FIR sample values in order to
speed up the linear interpolation. The length of each table is
, including the endpoint definition
.

Consider a sampling-rate conversion by the factor
.
For each output sample, the basic interpolation Eq.(4.13) is
performed. The filter table is traversed twice--first to apply the
left wing of the FIR filter, and second to apply the right wing.
After each output sample is computed, the time register is incremented
by
(*i.e.*, time is incremented by in
fixed-point format). Suppose the time register has just been
updated, and an interpolated output is desired. For
, output is computed via

where is the current input sample, and is the interpolation factor. When , the initial is replaced by , becomes , and the step-size through the filter table is reduced to instead of ; this lowers the filter cutoff to avoid aliasing. Note that is fixed throughout the computation of an output sample when but changes when .

When , more input samples are required to reach the end of the
filter table, thus preserving the filtering quality. The number of
multiply-adds per second is approximately
.
Thus the higher sampling rate determines the work rate. Note that for
there must be
extra input samples
available before the initial conversion time and after the final conversion
time in the input buffer. As , the required extra input
data becomes infinite, and some limit must be chosen, thus setting a
minimum supported
. For , only extra input samples are required on
the left and right of the data to be resampled, and the upper bound for
is determined only by the fixed-point number format, *viz.*,
.

As shown below, if denotes the word-length of the stored impulse-response samples, then one may choose , and to obtain effective bits of precision in the interpolated impulse response.

Note that rational conversion factors of the form , where
and is an arbitrary positive integer, do not use the linear
interpolation feature (because
). In this case our method reduces
to the normal type of bandlimited interpolator [97]. With the
availability of interpolated lookup, however, the range of conversion
factors is boosted to the order of
. *E.g.*, for
,
, this is about decimal digits of
accuracy in the conversion factor . Without interpolation, the
number of significant figures in is only about .

The number of zero-crossings stored in the table is an independent design parameter. For a given quality specification in terms of aliasing rejection, a trade-off exists between and sacrificed bandwidth. The lost bandwidth is due to the so-called ``transition band'' of the lowpass filter [362]. In general, for a given stop-band specification (such as ``80 dB attenuation''), lowpass filters need approximately twice as many multiply-adds per sample for each halving of the transition band width.

As a practical design example, we use in a system designed for
high audio quality at % oversampling. Thus, the effective FIR
filter is zero crossings long. The sampling rate in this case would
be kHz.^{5.9}In the most straightforward filter design, the lowpass filter pass-band
would stop and the transition-band would begin at kHz, and the
stop-band would begin (and end) at kHz. As a further refinement,
which reduces the filter design requirements, the transition band is really
designed to extend from kHz to kHz, so that the half of it
between and kHz aliases on top of the half between and
kHz, thereby approximately halving the filter length required. Since the
entire transition band lies above the range of human hearing, aliasing
within it is not audible.

Using samples per zero-crossing in the filter table for the above example (which is what we use at CCRMA, and which is somewhat over designed) implies desiging a length FIR filter having a cut-off frequency near . It turns out that optimal Chebyshev design procedures such as the Remez multiple exchange algorithm used in the Parks-McLellan software [362] can only handle filter lengths up to a couple hundred or so. It is therefore necessary to use an FIR filter design method which works well at such very high orders, and the window method employed here is one such method.

It is worth noting that a given percentage increase in the original
sampling rate (``oversampling'') gives a larger percentage savings in
filter computation time, for a given quality specification, because the
added bandwidth is a larger percentage of the filter transition bandwidth
than it is of the original sampling rate. For example, given a cut-off
frequency of kHz, (ideal for audio work), the transition band
available with a sampling rate of kHz is about kHz, while a
kHz sampling rate provides a kHz transition band. Thus, a
% increase in sampling rate *halves* the work per sample in
the digital lowpass filter.

#### Choice of Table Size and Word Lengths

It is desirable that the stored filter impulse response be sampled
sufficiently densely so that interpolating linearly between samples
does not introduce error greater than the quantization error. It is
shown in [462] that this condition is satisfied
when the filter impulse-response table contains at least
entries per ``zero-crossing'', where is the
number of bits allocated to each table entry. (A later, sharper,
error bound gives that
is sufficient.) It is
additionally shown in [462] that the number of bits in the interpolation
between impulse-response samples should be near or more. With these
choices, the linear interpolation error and the error due to quantized
interpolation factors are each about equal to the coefficient
quantization error. A signal resampler designed according to these
rules will typically be limited primarily by the lowpass filter
*design*, rather than by quantization effects.

### Summary of Windowed Sinc Interpolation

The digital resampling method described in this section is convenient for bandlimited interpolation of discrete-time signals at arbitrary times and/or for arbitrary changes in sampling rate. The method is well suited for software or hardware implementation, and widely used free software is available.

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