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 6 shows the time register *t*, and
Figure 7 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 *n* of
samples into the input signal buffer, the middle field is an initial
index *l* into the filter coefficient table *h*(*l*), and the rightmost
field is interpreted as a number between 0 and 1 for doing
linear interpolation between samples *l* and *l*+1 (initially) of the
filter table. The concatenation of *l* and are called
which is interpreted as the position of the current time
between samples *n* and *n*+1 of the input signal.

Let the three fields have *n*_{n}, *n*_{l}, and bits,
respectively. Then the input signal buffer contains *N*=2^{nn}
samples, and the filter table contains *L*=2^{nl} ``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 *t*_{c}=0.5/*f*_{c}, where
*f*_{c} is the lowpass cutoff frequency in cycles per sample.) For
example, to use the ideal lowpass filter, the table would contain
.

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 [#!RabinerAndGold!#]). Specifically, if
*h*(*l*),
, denotes a
symmetric finite impulse response, then the *right wing* of *h*
is defined as the set of samples *h*(*l*) for
. By
symmetry, the *left wing* can be reconstructed as *h*(-*l*)=*h*(*l*),
.

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 *N*_{h}=*LN*_{z}+1, including the endpoint
definition
.

Consider a sampling-rate conversion by the factor
.
For each output sample, the basic interpolation Eq.(1) 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 *t* has just been
updated, and an interpolated output *y*(*t*) is desired. For
, output is computed via

v |
(3) | ||

P |
1-P |
(4) | |

y(t) |
(5) |

where *x*(*n*) is the current input sample, and is the
interpolation factor. When , the initial *P* is replaced by
, 1-*P* becomes
, and the step-size
through the filter table is reduced to instead of *L*; 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 *N*_{z} 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 *n*_{c} denotes the word-length of the stored
impulse-response samples, then one may choose *n*_{l}=1+*n*_{c}/2, and
to obtain *n*_{c}-1 effective bits of precision in the
interpolated impulse response.

Note that rational conversion factors of the form , where
*L*=2^{nl} and *M* 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 [#!Crochiere!#]. With the
availability of interpolated lookup, however, the range of conversion
factors is boosted to the order of
. E.g., for
,
, this is about 5.1 decimal digits of
accuracy in the conversion factor . Without interpolation, the
number of significant figures in is only about 2.7.

The number *N*_{z} 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 *N*_{z} and sacrificed bandwidth.
The lost bandwidth is due to the so-called ``transition band'' of the
lowpass filter [#!RabinerAndGold!#]. 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 *N*_{z}=13 in a system designed for
high audio quality at 20% oversampling. Thus, the effective FIR
filter is 27 zero crossings long. The sampling rate in this case would
be 50 kHz.^{1}In the most straightforward filter design, the lowpass filter pass-band
would stop and the transition-band would begin at 20 kHz, and the
stop-band would begin (and end) at 25 kHz. As a further refinement,
which reduces the filter design requirements, the transition band is really
designed to extend from 20 kHz to 30 kHz, so that the half of it
between 25 and 30 kHz aliases on top of the half between 20 and 25
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 512 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 27 x 512 = 13824 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 [#!RabinerAndGold!#] 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 20 kHz, (ideal for audio work), the transition band
available with a sampling rate of 44 kHz is about 2 kHz, while a
48 kHz sampling rate provides a 4 kHz transition band. Thus, a
10% increase in sampling rate *halves *the work per sample in
the digital lowpass filter.

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