Patent Publication Number: US-7596490-B2

Title: Low bit-rate audio encoding

Description:
The present invention relates to encoding and decoding of broadband signals, in particular audio signals. The invention relates both to the encoder and the decoder, and to an audio stream encoded according to the invention and a data storage medium on which such an audio stream has been stored. 
   When transmitting broadband signals, e.g. audio signals such as speech, compression or encoding techniques are used to reduce the bandwidth or bit rate of the signal. 
     FIG. 1  shows a known parametric encoding scheme, in particular a sinusoidal encoder, which is used in the present invention, and which is described in WO 01/69593. In this encoder, an input audio signal x(t) is split into several (possibly overlapping) time segments or frames, typically of duration 20 ms each. Each segment is decomposed into transient, sinusoidal and noise components. It is also possible to derive other components of the input audio signal such as harmonic complexes, although these are not relevant for the purposes of the present invention. 
   In the sinusoidal analyzer  130  of  FIG. 1  the signal x 2  for each segment is modeled using a number of sinusoids represented by amplitude, frequency and phase parameters. This information is usually extracted for an analysis time interval by performing a Fourier transform (FT) which provides a spectral representation of the interval including: frequencies, amplitudes for each frequency, and phases for each frequency, where each phase is “wrapped”, i.e. in the range {−π;π}. Once the sinusoidal information for a segment is estimated, a tracking algorithm is initiated. This algorithm uses a cost function to link sinusoids in different segments with each other on a segment-to-segment basis to obtain so-called tracks. The tracking algorithm thus results in sinusoidal codes C S  comprising sinusoidal tracks that start at a specific time instance, evolve for a certain duration of time over a plurality of time segments and then stop. 
   In such sinusoidal encoding, it is usual to transmit frequency information for the tracks formed in the encoder. This can be done in a simple manner and with relatively low costs, since tracks only have slowly varying frequency. Frequency information can therefore be transmitted efficiently by time differential encoding. In general, amplitude can also be encoded differentially over time. 
   In contrast to frequency, phase changes more rapidly with time. If the frequency is constant, the phase will change linearly with time, and frequency changes will result in corresponding phase deviations from the linear course. As a function of the track segment index, phase will have an approximately linear behavior. Transmission of encoded phase is therefore more complicated. However, when transmitted, phase is limited to the range {−π;π}, i.e. the phase is “wrapped”, as provided by the Fourier transform. Because of this modulo 2π representation of phase, the structural inter-frame relation of the phase is lost and, at first sight appears to be a random variable. 
   However, since the phase is the integral of the frequency, the phase is redundant and needs, in principle, not be transmitted. This is called phase continuation and reduces the bit rate significantly. 
   In phase continuation, only the first sinusoid of each track is transmitted in order to save bit rate. Each subsequent phase is calculated from the initial phase and frequencies of the track. Since the frequencies are quantized and not always very accurately estimated, the continuous phase will deviate from the measured phase. Experiments show that phase continuation degrades the quality of an audio signal. 
   Transmitting the phase for every sinusoid increases the quality of the decoded signal at the receiver end, but it also results in a significant increase in bit rate/bandwidth. Therefore, a joint frequency/phase quantizer, in which the measured phases of a sinusoidal track having values between −π and π are unwrapped using the measured frequencies and linking information, results in monotonically increasing unwrapped phases along a track. In that encoder the unwrapped phases are quantized using an Adaptive Differential Pulse Code Modulation (ADPCM) quantizer and transmitted to the decoder. The decoder derives the frequencies and the phases of a sinusoidal track from the unwrapped phase trajectory. 
   In phase continuation, only the encoded frequency is transmitted, and the phase is recovered at the decoder from the frequency data by exploiting the integral relation between phase and frequency. It is known, however, that when phase continuation is used, the phase cannot be perfectly recovered. If frequency errors occur, e.g. due to measurement errors in the frequency or due to quantization noise, the phase, being reconstructed using the integral relation, will typically show an error having the character of drift. This is because frequency errors have an approximately random character. Low-frequency errors are amplified by integration, and consequently the recovered phase will tend to drift away from the actually measured phase. This leads to audible artifacts. 
   This is illustrated in  FIG. 2   a  where Ω and ψ are the real frequency and real phase, respectively, for a track. In both the encoder and decoder frequency and phase have an integral relationship as represented by the letter “I”. The quantization process in the encoder is modeled as an added noise n. In the decoder, the recovered phase {circumflex over (ψ)} thus includes two components: the real phase ψ and a noise component ε 2 , where both the spectrum of the recovered phase and the power spectral density function of the noise ε 2  have a pronounced low-frequency character. 
   Thus, it can be seen that in phase continuation, since the recovered phase is the integral of a low-frequency signal, the recovered phase is a low-frequency signal itself. However, the noise introduced in the reconstruction process is also dominant in this low-frequency range. It is therefore difficult to separate these sources with a view to filtering the noise n introduced during encoding. 
   In conventional quantization methods, frequency and phase are quantized independent of each other. In general, a uniform scalar quantizer is applied to the phase parameter. For perceptual reasons the lower frequencies should be quantized more accurately than the higher frequencies. Therefore the frequencies are converted to a non-uniform representation using the ERB or Bark function and then quantized uniformly, resulting in a non-uniform quantizer. Also physical reasons can be found: in harmonic complexes, higher harmonic frequencies tend to have higher frequency variations than the lower frequencies. 
   When the frequency and phase are quantized jointly, frequency dependent quantization accuracy is not straightforward. The use of a uniform quantization approach results in a low quality sound reconstruction. 
   The choice of initial quantization accuracy, i.e. the quantization accuracy, which is also referred to as the quantization grid, that is used for quantizing the first element of a track, used in the phase ADPCM quantizer, is a balance between the following two cases: 
   the speed with which an unwrapped phase that is difficult to predict can be followed. An example of this is a track whose frequency is changing rapidly; and 
   the accuracy with which an unwrapped phase that is easy to predict can be followed. An example of this is a track whose frequency is nearly constant. 
   If the initial quantization grid is too fine, the phase ADPCM quantizer may be incapable of following the unwrapped phase when it is difficult to predict. If this is the case, large quantization errors are made in a track, and audible distortions are introduced. This leads to an increase in bit rate. If, on the other hand, the initial quantization grid is too coarse, switching-on oscillations can occur in easily predictable tracks, as indicated in  FIG. 7 , where the frequency of the original track changes step-like. In this Figure, the original frequency is estimated with an accuracy of about 1.9 Hz. The oscillations of the estimated frequency can be audible, which is undesired. 
   The invention provides a method of encoding a broadband signal, in particular an audio signal such as a speech signal, using a low bit-rate. In the sinusoidal encoder a number of sinusoids are estimated per audio segment. A sinusoid is represented by frequency, amplitude and phase. Traditionally, phase is quantized independent of frequency. The invention gives a significant improvement in decoded signal quality, especially for low bit-rate quantizers. 
   According to the invention, a track is encoded with a suitable initial quantization grid that is chosen among a set of possible initial grids. These initial grids vary from fine to coarse. Good results are obtained with just two possible initial grids, but several grids can be used. If, in a series of time segments the frequency variation in a particular track is smaller than a predetermined value, the track is quantized using a finer quantization grid. This method avoids the problem of oscillations in  FIG. 7 . Information regarding the choice of the initial grid needs to be sent to the decoder. 
   This results in the advantage of transmitting phase information with a low bit rate while still maintaining good phase accuracy and signal quality at all frequencies. The advantage of this method is improved phase accuracy and thus improved sound quality, especially when only a small number of bits are used for quantizing the phase and frequency values. On the other hand, a required sound quality can be obtained using fewer bits. 

   
       FIG. 1  shows a prior art audio encoder in which an embodiment of the invention is implemented; 
       FIG. 2   a  illustrates the relationship between phase and frequency in prior art systems; 
       FIG. 2   b  illustrates the relationship between phase and frequency in audio systems according to the present invention; 
       FIGS. 3   a  and  3   b  show a preferred embodiment of a sinusoidal encoder component of the audio encoder of  FIG. 1 ; 
       FIG. 4  shows an audio player in which an embodiment of the invention is implemented; and 
       FIGS. 5   a  and  5   b  show a preferred embodiment of a sinusoidal synthesizer component of the audio player of  FIG. 4 ; 
       FIG. 6  shows a system comprising an audio encoder and an audio player according to the invention; and 
       FIG. 7  illustrates an example of an original frequency track and two estimations by the phase ADPCM quantizer with different quantization grids. 
   

   Preferred embodiments of the invention will now be described with reference to the accompanying drawings wherein like components have been accorded like reference numerals and, unless otherwise stated, perform like functions. In a preferred embodiment of the present invention, the encoder  1  is a sinusoidal encoder of the type described in WO 01/69593,  FIG. 1 . The operation of this prior art encoder and its corresponding decoder has been well described and description is only provided here where relevant to the present invention. 
   In both the prior art and the preferred embodiment of the present invention, the audio encoder  1  samples an input audio signal at a certain sampling frequency resulting in a digital representation x(t) of the audio signal. The encoder  1  then separates the sampled input signal into three components: transient signal components, sustained deterministic components, and sustained stochastic components. The audio encoder  1  comprises a transient encoder  11 , a sinusoidal encoder  13  and a noise encoder  14 . 
   The transient encoder  11  comprises a transient detector (TD)  110 , a transient analyzer (TA)  111  and a transient synthesizer (TS)  112 . First, the signal x(t) enters the transient detector  110 . This detector  110  estimates if there is a transient signal component and its position. This information is fed to the transient analyzer  111 . If the position of a transient signal component is determined, the transient analyzer  111  tries to extract (the main part of) the transient signal component. It matches a shape function to a signal segment preferably starting at an estimated start position, and determines content underneath the shape function, by employing for example a (small) number of sinusoidal components. This information is contained in the transient code C T , and more detailed information on generating the transient code C T  is provided in WO 01/69593. 
   The transient code C T  is furnished to the transient synthesizer  112 . The synthesized transient signal component is subtracted from the input signal x(t) in subtractor  16 , resulting in a signal x 1 . A gain control mechanism GC ( 12 ) is used to produce x 2  from x 1 . 
   The signal x 2  is furnished to the sinusoidal encoder  13  where it is analyzed in a sinusoidal analyzer (SA)  130 , which determines the (deterministic) sinusoidal components. It will therefore be seen that while the presence of the transient analyzer is desirable, it is not necessary and the invention can be implemented without such an analyzer. Alternatively, as mentioned above, the invention can also be implemented with for example a harmonic complex analyzer. In brief, the sinusoidal encoder encodes the input signal x 2  as tracks of sinusoidal components linked from one frame segment to the next. 
   Referring now to  FIG. 3   a , in the same manner as in the prior art, in the preferred embodiment, each segment of the input signal x 2  is transformed into the frequency domain in a Fourier transform (FT) unit  40 . For each segment, the FT unit provides measured amplitudes A, phases φ and frequencies ω. As mentioned previously, the range of phases provided by the Fourier transform is restricted to −π≦φ&lt;π. A tracking algorithm (TA) unit  42  takes the information for each segment and by employing a suitable cost function, links sinusoids from one segment to the next, so producing a sequence of measured phases φ(k) and frequencies ω(k) for each track. 
   In contrast to the prior art, the sinusoidal codes C S  ultimately produced by the analyzer  130  include phase information, and frequency is reconstructed from this information in the decoder. 
   As mentioned above, however, the measured phase is wrapped, which means that it is restricted to a modulo 2π representation. Therefore, in the preferred embodiment, the analyzer comprises a phase unwrapper (PU)  44  where the modulo 2π phase representation is unwrapped to expose the structural inter-frame phase behavior ψ for a track. As the frequency in sinusoidal tracks is nearly constant, it will be seen that the unwrapped phase ψ will typically be a nearly linearly increasing (or decreasing) function and this makes cheap transmission of phase, i.e. with low bit rate, possible. The unwrapped phase ψ is provided as input to a phase encoder (PE)  46 , which provides as output quantized representation levels r suitable for being transmitted. 
   Referring now to the operation of the phase unwrapper  44 , as mentioned above, instantaneous phase ψ and instantaneous frequency Ω for a track are related by:
 
ψ( t )=∫ T     0     l Ω(τ) dτ+ψ ( T   0 )  (1)
 
where T 0  is a reference time instant.
 
   A sinusoidal track in frames k=K, K+1 . . . K+L−1 has measured frequencies ω(k) (expressed in radians per second) and measured phases φ(k) (expressed in radians). The distance between the centers of the frames is given by U (update rate expressed in seconds). The measured frequencies are supposed to be samples of the assumed underlying continuous-time frequency track Ω with ω(k)=Ω(kU) and, similarly, the measured phases are samples of the associated continuous-time phase track ψ with φ(k)=ψ(kU) mod (2π). For sinusoidal encoding it is assumed that Ω is a nearly constant function. 
   Assuming that the frequencies are nearly constant within a segment Equation 1 can be approximated as follows: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         ψ 
                         ⁡ 
                         
                           ( 
                           kU 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ∫ 
                             
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               U 
                             
                             kU 
                           
                           ⁢ 
                           
                             
                               Ω 
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                                 ( 
                                 t 
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                             ⁢ 
                             
                               ⅆ 
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                           ψ 
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                             ( 
                             
                               
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                               ⁢ 
                               U 
                             
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                       ≈ 
                         
                       ⁢ 
                       
                         
                           
                             { 
                             
                               
                                 ω 
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                                   k 
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                 ( 
                 2 
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   It will therefore be seen that knowing the phase and frequency for a given segment and the frequency of the next segment, it is possible to estimate an unwrapped phase value for the next segment, and so on for each segment in a track. 
   In the preferred embodiment, the phase unwrapper determines an unwrap factor m(k) at time instant k:
 
ψ( kU )=φ( k )+ m ( k )2π  (3)
 
   The unwrap factor m(k) tells the phase unwrapper  44  the number of cycles which has to be added to obtain the unwrapped phase. 
   Combining equations 2 and 3, the phase unwrapper determines an incremental unwrap factor e(k) as follows:
 
2 πe ( k )=2 π{m ( k )− m ( k− 1)}={ω( k )+ω( k− 1)} U/ 2−{φ( k )−φ( k− 1)}
 
where e should be an integer. However, due to measurement and model errors, the incremental unwrap factor will not be an integer exactly, so:
 
 e ( k )=round([{ω( k )+ω( k− 1)} U/ 2−{φ( k )−φ( k− 1)}]/(2π))
 
assuming that the model and measurement errors are small.
 
   Having the incremental unwrap factor e, the m(k) from equation (3) is calculated as the cumulative sum where, without loss of generality, the phase unwrapper starts in the first frame K with m(K)=0, and from m(k) and φ(k), the (unwrapped) phase ψ(kU) is determined. 
   In practice, the sampled data ψ(kU) and Ω(kU) are distorted by measurement errors:
 
φ( k )=ψ( kU )+ε 1 ( k ),
 
ω( k )=Ω( kU )+ e   2 ( k ),
 
where ε 1  and ε 2  are the phase and frequency errors, respectively. In order to prevent the determination of the unwrap factor becoming ambiguous, the measurement data needs to be determined with sufficient accuracy. Thus, in the preferred embodiment, tracking is restricted so that:
 
δ( k )= e ( k )−[{ω( k )+ω( k− 1)} U/ 2−{φ( k )−φ( k− 1)}]/(2π)&lt;δ 0  
 
where δ is the error in the rounding operation. The error δ is mainly determined by the errors in ω due to the multiplication with U. Assume that ω is determined from the maxima of the absolute value of the Fourier transform from a sampled version of the input signal with sampling frequency F s  and that the resolution of the Fourier transform is 2π/L a  with L a  the analysis size. In order to be within the considered bound, we have:
 
   
     
       
         
           
             
               L 
               a 
             
             U 
           
           = 
           
             δ 
             0 
           
         
       
     
   
   That means that the analysis size should be few times larger than the update size in order for unwrapping to be accurate, e.g., setting δ 0 =¼, the analysis size should be four times the update size (neglecting the errors ε 1  in the phase measurement). 
   The second precaution, which can be taken to avoid decision errors in the round operation, is to defining tracks appropriately. In the tracking unit  42 , sinusoidal tracks are typically defined by considering amplitude and frequency differences. Additionally, it is also possible to account for phase information in the linking criterion. For instance, we can define the phase prediction error ε as the difference between the measured value and the predicted value {tilde over (φ)} according to:
 
ε={φ( k )−{tilde over (φ)}( k )} mod 2π
 
where the predicted value can be taken as:
 
{tilde over (φ)}( k )=φ( k− 1)+{ω( k )−ω( k− 1)} U/ 2
 
   Thus, preferably the tracking unit  42  forbids tracks where ε is larger than a certain value (e.g. ε&gt;π/2), resulting in an unambiguous definition of e(k). 
   Additionally, the encoder may calculate the phases and frequencies such as will be available in the decoder. If the phases or frequencies which will become available in the decoder differ too much from the phases and/or frequencies such as are present in the encoder, it may be decided to interrupt a track, i.e. to signal the end of a track and start a new one using the current frequency and phase and their linked sinusoidal data. 
   The sampled unwrapped phase ψ(kU) produced by the phase unwrapper (PU)  44  is provided as input to phase encoder (PE)  46  to produce the set of representation levels r. Techniques for efficient transmission of a generally monotonically changing characteristic such as the unwrapped phase are known. In the preferred embodiment,  FIG. 3   b , Adaptive Differential Pulse Code Modulation (ADPCM) is employed. Here, a predictor (PF)  48  is used to estimate the phase of the next track segment and encode the difference only in a quantizer (Q)  50 . Since ψ is expected to be a nearly linear function and for reasons of simplicity, the predictor  48  is chosen as a second-order filter of the form:
 
 y ( k+ 1)=2 x ( k )− x ( k− 1)
 
where x is the input and y is the output. It will be seen, however, that it is also possible to take other functional relations (including higher-order relations) and to include adaptive (backward or forward) adaptation of the filter coefficients. In the preferred embodiment, a backward adaptive control mechanism (QC)  52  is used for simplicity to control the quantizer  50 . Forward adaptive control is also possible as well but would require extra bit rate overhead.
 
   As will be seen, initialization of the encoder (and decoder) for a track starts with knowledge of the start phase φ(0) and frequency ω(0). These are quantized and transmitted by a separate mechanism. Additionally, the initial quantization step used in the quantization controller  52  of the encoder and the corresponding controller  62  in the decoder,  FIG. 5   b , is either transmitted or set to a certain value in both encoder and decoder. Finally, the end of a track can either be signaled in a separate side stream or as a unique symbol in the bit stream of the phases. 
   The start frequency of the unwrapped phase is known, both in the encoder and in the decoder. On basis of this frequency, the quantization accuracy is chosen. For the unwrapped phase trajectories beginning with a low frequency, a more accurate quantization grid, i.e. a higher resolution, is chosen than for an unwrapped phase trajectory beginning with a higher frequency. 
   In the ADPCM quantizer, the unwrapped phase ψ(k), where k represents the number in the track, is predicted/estimated from the preceding phases in the track. The difference between the predicted phase {tilde over (ψ)}(k) and the unwrapped phase ψ(k) is then quantized and transmitted. The quantizer is adapted for every unwrapped phase in the track. When the prediction error is small, the quantizer limits the range of possible values and the quantization can become more accurate. On the other hand, when the prediction error is large, the quantizer uses a coarser quantization. 
   The quantizer Q in  FIG. 3   b  quantizes the prediction error Δ, which is calculated by:
 
Δ( k )=ψ( k )−{tilde over (ψ)}( k )
 
   The prediction error Δ can be quantized using a look-up table. For this purpose, a table Q is maintained. For example, for a 2-bit ADPCM quantizer, the initial table for Q may look like the table shown in Table 1. 
   
     
       
         
             
           
             
               TABLE 1 
             
           
          
             
                 
             
             
               Quantization table Q used for first continuation. 
             
          
         
         
             
             
             
          
             
               Index i 
               Lower boundaries bl 
               Upper boundary bu 
             
             
                 
             
          
         
         
             
             
             
          
             
               0 
               −∞ 
               −3.0 
             
             
               1 
               −3.0 
               0 
             
             
               2 
               0 
               3.0 
             
             
               3 
               3.0 
               ∞ 
             
             
                 
             
          
         
       
     
   
   The quantization is done as follows. The prediction error A is compared to the boundaries b, such that the following equation is satisfied:
 
bl 1 &lt;Δ≦bu 1  
 
   From the value of i, that satisfies the above relation, the representation level r is computed by r=i. 
   The associated representation levels are stored in representation table R, which is shown in Table 2. 
   
     
       
         
             
           
             
               TABLE 2 
             
           
          
             
                 
             
             
               Representation table R used for first continuation 
             
          
         
         
             
             
             
             
          
             
                 
               Representation level r 
               Representation table R 
               Level type 
             
             
                 
                 
             
          
         
         
             
             
             
             
          
             
                 
               0 
               −3.0 
               Outer level 
             
             
                 
               1 
               −0.75 
               Inner level 
             
             
                 
               2 
               0.75 
               Inner level 
             
             
                 
               3 
               3.0 
               Outer level 
             
             
                 
                 
             
          
         
       
     
   
   The entries of tables Q and R are multiplied by factor c for the quantization of the next sinusoidal component in the track.
 
 Q ( k+ 1)= Q ( k )· c  
 
 R ( k+ 1)= R ( k )· c  
 
   During the decoding of a track, both tables are scaled according to the generated representation levels r. If r is either 1 or 2 (inner level) for the current sub-frame, then the scale factor c for the quantization table is set to:
 
c=2 −1/4  
 
   Since c&lt;1, the frequency and phase of the next sinusoid in a track becomes more accurate. If r is 0 or 3 (outer level), the scale factor is set to:
 
c=2 1/2  
 
   Since c&gt;1, the quantization accuracy for the next sinusoid in a track decreases. Using these factors, one up-scaling can be made undone by two down-scalings. The difference in upscale and downscale factors results in a fast onset of an up-scaling, whereas a corresponding downscaling requires two steps. 
   In order to avoid very small or very large entries in the quantization table, the adaptation is only done if the absolute value of the inner level is between π/64 and 3π/4. In that case c is set to 1. 
   In the decoder only table R has to be maintained to convert to received representation levels r to a quantized prediction error. This de-quantization operation is performed by block DQ in  FIG. 5   b.    
   Using the above settings, the quality of the reconstructed sound needs improvement. In accordance with the invention, different initial tables for unwrapped phase tracks, depending on the start frequency, are used. Hereby a better sound quality is obtained. This is done as follows. The initial tables Q and R are scaled on basis a first frequency of the track. In Table 3, the scale factors are given together with the frequency ranges. If the first frequency of a track lies in a certain frequency range, the appropriate scale factor is selected, and the tables R and Q are divided by that scale factor. The end-points can also depend on the first frequency of the track. In the decoder, a corresponding procedure is performed in order to start with the correct initial table R. 
   
     
       
         
             
           
             
               TABLE 3 
             
           
          
             
                 
             
             
               Frequency dependent scale factors and initial tables 
             
          
         
         
             
             
             
             
          
             
                 
               Scale 
                 
                 
             
             
               Frequency range 
               factor 
               Initial table Q 
               Initial table R 
             
             
                 
             
          
         
         
             
             
             
             
             
          
             
               0-500 
               Hz 
               8 
               −∞ −0.19 0 0.19 ∞ 
               −0.38 −0.09 0.09 0.38 
             
             
               500-1000 
               Hz 
               4 
               −∞ −0.37 0 0.37 ∞ 
               −0.75 −0.19 0.19 0.75 
             
             
               1000-4000 
               Hz 
               2 
               −∞ −0.75 0 0.75 ∞ 
               −1.5 −0.38 0.38 1.5 
             
             
               4000-22050 
               Hz 
               1 
               −∞ −1.5 0 1.5 ∞ 
               −3 −0.75 0.75 3 
             
             
                 
             
          
         
       
     
   
   Table 3 shows an example of frequency dependent scale factors and corresponding initial tables Q and R for a 2-bit ADPCM quantizer. The audio frequency range 0-22050 Hz is divided into four frequency sub-ranges. It is seen that the phase accuracy is improved in the lower frequency ranges relative to the higher frequency ranges. 
   The number of frequency sub-ranges and the frequency dependent scale factors may vary and can be chosen to fit the individual purpose and requirements. Like described above, the frequency dependent initial tables Q and R in table 3 may be up-scaled and down-scaled dynamically to adapt to the evolution in phase from one time segment to the next. 
   In e.g. a 3-bit ADPCM quantizer, the initial boundaries of the eight quantization intervals defined by the 3 bits can be defined as follows: 
   Q={−∞−1.41−0.707−0.35 0 0.35 0.707 1.41 ∞}, and can have minimum grid size π/64, and a maximum grid size π/2. The representation table R may look like: 
   R={−2.117, −1.0585, −0.5285, −0.1750, 0.1750, 0.5285, 1.0585, 2.117}. A similar frequency dependent initialization of the table Q and R as shown in Table 3 may be used in this case. 
   From the sinusoidal code C S  generated with the sinusoidal encoder, the sinusoidal signal component is reconstructed by a sinusoidal synthesizer (SS)  131  in the same manner as will be described for the sinusoidal synthesizer (SS)  32  of the decoder. This signal is subtracted in subtractor  17  from the input x 2  to the sinusoidal encoder  13 , resulting in a remaining signal x 3 . The residual signal x 3  produced by the sinusoidal encoder  13  is passed to the noise analyzer  14  of the preferred embodiment which produces a noise code C N  representative of this noise, as described in, for example, international patent application No. PCT/EP00/04599. 
   Finally, in a multiplexer  15 , an audio stream AS is constituted which includes the codes C T , C S  and C N . The audio stream AS is furnished to e.g. a data bus, an antenna system, a storage medium etc. 
     FIG. 4  shows an audio player  3  suitable for decoding an audio stream AS′, e.g. generated by an encoder  1  of  FIG. 1 , obtained from a data bus, antenna system, storage medium etc. The audio stream AS′ is de-multiplexed in a de-multiplexer  30  to obtain the codes C T , C S  and C N . These codes are furnished to a transient synthesizer  31 , a sinusoidal synthesizer  32  and a noise synthesizer  33  respectively. From the transient code C T , the transient signal components are calculated in the transient synthesizer  31 . In case the transient code indicates a shape function, the shape is calculated based on the received parameters. Further, the shape content is calculated based on the frequencies and amplitudes of the sinusoidal components. If the transient code C T  indicates a step, then no transient is calculated. The total transient signal y T  is a sum of all transients. 
   The sinusoidal code C S  including the information encoded by the analyzer  130  is used by the sinusoidal synthesizer  32  to generate signal y S . Referring now to  FIGS. 5   a  and  5   b , the sinusoidal synthesizer  32  comprises a phase decoder (PD)  56  compatible with the phase encoder  46 . Here, a de-quantizer (DQ)  60  in conjunction with a second-order prediction filter (PF)  64  produces (an estimate of) the unwrapped phase {circumflex over (ψ)} from: the representation levels r; initial information {circumflex over (φ)} (0), {circumflex over (ω)}(0) provided to the prediction filter (PF)  64  and the initial quantization step for the quantization controller (QC)  62 . 
   As illustrated in  FIG. 2   b , the frequency can be recovered from the unwrapped phase {circumflex over (ψ)} by differentiation. Assuming that the phase error at the decoder is approximately white, and since differentiation amplifies the high frequencies, the differentiation can be combined with a low-pass filter to reduce the noise and, thus, to obtain an accurate estimate of the frequency at the decoder. 
   In the preferred embodiment, a filtering unit (FR)  58  approximates the differentiation, which is necessary to obtain the frequency {circumflex over (ω)} from the unwrapped phase by procedures as forward, backward or central differences. This enables the decoder to produce as output the phases {circumflex over (ψ)} and frequencies {circumflex over (ω)} usable in a conventional manner to synthesize the sinusoidal component of the encoded signal. 
   At the same time, as the sinusoidal components of the signal are being synthesized, the noise code C N  is fed to a noise synthesizer NS  33 , which is mainly a filter, having a frequency response approximating the spectrum of the noise. The NS  33  generates reconstructed noise y N  by filtering a white noise signal with the noise code C N . The total signal y(t) comprises the sum of the transient signal y T  and the product of any amplitude decompression (g) and the sum of the sinusoidal signal y S  and the noise signal y N . The audio player comprises two adders  36  and  37  to sum respective signals. The total signal is furnished to an output unit  35 , which is e.g. a speaker. 
     FIG. 6  shows an audio system according to the invention comprising an audio encoder  1  as shown in  FIG. 1  and an audio player  3  as shown in  FIG. 4 . Such a system offers playing and recording features. The audio stream AS is furnished from the audio encoder to the audio player over a communication channel  2 , which may be a wireless connection, a data bus  20  or a storage medium. In case the communication channel  2  is a storage medium, the storage medium may be fixed in the system or may also be a removable disc, a memory card or chip or other solid-state memory. The communication channel  2  may be part of the audio system, but will however often be outside the audio system. 
   The encoded data from several consecutive segments are linked. This is done as follows. For each segment a number of sinusoids are determined (for example using an FFT). A sinusoid consists of a frequency, amplitude and phase. The number of sinusoids per segment is variable. Once the sinusoids are determined for a segment, an analysis is done to connect to sinusoids from the previous segment. This is called ‘linking’ or ‘tracking’. The analysis is based on the difference between a sinusoid of the current segment and all sinusoids from the previous segment. A link/track is made with the sinusoid in the previous segment that has the smallest difference. If even the smallest difference is larger than a certain threshold value, no connection to sinusoids of the previous segment is made. In this way a new sinusoid is created or “born”. 
   The difference between sinusoids is determined using a ‘cost function’, which uses the frequency, amplitude and phase of the sinusoids. This analysis is performed for each segment. The result is a large number of tracks for an audio signal. A track has a birth, which is a sinusoid that has no connection with sinusoids from the previous segment. A birth sinusoid is encoded non-differentially. Sinusoids that are connected to sinusoids from previous segments are called continuations and they are encoded differentially with respect to the sinusoids from the previous segment. This saves a lot of bits, since only differences are encoded and not absolute values. 
   In accordance with the invention, if e.g. a set of two possible initial grids is used for each track, one bit has to be transmitted to the decoder indicating which one of the two initial grids was actually used. In the encoder, the frequencies along a track are examined to determine a frequency difference that is compared to a predetermined threshold. If the difference exceeds the threshold, a coarse grid is chosen, otherwise a finer grid is chosen. The frequency difference can be the numerical difference between frequencies or another statistical quantity than the difference, such as the standard deviation. 
   This improves the audio quality. Correspondingly, if a set of four possible initial grids is used for each track, two bits have to be transmitted to the decoder indicating which one of the four initial grids was used, etc. Typically, a bit rate of 300 bits/s is associated with this method, for the encoder described in [1] operating at a bit rate of 12500 bit/s. However the bit rate can be reduced by the following method of the invention, whilst the audio quality is maintained. 
   In the Encoder, Tracks that are Both: 
   a) at least a predetermined number of frames, e.g. 5 frames, long, and 
   b) have a difference between the highest and lowest frequency in the second up to the fifth frame that is smaller than a predetermined value, 
   are encoded with an initial quantization grid that is finer, e.g. two times finer, than the initial quantization grid that is used for the remaining tracks that do not fulfill the above two conditions a) and b). 
   Preferably, in frames that have at least one initialization of a track that is at least a predetermined number of frames, e.g. 5 frames, long, one of the following conditions will apply: 
   none of the tracks in the frame was encoded using a fine quantization grid. In this case a ‘0’ is sent to the decoder, and no further information needs to be sent to the decoder; or 
   at least one track was encoded using a fine quantization grid. In this case a ‘1’ is sent to the decoder, and for every track that is at least a predetermined number of frames, e.g. 5 frames, long, it is indicated whether it is encoded with a fine or a coarse initial quantization grid. The decoder can use the tracking information to determine which tracks have a length of at least the predetermined number of frames. 
   Applied in the encoder the above encoding method enables the decoder to decide if tracks were encoded with a fine or a coarse initial quantization grid. 
   When applying the method of the invention to the encoder described in [1], about 100 bit/s are required at a total bit rate of 12500 bit/s. The gain in bit rate between the bit-rate reduced version (100 bit/s) and the normal version (300 bit/s) of the method of the invention can increase substantially when more than two initial grids are employed. 
   REFERENCE 
   
       
       Gerard Hotho and Rob Sluijter. A low bit rate audio and speech sinusoidal coder for narrowband signals. In Proc. 1st IEEE Benelux workshop on MPCA-2002, pages 1-4, Leuven, Belgium, Nov. 15, 2002.