The process used for interpolation depends on the ratio of the output sampling rate to the input sampling rate.
The default interpolation filter is a linear phase FIR filter designed by applying a Kaiser window to an ideal lowpass filter response. The filter is characterized by a cutoff frequency, a window shape parameter, and the number of coefficients. The window shape parameter (alpha) controls the passband ripple and the stopband attenuation. For a fixed number of coefficients, decreasing ripple and increasing attenuation (larger alpha) come at the expense of a wider transition width.
The cutoff of the default interpolation filter depends on the input and output sampling rates. Let fsi be the sampling rate of the input signal and fso be the sampling rate of the output signal.
The default design aims for an 80 dB stopband attenuation and a transition width which is 15% of the cutoff frequency. The attenuation directly determines alpha. The value of alpha together with the transition width determines the number of filter coefficients.
The parameters of the interpolating filter can also be set by the user. The design parameters are the interpolation factor, the filter cutoff frequency, the Kaiser window parameter, and the number of filter coefficients. The following table shows the effect of changing the Kaiser window parameter.
stopband alpha transition passband
attenuation width D ripple
30 dB 2.210 1.536 +/- 0.270 dB
40 dB 3.384 2.228 +/- 0.0864 dB
50 dB 4.538 2.926 +/- 0.0274 dB
60 dB 5.658 3.621 +/- 0.00868 dB
70 dB 6.764 4.317 +/- 0.00275 dB
80 dB 7.865 5.015 +/- 0.00089 dB
90 dB 8.960 5.712 +/- 0.00027 dB
100 dB 10.056 6.408 +/- 0.00009 dB
The transition width parameter D = (N-1) dF, where dF is the transition width
normalized by the filter sampling rate. Consider interpolating from 8 kHz to
44.1 kHz. The filter will be designed for a sampling rate of 80 kHz. The
cutoff of the filter will be 4 kHz. The stopband attenuation is to be 80 dB.
The attenuation requirement gives alpha=7.865. The parameter D corresponding
to this value of alpha is 5.015. A transition width which is 15% of the
cutoff corresponds to a width of 600 Hz. This is a normalized transition
width of dF = 600/80000 = 0.0075. Solving for N, gives 670 coefficients.
It is common to choose N to be odd, and furthermore for N to be of the form
2*Ir*M+1, where Ir is the interpolation factor (here 10). Such a time
response has M sidelobes on either side of the reference point. In this
example, we can choose M = 34, giving N = 681 coefficients.
If we designate the interpolation factor for the interpolation filter as Ir, about 1/Ir of the coefficients are used to calculate each output sample. The number of coefficients needed for a given value of alpha and given transition width is proportional to Ir. Increasing Ir improves the accuracy of the linear interpolation step and increases the total number of filter coefficients, but does not increase the computation effort time for the filtering operation.
For the transition width expressed as a percentage of the cutoff frequency, the number of coefficients needed to calculate each output sample is approximately 2D/P where P is the fractional bandwidth (e.g. 0.15 for a 15% transition width). The number of coefficients (rounded up) used to calculate each interpolated point is shown in the following table.
stopband alpha transition no. coeffs per output
attenuation width D 15% trans. 25% trans.
30 dB 2.210 1.536 22 14
40 dB 3.384 2.228 31 19
50 dB 4.538 2.926 41 25
60 dB 5.658 3.621 50 30
70 dB 6.764 4.317 59 36
80 dB 7.865 5.015 68 42
90 dB 8.960 5.712 78 47
100 dB 10.056 6.408 87 53
On a medium speed workstation (Sun Sparc 4), with the default filter (15% transition width), this program generates about 60,000 output samples per CPU second for ordinary interpolation and about half that number for the general case.
The accuracy of the sample rate operation depends on the frequency content of the input signal. Consider changing the sampling rate for a speech file with a 8000 Hz sampling rate. The default filter uses a cutoff frequency of 4000 Hz with a transition width of 600 Hz. The filter passband extends to 3700 Hz and the stopband starts at 4300 Hz. The interpolation will be imperfect in that (1) high frequencies falling in the lower part of the transition band will be attenuated and (2) aliased frequencies falling in the upper part of the transition band will be only be partially attenuated. If the input signal has little energy above 3700 Hz, then the error due to both factors will be small. Tests on speech files indicate that the signal-to-distortion ratios after interpolation (say from 8000 Hz to 8001 Hz) range from 46 to 77 dB. The poorest SDR occurs for signals that have significant energy above 3700 Hz. For a fixed stopband attenuation, the SDR can be improved by increasing the number of filter coefficients to affect a decrease in the transition band width.
The interpolation filter can also be read in as a filter file. For such a filter, the filter interpolation factor must be specified.
The output sample positions are determined by the output sampling rate and a sample offset parameter. The sample offset determines the position of the first output sample relative to the input samples. The default is that the first output sample coincides with the first input sample. The number of samples in the output file can also be specified. The default is to make the time corresponding to the end of the output (rounded to the nearest sample) be the same as the time corresponding to the end of the input.
Filter file:
file="file_name" Input filter file name. If specified, the filter
coefficients are read from the named file.
ratio=Ir Filter interpolation factor
delay=Del Filter delay in units of filter samples (default
(N-1)/2, where N is the number of coefficients for
symmetrical filters). The delay can be specified
as a single number or as a ratio. The filter delay
must be supplied for non-symmetrical filters.
Windowed lowpass:
ratio=Ir Filter interpolation factor. The default depends
on the ratio of output sampling frequency to
input sampling frequency. This parameter can be
specified as a single number or as a ratio.
cutoff=Fc Filter cutoff in normalized frequency relative to
the filter interpolation factor (0 to Ir/2). This
value can be specified as a single number or as a
ratio. The default cutoff frequency is determined
from the the input and output sampling rates. For
an increase in sampling rate, it is set to 0.5.
For a decrease in sampling rate it is set to
0.5*fso/fsi.
atten=A Filter stopband attenuation in dB. The attenuation
must be at least 21 dB. The default is 80. The
attenuation is an alternate way to specify the
Kaiser window parameter alpha.
alpha=a Kaiser window parameter. Zero corresponds to a
rectangular window (stopband attenuation 21 dB).
The default is 7.865 corresponding to a stopband
attenuation of 80 dB.
N=Ncof Number of filter coefficients. The default is
to chose the number of coefficients to give a
transition band which is 15% of the cutoff
frequency.
span=Wspan Window span. The default window span is equal to
the number of filter coefficients minus one.
offset=Woffs Window offset in units of filter samples. This is
the offset of the first filter sample from the
beginning of the window. The default is a
fractional value determined from the fractional
part of the input sample offset value.
gain=g Passband gain. The default gain is equal to
the filter interpolation factor. This choice
reproduces signals within the passband with the
correct amplitude.
write="file_name" Output filter file name. If specified, the filter
coefficients are written to the named file.
"mu-law8" - 8-bit mu-law data "A-law8" - 8-bit A-law data "unsigned8" - offset-binary 8-bit integer data "integer8" - two's-complement 8-bit integer data "integer16" - two's-complement 16-bit integer data "float32" - 32-bit IEEE floating-point data "text" - text dataThe data formats available depend on the output file type. AFsp (Sun) audio files:
mu-law, A-law, 8-bit integer, 16-bit integer, floatRIFF WAVE files:
mu-law, A-law, offset-binary 8-bit integer, 16-bit integerAIFF-C audio files:
mu-law, A-law, 8-bit integer, 16-bit integerHeaderless files:
all data formats
"AFsp", "Sun" or "sun" - AFsp (Sun) audio file "WAVE" or "wave" - RIFF WAVE file "AIFF-C" - AIFF-C audio file "raw" or "raw_native" - Headerless file (native byte order) "raw_swap" - Headerless file (byte swapped) "raw_big-endian" - Headerless file (big-endian byte order) "raw_little-endian" - Headerless file (little-endian byte order)
For AFsp output files, the audio file header contains an information string.
Standard Header Information:
date:1994/01/25 19:19:39 UTC date
user:kabal@aldebaran user
program:ResampAudio program name
This information can be changed with the header information string which is
specified as one of the command line options. Structured information records
should adhere to the above format with a named field terminated by a colon,
followed by numeric data or text. Comments can follow as unstructured
information. For the purpose of this program, records are terminated by
newline characters. However in the header itself, the newline characters are
replaced by nulls. To place a newline character into the header, escape
the newline character by preceding it with a '\' character. If the first
character of the user supplied header information string is a newline
character, the header information string is appended to the standard header
information. If not, the user supplied header information string replaces
the standard header information.
ResampAudio -i 1 abc.au new.au
ResampAudio -i 1 -a -1/8 abc.au new.au
ResampAudio -s 8001 abc.au new.au
ResampAudio -i 6 abc.au new.au
"Format, Start, Sfreq, Swapb, Nchan, ScaleF"
"undefined" - Headerless files will be rejected "mu-law8" or "MU-LAW8" - 8-bit mu-law data "A-law8" or "A-LAW8" - 8-bit A-law data "unsigned8" or "UNSIGNED8" - offset-binary 8-bit integer data "integer8" or "INTEGER8" - two's-complement 8-bit integer data "integer16" or "INTEGER16" - two's-complement 16-bit integer data "float32" or "FLOAT32" - 32-bit floating-point data "text" or "TEXT" - text data
"native" - no byte swapping "little-endian" - file data is in little-endian byte order "big-endian" - file data is in big-endian byte order "swap" - swap the data bytes as the data is read
"undefined, 0, 8000., native, 1, 1.0"