Patent Publication Number: US-2013253917-A1

Title: Psychoacoustic filter design for rational resamplers

Description:
TECHNICAL FIELD 
     The present document relates to the design of anti-aliasing and/or anti-imaging filters for resamplers using rational resampling factors. In particular, the present document relates to a method for designing such filters having a reduced number of filter coefficients or an increased perceptual performance, as well as to the filters designed using such a method. 
     BACKGROUND 
     Different audio formats may require different sampling rates (e.g. 32 kHz, 44.1 kHz or 48 kHz). In order to transfer an audio signal at a first sampling rate (e.g. at 32 kHz) to an audio signal at a second sampling rate (e.g. at 48 kHz) rational resamplers may be used. The resampling of audio by rational factors typically introduces imaging/aliasing artifacts into the resampled audio signal. An anti-imaging/anti-aliasing filter may be used to suppress the unwanted images and/or aliases of the audio signal. The present document describes anti-imaging/anti-aliasing filters used in rational resamplers. Furthermore, the present document describes a method for designing such anti-imaging/anti-aliasing filters. In particular, filter design methods (and resulting filters) are described which take into account psychoacoustic constraints, in order to provide filters having a reduced number of filter coefficients, while providing subjectively unchanged or similar audio quality of the resampled audio signal. Vice versa, the filter design methods may be used to design filters with a given number of filter coefficients, which provide an improved audio quality compared to filters designed in accordance to conventional filter design methods. 
     As a consequence of designing improved anti-imaging/anti-aliasing filters, the complexity of rational resamplers may be decreased, while maintaining a given subjective audio quality. Vice versa, the audio quality of the rational resamplers may be increased, while maintaining the rational resamplers at a given computational complexity. 
     SUMMARY 
     According to an aspect, a method for designing a filter, e.g. an anti-aliasing and/or anti-imaging filter, is described. The filter may be a digital filter comprising a set of filter coefficients. In the following, the filter will be referred to as anti-aliasing filter (even though the filter may also remove imaging effects). The resulting filter may be configured to reduce imaging and/or aliasing of an output audio signal at an output sampling rate fs out . The output audio signal is a resampled version of an input audio signal at an input sampling rate fs in . The ratio of the output sampling rate fs out  and the input sampling rate fs in  is a rational number N/M, wherein N&gt;0, M&gt;0. Without loss of generality, N and M should be mutually prime. In an embodiment, neither N, nor M are equal to 1, meaning that the resampler is neither a pure upsampler by an integer factor N, nor a pure downsampler by an integer factor M. In other words, the resampler comprises an upsampling component by a factor N&gt;1, and a downsampling component by a factor M&gt;1. In yet other words, the fraction N/M may not be an integer value, and the fraction M/N may not be an integer value. By way of example, for fs in =40 kHz, fs out =48 kHz, N=6, M=5; or for fs in =32 kHz, fs out =48 kHz, N=3, M=2; or for fs in =44.1 kHz, fs out =48 kHz, N=160, M=147; or for fs in =32 kHz, fs out =44.1 kHz, N=441, M=320; or vice versa, i.e. for fs in =48 kHz, fs out =40 kHz, N=5, M=6; or for fs in =48 kHz, fs out =32 kHz, N=2, M=3; or for fs in =48 kHz, fs out =44.1 kHz, N=147, M=160; or for fs in =44.1 kHz, fs out =32 kHz, N=320, M=441. 
     The filter may be operated at an upsampled sampling rate which equals N times the input sampling rate fs in . The upsampled sampling rate also equals M times the output sampling rate fs out . 
     The method for designing the filter may comprise the step of selecting a pass band edge of the frequency response of the filter. The selection of the pass band edge may comprise the selection of the frequency interval of the pass band. Frequencies smaller than the pass band edge are comprised within the pass band. The method may comprise the step of selecting a stop band edge of the frequency response of the filter. The selection of the stop band edge may comprise the selection of the frequency interval of the stop band. Frequencies greater than the stop band edge are comprised within the stop band. 
     The method may comprise the step of selecting an allowed deviation of the frequency response of the filter within the stop band. The allowed deviation indicates a deviation of the frequency response of the filter from a predetermined attenuation within the stop band. Typically, the predetermined attenuation of the filter within the stop band is 0 (i.e. −inf dB). As such, the allowed deviation may specify the tolerable deviation of the stop band attenuation from the ideal stop band attenuation at the predetermined value 0. In other words, a target frequency response of the filter may be determined. The target frequency response may specify a pass band attenuation (e.g. a value of 1 or 0 dB), a stop band attenuation (e.g. a value of 0 or −inf dB), a pass band edge and/or a stop band edge. The allowed deviation may be the allowed deviation of the frequency response of the filter from the target frequency response. 
     The resulting filter may be a low pass filter with a pass band covering a frequency interval in the frequency range of 0 kHz to the pass band edge. In such a case, the stop band of the resulting filter would cover the frequency interval above the stop band edge, wherein the stop band edge corresponds to a higher frequency than the pass band edge. 
     The allowed deviation of the frequency response of the filter within the stop band may be selected based on a perceptual frequency response indicative of an auditory spectral sensitivity. The perceptual frequency response may indicate the sensitivity of an average listener to particular frequencies of an audio signal. In other words, the perceptual frequency response may indicate how well certain frequencies of an audio signal are perceived by a listener. The perceptual frequency response may be associated with a first perceptual frequency response. The first perceptual frequency response may correspond to or may be indicative of a scaled version of an absolute threshold of hearing curve. The scaling may depend on the desired degree of rejection of the stop band. In particular, the absolute threshold of hearing (ATH) curve may be scaled such that the lowest absolute threshold of the ATH curve or an average value of the scaled ATH curve corresponds to a pre-determined degree of attenuation (e.g. −90 dB) of the target frequency response. 
     The step of selecting an allowed deviation of the frequency response of the filter within the stop band may comprise the step of selecting the allowed deviation based on images and/or mirrored images of the first perceptual frequency response. Images of the first perceptual frequency response may be copies of the first perceptual frequency response, possibly transposed to other frequency intervals. Mirrored images of the first perceptual frequency response may be mirrored versions of the first perceptual frequency response, possibly transposed to other frequency intervals. Typically, the images and/or mirrored images are transposed or shifted by the output sampling rate and/or a multiple thereof. 
     As indicated above, the resulting filter may be operated at an upsampled sampling rate M*fs out . As such, a spectrum of the (upsampled) output audio signal may cover a frequency range from 0 to M*fs out /2. As a result of a downsampling operation by the factor M generating the output audio signal, a portion of the spectrum covering the frequency range [(m−1)*fs out /2,(m+1)*fs out /2] for m=2,4, . . . , M, may be shifted to the baseband [−fs out /2,+fs out /2], thereby creating aliasing artifacts in the output audio signal. These artifacts are perceived by a human listener in accordance to an auditory spectral sensitivity reflected in the perceptual frequency response. 
     In order to reflect the shifting of high frequency ranges into the baseband, the first perceptual frequency response covering a frequency range of [0,+fs out /2], as well as a mirrored image of the first perceptual frequency response covering a frequency range of [−fs out /2,0] may be shifted to the frequency ranges [(m−1)*fs out /2,(m+1)*fs out /2] for m=2,4, . . . , M, thereby creating the images and/or mirrored images of the first perceptual frequency response. These images and/or mirrored images are symmetrical with respect to a frequency derived from the output sampling rate fs out . In particular, these images and/or mirrored images are symmetrical with respect to the output sampling rate fs out  and/or a multiple thereof. In other words, some of these images and/or mirrored images may be symmetrical with respect to a symmetry axis corresponding to the output sampling rate fs out  and/or a multiple thereof. 
     As such, the first perceptual frequency response may cover a frequency interval from 0 kHz to half the output sampling rate (i.e. [0, +fs out /2]) or a part of this frequency interval. Furthermore, a baseband mirrored image of the first perceptual frequency response (i.e. a mirrored image of the first perceptual frequency response in the baseband, mirrored along the symmetry axis at 0 kHz) may cover a frequency interval from 0 kHz to minus half the output sampling rate (i.e. [−fs out /2,0]) or a part of this latter frequency interval. 
     The images of the first perceptual frequency response which are used for the selection of the allowed deviation of the frequency response of the filter within the stop band may correspond to the first perceptual frequency response and/or the baseband mirrored image of the first perceptual frequency response shifted by the output sampling rate and/or a multiple thereof. 
     The selection of the allowed deviation of the frequency response of the filter within the stop band may comprise the step of setting the allowed deviation within a given frequency interval equal to the images of the first perceptual frequency response (and/or its baseband mirrored image) within the given frequency interval. In other words, the allowed deviation of the frequency response within a given frequency interval may be set equal to the images and/or mirrored images of the first perceptual frequency response within the given frequency interval. The given interval may correspond to the frequency intervals within the stop band outside the “don&#39;t care” intervals specified below. 
     The perceptual frequency response may be associated with a second perceptual frequency response. The second perceptual frequency response may comprise a scaled relative masking threshold curve indicative of the masking by a neighbouring masker frequency. In other words, the second perceptual frequency response may reflect the fact that a signal at a masker frequency masks signals at frequencies within the vicinity of the masker frequency. The relative masking threshold curve may indicate the threshold of hearing a frequency in the vicinity of the masker frequency. Due the masking effect of the masker frequency, the threshold of hearing may be increased in the vicinity of the masker frequency. 
     As a result of an upsampling operation, images of a baseband masker frequency may be created in the intermediate, i.e. upsampled, frequency domain. Some of these images may be aliased back to the baseband masker frequency during the downsampling operation. The images of the baseband masker frequency in the intermediate frequency domain which meet this condition may be referred to as maskee frequencies as their aliases may be masked by the baseband masker frequency. In other words, the baseband masker frequency (and/or the maskee frequency) may meet the self masking condition that the maskee frequency in the intermediate frequency domain corresponds to a baseband masker frequency of the input audio signal in the frequency range of [−fs in /2, fs in /2] shifted by fs in  or a multiple thereof; and that the maskee frequency aliases to the output audio signal at plus and/or minus the baseband masker frequency. 
     The baseband masker frequency may meet the condition that the absolute value of the baseband masker frequency corresponds to the absolute value of (n*fs in /2−m*fs out /2), for at least some of n=1, . . . , N and m=1, . . . , M. In other words, for the baseband masker frequencies the condition 
       |f|=| n·fs   in /2 −m·fs   out   /2|, with   n= 1 , . . . , N, m= 1 , . . . , M,    
     may be met for at least some of the possible values of n and m. In a similar manner, the maskee frequency in the intermediate frequency domain may correspond to n*fs in /2+m*fs out /2, for at least some of n=−N, . . . , N and m=−M, . . . , M, i.e. the maskee frequency may meet the condition 
         f=n·fs   in /2 +m·fs   out /2, with  n=−N, . . . , N, m=−M, . . . , M,    
     for at least some of the possible values of n and m. 
     The above self masking conditions may be used to identify one or more maskee frequencies in the intermediate domain and one or more corresponding masker frequencies in the baseband. The one or more maskee frequencies meeting the self masking condition may correspond to a maximum of a scaled relative masking threshold curve. I.e. for these maskee frequencies the masking caused by the baseband masker frequency may be maximal. In addition, if a frequency in the intermediate domain approximately fulfills the above condition, then the alias of this frequency will typically be close to the baseband masker frequency, and can be subject to masking by that baseband masker frequency. This masking of frequencies in the vicinity of the maskee frequencies in the intermediate domain may be modeled by the progression of the scaled relative masking threshold curve. 
     In an embodiment, the second perceptual frequency response comprises a scaled relative masking threshold curve for each maskee frequency meeting the self masking condition. The overall perceptual frequency response used for the determination of the allowed deviations of the frequency response may correspond to a combination, e.g. a maximum, of the first perceptual frequency response and the second perceptual frequency response. 
     The step of selecting the allowed deviation of the frequency response of the filter within the stop band may comprise the step of partitioning the stop band into a plurality of frequency intervals comprising one or more “don&#39;t care” intervals. The allowed deviation may take on arbitrary or undefined values within a “don&#39;t care” interval. In other words, the allowed deviation of the frequency response may be unconstrained or undefined within a “don&#39;t care” interval. The one or more “don&#39;t care” intervals may comprise one or more first “don&#39;t care” intervals associated with frequencies for which a spectrum of the input audio signal is below a pre-determined input energy threshold. By way of example, the input audio signal may be bandwidth limited to a frequency fx lower than the Nyquist frequency fs in /2. As a result, the spectrum of the input audio signal may be below the input energy threshold in the frequency interval [fx,fs in /2] (as well as in the mirrored frequency interval [−fs in /2,−fx]. The one or more first “don&#39;t care” intervals may be associated with the frequencies of the frequency interval [fx,fs in /2] (as well as with the frequencies of the mirrored frequency interval [−fs in /2,−fx]). 
     The one or more first “don&#39;t care” intervals may be symmetrical with respect to a frequency derived from the input sampling rate fs in . In particular, the one or more first “don&#39;t care” intervals may be symmetrical with respect to the input sampling rate fs in  and/or a multiple thereof. In the above example, the frequency interval [fx,fs in /2], as well as the mirrored image [−fs in /2,−fx] may constitute first “don&#39;t care” intervals associated with the input audio signal. As a result of the up-by-N upsampling operation, further images and mirrored images of these first “don&#39;t care” intervals may be created at frequency intervals [fx+n*fs in ,fs in /2+n*fs in ], as well as [−fs in /2+n*fs in ,−fx+n*fs in ], for n=1, . . . , N/2. These further images and mirrored images may also constitute first “don&#39;t care” intervals. As a result of the shift operation by fs in  some of the first “don&#39;t care” intervals are symmetrical with respect to the input sampling rate fs in  and/or a multiple thereof. 
     The one or more “don&#39;t care” intervals may comprise one or more second “don&#39;t care” intervals associated with frequencies for which the perceptual frequency response exceeds a pre-determined perceptual threshold. The one or more second “don&#39;t care” intervals may correspond to frequencies for which the images and/or mirrored images of the perceptual frequency response exceed the pre-determined perceptual threshold. The perceptual frequency response may take on values indicating a low auditory spectral sensitivity for certain frequency intervals. If the perceptual frequency response exceeds the pre-determined perceptual threshold, i.e. if the perceptual frequency response indicates an auditory sensitivity lying below a pre-determined sensitivity threshold, it may be beneficial to remove any constraints on the target frequency response, thereby increasing the degrees of freedom for the filter design. As such, further “don&#39;t care” intervals (i.e. the second “don&#39;t care” intervals) may be defined. 
     As indicated above, the method may comprise the step of selecting a pass band edge and/or a stop band edge of the frequency response of the filter. The pass band edge and/or the stop band edge may be based on the lower one of the input sampling rate fs in  and the output sampling rate fs out . In particular, the pass band edge and/or the stop band edge may be set to the lower one of the Nyquist rate fs in /2 of the input audio signal and the Nyquist rate fs out /2 of the output audio signal. Alternatively or in addition, the pass band edge and/or the stop band edge may be selected based on the bandwidth of the input audio signal. The pass band is positioned at frequencies lower than the pass band edge, and the stop band is positioned at frequencies higher than the stop band edge. 
     The method may comprise the step of determining coefficients of the filter such that the frequency response of the filter is fitted to the allowed deviation of the frequency response. The step of determining coefficients of the filter may comprise the step of fitting the frequency response of the filter to the allowed deviation using a maximum absolute difference criteria or a least mean square criteria. In particular, the coefficients of the filter may be determined using a Remez exchange algorithm or Parks-McClellan algorithm. 
     The Parks-McClellan algorithm minimizes the maximum of an approximation error function, wherein the approximation error function is based on the different between the frequency response of the filter and the predetermined attenuation within the stop band. Typically, the approximation error function is weighted. The weights may be proportional to the inverse of the allowed deviation of the frequency response of the filter. 
     In an embodiment, the step of determining coefficients of the filter comprises the step of fitting the frequency response of the filter to the allowed deviation of the frequency response outside of the one or more “don&#39;t care” intervals. As indicated above, the allowed deviation may take on arbitrary or undefined values within the one or more “don&#39;t care” intervals. As such, the fitting to the frequency response to the allowed deviation may be performed by imposing no constraints on the frequency response of the filter within the one or more “don&#39;t care” intervals. In the context of the Parks-McClellan algorithm, the “don&#39;t care” intervals may be taken into account by ignoring the approximation error function within the “don&#39;t care” intervals. In other words, the maximum approximation error function would not be minimized within the “don&#39;t care” intervals. 
     The method may comprise the step of selecting an allowed deviation of the frequency response of the filter within the pass band. The allowed deviation may indicate the deviation of the magnitude of the frequency response from a predetermined pass band attenuation, which is typically 1 (i.e. 0 dB). The allowed deviation may be a fixed, i.e. frequency independent, allowed deviation within the pass band. 
     According to further aspect a filter is described, wherein the filter may be designed in accordance to the design method and any related feature outlined in the present document. 
     According to another aspect a filter is described, wherein the filter is configured to reduce imaging and/or aliasing of an output audio signal at an output sampling rate fs out . The output audio signal may be a resampled version of an input audio signal at an input sampling rate fs in . The ratio of the output sampling rate fs out  and the input sampling rate fs in  may be a rational number N/M, as outlined above. The filter may operate at an upsampled sampling rate which equals N times the input sampling rate fs in . As indicated above, the upsampled sampling rate may also be equal to M times the output sampling rate fs out . The filter may comprise a pass band and a stop band. Furthermore, the filter may have a have a pass band edge and/or a stop band edge (or a cut off frequency) based on the lower one of the input sampling rate and the output sampling rate. 
     The frequency response of the filter within the stop band may be associated with a perceptual frequency response indicative of an auditory spectral sensitivity. As outlined above, the perceptual frequency response may be associated with a first frequency response comprising a scaled and/or shifted version of the absolute threshold of hearing (ATH) curve. In particular, the frequency response of the filter within the stop band may be associated with images and/or mirrored images of the first perceptual frequency response. These images and/or mirrored images may be symmetrical with respect to a frequency derived from the output sampling rate fs out . As such, the frequency response of the filter within the stop band may be associated with the first perceptual frequency response covering a frequency interval of [0, +fs out /2], as well as a mirrored image of the first perceptual frequency response covering a frequency interval of [−fs out /2, 0]. In particular, the frequency response of the filter within the stop band may be associated with images of these first perceptual frequency responses centered at the output sampling rate fs out  and/or multiples thereof, i.e. images at [(m−1)*fs out /2,(m+1)*fs out /2] for m=2,4, . . . , M. 
     Alternatively or in addition the frequency response in the stop band may be associated with a second perceptual frequency response comprising a scaled relative masking threshold curve indicative of the masking (by a masker frequency) of neighbouring frequencies. In particular, the overall perceptual frequency response may be a combination of the first and second perceptual frequency response. 
     The frequency response of the filter may be fitted to the perceptual frequency response using a maximum absolute difference criteria. In an embodiment, the frequency response of the filter does not exceed the perceptual frequency response within selected frequency intervals, e.g. frequency intervals outside of the above mentioned “don&#39;t care” intervals. In other words, the attenuation of the filter may not exceed the attenuation defined by the perceptual frequency response within selected frequency intervals. 
     According to a further aspect, a method for resampling an input audio signal at an input sampling rate fs in  to an output audio signal at an output sampling rate fs out  is described. The ratio of the output sampling rate fs out  and the input sampling rate fs in  may be a rational number N/M. The method may comprise the step of providing a set of coefficients of a filter. The filter may be any filter described in the present document, e.g. any filter designed according to a method outlined in the present document. The method may proceed in selecting a first subset of coefficients from the set of coefficients. This first subset may comprise a first coefficient of the set and additional coefficients of the set following the first coefficient by multiples of N. In other words, every N th  coefficient (starting from the first coefficient) of the set of coefficients may be selected for the first subset of coefficients. 
     The method may further comprise the step of determining a first sample of the output audio signal based on the first subset of coefficients and a first plurality of samples of the input audio signal. In other words, a first sample of the output audio signal may be determined by filtering a first plurality of samples of the input audio signal using a filter based on the first subset of coefficients. 
     In order to determine a second sample of the output audio signal, the method may comprise the step of selecting a second coefficient of the set based on the first coefficient and M. The method may proceed in selecting a second subset of coefficients from the set of coefficients, wherein the second subset comprises the second coefficient and coefficients of the set following the second coefficient by multiples of N. In other words, the method may proceed in selecting a second subset comprising a shifted subset of filter coefficients. Finally, the method may determine the second sample of the output audio signal directly following the first sample, based on the second subset of coefficients and a second plurality of samples of the input audio signal. 
     In other words, the samples of the output audio signal may be determined using a polyphase finite impulse response implementation of the psychoacoustic filter described in the present document. 
     According to another aspect a resampler configured to generate an output audio signal at an output sampling rate fs out  from an input audio signal at an input sampling rate fs in  is described. The ratio of the output sampling rate fs out  and the input sampling rate fs in  may be a rational number N/M. The resampler may comprise a filter according to any of the aspects outlined in the present document. The filter comprises a set of coefficients. Furthermore, the resampler may comprise a coefficient selection unit configured to select a subset of coefficients from the set of coefficients. The selection of the subset may be performed as outlined above in the context of the first and/or second subset. In addition, the resampler may comprise a filtering unit configured to generate a sample of the output audio signal from a plurality of samples of the input audio signal using the subset of coefficients. 
     According to a further aspect, a software program is described. The software program may be adapted for execution on a processor and for performing the aspects and features outlined in the present document when carried out on a computing device. 
     According to another aspect, a storage medium comprising a software program is described. The software program may be adapted for execution on a processor and for performing the aspects and features outlined in the present document when carried out on a computing device. 
     According to a further aspect, a computer program product is described. The computer program product may comprise executable instructions for performing the aspects and features outlined in the present document when executed on a computer. 
     It should be noted that the methods and systems including its preferred embodiments as outlined in the present document may be used stand-alone or in combination with the other methods and systems disclosed in this document. Furthermore, all aspects of the methods and systems outlined in the present document may be arbitrarily combined. In particular, the features of the claims may be combined with one another in an arbitrary manner. 
    
    
     
       SHORT DESCRIPTION OF THE FIGURES 
       The methods and systems described in the present document are explained below in an exemplary manner with reference to the accompanying drawings, wherein 
         FIG. 1  illustrates a conceptual diagram of an example resampler; 
         FIG. 2  depicts a spectrum of an example upsampled input audio signal comprising two sinusoids; 
         FIG. 3  shows a bifrequency map indicating the imaging/aliasing contributions of example signals; 
         FIG. 4   a  illustrates example frequency intervals of an input audio signal not contributing to imaging/aliasing; 
         FIG. 4   b  shows an example target frequency response and an example frequency response of an anti-imaging/anti-aliasing filter; 
         FIG. 5   a  depicts an example absolute threshold of hearing curve; 
         FIG. 5   b  shows frequency intervals of an example output audio signal, wherein signal components of the output audio signal in the illustrated frequency intervals are not perceived by a human listener; 
         FIG. 5   c  shows an example target frequency response and an example frequency response of an anti-imaging/anti-aliasing filter taking into account psychoacoustic aspects; 
         FIG. 6  illustrates a comparison of example frequency responses of anti-imaging/anti-aliasing filters which do and do not take into account psychoacoustic aspects while having the same number of coefficients; 
         FIG. 7  illustrates an example spectrum of a resampled audio signal comprising a sinusoid at varying frequencies; 
         FIG. 8  shows an example triangle wave function used to illustrate imaging subject to upsampling; 
         FIG. 9  illustrates an example bifrequency map highlighting self-masking points; 
         FIG. 10  shows an example frequency response of a psychoacoustic upsampling filter; 
         FIG. 11   a  illustrates a mapping of a linear frequency scale to the Bark scale; 
         FIG. 11   b  illustrates an example self-masking threshold curve; 
         FIG. 11   c  illustrates the allowed deviations of an example resampling filter, due to self-masking and due to absolute threshold of hearing; 
         FIG. 12  shows an example frequency response of a psychoacoustic filter which takes into account the absolute threshold of hearing and the self-masking threshold; and 
         FIG. 13  shows a block diagram of an example method for designing a psychoacoustic filter for the resampling of audio signals. 
     
    
    
     DETAILED DESCRIPTION OF THE FIGURES 
       FIG. 1  illustrates conceptually a rational resampler  100 . The rational resampler  100  comprises an up-by-N upsampler  101  which converts a sample of an input audio signal  110  into N samples of an upsampled audio signal  111 . This may be achieved by inserting N−1 zeros between two samples of the input audio signal  110 . Subsequently, the upsampled audio signal is filtered by an anti-aliasing/anti-imaging filter  102  with transfer function H(z). This results in a filtered audio signal  112 . Finally, the filtered audio signal  112  is passed to a down-by-M decimator or downsampler  103  which only retains every M th  sample of the filtered audio signal  112 , to thereby provide the resampled (or output) audio signal  113 . In case of a resampling of an input audio signal  110  at a sampling rate of 32 kHz to an output audio signal  113  at a sampling rate of 48 kHz, N is 3 and M is 2. In case of a resampling of an input audio signal  110  at a sampling rate of 44.1 kHz to an output audio signal  113  at a sampling rate of 48 kHz, N is 160 while M is 147. 
     It should be noted that the filter  102  runs at an intermediate frequency (IF) at N times the input sampling rate or at M times the output sampling rate (e.g. IF=M*48 kHz for the above mentioned cases). This means that the anti-aliasing filters  102  typically operate at high sampling rates, such that a reduction of the number of computational filter operations is desirable. In other words, it is desirable to reduce the number of required coefficients of the anti-aliasing filter  102 , in order to reduce the overall computational complexity of the rational resampler  100 . 
     The filters may be realized as a polyphase FIR (Finite Impulse Response) implementation. Such an implementation exploits the fact that the upsampled audio signal  111  which is filtered by filter  102  comprises N−1 zeros between the samples of the input audio signal  110 . Consequently, the “zero” multiplications and additions can be omitted. Furthermore, a polyphase implementation exploits the fact that due to the subsequent down-by-M decimator  103 , only every M th  sample of the filtered audio signal  112  needs to be determined. By exploiting this information during the filter implementation, the number of multiplication and/or adding operations can be significantly reduced, thereby reducing the computational complexity of the rational resampler  100 . Nevertheless, it is desirable to further reduce the computational complexity or to further improve the perceptual performance of the resampler  100 . 
     As indicated above, the resampling operation creates imaging and/or aliasing artifacts in the output audio signal  113  if no anti-aliasing filter  102  is used. These imaging and/or aliasing artifacts are created as a result of the upsampling  101  and downsampling  103  operations. This is illustrated in the frequency spectrum shown in  FIG. 2 , where the spectrum  200  of an example input audio signal  110  at an input sampling rate or sampling frequency fs in =40 kHz is depicted subsequent to upsampling  101  by a factor N=6. The original input audio signal  110  comprises two sinusoids  201  at 2 and 3 kHz, respectively. It can be seen that as a result of the upsampling operation  101 , various images  202 , . . . ,  206  of the two sinusoids  201  have been created in the spectrum of the upsampled audio signal  111 . 
     The input audio signal  110  has an input sampling rate fs in =40 kHz, i.e. the Nyquist frequency of the input audio signal  110  is at fs in /2=20 kHz. As a result of the upsampling operation  101 , the upsampled audio signal  111  has an upsampled sampling rate of N×fs in =240 kHz, i.e. a Nyquist frequency of 120 kHz. The images of the sinusoids  201  at 2/3 kHz can be found at 40 kHz±2/3 kHz (reference numerals  202 ,  203 ), at 80 kHz±2/3 kHz (reference numerals  204 ,  205 ) and at 120 kHz−2/3 kHz (reference numeral  206 ). As such, the upsampled audio signal  111  comprises frequency components which exceed the Nyquist frequency of 20 kHz of the input audio signal  110 . 
     If it is assumed that the input audio signal  110  at input sampling rate fs in =40 kHz is to be resampled to an output audio signal  113  at output sampling rate fs out =48 kHz, the downsampler  103  has to perform a downsampling by a factor M=5. However, due to the fact that the upsampled audio signal  111  comprises frequency components which exceed the Nyquist frequency fs out /2=24 kHz of the output audio signal  113  (see the sinusoid images  202 , . . . ,  206 ), so called aliasing occurs, thereby creating undesirable contributions of the sinusoid images  202 , . . . ,  206  to the spectrum of the output audio signal  113 . 
     In order to avoid these undesirable contributions to the output audio signal  113 , the upsampled audio signal  111  should be filtered using an anti-aliasing filter  102 . The filter  102  should ensure that the spectral images  202 , . . . ,  206  created during the upsampling operation  101  do not cause aliasing during the downsampling operation  103 . This can be ensured by using a low pass filter having a cut-off frequency or a pass band edge/stop band edge which corresponds to the lower one of fs out /2 and fs in /2, i.e. which corresponds to the lower one of the Nyquist frequency of the input audio signal  110  and the output audio signal  113 . 
     The anti-aliasing filters  102  are usually specified by one or more filter design parameters. Typically, the most important design parameters for this type of filters are “stop band rejection”, “pass band edge”, and “pass band ripple” (in particular with regards to the signal processing involved). These three design parameters may have an influence on the number of filter coefficients (i.e. the length) of the anti-aliasing filter  102 , and therefore on the complexity of the rational resampler  100 . Consequently, a trade-off between the imposed filter design parameters and the length of the anti-aliasing filter  102  must be found. By way of example, the pass band ripple may be set at 0.1 dB and the available cycle budget (i.e. the available number of filter coefficients) may allow for a stop band rejection of around −50 dB. 
     In the following, different aspects are described which should be taken into account when designing an appropriate anti-aliasing filter  102 . For this purpose, reference is made to  FIG. 3  where an example bifrequency map  300  for a rational resampler  100  is depicted. The resampler  100  is configured for an input signal  110  at an input sampling rate fs in −40 kHz and an output signal  113  at an output sampling rate fs out =48 kHz. The bifrequency map  300  is a means for illustrating the relationship between the frequency components of the input signal  110  (shown along the abscissa of the map  300 ) and the frequency components of the output signal  113  (shown along the ordinate of the map  300 ). As such, the bifrequency map  300  can be used to illustrate the complex transfer function of the rational resampler  100 . 
     In column  310  of the bifrequency map  300  it can be seen that the frequency component at 3 kHz of the input signal  110  creates an intended frequency component  311  at 3 kHz in the output signal  113 . However, due to imaging and aliasing effects, the frequency component at 3 kHz of the input signal  110  also creates frequency components  312 ,  313 ,  314 ,  315 ,  316  at other frequencies of the output signal  113 . These latter frequency components may be perceived as artifacts within the output signal  113 . 
     In a similar manner, it can be seen in line  320  of the bifrequency map  300  that the frequency component at 7 kHz of the output signal  113  receives an intended contribution from the frequency component  321  of the input signal  110 . However, the frequency component at 7 kHz of the output signal  113  also receives contributions  322 ,  323 ,  324 ,  325  from other frequencies of the input signal  110 . These latter contributions may result in audible artifacts of the output signal  113 . 
     As such, the bifrequeny map  300  may be used to illustrate how the different frequency components of the output signal  113  are influenced by the frequency components of the input signal  110 . Consequently, the bifrequency map  300  may also be used to identify certain frequency ranges of the input signal  110  which do not influence the output signal  113 . This knowledge on frequency ranges of the input signal  110  not influencing the output signal  113  may be taken into account during the design of the anti-aliasing filter  112 . As a result, the performance of the filter  112  may be improved and/or the length/complexity of the filter  112  may be reduced. 
     This is illustrated in  FIG. 4   a , where a certain frequency interval  401  of the input signal  110  is highlighted within a bifrequency map  400 . It can be seen that if the input signal  110  does not comprise any signal components in the frequency range  401  of 17 kHz to 20 kHz, there is no contribution of these components of the input signal  110  to the frequency components of the output signal  113 . This information can be used in the filter design as shown in  FIG. 4   b . In other words, if the input audio signal  110  has been band-limited to 17 kHz or less, the filter  112  can be designed with additional degrees of freedom, due to the fact that no frequency components are to be expected in the frequency range of 17 kHz to 20 kHz of the input signal  110 . 
     It can be seen in the graph  410  of  FIG. 4   b  that the total frequency range of the upsampled signal  111  (i.e. the frequency range of 0 kHz to 120 kHz) can be partitioned into frequency ranges  412 ,  413 ,  414  where the frequency response of the filter  102  can take on any form (so called “don&#39;t care” intervals), as well as into frequency ranges  411 ,  415 ,  416 ,  417  where the frequency response of the filter  102  should take on a pre-determined form (so called “care” intervals). The “don&#39;t care” intervals  412 ,  413 ,  414  correspond to the frequency range of 17 kHz to 20 kHz (i.e. the frequency range within which the input signal  110  has no contributions), as well as to the images of this frequency range (i.e. 20 kHz to 23 kHz, 57 kHz to 60 kHz, 60 kHz to 63 kHz, 97 kHz to 100 kHz, 100 kHz to 103 kHz). As such, the information regarding the (missing) contributions of the input signal  110  can be used in order to provide additional degrees of freedom during the design of the filter  102 . 
     The graph  420  illustrates the constraints or parameters imposed during the design of the filter  102 . In the illustrated case, the order of the filter  102  (i.e. the number of filter coefficients) was set to 60. The stop band suppression or attenuation was set to −28 dB within the “care” intervals  415 ,  416 ,  417  (constraints  425 ,  426 ,  427 , respectively). The “care” interval  411  corresponds to the desired pass band  421  of the filter  102  (with no attenuation, i.e. a pass band attenuation of 0 dB). No constraints or parameters were imposed for the “don&#39;t care” intervals  412 ,  413 ,  414 . 
     As such, a set of constraints on the target frequency response of filter  102  can be formulated. These constraints apply to the “care” intervals  411 ,  415 ,  416 ,  417 , whereas the target frequency response of the filter  102  can taken on any form within the “don&#39;t care” intervals  412 ,  413 ,  414 . The filter coefficients of a filter  102  meeting or approximating these requirements may be determined using filter design methods such as the Parks-McClellan algorithm. This algorithm determines the set of filter coefficients that minimize the maximum deviation from the target frequency response. 
     The Parks-McClellan algorithm is directed at minimizing the maximum of an approximation error E(f) given by 
         E ( f )= W ( f )| D ( f )− H ( f )|,
 
     wherein D(f) is the desired form of the low pass filter  102 , i.e. the target frequency response, and is typically given by 
     
       
         
           
             
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     with f p  being the pass band edge and f s  being the stop band edge. As outlined above, other attenuation values may be defined in the pass band and/or the stop band. W(f) is a frequency dependent weighting function of the approximation error. H (f) is given by 
     
       
         
           
             
               
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     and relates to the frequency response of the filter  102  by exp(−j2πnf)H(f). The filter coefficients h k  of filter  102  are given by 
         h   k   =h   2n-k   ;d   n-k =2 h   k   ,k= 0 , . . . , n− 1 ;d   0   =h   n . 
     The Parks-McClellan (Remez exchange) algorithm comprises the following steps: 
     1) Initialization: Choose an extremal set of frequencies {f (0) }. 
     2) Finite Set Approximation (at iteration m): Calculate the best Chebyshev (i.e. minmax) approximation on the present extremal set, giving a derivation value δ (m)  for the minmax error on the present extremal set. 
     3) Interpolation: Calculate the error function E(f) over the entire set of frequencies Ω using step (2). 
     4) Look for local maxima of E (m) (f) on the set of frequencies Ω. 
     5) If max fεΩ  E (m) (f)&gt;δ (m) , then update the extremal set to {f (m+1) } by picking new frequencies where E (m) (f) has its local maxima. Make sure that the error alternates on the ordered set of frequencies Ω as described in (4) and (5). Return to Step 2 and iterate. 
     6) If max fεΩ  E (m) (f)≦δ (m) , then the algorithm is complete. Use the set {f (m) } and the interpolation formula to compute an inverse discrete Fourier transform to obtain the filter coefficients. 
     Details on the Parks-McClellan algorithm are outlined in T. Parks, J. McClellan, “Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase”, IEEE Transactions on Circuit Theory, Vol. CT-19, No.2, March 1972, which is incorporated by reference. 
     The fact that certain frequency intervals of the target frequency response D(f) are not specified, i.e. the fact that the target frequency response of the filter  102  comprises “don&#39;t care” intervals  412 ,  413 ,  414 , typically leads to shorter filters  102  for achieving the target frequency response or to filters  102  of a given length achieving an improved approximation of the target frequency response of the filter  102  within the “care” intervals  411 ,  415 ,  416 ,  417 . The “don&#39;t care” intervals  412 ,  413 ,  414  may be taken into account within the Parks-McClellan algorithm by ignoring the approximation error E(f) within the “don&#39;t care” intervals  412 ,  413 ,  414 . In other words, the approximation error E(f) exceeding the derivation value δ within a “don&#39;t care” interval would not trigger a further iteration of the algorithm. 
     It should be noted that other filter design methods may be used to determine a filter  102  approximating the target frequency response. 
     The frequency response  430  of the resulting filter  102  is depicted in the graph  420  of  FIG. 4   b . It can be seen that the target frequency response D(f) (reference numerals  421 ,  425 ,  426 ,  427 ) is well met within the “care” intervals  411 ,  415 ,  416 ,  417 . On the other hand, it can be seen that within the “don&#39;t care” intervals  412 ,  413 ,  414  the frequency response  430  takes on arbitrary values and partially falls short of the stop band suppression  425 ,  426  imposed on neighboring “care” intervals  415 ,  416  (see reference numeral  431 ). 
     Alternatively or in addition, other aspects may be taken into consideration when designing the anti-aliasing filter  102 . In particular, audio perceptual aspects, e.g. the absolute threshold of hearing (ATH), may be taken into consideration.  FIG. 5   a  illustrates an example curve  505  of the ATH measured in dB SPL (Sound pressure level) across a frequency range of 0 Hz to approximately 20 kHz. It can be seen that in the lower frequency range  506  (approx. 10 Hz and below) and in the higher frequency range  507  (approx 17 kHz and higher) the ATH is very high indicating that a human is only able to perceive sounds in the frequency ranges  506 ,  507  at significant sound pressure levels. 
     The ATH curve  505  may be approximated by a mathematical equation, e.g. by the equation proposed by Terhardt: 
     
       
         
           
             
               
                 
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     wherein the frequency f is measured in Hz. 
       FIG. 5   b  illustrates the frequency intervals  506 ,  507  corresponding to high values of ATH in the bifrequency map  500 . It can be seen that these frequency ranges  506 ,  507  of the output signal  113  may have high imaging/aliasing contributions from the input signal  110 , without being audible to a listener of the output signal  113 . In other words, due to the high absolute threshold of hearing in the frequency ranges  506 ,  507 , imaging/aliasing artifacts within the output signal  113  are of reduced importance to the perceived quality of the output signal  113 . It is proposed to use this knowledge during the design of the anti-imaging/anti-aliasing filter  102 . 
       FIG. 5   c  shows how this information regarding the absolute threshold of hearing can be taken into account during the design of the anti-imaging/anti-aliasing filter  102 . In a similar manner to  FIG. 4   b , the graph  410  shows the “don&#39;t care” intervals  412 ,  413 ,  414  which are due to missing frequency components of the input signal  110 . Furthermore, the “care” interval  411  for the pass band and the “care” intervals  415 ,  416 ,  417  for the stop band are illustrated. It should be noted that the “don&#39;t care” intervals and the “care” intervals can take on various forms. In particular, if the input signal  110  covers the complete Nyquist range (up to 20 Hz), there may be no “don&#39;t care” intervals  412 ,  413 ,  414 . 
     The graph  520  of  FIG. 5   c  illustrates the allowed deviation from the target frequency response of the filter  102 . The target frequency response D(f) defined by a certain pass band attenuation (e.g. 0 dB) and a certain stop band attenuation (e.g. −100 dB). The allowed deviation indicates how much the frequency response of the filter  102  may deviate from the target frequency response. The allowed deviation may be used as an alleviated constraint during the design of the filter  102 . In particular, the allowed deviation may be taken into account within the weight function W(f) of the above mentioned approximation error E(f). Even more particularly, the weight function W(f) may be proportional to the inverse of the allowed deviation. 
     I.e. the graph  520  illustrates the constraints which are used during the design of the filter  102 . In a similar manner to  FIG. 4   b , no constraints are imposed on the frequency response of the filter  102  within the “don&#39;t care” intervals  412 ,  413 ,  414 . In order to take into account the frequency evolution of the absolute threshold of hearing, the allowed deviation from the target frequency response within the example “care” interval  415  is associated with the ATH curve  505 . As such, the segments  525 - 1 ,  525 - 2 ,  525 - 3  of the allowed deviation from the target frequency response of filter  102  within the “care” interval  415  are derived from the ATH curve  505 . 
     It should be noted that frequency ranges for which the ATH value exceeds a certain level, i.e. frequency ranges which cannot reasonably comprise audible frequency components of the output signal  113 , may be declared as “don&#39;t care” intervals, thereby removing further constraints during the design of filter  102  in this particular frequency range. This is illustrated in segment  525 - 1  (associated with a frequency range of approx 20-24 kHz), where the ATH value is very high. Consequently, the number of degrees of freedom on the frequency response of filter  102  can be increased. 
     As illustrated in  FIG. 5   b , the absolute threshold of hearing affects the way that frequencies of the output signal  113  are perceived by a listener. Consequently, the ATH curve  505  should be reflected in the relevant frequency diagram of the output signal  113 . This is illustrated in graph  520  of  FIG. 5   c , where the dotted lines  528 ,  529  reflect the frequencies corresponding to 2× and 4× the Nyquist frequency of 24 kHz of the output signal  113 . These dotted lines  528 ,  529  are minoring lines of the images of the upsampled signals  111  or  112  prior to the down-sampling of the down-by-M decimator  103  (similar to the mirroring lines at 40 kHz and 80 kHz depicted in  FIG. 2 ). 
     In view of the above, the ATH curve  505  is fitted into the frequency diagram of the upsampled output signal while taking into account the images created due to the (imaginary) up-by-M upsampling (i.e. prior to the down-by-M decimation). As such, the allowed deviation from the target frequency response of the filter  102  in the “care” frequency interval  415 ,  416 ,  417  is derived from images of the ATH curve  505  in the frequency diagram of the upsampled output signal, i.e. in the frequency diagram mirrored in accordance to the Nyquist frequency of the output signal  113 . This is shown in  FIG. 5   c  where the segment  525 - 2  of the allowed deviation from the target frequency response  531  corresponds to a scaled and mirrored version of the ATH curve  505  ranging from frequencies 0 kHz to approx. 20 kHz. The segment  525 - 3  of the allowed deviation from the target frequency response  532  corresponds to a scaled version of the ATH curve  505  at the frequencies 0 kHz to 17 kHz. 
     Segment  525 - 3  is adjacent to segment  523  corresponding to the “don&#39;t care” interval  513 . The fact that the target frequency response  532  is left blank within the segment  523  indicates that no constraints are imposed on the target frequency response  532  within the “don&#39;t care” interval  513 . 
     Overall, allowed deviations from the target frequency response  531 ,  532  in the stop band are obtained which is made up of a succession of scaled and possibly mirrored images of the ATH curve  505 . These allowed deviations from the target frequency response  531 ,  532  may be interrupted by “don&#39;t care” frequency intervals. Using the allowed deviation from the target frequency response  531 ,  532  as an input to filter design methods such as the Parks-McClellan algorithm provide the coefficients of an anti-aliasing filter  102 . The resulting frequency response  430  of filter  102  is shown in graph  520  of  FIG. 5   c . It can be seen that the stop band rejection in the proximity of the mirroring lines  528 ,  529  is very high, thereby significantly attenuating the images and aliases in the frequency ranges which are well perceived by a human being. At the same time, the stop band rejection is relatively weak for frequency ranges which are not well perceived by a human being. 
     In  FIG. 6  a comparison between the anti-aliasing filter  601  designed using a allowed deviation derived from the ATH curve  505  and a conventional anti-aliasing filter  611  is shown. Both filters  601 ,  611  have the same number of filter coefficients (60 coefficients). It can be seen that filter  601  exhibits a significantly higher stop band rejection than filter  611  in frequency ranges associated with low ATH values (i.e. in frequency ranges associated with a relatively high auditory sensitivity). As a result, the perceived quality of the resampled output signal is improved when using filter  601 . 
     The effect of the psychoacoustic resampler  100  using an anti-aliasing filter  601  can be seen in  FIG. 7 , where the frequency diagram  700  of a resampled output signal  113  is depicted. The abscissa indicates the time. The input signal  110  corresponds to a sinusoid which changes from a frequency of 0 kHz (at the time of 0 seconds) to 17 kHz (at the time of 10 seconds). As such, the abscissa actually indicates the frequency of the input signal  110 . The ordinate indicates the frequency spectrum of the resampled output signal  113  at a particular time instant. It can be seen that the frequency spectrum of the output signal  113  mainly comprises the sinusoid  701  of the input signal  110 . In addition, the output signal  113  comprises frequency components  702  at higher frequencies above 15 kHz. These frequency components  702  are due to imaging/aliasing of the resampler  100 . However, due to the psychoacoustic design of the anti-aliasing filter  102  having the frequency response  601  these frequency components  702  are not perceived by a human listener. 
     In the following further aspects are outlined which may be taken into consideration when designing the anti-aliasing filter  102 . For this purpose, the imaging and aliasing caused by rational resampling is analyzed in further detail from a mathematical perspective. 
     As outlined above, fs in  and fs out  are the input and output sampling rates, respectively. It has been outlined in the context of  FIG. 1  that the resampling by rational ratios M/N=fs in /fs out  (with M, N being mutually prime), can be implemented by an upsampling by N step  101 , followed by a filter  102 , followed by a downsampling by M step  103 . In the course of upsampling, N−1 images of the original signal are created, at multiples of the original sampling rate fs in . That is, a component at baseband frequency f bb ε[−fs in /2;fs in   /2] will be imaged to components at frequencies f   bb +n·fs in  (n=1 . . . N−1). If one were to downsample by the same ratio M=N , all these images would alias back into the original component, i.e. upsampling and downsampling (by the same factor M=N) are inverse processes. 
     A triangle wave function T(x)=|frac(x+½)−½| may be defined, where the function “frac(.)” denotes the fractional part of its argument. Such a triangle wave function is illustrated in  FIG. 8 . The triangle wave function T(x) can be used to illustrate the imaging and aliasing operations of the upsampling and the downsampling, respectively. In particular, the triangle wave function T(x) can be used to illustrate that frequencies in the baseband f bb ε[−fs in /2;fs in /2], which correspond to the abscissa interval x bb ε[−0.5;0.5] of the triangle wave function T(x), are imaged to frequencies f bb +n·fs in , which correspond to x bb +n of the triangle wave function T(x). 
     On the other hand, looking at an image frequency f=f bb +n·fs in , the triangle wave function T(x) can be used to derive the baseband frequency f bb , that the image frequency f originated from 
         f   bb   =fs   in   ·T ( f/fs   in ).  (1)
 
     Likewise, during downsampling to the output sampling rate fs out , the image frequency f aliases back into the baseband via the function f al =fs out ·T(f/fs out ), wherein f al  is the alias component (in the baseband of the output signal  113 ) originating from image frequency f. Varying the parameter f and plotting the periodically varying baseband component f bb  against the alias component f al  produces the Lissajous-like  FIG. 9  (also referred to herein as the bifrequency plot). 
     The bifrequency plot  900  of  FIG. 9  is illustrated for the example of fs in =40 Hz and fs out =48 Hz , i.e. for a rational resampler with N=6 and M=5 . The bifrequency plot  900  shows several aspects of the resampling process. First, it shows that for every input frequency (plotted along the abscissa), there are N−1=6−1=5 images. The images of each input frequency can be seen when following the diagonal line  901  starting at the point (0 kHz, 0 kHz) up to the point (20 kHz, 20 kHz), then turning left and following the line  901  along the arrow to point (16 kHz, 24 kHz), then turning left along the arrow to point (0 kHz, 8 kHz), and so on, to the point (0 kHz, 24 kHz). The line  901  represents the frequency axis of the upsampled signal  111  and goes from 0 kHz at the beginning to 120 kHz at the end. As such, line  901  corresponds to the frequency axis of  FIG. 2  folded into the bifrequency plot  900 . As shown in  FIG. 3 , the upsampling by N creates 5 images (e.g. images  312 ,  313 ,  314 ,  315 ,  316 ) from an input frequency  310 , i.e. an input frequency  310  appears N times in the upsampled signal  111 . In particular, images of an input frequency f bb  appear at the frequencies f bb +n·fs, for n=1, . . . , N/2−1 and −f bb +n·fs in  for n=1, . . . , N/2. 
     The bifrequency plot  900  also shows that every frequency in the output domain can be an alias of M=5 frequencies in the input domain, one being the same frequency in the input signal itself. This had already been shown in  FIG. 3 , where frequency  321  in the output domain receives aliasing contributions from input frequencies  322 ,  323 ,  3224 ,  325  plus input frequency  321  (corresponding to the output frequency  321 ). 
     Furthermore, points  911 ,  912 ,  913 ,  914  can be identified in the bifrequency map  900 , where the alias component of an image of the input frequency coincides with the input frequency. These points  911 ,  912 ,  913 ,  914  may be referred to as self-masking points. By way of example, point  911  is positioned at coordinates (4 kHz, 4 kHz) in the bifrequency diagram  900 . Line  901  (corresponding to the frequency axis of the upsampled signal  111 ) traverses point  911  twice, a first time at 4 kHz and a second time at 44 kHz. This means that not only the original input frequency f bb =4 kHz , but also its image f bb +fs in =44 kHz contributes to the output frequency f=4 kHz. In a similar manner, at point  912 , not only the original input frequency f bb =8 kHz , but also its image f bb +2fs in =88 kHz contributes to the output frequency f=8 kHz . In view of the fact that the input audio signals are real signals, their frequency spectrum is symmetrical. Consequently, not only the original input frequency f bb =16 kHz , but also its image −f bb +2fs in =64 kkHz contributes to the output frequency f=16 kHz (point  914 ), as well as, not only the original input frequency f bb =12 kHz , but also its image −f bb +3fs in =108 kkHz contributes to the output frequency f=12 kHz (point  913 ). 
     In addition, one can identify points where two aliases which are due to images of the same input frequency coincide with one another (while not coinciding with the original input frequency). These frequencies may require particular attenuation by the filter  102  because the two aliasing components might constructively interfere. These points may be referred to as self interference points. By way of example, point  921  at frequency coordinates (4 kHz, 12 kHz) is traversed twice by the frequency axis  901  of the upsampled signal  111 , once at 36 kHz (i.e. at −f bb +fs, with f bb =4 kHz) and a second time at 84 kHz (i.e. f bb +2fs in  with f bb =4 kHz). As can be seen, the two images of the baseband frequency at 4 kHz contribute to the output frequency at 12 kHz, i.e. the images of the baseband frequency f bb  contribute to an output frequency f al  which is different from f bb . 
     In order to further analyze the generation of images during the upsampling operation  101 , it is assumed in the following that M=1. As such, the relation between the input sampling rate fs in  and the output sampling rate is fs out =N·fs in ·Let τ abs (f) be the absolute threshold of hearing (ATH) at frequency f, such that a tone with a signal level lower than τ abs (f) will not be audible. A mathematical approximation of the ATH curve  505  has been provided in the context of  FIG. 5   a.    
     Considering the images at frequencies f=n fs in /2, n=1 . . . N, it is the purpose of the anti-imaging (upsampling) filter  102  to reduce the images to a level that will not be audible. To determine whether an image with energy level L at frequency f will be audible, the signal level L at frequency f can be compared to the ATH curve τ abs (f). If L&lt;τ abs (f), the image will not be audible. 
     At the time of designing the filter  102 , the signal level of the baseband audio signal  110  it typically not known, and thus the signal level L of the images is not known. The higher the signal level of the input signal  101 , the more attenuation is required by filter  102 . However, the assumption can be made that the signal level of the baseband audio signal is below the threshold of pain τ p (f bb ) (where f bb  is given by equation 1). By way of example, the threshold of pain τ p (f bb ) can be approximated by a frequency independent constant of 120 dB SPL. In view of the above assumptions, the attenuation of the filter should be equal to or better than τ abs (f)/τ p (f bb ), i.e. the magnitude of the frequency response of the filter  102  should be 
         |H ( f )| 2 &lt;τ abs ( f )/τ p ( f   bb ).  (2)
 
     A possible way of designing such a filter  102  may be the above described Parks-McClellan algorithm, with pass band gain of 1 and a stop band gain of 0, i.e. with a target frequency response 
     
       
         
           
             
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     as outlined above. The linear error weighting function W (f) within the stop band may be set To 
         W ( f )=√{square root over (τ p ( f   bb )/τ abs ( f ))}{square root over (τ p ( f   bb )/τ abs ( f ))}.
 
       FIG. 10  shows the frequency response of a psychoacoustic filter  102  obtained in accordance with the above mentioned method. The filter is used for removing the images of input signals  110  with an input sampling rate fs in =12 kHz and an upsampling factor N=4. The pass band weighting has been set such that the pass band ripple stays below 0.1 dB. Furthermore, the error weighting function has been modified such that in the frequency interval above approx. 18 kHz, no unconstrained amplification occurs due to the high values of the ATH curve  505  (possibly except for the “don&#39;t care” region, where no energy can be aliased). In addition, it has been assumed that the audio signal is already band limited to below 5.5 kHz (i.e. at 90% of the Nyquist rate), thereby providing “don&#39;t care” intervals. 
     Likewise, when downsampling an input signal by a factor M (i.e. N=1) from an input sampling rate fs in  to an output sampling rate fs out =fs in /M, the threshold of hearing curve  505  may be used to derive the allowed deviations from the ideal stop band suppression. However, for audio applications, the audio input signal  110  is already in the audible frequency range. Consequently, also the downsampled output signal  113  is in the audible frequency range. Therefore, the potential of using the high thresholds of hearings for high frequencies in a pure downsampling scenario is limited. 
     As has been outlined in the context of  FIG. 9 , certain input frequencies f bb  create images f bb +nfs in  and/or −f bb +nfs in  during the upsampling operation  101  which are aliased back to the same output frequencies f al , with f al =f bb , during the downsampling operation  103 . As such, the alias component of the output signal is directly dependent on the input signal components at the same frequency. This observation may be used to exploit perceptual self-masking effects during the filter design. 
     Let μ(f, f 0 ) be the relative masking threshold afforded by a single tone with signal level L at frequency f 0 . That is, a tone with level L′ at frequency f where L′&lt;L·μ(f, f 0 ) will not be audible. An approximation of the relative masking threshold curve (also referred to as tone masking tone, TMT) may be given by 
     
       
         
           
             
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     wherein the Bark scale can be approximated by 
       Bark( f )=13·atan(0,76·10 −3   f )+3,5·atan(0,13·10 −3   f ) 2 .
 
     The Bark scale is illustrated in  FIG. 11   a  and the relative masking threshold curve  1100  for relative frequency deviations from the masker frequency f 0  is illustrated in  FIG. 11   b . It can be seen that a frequency f which is close to the masker frequency f 0  is only audible, if its signal level exceeds the signal level L at frequency f 0  minus 27 dB. 
     In the following, a signal component of the upsampled signal  110  at frequency f ε [0; N·fs in /2] in the intermediate, i.e. upsampled, frequency domain is considered. This component at frequency f is an image of the baseband signal component at f bb =fs in ·T(f/fs in ). During the downsampling process, it will be aliased to a component at frequency f al =fs out ·T(f/fs out ) of the output signal  113 . 
     Let L al =L·|H(f)| 2  be the signal level of the alias component at frequency f al . The alias component at f al  may be subject to masking from the component at f bb  that it originated from via the upsampling-filtering-downsampling process. This may be the case, if the alias component at f al  originated from an input signal component at f bb , wherein f bb ≈f al . In order to exploit the masking from the component at f bb , i.e. in order to ensure that the alias component at f al  is not audible the signal level of the alias component should be L al =L·|H| 2 ≦L·μ(f al , f bb ), i.e. 
         |H|   2 ≦μ( f   al   ,f   bb ).  (3)
 
     As can be seen in  FIG. 11   b , the relative masking threshold curve μ(f al, f   bb ) will reach its maximum of −27 dB at the points of self masking, i.e. at points where an alias component at f al  is masked by a frequency component which originates from a baseband component at f bb , with f al =f bb . 
     As outlined in the context of  FIG. 9 , self-masking occurs in the vicinity of the frequencies of the self masking points  911 ,  912 ,  913 ,  914 . As has been outlined above, the self masking points  911 ,  912 ,  913 ,  914  have the characteristic that an image of the spectrum of the input signal at a given frequency aliases to the spectrum of the output signal at the same given frequency. As such, the aliased image is subjected to the masking of the original spectrum of the input signal at the given frequency. This finding can be used for defining allowed deviations of the frequency response of the anti-aliasing filter  102  during the filter design. In particular, the attenuation performed by the filter  102  on the respective images of the original spectrum can be reduced due to the self-masking effect. 
     This is illustrated in  FIG. 11   c , where the allowed deviations for an example filter  102  in a 40 kHz to 48 kHz resampler are illustrated.  FIG. 11   c  illustrates the allowed deviations  1101 ,  1102 ,  1103 ,  1104  which are due to the self-masking effect. It can be seen that the allowed deviations  1101 ,  1102 ,  1103 ,  1104  are positioned around the frequencies 44 kHz, 64 kHz, 88 kHz, 108 kHz, respectively. These frequencies in the intermediate domain correspond to the frequencies associated with the self-masking points  911 ,  914 ,  912 ,  913 , respectively. These frequencies may be referred to as the maskee frequencies (in the intermediate frequency domain). The frequency which a maskee frequency aliases to, may be referred to as the corresponding baseband masker frequency. The form of the allowed deviations  1101 ,  1102 ,  1103 ,  1104  corresponds to the relative self-masking threshold curve illustrated in  FIG. 11   b , however, illustrated on a linear scale instead of a Bark scale. 
     Furthermore,  FIG. 11   c  illustrates the allowed deviations  1111 ,  1112 ,  1113 ,  1114 ,  1115  which are due to the ATH curve  505 . As such, various perceptual contributions to the allowed deviations of a frequency response from the target frequency response have been identified. The allowed deviations due to the ATH curve  505  may be described using equation (2), whereas the allowed deviations due to the self-masking threshold curve  1100  may be described using equation (3). Making the assumption that the alias will be inaudible if its signal level is below either the absolute threshold of hearing or the masking threshold, provides the following constraint on the magnitude of the transfer function (i.e. the allowed deviations from the target frequency response) 
         |H ( f ) 2 ≦max(τ abs ( f   al )/τ p ( f   BB ),μ( f   al   , f   BB )).
 
     The stop band gain may be restricted to unity or lower in order to avoid amplification of aliases.  FIG. 12  illustrates the resulting filter frequency response  1201  using the above mentioned function of allowed deviations  1202 . The illustrated filter  102  requires 10 taps per phase (i.e. 60 coefficients overall). It can be seen that the frequency response  1201  follows well the allowed deviations  1202  provided by the perceptual frequency response associated with the ATH curve  505  and the self-masking threshold curve  1100 . 
     In  FIG. 13  an example method  1300  for designing a psychoacoustic filter for the re-sampling of audio signals is illustrated. The method  1300  comprises a plurality of steps  1301  to  1304  related to the specification of a target frequency response. In step  1301  a cut-off frequency (and/or a pass band edge/stop band edge) of the target frequency response is selected. As outlined in the present document, the cut-off frequency may be selected as the smaller one of the Nyquist rate of the input audio signal and the Nyquist rate of the output audio signal. Furthermore, in step  1302  a pass band of the target frequency response is selected. The selection of the pass band comprises the selection of the frequency range of the pass band, as well as the selection of a target attenuation of the target frequency response within the pass band. 
     In step  1303  “don&#39;t care” intervals of the stop band are identified. The “don&#39;t care” intervals may be due to frequency intervals of the input audio signal having low energy, i.e. having an energy value below an energy threshold. Furthermore, the “don&#39;t care” intervals may be due to spectral images and/or mirrored images of such low energy frequency ranges of the input audio signal. Alternatively or in addition, the “don&#39;t care” intervals may be due to frequency ranges associated with a low auditory spectral sensitivity of a human listener. As outlined in the present document, images and/or mirrored images of such frequency ranges may be selected as “don&#39;t care” intervals of the stop band. 
     In step  1304  images or mirrored images of a perceptual frequency response indicative of the auditory spectral sensitivity of a human listener are assigned to the stop band, in particular to the frequency ranges of the stop band outside of the “don&#39;t care” intervals. The perceptual frequency response may be associated with a scaled and/or shifted version of the absolute threshold of hearing curve, thereby attributing different degrees of attenuation to different frequencies of the stop band. Alternatively or in addition, the perceptual frequency response may be associated with the self-masking threshold curve at particular frequencies. 
     As a result of steps  1301  to  1304  a target frequency response and allowed deviations from this target frequency response of the psychoacoustic filter  102  have been determined. Within the “don&#39;t care” intervals, the allowed deviation from the target frequency response takes on arbitrary or undefined values. Outside the “don&#39;t care” intervals of the stop band, the allowed deviation from the target frequency response is associated with a perceptual frequency response indicative of the auditory spectral sensitivity of a human listener. In step  1305  the coefficients of the filter are determined using filter design methods such as a Parks-McClellan algorithm. Such filter design methods determine the filter coefficients such that the frequency response of the resulting filter is fitted to the target frequency response, while taking into account the allowed deviation from the target frequency response. 
     In the present document, a method and system for designing psychoacoustic anti-aliasing filters has been described. The resulting filters may be used to implement psychoacoustic resamplers  100  which perform rational resampling at reduced computational complexity and/or at improved perceptional quality. 
     The methods and systems described in the present document may be implemented as software, firmware and/or hardware. Certain components may e.g. be implemented as software running on a digital signal processor or microprocessor. Other components may e.g. be implemented as hardware and or as application specific integrated circuits. The signals encountered in the described methods and systems may be stored on media such as random access memory or optical storage media. They may be transferred via networks, such as radio networks, satellite networks, wireless networks or wireline networks, e.g. the internet. Typical devices making use of the methods and systems described in the present document are portable electronic devices or other consumer equipment which are used to store and/or render audio signals. The methods and system may also be used on computer systems, e.g. internet web servers, which store and provide audio signals, e.g. music signals, for download.