Lossless band splitting and band joining with all-pass filter

Abstract

To provide a method and a device for lossless band splitting and band joining of a stream of a signal sample using all-pass filtering.SOLUTION: A band split operation reformats the original stream into two intermediate streams that represent even and odd samples of the original stream, and then the matrix filters the two intermediate streams to provide two output substreams that represent the high and low frequency components of the original stream. Conversely, the band join arithmetic matrix filters two subband streams to provide two quantized intermediate substreams, and then provide the output stream by interleaving the filtered stream, and the intermediate substreams are even and odd samples of the output stream.SELECTED DRAWING: None

Description

Technical Field

[0001] The present invention relates to the processing of sampled signals, and more particularly to lossless band splitting and band joining of such signals.

Background Art

[0002] Many applications require that a sampled signal be split into two or more frequency bands to create sub-band signals that can be processed or transmitted separately at a lower sampling rate and then recombined to create a signal at the full sampling rate. Polyphase filtering networks (including quadrature mirror filters) for performing splitting and joining have been the subject of extensive research. Signal artifacts potentially introduced by band-splitting methods include passband ripple and aliasing, but for designs where the ripple is zero and for transmission applications where the sub-band signals are provided without being transformed by the final band-joining filter, designs are known where the alias products present in the sub-band signals are canceled in the final recombination.

[0003] The term "lossless" is often used in the communications literature to denote such designs, but in such literature, perfect arithmetic is assumed, and even designs so named may or may not provide exact reconstruction in the presence of arithmetic rounding errors. In this document, we adopt the syntax of the audio literature, where "lossless" implies exact one-bit reconstruction of a signal that has already been quantized. Thus, a lossless decoder must reverse any arithmetic errors or quantization made by the encoder.

[0004] The "lifting" technique has often been used to achieve lossless processing, and a band splitting / joining architecture using lifting is described in "Wavelet Transforms That Map Integers to Integers", Applied And Computational Harmonic Analysis 5, 332-369 (1998) by A. R. Calderbank, I. Daubechies, W. Sweldens, and B-L. Yeo, particularly FIGS. 4 and 5 therein. For an encoder to separate a sampled signal into low frequency (LF) and high frequency (HF), and then for the corresponding decoder to combine those bands, such an architecture generally requires each of the encoder and decoder to implement two finite impulse response (FIR) filters. The filters can be inconveniently long and may require a number of taps that is inversely proportional to the width of the transition between the LF and HF bands. Also, a 2FIR design does not provide LF and HF responses that are mirror images about half the Nyquist frequency, and at least three FIR filters are required for each of the encoder and decoder to achieve higher symmetry.

[0005] Another type of band splitting and joining in the communication literature uses IIR filtering. IIR filters can generally achieve a steeper slope than FIR filters with a given number of arithmetic operations, but the IIR band splitting and joining filters in the literature do not achieve lossless reconstruction. For example, in "Efficient Design of Low Delay IIR QMF Banks for Speech Subband Coding" by Kleinmann T and Lacroix A in Proceedings of EUSIPCO-96 Eighth European Signal Processing Conference Trieste, Italy, 10-13 September 1996, the reconstructed amplitude response is flat, but the group delay increases around the crossover frequency. Thus, this scheme does not become lossless even if implemented without quantization error.

SUMMARY OF THE INVENTION

PROBLEMS TO BE SOLVED BY THE INVENTION

[0006] Therefore, what is needed is an economical IIR architecture that provides lossless reconstruction. For applications where the encoder transmits the LF and HF bands separately to consumer products, it is particularly desirable to minimize the computational complexity of the decoder.

MEANS FOR SOLVING THE PROBLEM

[0007] In a first aspect, the present invention provides a method of splitting an original stream of quantized signal samples having an original sample rate into two output sub-streams of quantized signal samples having half of the original sample rate, the two output sub-streams representing higher and lower frequency components of the original stream respectively, the method comprising the steps of reformatting the original stream into two intermediate streams representing even and odd samples of the original stream respectively, and providing the two output sub-streams by filtering and matrix operations on the two intermediate streams, the steps of performing the filtering and matrix operations comprising creating a quantized signal having samples using a quantizer, creating the samples of the quantized signal in reverse time order, and creating the samples of the quantized signal based on feedback obtained from previously created samples of the quantized signal, each output sub-stream being associated with each intermediate stream by a respective transfer function having a maximum phase pole.

[0008] Feedback is used to create poles in the transfer function, which allows for frequency discrimination with coefficients of 2, 3. Making the poles maximum phase improves the prior art of Kleinmann and Lacroix by allowing a causal band joiner to remove phase distortion. The reverse direction operation on the samples allows for stable implementation of filtering with maximum phase poles.

[0009] Preferably, for any output sub-stream, the transfer functions from both intermediate sub-streams have the same DC gain magnitude. This ensures that the use of sum and difference matrix operations ensures that the band splitter directs DC purely to one output and the Nyquist frequency purely to the other output.

[0010] According to an embodiment, the step of performing the filtering and matrix operation includes processing overlapping blocks of samples of the two intermediate sub-streams, discarding the last part of each processed block of samples corresponding to the overlap with other blocks, and combining the remaining parts of each processed block of samples.

[0011] In this way, the band splitter according to the present invention can process audio in the forward direction as a whole and can perform operations on blocks in the reverse time direction locally. The overlap and discard enable the transient states that occur when the processing of each block starts to dissipate before reaching the section that affects the band splitter output.

[0012] Preferably, the two output sub-streams jointly contain the information required to enable the original quantized stream to be accurately restored by a properly initialized band joiner.

[0013] In this way, the operation can be accurately reversed, enabling a system with band split, lossless transmission of each band, and band join to be lossless as a whole.

[0014] It is preferable that two separate input streams do not create the same output sub-stream and residual state in the filter.

[0015] In this way, the information about the signal samples is not lost in the operations of the band splitter, because each possible set of outputs is created by at most one stream of the inputs. As a result, the band splitter can be described as lossless. The filter state needs to be included in the comparison because filtering spreads the influence of the input over time.

[0016] According to an embodiment, the steps of the filtering and matrix operations include creating two filtered intermediate streams by filtering two intermediate streams, and creating the two output sub-streams by performing matrix operations on the filtered intermediate streams.

[0017] In this way, the implementation example can use two filtering operations to improve the implementation efficiency with simple matrix operations. However, the trade-off may, in some cases, result in the opposite, and a lower-order all-pass filter will be used.

[0018] Preferably, performing the matrix operations is executed using sum and difference matrices.

[0019] Preferably, the output sub-stream is obtained from the quantized signal by an inverse-transformable linear process without further quantization.

[0020] In this way, the band splitter can operate using only one quantization in the signal path, and the quantization noise at the band splitter output is lower.

[0021] Since the feedback is obtained from the quantization, subsequent processes can clearly determine the quantized signal, and thus can determine the feedback from the output sub-stream. Knowledge of the feedback is important for the state variables in the band joiner to accurately follow those in the band splitter.

[0022] The present invention in a second aspect provides a band splitter configured to execute the method of the first aspect.

[0023] The present invention in a third aspect provides a recording medium including data obtained based on the high-frequency output and the low-frequency output of the band splitter according to the second aspect.

[0024] This way, the recorded medium can be provided to both consumers who use a band joiner to reconstruct a full-bandwidth audio copy and consumers who enjoy reduced-bandwidth audio without using a band joiner.

[0025] The present invention in a fourth aspect provides a method of combining two sub-band streams of quantized signal samples, each having a sub-band sample rate, the method providing an output stream of quantized signal samples having twice the sub-band sample rate, the output stream having higher and lower frequency components respectively represented by the two sub-band streams, the method comprising providing two quantized intermediate sub-streams by performing matrix operations and filtering on the two sub-band streams, and interleaving the two quantized intermediate sub-streams to provide the output stream such that the intermediate sub-streams are respectively even and odd samples of the output stream, each intermediate sub-stream being associated with a respective sub-band stream by a respective transfer function that is an infinite impulse response (IIR) including a maximum phase zero, and the steps of matrix operations and filtering being configured to ensure that the output stream includes information required to enable the quantized signal samples of each sub-band stream to be accurately restored by a band splitter that is appropriately initialized, including quantization.

[0026] This way, the operation of the prior art Kleinmann and Lacroix band joiner is improved to ensure that it can accurately invert the operation of the band splitter according to the present invention. First, the phase distortion of the combination of the prior art band splitter and band joiner is removed. Second, although the sub-band streams created by the band splitter according to the present invention necessarily contain quantization noise, appropriate quantization in the band joining method cancels out the noise introduced by the band split quantization rather than adding additional noise.

[0027] Preferably, for any sub-band stream, the transfer functions for both intermediate streams have the same DC gain magnitude. In this way, the use of the sum and difference matrices can ensure that the DC at the output comes purely from one input and the Nyquist frequency comes purely from the other output.

[0028] Note that to aid understanding, if the operation of the band joiner can be inverted by a subsequent band splitter, the operation of the band splitter can also be inverted by the same band joiner placed behind the band splitter.

[0029] In one embodiment, the step of performing matrix operations and filtering on the two sub-band streams includes performing matrix operations on the two sub-band streams to create two matrix-operated sub-streams, and creating the two quantized intermediate sub-streams by filtering the two matrix-operated sub-streams with two different quantization filters respectively.

[0030] In this way, the implementation example can obtain higher implementation efficiency by using simple matrix operations and two filtering operations. However, the trade-off may, in some cases, result in the opposite outcome and require the use of a lower-order all-pass filter.

[0031] In one embodiment, the step of performing matrix operations includes quantization. This operates to invert a less preferred band splitter embodiment where the matrix operation is performed after filtering and incorporates additional quantization.

[0032] Preferably, the quantization included within the signal processing loop is performed by a vector quantizer.

[0033] In this way, the band joiner can losslessly invert the operation of a preferred embodiment of a band splitter where its output substream is obtained from a quantized signal without further quantization.

[0034] Preferably, the filtering step is characterized by two different all - pass responses.

[0035] In this way, the discrimination of the band splitter is derived from the range of 90 - degree differential phase shifts exhibited by the two all - passes. This leads to effective discrimination with few coefficients.

[0036] In one embodiment, the first all - pass response has coefficients within 1.0 and 0.527864045 -15 and the second all - pass response has coefficients within 1.0 and 0.105572809. -15

[0037] In one embodiment, the first all - pass response has coefficients within 1.0, within 0.3644245374 -15 and within 0.01036373471, -15 and the second all - pass response has coefficients within 1.0, within 0.8365625224 -15 and within 0.09327361235. -15

[0038] In this way, a ripple - free band splitter transfer function suitable for applications where band - split audio is listened to is obtained from a first - order or second - order all - pass. Practical realizations require rounding non - unity coefficients, but the -15 tolerance of 2 corresponds to rounding to the common coefficient size of signed 16 - bit.

[0039] In a fifth aspect, the present invention provides a band splitter, an input configured to receive an input stream of signal samples at a sample rate, two outputs configured to supply two output streams, each output stream having a sampling rate that is half of the sampling rate of the input stream, a deinterleaving unit having an input, two outputs, and the two outputs, the input of the deinterleaving unit being coupled to the input of the band splitter, the outputs of the deinterleaving unit including the even samples and the odd samples of the input stream respectively, a deinterleaving unit, two all-pass filters each having a first input and an output, the first input of each all-pass filter being coupled to the respective output of the deinterleaving unit, an all-pass filter, and a lossless sum and difference unit having two inputs and two outputs, each of the inputs to the sum and difference unit being coupled to the respective output of the two all-pass filters, and each of the outputs of the sum and difference unit being coupled to the respective output of the band splitter, each all-pass filter being configured to receive samples of the input stream in reverse chronological order, to provide a band splitter.

[0040] In this way, the operations of taking the sum and difference of the all-pass filters enable good discrimination with few coefficients. The operation in reverse chronological order enables an all-pass filter in which the maximum phase poles are stably realized. These can be inverted by a causal all-pass filter having minimum phase poles within the corresponding band joiner, and no phase or amplitude error occurs due to splitting into bands.

[0041] In one embodiment, the band splitter also includes a quantizer, and each all-pass filter has a second input configured to receive feedback obtained from the outputs of the sum and difference unit, whereby the sum and difference unit is integrated within the filter.

[0042] Preferably, each all-pass filter has a second input configured to receive feedback obtained from the outputs of the sum and difference units, whereby the sum and difference units are integrated within the filter.

[0043] This allows the band splitter to operate with only one quantization in the signal path, enabling a lower noise approximation for each of the high-frequency and low-frequency components of the original signal at the output of the band splitter.

[0044] Preferably, the band splitter also includes a quantizer, and each all-pass filter includes the previously received samples of the input stream, and the feedback samples previously received by the second input of the all-pass filter, and the current sample received after the previously received input sample, and is configured to supply an output sample equal to the quantized sum of the linear combination of the samples of the input stream up to the current sample.

[0045] In one embodiment, one of the two filters is characterized by an infinite impulse response (IIR) having coefficients 340 / 32768 and 11941 / 32768, and the other all-pass filter is characterized by an IIR having coefficients 3056 / 32768 and 27412 / 32768.

[0046] In this way, the coefficients are used for a second-order all-pass that approximates a ripple-free band splitter transfer function. These values are rounded for a fixed-point implementation with 16-bit coefficients.

[0047] In a preferred embodiment, the band splitter comprises a blocking unit having an input and an output, and a combining unit having an input, wherein the blocking unit receives a stream of samples provided to its input, splits the stream into overlapping blocks of samples, each block having a start point and an end point, and supplies the overlapping blocks at its output, the output of the blocking unit being coupled to the first input of the all-pass filter, the all-pass filter being configured to process in reverse time order within each overlapping block of samples and supply the processed blocks of samples at its output, the output of the all-pass filter being coupled to the input of the combining unit, the combining unit being configured to receive the overlapping processed blocks of samples provided to its input, discard the overlapping portions from each processed block, and combine the remaining portions to supply a continuous stream of processed samples.

[0048] This allows each block of samples to be processed in reverse time, enabling the maximum phase pole to be stably realized. However, consecutive blocks can be processed in normal order, allowing the band split to proceed with a finite look-ahead. The overlap and discard give time for the transients that occur when the band splitter rises in each block to disappear before processing the samples that contribute to the output. Due to the reverse-time processing, these transients occur at the end of each block.

[0049] The present invention in the sixth aspect provides a band joiner, comprising two inputs configured to receive first and second streams of input quantization signals, an output configured to supply an output stream having a sampling rate that is twice the sampling rate of each of the input streams, a sum and difference unit having two inputs and two outputs configured as a sum output and a difference output respectively, two all-pass filters each having a first input and an output, and an interleaving unit having two inputs and two outputs, wherein the inputs of the sum and difference unit are connected to the inputs of the band joiner, the first input of each of the two all-pass filters is connected to the sum output and the difference output of the sum and difference unit respectively, the inputs of the interleaving unit are coupled to the outputs of the all-pass filters, the output of the interleaving unit is coupled to the output of the band joiner, and the band joiner is lossless.

[0050] In this way, the lossless characteristic improves the operation of the Kleinmann and Lacroix band joiner by enabling a system consisting of the band joiner according to the present invention and the band splitter according to the present invention to accurately replicate the input to the band joiner. Thus, not only is the phase distortion of Kleinmann and Lacroix removed, but also the noise introduced by the quantization of the band splitter is removed by the quantization of the band joiner.

[0051] In an embodiment, the sum and difference unit scales one of its inputs by a factor of two before taking the sum and difference.

[0052] In this way, the sum and difference matrices can tolerate a factor-of-two difference in gain at the two inputs resulting from a band splitter using a unit determinant for the sum and difference unit.

[0053] Preferably, the band joiner further comprises a quantizer, and each all-pass filter includes samples previously received by the first input of the all-pass filter and current samples received after the previously received samples, and is configured to supply an output equal to the quantized sum of the linearly combined previously supplied output samples and input samples up to the current sample.

[0054] Preferably, the quantizer is a vector quantizer configured to jointly quantize within both all-pass filters.

[0055] In this way, the band joiner can invert the operations of the band joiner operating in a preferred low-noise mode using a single quantization instead of separate quantizations in matrix operations and filtering.

[0056] Preferably, the band joiner comprises a vector quantizer having two inputs and two outputs, the inputs of the vector quantizer are connected to the respective outputs of the two all-pass filters, the outputs of the vector quantizer are connected to the output of the band joiner, and each all-pass filter has a second input configured to receive feedback obtained based on the output of the vector quantizer.

[0057] Preferably, the band joiner further comprises a quantizer, and each all-pass filter includes samples previously received by the first input of the all-pass filter and current samples received after the previously received samples, and is configured to supply an output equal to the quantized sum of the linearly combined previously supplied samples of feedback and input samples up to the current sample.

[0058] Preferably, the band joiner is configured such that by processing a pair of signals created by a band splitter, the output of the band joiner is a lossless replica of the stream of signal samples received by the band splitter.

[0059] Doing so clearly provides the advantage that the lossless operation of the band joiner has no phase distortion as outlined above and no net quantization noise.

[0060] Preferably, the band joiner includes an all-pass filter having state variables, and if the band joiner is operated twice to provide a first output stream and a second output stream, and the initialization of the state variables is the same, but the input streams received on two occasions are different, then there is a difference between the first output stream and the second output stream, or a difference between the states of the filter after each operation.

[0061] Doing so establishes that the band joiner does not lose information because different inputs can be distinguished after the operation and thus it is lossless.

[0062] In one embodiment, the first all-pass filter is characterized by an IIR having coefficients 340 / 32768 and 11941 / 32768, and the second all-pass filter is characterized by an IIR having coefficients 3056 / 32768 and 27412 / 32768.

[0063] Doing so, the coefficients are used for second-order all-pass filters that approximate the ripple-free band splitter transfer function. These values are rounded for a fixed-point implementation using 16-bit coefficients.

[0064] A seventh aspect of the invention is an encoder comprising a lossless band splitter, and Provided is a transmission system comprising a decoder with a lossless band joiner, wherein the band splitter and the band joiner each include an all-pass filter with a dithered quantizer, and the transmission system also provides synchronized dithering for the quantizer in the band splitter and the quantizer in the band joiner.

[0065] In this way, the quantization of the band splitter can benefit from the use of dithering while the synchronization preserves the lossless behavior of the combined system. These quantizations are audible if the band split signals are listened to directly.

[0066] As will be understood by those skilled in the art, the present invention provides techniques and apparatus for lossless band splitting and band joining of sampled signals that provide lossless reconstruction. Further variations and modifications will become apparent to those skilled in the art in view of the present disclosure.

Brief Description of the Drawings

[0067] Examples of the present invention will be described in detail with reference to the accompanying drawings.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5A

Figure 5B

Figure 6

Figure 7

Figure 8

Figure 9A

Figure 9B

Figure 10

Figure 11

DETAILED DESCRIPTION OF THE INVENTION

[0068] All-pass with time reversal The prior art structure of FIG. 1 reproduced from the above-mentioned paper by Kleinmann and Lacroix is designed to split the input sampled signal 11 into two sub-band signals 9 and 10 sampled at half the original rate, and then recombine them to produce the output signal 12. Typically, the sub-band signal 9 is an "LF" signal predominantly containing low-frequency information from the input signal 11, and the sub-band signal 10 is an "HF" signal predominantly containing high-frequency information from the input signal 11.

[0069] Note that the sum and difference unit 3 reverses the effect of the sum and difference unit 2 and preserves it for a two-fold overall scaling. Units 2 and 3 may be the same. Thus, the operation of FIG. 1 can be written as follows.

[0070] · Signal 11 is separated into even and odd sample streams by the deinterleaving unit 1.

[0071] · The even samples are filtered by filter 5 having a transfer function E 0 and the odd samples are filtered by filter 6 having a transfer function E 1 .

[0072] · The two sum and difference units 2 and 3 both have a null effect save for scaling by 2.

[0073] · Next, the even samples are filtered by filter 7 having a transfer function E 1 and the odd samples are filtered by filter 8 having a transfer function E 0 .

[0074] · The even and odd sample streams are recombined in the interleaving unit 4.

[0075] Thus, the even samples from the deinterleaving unit are filtered by E 0 and then by E 1 , while the odd samples are filtered by E 1 and then by E 0 . Since the filtering is commutative, it is clear that the total effect of FIG. 1 is to scale stream 11 by a factor of 2 in amplitude and filter with the transfer function E 0 .E 1 . There is also a 1-sample delay introduced by the z -1 elements in the deinterleaving and interleaving units.

[0076] If filters 5 and 6 are straight-through paths, i.e., if E 0 = 1 and E 1If it is = 1, signal 10 will have a zero response to the zero - frequency signal component of input 11. Similarly, signal 9 will have a zero response to the original signal component at the Nyquist frequency of signal 11, i.e., half of the sampling frequency. Thus, very low and very high frequencies will be separated. Other frequencies will be separated imperfectly, which is due to the frequency - dependent phase shift created by the "z -1 " delay in the de - interleaving unit. The purpose of filters 5 and 6 is to make it so that a good discrimination between high and low frequencies is maintained over a fairly wide bandwidth by approximately compensating for this phase shift.

[0077] Thus, response E 0 should provide a phase shift corresponding to a phase shift of E 1 that is approximately equal to a delay of one sample period of signal 11 at low frequencies. E 0 and E 1 are realized at half of the original sample frequency. Thus, they must be designed as a pair of all - pass filters whose phase difference is approximately equal to half of the sample frequency at the local sampling frequency. We will present a suitable design shortly, but we first need to touch on the problem that the band - splitter and band - joiner shown in Figure 1 are all - pass and thus have a transfer function (E 0 .E 1 ) that introduces phase distortion. This problem was recognized in the paper by Kleinmann and Lacroix referenced above, but in the practice of communication, some residual phase distortion was considered acceptable and a completely lossless solution was not sought.

[0078] Conceptually, the unwanted transfer function (E 0 .E 1 ) can be corrected using the inverse filter (E 0 .E 1 ). -1 Ignoring for the time being the fact that it is a rather practical difficulty for this inverse filter to be non-causal, in FIG. 2, if the conceptual inverse filter (E 0 .E 1 ) -1 is incorporated into filters 5' and 6', then they will in turn have the conceptual responses E 1 -1 and E 0 -1 respectively.

[0079] A design technique suitable for generating a pair of all-pass filters whose sum and difference exhibit Butterworth, Chebyshev, or elliptical responses is described in "A New Approach to the Realization of Low Sensitivity IIR Digital Filters" by P. P. Vaidyanathan, S. K. Mitra, and Y. Neuvo, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 2, pp. 350-361, April 1986.

[0080] For audio applications where zero ripple is desirable and sharp corners are not, we have found that the following filters are suitable.

[0081] First order

[0082]

Number

[0083] Second order

[0084]

Number

[0085] Here and hereinafter in this written document, z -1represents the delay of one sample at the sub-band sample rate, which is appropriate to implement but different from the notation used by Kleinmann and Lacroix.

[0086] Insert the scale factor 1 / 2, and the low-pass and high-pass responses are given by lopass = (E 1 -1 +E 0 -1 ) / 2, hipass = (E 1 -1 -E 0 -1 ) / 2. It is well known that the time-reverse of an all-pass filter is also its inverse. This can be verified, for example, by substituting z -1 with z, which has the same effect as swapping the numerator and denominator.

[0087] Note that the reverse-time processing is not necessarily unrealistic. In some consumer applications, the encoder separates the audio signal into LF and HF components, which are transmitted separately and combined at the consumer's decoder. Since the pre-encoding of the audio track is usually performed as a file-level process, the reverse-time processing is conceptually no more difficult than the forward-time processing. Thus, the non-causal all-pass filters E 1 -1 and E 0 -1 can be realized as causal filters in reverse time, and lopass = Rev (E 1 +E 0 ) / 2, hipass = Rev (E 1 -E 0 ) / 2.

[0088] The resulting low-pass and high-pass responses are shown in FIGS. 3 for a first-order filter and in FIG. 4 for a second-order filter. The frequencies are scaled, so f = 1 is the crossover frequency, which is equal to the sub-band Nyquist frequency, and f = 2 is equal to the original Nyquist frequency. In this design, the total power is conserved, and the low-pass and high-pass curves are symmetric about f = 1 and are each -3 dB. The first-order high-pass in FIG. 3 attenuates by 38 dB at f = 0.5 and by 70 dB at f = 0.25. The second-order high-pass in FIG. 4 attenuates by 69 dB at f = 0.5 and by 126 dB at f = 0.25. These attenuations would be considered significant given the low computational cost of these designs.

[0089] Assuming exact arithmetic throughout, appropriate initialization would cause the above prescription to provide exact reconstruction by the band joiner of the signal presented to the band splitter. We now consider how filtering can be done when using quantized arithmetic.

[0090] Lossless minimum-phase IIR filtering The well-known "Direct form I" realization of the minimum-phase IIR filter is easily made lossless as shown in "Lossless Coding Method for Waveform Data" of WO 96 / 37048. FIGS. 6c and 6d of this document are reproduced here as FIGS. 5A and 5B respectively. Other figures of this document show several other topologies with the same or similar functions. FIG. 5A shows a first-order lossless IIR filter having a z-transform (1 + A(z -1 )) / (1 + B(z -1 )) and FIG. 5B shows the corresponding inverse filter having a z-transform (1 + B(z -1 )) / (1 + A(z -1 ))

[0091] The input to FIG. 5A is assumed to be quantized at a certain step size, and quantizer 20 quantizes at the same step size, thus ensuring that the output is similarly quantized. The coefficients of filters 21 and 22 have a finite word length, and quantizer 20 also ensures that the recirculating signal does not arbitrarily obtain an arbitrarily long word length through multiplication repeated by the fractional coefficients in filter 22.

[0092] The operations of FIG. 5A are deterministic. Assuming that the input is already quantized as described in WO 96 / 37048 and that the state variables in filters 21' and 22' are initialized to the same values as the state variables of filters 21 and 22, the cascaded connection of FIGS. 5A and 5B reproduces an exact copy of the input to FIG. 5A at the output of FIG. 5B. Depending on the design, this initialization is explicitly performed, while in other designs, depending on the probabilistic convergence between the states of the two filters, it is accepted that the reproduction is not lossless until such convergence is obtained, accepting that such convergence is not obtained until such convergence is obtained.

[0093] Inverse-time realization of non-causal IIR filters We will now show in more detail how a first-order all-pass filter E 0 and its inverse E 0 -1 can be realized. Here

[0094]

Number

[0095] and more simply

[0096]

Number

[0097] where k = 0.527864045, and in particular, the denominator of E 0 is minimum-phase, and thus E 0Ensure that |k| < 1 so that it is a causal filter that is stable and can be realized by standard means.

[0098] We consider the LF path of an encoding-decoding application where an input sequence of sample values {x i} is presented to E 0 -1 in the encoder, a transmitted sequence {y i} is created, and this is then presented to E 0 in the decoder. We require that the output of E 0 be the same input sequence {x i}, which is expressed by the following recursive relation.

[0099]

Number

[0100] To infer the operation of the E 0 -1 filter in the encoder,

[0101]

Number

[0102] Solve. Causality requires that the calculation of the values {y i} be performed in decreasing order of i as indicated by the notation i = n.. 1, which reflects the reverse-time realization of the filter E 0 -1 . To initialize the calculation, the encoder requires the values of y i , i = 1.. n together with the given signal values {x n}. The value of y n can be arbitrarily chosen, for example, as zero. The decoder also requires initialization, and as a simple method, the encoder provides the original values x i , i = 1.. n together with the filtered values {y 1is to transmit. Then, the decoder uses x as the state initialization for the remaining calculations to be executed forward from i = 2, and directly uses x as its first output value. 1 along with using x 1 directly as its first output value.

[0103] Given such initialization, then the decoder can accurately reconstruct the original signal {x i}, although it is subject to the effects of computational rounding errors and truncation of word lengths during transmission. E 1 and E 1 -1 To achieve, a method exactly similar with k = 0.1055728090 can be used.

[0104] Lossless inverse time processing For lossless processing, we assume that the result of multiplying the quantized input sequence {x i} by the fractional coefficient must be quantized. The above recursive relation is replaced as follows.

[0105]

Number

[0106] Here, Q i represents quantization with the same step size as the input sequence {x i}. The transmitted sequence {y i} then also includes values quantized to the same step size. The subscript "i" in "Q i " emphasizes that the quantization Q can be different for each sample, for example, in a dithered quantizer. However, in an encoder-decoder pair, each Q i in the encoder must be identical to the corresponding Q i in the decoder, which is usually achieved by the same pseudo-random number sequence generator synchronized between the encoder and the decoder in the case of dithering.

[0107] It is not required that the quantized value be an integer multiple of the step size. That is, it may be advantageous to use a quantizer with a random offset as described in the co-pending patent application PCT / GB2015 / 050910. As another generalization, if Q I is a vector quantizer, the signals {x i} and {y i} can be vector quantized.

[0108] Blockwise inverse-time encoder processing In both the unquantized case and the lossless case, the exact reconstruction of the complete output sequence {x n} requires, for example, the initialization of the decoder state by the value x 1 .

[0109] In an unquantized process using exact arithmetic, if an exact initialization cannot be provided, a transient error proportional to the impulse response of E 0 occurs, which is a decaying exponential function when E 0 is first order, and more generally, a linear combination including a decaying sine wave. This transient error decreases rapidly as i increases and generally becomes insignificant after 2 or 3 samples or 20 - 30 samples.

[0110] In a "lossless" quantized process, an inaccurate initialization results in a similar initial transient error. Once the transient state has settled, this error appears like noise until the state of the decoding filter E 0 synchronizes with the encoding state. For a higher order filter, this state synchronization may never occur, but for the second order filter E 0 considered in this document, synchronization is not achieved even after 120 sample periods after the initial transient state has settled and the error has become noise-like when using a quantizer with appropriate dithering. -12We speculated that it exists with a probability less than. For the second-order filters discussed here, the initial transient state takes about 30 samples to attenuate by 96 dB and about 45 samples to attenuate by 144 dB. Thus, these filters can be said to settle into a state independent of initialization after 165 samples with almost complete certainty.

[0111] The reason for this is that it can be applied here to inverse-time filtering. If a block of 1165 samples taken from the start of a longer file is filtered in inverse time, the first 1000 filtered samples will thus be the same with almost complete certainty, because the first 1000 samples of the entire file are filtered in inverse time. Therefore, inverse-time filtering of the entire file is unnecessary, and this file can be processed in overlapping blocks of at least 165 samples. These blocks can be processed in any order, particularly in the forward direction, or in parallel, with inverse-time filtering being used within each block and the last 165 samples of each block being discarded. This principle allows for live processing of the sample stream, although it is subject to the effects of the delay introduced by block processing and overlap.

[0112] The estimate of 165 samples is that two quantized filters are 2 15Based on the extrapolation of Figure 6 for 100,000 trials initialized in states corresponding to signal values that are randomly chosen and of different orders of quantization steps. This filter is second-order and has coefficients k1 = 0.8365625224 and k2 = 0.09327361235 as given before, and each of their respective quantizers is dithered with the same "RPDF" dither having a rectangular probability density function and a peak-to-peak amplitude equal to 1 quantization step. Figure 6 is a histogram of the time it takes for the two quantizers to align. The vertical axis is the logarithm of the number of trials with a base of 10, and the horizontal axis is the time in the sample period. In most trials, the quantizers take about 30 samples to synchronize, and the number of unsynchronized ones then decreases by about one-tenth every 10 sample periods.

[0113] Second-order recursive relation For reference, the recursive relation shown previously is also extended to second-order filtering. As an example, E 0 The numerical representation of is

[0114]

Number

[0115] and this is

[0116]

Number

[0117] which can be expressed as, where k 1 = 0.3644245374 and k 2 = 0.01036373471.

[0118] Here the decoding and encoding equations are

[0119]

Number

[0120] and are, respectively, the conceptual filters E 0 and E 0 -1 corresponding to. The initialization conditions of the encoder are such that any convenient value like zero can be used for the quantities y n-1 and y n which are referenced but not computed. The encoder initializes the decoder by sending the original values x 1 and x 2 along with the filtered values {y i , i = 1..n}. Then the decoder directly uses x 1 and x 2 as its first two output values and also uses them as the state initialization for the remaining calculations made from i = 3 onwards.

[0121] The initialization can, alternatively, be omitted if an exact reconstruction is not required for the first few decoded samples.

[0122] Lossless sum and difference Figure 2 shows. The sum and difference networks 2 in the band splitter and the inverse sum and difference networks 3 in the band joiner are shown. The band splitter and the band joiner are redrawn in Figures 7 and 8, showing the lossless all - pass network 16 incorporating lossless all - pass.

[0123] In the above discussion about implementing non - causal filters, we were satisfied with the configurations of units 2 and 3 to introduce a scaling by 2. However, if we go further for lossless operation, this factor of 2 becomes cumbersome to use because we require that the inputs to filters 7 and 8 be exact lossless replicas of the outputs from filters 5’ and 6’. We present several ways to handle this problem.

[0124] The simplest approach is to incorporate the scaling by 2 into unit 3 such that it becomes the exact inverse of unit 2.

[0125] Therefore, unit 2

[0126]

Number

[0127] calculates, and unit 3

[0128]

Number

[0129] calculates, which is a replication of unit 2 combined with scaling by 0.5.

[0130] However, this implementation is not easy to use as part of a system involving lossless compression of Lf and Hf signals, because E and O are independently quantized values, while L and H are different. Transfer function with determinant -2

[0131]

Number

[0132] Therefore, there is common information in the outputs of L and H (they have a common lsb), and any lossless compression would be inefficient if this redundancy were not exploited. However, exploiting this interesting redundancy is a cumbersome requirement to impose.

[0133] To avoid this problem, the sum and difference unit 2 preferably has a determinant of ±1, and as a reasonable choice, the sum and half the difference are used as follows.

[0134]

Number

[0135] Therefore, unit 3 calculates the following.

[0136]

Number

[0137] The calculation of 0.5(O - E) requires quantization, which introduces additional noise into the Hf output of the band splitter, L = E + O H = O - Q(0.5L) can be done in a lossless way.

[0138] The inverse operation for unit 3 is O = H + Q(0.5L) E = L - O is.

[0139] Integration of all-pass lossless sum and difference By integrating all-pass filtering with the sum and difference operations, it is further possible to reduce the amount of quantization noise in the Lf output. This is particularly beneficial in the system described in WO2013186561, which may be heard by those who do not have access to the bandwidth-expanded data in the Lf output of the band splitter. This also eliminates the need for additional quantization in the Hf audio path.

[0140] This is shown in FIGS. 7 and 8, where the sum and difference operations 2 and 14 are

[0141]

Number

[0142] intended to be realized by. And the inverse sum and difference operations 3, 13, and 15 are

[0143]

Number

[0144] It is intended to achieve. In contrast to the last part, these can now be performed with exact arithmetic.

[0145] Figure 7 shows the reorganization of 5', 6' and 2 in the band splitter. Filter 16 replaces 5' with an all-pass

[0146]

Number

[0147] but quantization is deferred until after the sum and difference operations 2, and feedback is taken therefrom after the additional inverse sum and difference operations. Similarly, filter 17 replaces 6'. The net effect is that vector quantization is performed inside both all-passes and the Lf and Hf signals are quantized separately.

[0148] Figure 8 shows the operation of the band joiner. If Figure 8 is given the output of Figure 7, operation 3 repeats operation 15 of Figure 7, ensuring that the inputs to filters A(z) and B(z) replicate their inputs in Figure 7. If the previous output of Figure 8 has made a copy of the input to Figure 7 and assuming that quantization 31 reduces the same quantization error that quantization 30 added, we can inductively conclude that Figure 8 performs exactly the reverse of the operation of Figure 7.

[0149] Here we consider what conditions must be met for quantizer 31 in order for the quantization error of quantizer 30 to be nullified.

[0150] First, we need to consider the case where the two output values are equidistant from the input value. If quantizer 30 rounds the tie towards -∞, then quantizer 31 must round the tie towards +∞ (since the band joiner quantizer is in the main signal path here, not a quantization side-chain modification as in the cases of Figures 5A and 5B).

[0151] Second, if it is assumed that the inputs and outputs in FIG. 7 are quantized to multiples of the step size Δ, the same also applies to the output from FIG. 8. However, they are obtained from the output of the quantizer 31 by the following sum - inverse and difference matrices.

[0152]

Number

[0153] Rewriting gives the following.

[0154]

Number

[0155] If both E and O are even multiples of Δ, or both are odd multiples of Δ, then L is an even multiple of Δ and H is a multiple of Δ. However, if E and O have opposite parities, then L is an odd multiple of Δ and H is an odd multiple of Δ / 2.

[0156] Therefore, the band joiner in FIG. 8 first quantizes L, and then, depending on whether L is even or odd, quantizes H to a multiple of Δ or a multiple of Δ plus Δ / 2, respectively.

[0157] One way to do this is to add half of the quantized value of L before using a quantizer that quantizes Q H to integer values of Δ, and then subtract it again later. This extension of operation 31 is shown in FIG. 9A and also extends the following operation 13 that realizes the sum - inverse and difference.

[0158]

Number

[0159] The operation at 13 of adding 0.5L cancels the operation at H of subtracting it, and the combined operation simplifies as shown in Figure 9B.

[0160] The alternative perspective shown in Figure 10 is that the operations 14, 31, and 13 of Figure 8 form a vector quantizer 32 that realizes the quantization shown in Figure 11. The dots are the quantized outputs in the EO space. The diagonal squares are the regions quantized to their respective output values. The L and H axes are also shown, and for these axes, the quantization regions are squares and the axes are aligned. However, the alternating L rows are shifted to form a brick pattern.

[0161] Variants of arithmetic operations It will be understood that there are many ways to reconfigure the operations without affecting the essence of the present invention.

[0162] For example, Figure 8 represents the signal path from the input to the quantizer input as follows.

[0163] [Number]

[0164] When multiplied together

[0165] [Number]

[0166] results in. Two filters with sum and difference operations are transformed into four filters with the relevant coefficients on all four paths between L / H and Q L in / Q H in and. Clearly, the essence of the present invention is invariant under such transformations.

Claims

1. 1. A method for combining two subband streams of quantized signal samples, comprising the steps of: each band stream having a subband sample rate; the two subband streams are obtained by applying a lossless band splitting method to a quantized audio signal, The method includes providing an output stream of quantized signal samples having twice a subband sample rate, the output stream having higher and lower frequency components respectively represented by the two subband streams; The method comprises: performing matrixing and filtering on the two subband streams to provide two quantized intermediate substreams; interleaving the two quantized intermediate sub-streams to provide the output stream, such that the intermediate sub-streams are even and odd samples, respectively, of the output stream; Including, Each intermediate substream is related to a respective subband stream by a respective transfer function that is an infinite impulse response (IIR) that includes a maximum phase zero; the steps of matrixing and filtering include quantization that allows the two subband streams of quantized signal samples to be exactly reconstructed applying the lossless band splitting method to the output stream. method.

2. For any given subband stream, the transfer functions for both intermediate streams have the same DC gain magnitude. The method of claim 1.

3. The step of performing a matrix operation and filtering on the two subband streams includes: matrixing the two subband streams to produce two matrixed substreams; 2. The method of claim 1, comprising producing the two quantized intermediate sub-streams by filtering the two matrixed sub-streams with two different quantization filters, respectively.

4. The step of performing a matrix operation includes quantization. The method according to claim 3.

5. The filtering step includes quantization performed by a vector quantizer that jointly quantizes across the two different quantization filters. The method according to claim 3.

6. Each of the transfer functions from each of the two subband streams to each of the two quantized intermediate substreams is an all-pass response. The method of claim 1.

7. The first all-pass response is a square root of 1.0 and 0.527864045. -15 The second allpass response has coefficients in the range of 1.0 and 0.105572809. -15 With coefficients in The method according to claim 6.

8. The first all-pass response is 2, 1.0, 0.3644245374. -15 Within, and 0.01036373471 of 2 -15 The second allpass response has coefficients in the range 1.0, 0.8365625224. -15 Within, and 0.09327361235 of 2 -15 With coefficients in The method according to claim 6.

9. A band joiner configured to carry out the method of any one of claims 1-8.

10. A computer program arranged to cause a processor to carry out the method according to any one of claims 1 to 8 when said computer program is executed by said processor.

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