- Research
- Open Access

# Practical design of delta-sigma multiple description audio coding

- Jack Leegaard
^{1}, - Jan Østergaard
^{1}Email author, - Søren Holdt Jensen
^{1}and - Ram Zamir
^{2}

**2014**:16

https://doi.org/10.1186/1687-4722-2014-16

© Leegaard et al.; licensee Springer. 2014

**Received:**17 December 2013**Accepted:**31 March 2014**Published:**22 April 2014

## Abstract

It was recently shown that delta-sigma quantization (DSQ) can be used for optimal multiple description (MD) coding of Gaussian sources. The DSQ scheme combined oversampling, prediction, and noise-shaping in order to trade off side distortion for central distortion in MD coding. It was shown that asymptotically in the dimensions of the resampling, prediction, and noise-shaping filters as well as asymptotically in the quantizer dimensions, all rate-distortion points on the symmetric quadratic Gaussian MD rate-distortion function could be achieved. In this work, we show that this somewhat theoretical framework is suitable for practical low-delay MD audio coding. In particular, we design a practical MD audio coder with two descriptions and provide simulations on real audio data. The simulations demonstrate that even when using low-dimensional noise-shaping, prediction, and resampling filters, it is possible to obtain good quality audio in the presence of packet losses. Simulations on real audio reveal that, contrary to existing designs, it is straightforward to obtain a large number of trade-off points between side distortion and central distortion, which makes the proposed coder suitable for a wide range of applications.

## Keywords

- Audio coding
- Multiple descriptions
- Noise shaping
- Predictive coding

## 1 Introduction

There is a growing interest in achieving reliable streaming of high-quality audio over networks for digital audio broadcast services, internet radio, youtube, and similar multimedia streaming services. High-quality streaming can be achieved by using, e.g., error-correcting codes and by allowing large delays, large bandwidths, or dedicated/prioritized networks. However, for certain applications, long delays cannot be tolerated. For example, for interactive services such as voice over IP or musicians playing together via the internet, it is crucial that the delay is kept at a minimum. Indeed, for the latter case, it has been noted that delays less than 5 ms are often required [1].

Conventional broadband connections in homes are generally asymmetric in the sense that their downlink capacity is much greater than their corresponding uplink capacity. While this is good for common internet usage such as browsing, it is not ideal for interactive high-quality streaming services, where instead a more symmetric strategy would be advantageous. To reduce the required bandwidth for audio streaming services, it is common to exploit efficient audio compression methods. The *de facto* standard for lossy compression of music is the family of advanced audio coding (AAC) algorithms, which have been standardized by ISO and IEC as part of the MPEG-2 and MPEG-4 specifications [2–4]. AAC is used for audio compression for digital TV as well as digital audio broadcast (DAB) in several countries. AAC achieves better quality than MP3 and allows for high sampling rates and multiple channels [5]. It is based on the modified discrete cosine transform (MDCT), which is able to provide a high-frequency resolution by using long delays [5]. For low-delay coding, transform coders are generally not as efficient as parametric (model based) coders [6]. Recently, low-delay parametric audio coders based on linear prediction [6, 7] and generalized noise-shaped quantization [8] have been proposed. With such techniques, it is possible to achieve delays less than 5 ms while maintaining high-quality audio. In fact, even a few sample delays can be achieved by compromising the efficiency of the perceptual model [6–8].

In order to achieve a certain degree of robustness towards packet dropouts or channel failures, it is possible to use error-correcting codes. If the tolerable delay or bandwidth is large enough, then these codes can be extremely efficient. An alternative is to use joint source-channel coding techniques, where a certain amount of source redundancy is introduced to help the channel/source decoder at the receiving end. A particular case is multiple description (MD) coding [9], where the source is encoded into multiple (partially) redundant packets, which can be decoded independently of each other. Moreover, if several packets are received, then they are able to refine each other and, thus, improve the reconstruction quality. MDs generalize the concepts of repetition coding and layered source coding (i.e., successive refinements), where in the former case, a packet is simply duplicated, and in the latter case, packets are (nearly) independent of each other^{a}. In [10], it was shown that a tolerable music quality could be achieved even on unreliable networks having more than 30% packet dropouts by using MDs audio coding between two and four descriptions (packets). The MD audio coding schemes presented in [10] and [11] are both based on transform coders and therefore not able to achieve ultra low delay. Recently, a predictive strategy for high-quality audio MD coding was presented in [12] and a noise-shaped strategy in [8], which are both able to achieve very low delays.

In this paper, we construct an efficient high-quality low-delay MD coder based on the principles of oversampling, predictive coding, and noise shaping. Specifically, we utilize the theoretic construction proposed for Gaussian sources in [13–15] and show that it naturally extends to real audio signals. We restrict attention to two descriptions and a symmetric setup, where the rate-distortion performance of the individual descriptions are identical. It is worth emphasizing that contrary to Gaussian source coding, it is crucial in audio coding that the temporal envelope is kept smooth when filter parameters are updated. This complicates the transition from theory to practice. The contributions of this paper are therefore dedicated to the practical design of the coder, and we refer the reader to the aforementioned literature for the theoretical foundations. The present paper discusses the design of filters for resampling, prediction, and noise shaping, in addition to coding the parameters^{b}. The proposed design is evaluated on several audio signals sampled at 48 kHz, and the performance due to receiving different subsets of descriptions is assessed. Moreover, the proposed coder is simulated in an environment with packet losses. It is shown that good quality music is achievable with delays less than 5 ms.

In comparison, the low-delay noise-shaped coder in [8] reveals a significant reduction of the coding rate, which is mainly due to the inclusion of individual description prediction loops.

## 2 Background

The MD predictive noise-shaped coder proposed in [13–15] consists of *sampling rate conversion, encoders, noise-shaping, and decoding*. We briefly describe these components below and refer to [13–15] for further details.

### 2.1 Sampling rate conversion

In the MD noise-shaped predictive coder proposed in [13–15], the original signal *x*(*n*) is first oversampled by a factor of two to obtain *x*_{
u
p
}(*m*) as shown in Figure 1. We use the indices *n* and *m* to refer to samples of signals in the original and upsampled domains, respectively. Sampling rate conversion of discrete-time audio signals has been widely studied in the audio engineering literature, cf. [16–18] to name a few. Theoretically, if the signal is bandlimited, then changing its sampling rate is a reversible process, as long as the resulting sample rate is greater than the Nyquist frequency of the signal, i.e., greater than twice the signals’ bandwidth.

### 2.2 Encoder

The upsampled signal is split into even and odd samples (see Figure 1), and the even (odd) stream is fed to the even (odd) encoder. Each encoder can be cast in the framework of ‘noisy’ single-description prediction, which in case of Gaussian signals was treated in [19]. In [19], it was shown that optimal encoding is achieved with minimum mean square error (MMSE) prediction. This result was used in [14] to show that the optimal encoders should be MMSE prediction filters for the colored ‘even’ and ‘odd’ Gaussian signals, respectively. Interestingly, in [15], it was furthermore suggested that the encoders could actually be any existing standard compression schemes. Indeed, in [20], it was proposed to combine delta-sigma quantization with standard JPEG coding schemes to form compression algorithms for MD coding of still images. Similarly, in this work, we could choose to use e.g., standard AAC compression schemes. A key motivation for choosing a standard coder is of course the fact that one avoids the trouble of having to design an efficient audio coder, but perhaps more interestingly, the individual descriptions would then also be completely standard-compliant with existing technology. Only if both descriptions are to be jointly considered at the receiving end, the decoder needs to be slightly altered.

### 2.3 Noise shaping

*e*

_{even}(

*n*) and

*e*

_{odd}(

*n*) from the two encoders are interlaced to form the error signal

*e*(

*m*) having the same sampling frequency as

*x*

_{up}(

*m*). The error signal

*e*(

*m*) is filtered by the noise-shaping filter and then added to

*x*

_{up}(

*m*) and thereby closing the feedback loop. The purpose of the noise-shaping filter in conventional oversampled quantization (e.g., delta-sigma quantization) is to shape the noise away from the in-band spectrum and thereby reduce the energy of the noise, which is imposed upon the signal [21]. In MD coding, on the other hand, the purpose is to shape the noise so that a proper trade-off is achieved regarding the distortion due to receiving a single packet versus receiving both packets [13]. Indeed, it is illustrative to consider what happens when only a single packet is received. In this case, since we split the signal into even and odd samples, we have in fact downsampled the oversampled signal by a factor of two (without first applying an anti-aliasing filter). But since the source spectrum only covers half the frequency spectrum of

*x*

_{up}(

*m*) (due to oversampling), there will not be any source aliasing. On the other hand, due to interlacing the noise samples, the noise spectrum covers the full frequency range, see Figures Eight and Twelve in [13] for an illustration of the noise spectra. Thus, the noise spectrum will be aliased. In particular, the out-of-band noise spectrum will be aliased and superimposed upon the in-band source + noise spectrum. The effect is that no matter how the noise is shaped, the full noise spectrum will be imposed upon the source spectrum. On the other hand, if both packets are received, the oversampled signal can be recovered without noise aliasing and, thus, it is possible to apply a low-pass filter and get rid of the out-of-band noise. To summarize, on one hand, we would like to minimize the total noise energy in order to reduce the distortion when receiving only a single packet. On the other hand, we would like to put as much noise in the high-frequency spectrum and thereby reduce the amount of noise in the in-band spectrum, in order to minimize the distortion when receiving both packets. It is also interesting to note that the entropy rate of the quantizer, under high-resolution conditions, is independent of the noise-shaping filter and given by the ratio of the power ${\sigma}_{X}^{2}$ of the input signal and the power of the excitation noise (the input

*e*(

*m*) to the noise-shaping filter), i.e.,

### 2.4 Decoder

*x*

_{up}(

*m*). Then, a low-pass filter is applied to get rid of the out-of-band noise, and the signal is downsampled by two. We note that there is no noise-shaping loop at the decoder.

## 3 Practical construction of the MD audio coder

In the following subsections, the design of the individual parts of the coder is described.

### 3.1 Sampling rate conversion

Ideal oversampling can theoretically be obtained by inserting zeros between every sample of the original signal and then apply an ideal low-pass filter (i.e., convolution by an infinite-length sinc function). The resulting signal *x*_{up}(*m*) has twice as many samples as *x*(*n*), and for *m*=2*n* and *m* even, we have that *x*_{up}(*m*)=*x*(*n*) (if we ignore possible integer time-delays due to filtering). Thus, we recover the original signal simply by taking all the even samples of *x*_{up}(*m*). The odd samples of *x*_{up}(*m*) are phase-shifted versions of *x*(*n*). Of course, in practice, we cannot use ideal low-pass filtering since this will result in prohibitively large delays. Due to using finite filters, a certain degree of aliasing is unavoidable.

*h*(

*z*) with linear phase obtained via the window method (Chebyshev) as the interpolation filter [22]. Specifically, we insert zeros between every sample of

*x*(

*n*) and apply the filter to obtain the upsampled signal

*x*

_{up}(

*m*) [16]. Figure 3 shows the performance in MSE as a function of the filter length. The solid lines are for a unit-variance audio signal ‘Abba’

^{c}and the dashed lines (with circles) are for a zero-mean unit-variance white Gaussian signal. As foreshadowed above, the error on the even samples due to resampling is negligible for filter orders greater than

*N*=18. On the other hand, the odd samples are highly affected by the resampling operation, which is due to frequency aliasing

^{d}. An estimate of the power spectral density (PSD) of the Abba signal is shown in Figure 4. It is clear that it is not flat, and, thus, the impact of aliasing is much less than for the Gaussian case.

### 3.2 Closed-loop prediction

*x*

_{up}(

*m*) is to be split into its even and odd samples, which are denoted by

*z*

_{even}(

*n*) and

*z*

_{odd}(

*n*) in Figure 1. For clarity, we have redrawn the encoder with more details in Figure 5. The even signal

*z*

_{even}(

*n*) is now fed into a compression algorithm (encoder), and the compressed signal

*y*

_{even}(

*n*) is transmitted to the decoder. The odd signal

*z*

_{odd}(

*n*) is processed in a similar way. Thus, the even samples then constitute one of the packets in the MD coder, and the odd samples constitute the other packet. The number of samples to include in each packet depends upon several factors and will be treated in the sequel.

#### 3.2.1 Linear predictive coding

The encoders will in this work be given by forward linear prediction coding. In particular, in order to encode the even signal *z*_{even}(*n*), we design a linear predictor based on the even unquantized samples *z*_{even}(*n*). We use a forward linear predictor, which as usual is obtained by minimizing the prediction error in the least squares sense, cf. [23] for details. The predictor performs closed-loop prediction, i.e., the quantizer is contained within the prediction loop [19]. To do so, we consider a block of samples and use these for estimating the prediction filter. The filter needs to be encoded and transmitted to the decoder. Thus, there is a trade-off between the rate required for coding the filter coefficients, the update rate of the filter, and the rate required for coding the prediction error. A general approach to choosing a proper rate distribution between model parameters and signal was considered in [24].

#### 3.2.2 Coding the prediction error

*Δ*of the quantizer. To obtain the bitrates of the coder, we first run the predictor using a fixed step size

*Δ*on a large data set of mixed audio having a sampling frequency of 48 kHz. Then, a scalar (Huffman) entropy coder is designed on the quantized output of the predictor [26]. Thus, we are using a static and memoryless entropy coder. Finally, the predictor is tested on an audio segment (in this case, it consists of jazz music), which is not part of the training material. Figure 6 shows the resulting coding rate due to using a scalar uniform quantizer with a step-size

*Δ*followed by a scalar (Huffman) entropy coder. The corresponding MSE due to changing the step size of the quantizer is shown in Figure 7. In these simulations, we update the two linear predictive coding (LPC) filters once in each block of 128 samples. Since the audio signals have a sampling frequency of 48 khz, then if the bitrate is say 5 bits/sample, the resulting rate for coding the prediction error is 240 kbps per packet.

#### 3.2.3 Predictor order

In predictive audio coding, it is common to use predictors of orders greater than 10 [6]. However, in our case, the outer loop introduces noisy feedback, which to a certain degree reduces the predictor capabilities. For example, let *Δ*=0.01, and construct a 10th-order noise-shaping filter using the design in Equation 1 provided in the next subsection. Then, the performance in terms of rate and distortion of the predictor as a function of its order is shown in Figures 6 and 7. The bitrates illustrated in the figures correspond to the rates required for encoding the prediction residuals. The actual predictor coefficients have not been coded in these simulations. The simulations are repeated for a wide range of predictor orders. It may be noticed that increasing the order from 1 to 5 significantly decreases the required bitrate for coding the residual, whereas using an order above 10 does not lead to significant improvements. On the other hand, the resulting MSE is approximately unaffected by the predictor order.

### 3.3 Noise shaping

The purpose of the noise-shaping filter is to shape the quantization noise appropriately in the frequency domain [21]. Ideally, the frequency response of the noise-shaping filter should be a two-step function, which in the in-band frequency range has power *δ*^{−1} and in the out-of-band frequency range has power *δ*[13]. Thus, if both descriptions are received, one is able to filter out the out-of-band noise and thereby obtain a resulting noise power that is proportional to *δ*^{−1}. On the other hand, if only a single description is received, then due to aliasing, the resulting noise power is proportional to *δ*+*δ*^{−1}. Furthermore, fixing the levels as *δ* and *δ*^{−1}, respectively, guarantees that their geometric mean is one, which basically fixes the coding rate while allowing one to trade-off side distortion for central distortion [13].

*c*(

*z*) for any filter order

*p*was given in [13] as:

**c**=(

*c*

_{1},…,

*c*

_{ p })

^{ T }are the filter coefficients,

**g**=(sinc(1/2),sinc(2/2),…,sinc(

*p*/2))

^{ T }, and

**G**is the matrix with entries

**G**

_{i,j}=sinc((

*i*−

*j*)/2),

*i*,

*j*=1,…,

*p*. In (1),

*λ*denotes the trade-off between central and side distortion. Choosing

*λ*=1 indicates that the central and side distortion are given the same weight. In this case, the central distortion will on average be around 3 dB smaller than the side distortion. On the other hand, choosing

*λ*≪1 reduces the central distortion at the price of increasing the side distortion. This is illustrated in Figures 8 and 9 for the case of

*p*=10 and

*p*=30, respectively. In these simulations,

*Δ*∈{0.01,0.05,0.1}. It may be noticed that larger

*Δ*yields larger distortions as expected. It can also be seen that for large

*λ*, the central distortion is approximately −10 log10(

*λ*) dB smaller than the side distortion.

#### 3.3.1 Coding the predictor coefficients

*π*/64. The quantized coefficients are then split into three subvectors of length 3, 3, and 4, respectively. Finally, each subvector is independently vector Huffman coded. The resulting bitrates are shown in Tables 1 and 2, where LSF

_{ i }denotes the

*i*th subvector. In these simulations, the window size of the predictor is 128 samples. It may be noticed from Table 2 that the average coding rate is approximately 20.8 bits per LSF vector. Since the sampling frequency is 48 kHz and the block size is 128 samples, the resulting average bitrate for coding the LSF vectors is 7.8 kbps per packet.

**Bitrates for coding the (even/odd) subvectors of the LSF vector**

Audio | LSF | LSF | LSF |
---|---|---|---|

Jazz | 6.63 / 6.62 | 6.50 / 6.47 | 5.63 / 5.58 |

Harpsichord | 6.77 / 6.77 | 7.29 / 7.25 | 8.11 / 8.14 |

Speech | 7.99 / 8.00 | 7.35 / 7.32 | 6.97 / 7.01 |

Pop | 6.98 / 6.98 | 5.80 / 5.78 | 5.26 / 5.23 |

Rock | 6.92 / 6.93 | 7.86 / 7.85 | 7.85 / 7.86 |

Average | 7.06 / 7.06 | 6.96 / 6.93 | 6.77 / 6.76 |

**Bitrates (in bits/vector) for coding the (even/odd) LSF vectors**

Audio | LSF vector |
---|---|

Jazz | 18.76 / 18.67 |

Harpsichord | 22.17 / 22.17 |

Speech | 22.32 / 22.33 |

Pop | 18.04 / 17.99 |

Rock | 22.64 / 22.63 |

Average | 20.79 / 20.76 |

### 3.4 Decoding

**The 16 (next) states the decoder can enter depending upon the decoders (current) state information**

Current/next | Central | Even | Odd | None |
---|---|---|---|---|

Central | 1 | 2 | 3 | 4 |

Even | 5 | 6 | 7 | 8 |

Odd | 9 | 10 | 11 | 12 |

None | 13 | 14 | 15 | 16 |

#### 3.4.1 State 1

This is the case with no packet dropouts. The decoder simply processes the two descriptions as described in Subsection 2.4. Both descriptions are first individually reconstructed, then interlaced, and finally downsampled to produce a single high-quality reconstruction. Thus, the states of the LPC filters at the side decoders as well as the state of the low-pass filter at the central decoder are all properly updated, which results in smooth transitions between consecutive blocks.

#### 3.4.2 States 2 and 3

Assume that the decoder is in state 1 (i.e., it has received both packets) but then in the next time slot it only receives the odd packet and thereby enters state 3^{e}. Then, no new even LPC filter coefficients are received, and the even LPC filter state (memory) is therefore not properly updated. The odd samples are phase-shifted by a 1/2 sample compared to the original signal, and the odd predictor is therefore not identical to the even predictor. Moreover, since both packets are not received, the low-pass filter at the central decoder is not applied and its state (memory) is therefore not updated.

*λ*=1/100, and

*Δ*=1/100.

#### 3.4.3 State 13

In this state, all buffers are zero, which corresponds to the initial state of the system. The decoder is then operated as in state 1.

#### 3.4.4 States 14 and 15

As was the case for state 13, all buffers are also zero here. If the current state is 14 (15), the decoder is then in the next state operated as in state 2 (3).

#### 3.4.5 States 4, 8, 12, and 16

In these states, no packets are received by the decoder. We then simply replace both packets by zeros and update the states of the LPC filters and low-pass filter accordingly.

## 4 Simulation study

In this section, we provide simulation studies of the proposed coder. We simulate an environment with packet losses of 0.1*%*, 1%, and 10%. We restrict the quantization step sizes to *Δ*∈{0.01,0.05}, the block size upon which the predictor is used to {64,128,256,512,1024,2048}, and the LPC filter order to *p*_{lpc}∈{5,10}. Finally, in all simulations, the low-pass filters used for resampling are of order 200, the noise-shaping filter is of order 10, and the noise-shaping ratio *λ*=0.01.

### 4.1 Study 1

*rock, jazz, pop, speech*, and

*harpsichord*music, respectively. Each segment is sampled at 48 kHz and with a duration of 10 s. We use objective difference grades (ODG) instead of MSE in order to better reflect the perceived quality of the reconstructed audio signals. For an explanation of the relationship between the ITU-R 5-grade scale and ODG, see Table 4 and [30]. To obtain the ODG scores, we use a Matlab implementation of the PEAQ standard [31]. The resulting ODG are shown in Tables 5, 6, and 7. In the tables, we have averaged the ODG scores over all audio segments.

**Relationship between the ITU-R 5-grade scale and ODG**[30]

Impairment | ITU-R 5-grade scale | ODG |
---|---|---|

Imperceptible | 5.0 | 0.0 |

Perceptible but not annoying | 4.0 | -1.0 |

Slightly annoying | 3.0 | -2.0 |

Annoying | 2.0 | -3.0 |

Very annoying | 1.0 | -4.0 |

**Average ODG at 0.1% packet losses**

Block size | 64 | 128 | 256 | 512 | 1,024 | 2,048 |
---|---|---|---|---|---|---|

| ||||||

| -0.77 | -0.82 | -0.81 | -0.22 | -0.14 | -0.20 |

| -0.18 | -0.21 | -0.21 | -0.20 | -0.16 | -0.18 |

| ||||||

| -1.54 | -1.44 | -1.43 | -1.02 | -0.99 | -1.04 |

| -1.06 | -0.95 | -0.95 | -1.04 | -1.06 | -1.12 |

**Average ODG at 1% packet losses**

Block size | 64 | 128 | 256 | 512 | 1,024 | 2,048 |
---|---|---|---|---|---|---|

| ||||||

| -0.99 | -0.94 | -0.40 | -0.46 | -0.43 | -0.37 |

| -0.42 | -0.43 | -0.32 | -0.35 | -0.36 | -0.34 |

| ||||||

| -1.87 | -1.81 | -1.73 | -1.32 | -1.17 | -1.09 |

| -1.50 | -1.39 | -1.23 | -1.28 | -1.25 | -1.16 |

**Average ODG at 10% packet losses**

Block size | 64 | 128 | 256 | 512 | 1,024 | 2,048 |
---|---|---|---|---|---|---|

| ||||||

| -2.77 | -2.16 | -1.20 | -0.87 | -0.60 | -0.60 |

| -2.61 | -1.88 | -1.45 | -1.08 | -0.72 | -0.67 |

| ||||||

| -3.31 | -3.00 | -2.49 | -2.44 | -2.19 | -2.16 |

| -3.25 | -2.97 | -2.93 | -2.85 | -2.61 | -2.35 |

From the tables, it is clear that decreasing the packet loss rate or the step size of the quantizers increases the quality as expected. It is also interesting to note that using a longer block size appears to improve the performance.

### 4.2 Study 2

*Δ*=0.05 and

*Δ*=0.01, respectively. Interestingly, the bitrate (per sample) as well as the ODG are improving as a function of the block size upon which the predictor is applied. Intuitively, one would think that a fixed-order predictor would be better on shorter segments of the signal. We ascribe this phenomenon to the fact that the performance of the current predictor depends upon the predictor applied in the previous block due to the filter’s memory (i.e., we reuse the state of the past predictor). Thus, for short blocks, a substantial part of the prediction of the block is influenced by the history of the previous predictor. This phenomenon is particularly pronounced in the case of large packet-loss rates, where the ODG is significantly improved by going from block sizes of, e.g., 64 to 512 samples.

### 4.3 Comparison to existing works

*jazz*music signal has been used, and two packet loss scenarios have been simulated, a high loss (10% packet losses) and a low loss (1% packet losses) scenario. For the proposed coder, we vary

*Δ*in the interval 0.01 to 0.05 in steps of 0.01. The total bitrate consists of the rates required for coding the LSF coefficients as well as the prediction residual. It can be see in the figure that the proposed coder is able to efficiently exploit its prediction loops and thereby reduce the bitrate over what is possible with the MH design. In these simulations, the proposed coder uses a block size of 64 samples for the prediction. Further improvement is possible by increasing the block size.

## 5 Conclusions

We presented a practical design of a low-delay MD audio coder, which is able to provide a certain degree of robustness towards packet losses. The proposed coder combined oversampling and noise shaping with source prediction. The oversampling process creates two source descriptions in order to counteract possible packet losses on the network. The prediction loop removes source redundancy and thereby reduces the coding rate, whereas the noise-shaping process controls the distortion due to receiving subsets of the descriptions. The quantized prediction residual was entropy coded using a static and memoryless Huffman coder. In practical simulations on real audio, it was shown that it is enough to use LPC filters of order 10 (estimated from blocks of 64 samples), noise-shaping filters of order 10, resampling filters of order 200, and bitrates of approximately 4 bits per sample (per description) in order to achieve good quality (ODG better than -1) music in the presence of 1% packet losses.

## Endnotes

^{a} In layered source coding, the source is usually split into a base layer and at least one refinement layer. While the base layer can be used by itself, the refinement layers are usually no good without the base layer.

^{b} For reproducibility, the complete source code for the proposed coding scheme is electronically available at http://kom.aau.dk/~jo.

^{c} The audio signal ‘Abba’ is a 10-s clip of the song ‘Head Over Heals’ by Abba - sampled at 44.1 kHz.

^{d} In order to correctly estimate the error, we need to correct the phase shift of the odd samples. This is done by once more filtering the odd samples with the same filter. Of course, for subjective listening tests, we do not need to correct the phase.

^{e} The effect of receiving different numbers of descriptions from frame to frame, corresponds in some sense to (noisy) non-uniform sampling in MDs, cf. [34].

## Authors’ information

The work of J. Leegaard was performed while he was affiliated with the Department of Electronic System, Aalborg University. He is now with the Department of Architecture, Design and Media Technology, Aalborg University.

## Declarations

## Authors’ Affiliations

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