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/*

* Copyright 2002 Sun Microsystems, Inc. All rights reserved.

* Use is subject to license terms.

*/

#pragma ident "%Z%%M% %I% %E% SMI"

Bandlimited Interpolation

In bandlimited interpolation a windowed sinc function is used to

interpolate input samples and implement sample rate conversion.

More information on the algorithm can be found on the website at

http://ccrma-www.stanford.edu/~jos/resample. Choices to be made

during an implementation include (i) the type of window with which

to truncate the infinitely long sinc function (ii) the length of

the latter window (i.e. how many sinc zero-crossings to include)

(iii) the density of sinc function sampling (iv) the number of bits

used to encode each sinc coefficient (v) whether to interpolate

sinc coefficients during sample rate conversion.

In our implementation we (i) use a Kaiser window to truncate

the sinc function (ii) sample up to the 10th zero crossing (iii)

sample the interval between each zero crossing 256 times (iv)

use an integer (32 bits) to encode each coefficient (v) do not,

for performance reasons, perform coefficient interpolation during

conversion.

Using the above settings we end up with a conversion table which

contains (2 * 10 + 1) * 256 = 5376 entries. (We add some more to

simplify the implementation.) For performance reasons, we use two

copies of the table, one for up and one for downsampling. Each is

a reordering of the other. We also use separate functions for up

and downsampling mono and stereo files giving 4 resampling functions.

Each can handle, theoretically, any start/target sample rate

configuration and is accurate to 1Hz.

Upsampling

During conversion the sinc window is centered over the current output

sample location. We project upwards from input sample values under

the sinc "umbrella" and multiply and sum corresponding input samples

and sinc coefficients. The sinc function acts like a low pass filter

with a cut off frequency of Fs/2 where Fs is the input sample rate.

The number of multiply-adds is fixed, irrespective of the start and

target sample rates. Often an input sample location will not intersect

with a prestored coefficient value and as stated earlier we do not

interpolate but multiply by the nearest coefficient value.

Downsampling

Downsampling is not as straightforward as upsampling. The stored

sinc table must be traversed in such a way as to act like a low

pass filter but this time with a cut off frequency equal to Fs'/2

where Fs' is the target sample rate. To achieve this the sinc

coefficients are stretched over a longer interval of the input

signal (compared to upsampling) and traversed at a slower rate.

Compared to upsampling there are then more multiply-adds to be

performed but this is offset by the fact that there are fewer

output samples to be generated.

Notes

When resampling we are performing an interpolation. This interpolation

is imperfect and errors will occur. Furthermore, in our implementation

there is often no direct mapping from filter coefficents to input

samples and in this case we use the closest filter coefficient which

introduces more error. These errors lead to unwanted frequencies in

the audio output signal. Many of these could be removed or reduced by

(i) employing a greater sampling density over the stored sinc function,

thereby increasing the overall table size and lowering the error

introduced by a lack of filter coefficient interpolation or (ii)

maintaining the same overall table size but interpolating between

filter coefficients for increased accuracy at a computational cost.

Implementing either of these would have a significant effect on

output audio quality.

As is stands our current implementation should theoretically exhibit a

stopband rejection of 60dB and a transition bandwidth of ~34% (these

are theoretical values since the errors mentioned above drag quality

downwards). Our transition band is centered over Fs/2 with the result

that should there be significant energy in the upper ~17% of the spectrum

there will be a considerable artifact introduced in the output signal.

We take that chance. Stopband rejection can be increased at the expense

of transition bandwidth and transition bandwidth can be lessened by

using more sinc zero crossings (i.e. lengthening the filter kernel).