A Review of Joint Source-Channel Coding Fredrik Hekland1 Dept. of Electronics and Telecommunications Norwegian University of Science and Technology (NTNU) February 16, 2004

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[email protected]

Abstract This article presents methods for robust and efficient communication over wireless channels. Both benefits and limitations of the classical approach of separate source and channel coding are discussed. Joint source-channel coding (JSCC) is presented as the most promising scheme for communication of analogue sources over wireless channels, due to its ability to cope with varying channel qualities and to approach the theoretical bounds of transmission rates. Different schemes for implementing joint source-channel coding are presented, ranging from a traditional tandem structure with resource control to hybrid digital-analogue JSCC techniques providing both bandwidth reduction and expansion. The direct source-channel mapping schemes are shown to be good candidates for joint source-channel coding, providing both robustness and efficiency.

Part I

The Source-Channel Problem 1

Introduction

Communication is all about trying to convey as much information as possible over a given channel with as few errors as possible. Before 1948 it was generally conceived that it was impossible to attain arbitrarily low error rates when communicating in the presence of noise. This misconception was disproved by Shannon (1948) in his groundbreaking work “A Mathematical Theory of Communication”. Here, he introduced the concept of channel capacity and proved that as long as the transmission rate is below the channel’s capacity, reliable communication with a non-zero, but arbitrarily small error rate is possible. Furthermore, he presented a theorem stating that a source with entropy H can be reliably transmitted over a channel with capacity C as long as H ≤ C. In principle this can be achieved by first applying a source coder that reduces the rate of the source down to the minimum, called the entropy H, and subsequently applying a channel code. In the receiver, the channel decoder, which is unaware of the type of source, outputs the most probable codeword to the source decoder. Finally, the source coder reconstructs the source without any knowledge of the channel statistics. This independence between source and channel coder is the reason why this theorem is also known as the separation theorem; the simplifications it provides makes it prevalent in the communication community. However, this theorem does not hold in every situation. In those cases where it breaks down, there are usually systems with combined or joint source-channel coding that perform better. This article will discuss such joint coding schemes in the context of analogue sources, where errors in the transmission are allowed. In the first part of this article, the limitations of the separation theorem are described, and some earlier sourcechannel coding systems are mentioned. Furthermore, the need for explicit channel coding is examined. The second part describes proposed systems and concepts for joint source-channel coding, from purely digital systems to both hybrid digital-analogue and analogue systems.

1.1

Shortcomings of the Channel Coding Theorem

The fact that we can perform source coding independently of channel coding and vice versa simplifies the construction of the system since we can optimise the coders separately and still achieve optimality. Also, having either a different source or channel, we can change the affected coder while leaving the other unchanged. However, there are some drawbacks with this approach. First of all, we have to allow infinite complexity and delay in the coders in order to reach optimality. Evidently this is problematic for real-time communication. Secondly, the theorem is no longer valid for non-ergodic and multi-user channels, and in those cases we no longer have an optimal system. Thirdly, such systems tend to break down completely when the channel quality falls under a certain threshold, and the channel code is no longer capable of correcting the errors. This phenomenon is often referred to as the threshold effect. Thus, these systems are not robust with respect to changing channel qualities. Knowing that wireless channels have fluctuating channel qualities and high bit-error rates, trying to evade (or at least reduce) the threshold effect should be important when designing wireless communication systems. 1

1.2

Joint Source-Channel Coding

When operating under a delay-constraint on a time-varying channel, it is generally no longer optimal to regard the two coders separately and we have to jointly optimise the source coder and the channel coder and the result is some sort of joint source-channel coding (JSCC). This is a rather loose label that encompasses all coding techniques where the source and channel coders are not entirely separated. This article will not try to cover all aspects of joint source-channel coding techniques, but rather survey the latest developments in the field. For an overview of earlier works, a good starting point is the review article by Zahir Azami et al. (1996), which gives a brief overview of the most prominent JSCC techniques known at that time. It covers unequal error protection (UEP), index assignment (IA), co-optimised vector quantizing and IA (i.e. channel optimised vector quantizer ), and direct modulation organising schemes. Examples of the latter are modulation organised vector quantization (MOVQ) where the codewords are mapped directly into the modulation plane, and hierarchical modulation (also called multiresolution modulation). In the latter, the most important coarse information is best protected by mapping it to “modulation clouds” placed far apart. The less important detail information is mapped to points within the clouds of the coarse information, and is therefore less protected and requires a better channel to be decoded correctly.

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The roles of the coders in the system

In a communication system where the bandwidth and transmit power are set by regulation authorities, the channel capacity C is limited by the channel noise. This means that whenever the information source has an entropy H > C, we need to introduce a lossy source coder that reduces the entropy to a level below the capacity. Otherwise, we have no control over the distortion introduced in the communication process. The first step for a source coder is to remove all deterministic components, called redundancy. After having removed redundancy, we must somehow further reduce the amount of information if the channel cannot support the information rate. First, all non-perceptible parts (irrelevancy) are removed. Any further reduction will introduce perceptible distortion. Both removal of irrelevancy and the introduction of distortion mean that the source coder is lossy. Quantizers are the most well known devices for this purpose. When the source rate has been reduced appropriately, some sort of channel code is usually applied to protect the information1 . Usually, a tandem system with separate source and channel coders seeks to obtain error-free transmission. This is reasonable for digital sources, and also for entropy coded analogue sources since the entropy coding in the source coder usually breaks down in the case of bit errors. A question is whether this approach is optimal for analogue sources. After all, the source is analogue and the observer usually tolerates a certain amount of distortion. For certain sources and channels, there are simpler delay-free systems that perform optimally in a rate-distortion sense. See the article by Gastpar et al. (2003) for a discussion on the possibility of omitting channel coding. They give examples of single-letter codes for single-source systems that perform optimally in the rate-distortion sense. The main contribution in their article is a criterion for determining whether a single-letter code performs optimally for a source-channel pair. They also introduce the notion of univer1

Remember that the aggregate rate must still be below the channel capacity.

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sal source-channel codes which means that the code is optimal for any channel out of a class of channels, given the source. Such codes would be desirable since they provide “immunity” towards varying channel qualities. This implies that in some cases we can omit the explicit channel code. We rather utilise a source-channel pair that performs optimally on a range of channel qualities, thereby avoiding the delay introduced by the channel coder.

Part II

Joint Source-Channel Coding Schemes 3

Digital JSCC Techniques

In this section, both the source and channel representations are assumed to be digital (meaning a set of discrete representation values), in accordance with most communication systems existing today.

3.1

Simple resource control

The most obvious way to create a JSCC-system is to still use the tandem structure, but instead of fixing the rates to the coders, distribute the channel capacity to the source and channel coder according to some criterion. This scheme could, for example, grant more bits to the channel coder (and fewer to the source coder) when the channel is bad in order to avoid breakdown in the source decoder, and allocate more bits to the source coder when the channel is good in order to improve the quality. Although this not a “true” JSSC-system, since the coders are not matched in any sense, its parameters are at least modified as the channel quality is changing. Cheung and Zakhor (2000) use a 3D subband video coder paired with rate-compatible punctured convolutional (RCPC) codes to provide unequal error protection for the different subbands. The optimal allocation of source and channel bits in the different subbands is found by minimising the distortion functions of the subbands, given rate constraints for the source and channel coders.

3.2

Multiresolution Modulation

Zahir Azami et al. (1996) discussed the concept of hierarchical modulation, and by using this technique it is possible to protect important parts of the data better by organising the modulation space properly. The wavelet transform used in the JPEG2000 standard is a so-called multiresolution coder where the image is split into different bins in which information content ranges from coarse to fine. By pairing such a multiresolution source coder with a multiresolution modulation scheme, we could potentially obtain a system where the receiver decodes the received signal to a resolution/quality depending on the channel signal-to-noise ratio (CSNR). The better the CSNR, the better the decoded signal. Kozintsev and Ramchandran (1998) proposed such a system for image transmission over time-varying channels where subbands with coarse detail are mapped to the modulation clouds (coarse), thus providing good noise immunity, while subbands with finer detail are mapped to satellites within each cloud requiring better CSNR in order to be decoded. They provided an algorithm that jointly optimises the design of the multiresolution source codebook, multiresolution constellation, and the decoding

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strategy of optimally matching the source and channel resolution according to the channel quality. In this system, only the receiver needs to have knowledge of channel state information (CSI) in order to decode the signal. The sender does not need to know the state of the channel when encoding, so this scheme would fit nicely into a broadcast scenario where all the channels are different (with high probability) and there is no feedback channel. Zheng and Liu (1999) employed the same framework as Kozintsev and Ramchandran (1998) and proposed a joint power and rate allocation scheme where they optimise the allocated rate and power to the subbands with respect to the distortion. Of course, this requires that the sender has knowledge of the channel quality.

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Hybrid Digital-Analogue (HDA) JSCC Techniques

Instead of sending discrete symbols over a continuous channel, it might prove better to let all or parts of the system use analogue channel representation in order to improve robustness towards channel noise. It would also be possible to better match the input distribution to the channel noise distribution which is necessary to approach OPTA2 (Gastpar et al., 2003).

4.1

Graceful Degradation/Improvement

As mentioned in Section 1.1, systems adhering to the separation theorem suffer from the threshold effect. This means that they lack robustness when the channel quality is below their design CSNR. However, it must be noted that they are very robust above the threshold. A way to increase robustness is to reduce the number of quantization intervals, and thereby increase the distance between the decision lines of the quantization levels. This will of course increase the distortion, so to compensate for the coarser representation, the quantization error is sent along as an analogue symbol. This will increase the the required bandwidth since more symbols are sent over the channel. By knowing the channel, the power can be distributed among the two symbols in an optimal way, guaranteeing good decoded quality as long as the actual CSNR does not deviate too much from the design CSNR. Mittal and Phamdo (2002) proposed different systems for robust broadcasting systems based on the concept described above, where the digital part is a standard tandem encoder pair, and the difference between the original and a reconstruction of the source coded signal is sent as an analogue symbol using a linear coder. See Figure 1 for a block diagram of the conceptual system. These systems are nearly optimal and show better distortion region performance than both a time-sharing and a purely digital system. They also show that it is possible to design nearly robust3 HDA systems when the source bandwidth is not equal to the channel bandwidth. Another weakness with tandem codes is their inability to improve the decoded quality when the channel improves. Usually such systems are designed with a target bit-error rate (BER) given a worst-case CSNR. Below this level the system breaks down, but the quality does not improve if the CSNR increases. This is due to the irreversible distortion caused by the quantizing. However, it should be noted that the BER improves as the CSNR increases. Skoglund et al. (2002) proposed an HDA scheme using vector quantizing (VQ) with the same “quantizer+linear coder” structure 2 3

Optimal Performance Theoretically Attainable. They define a code to be nearly robust when it is not severely affected by the threshold effect.

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N1(k) X(k)

Source Encoder

Channel Encoder

Channel Decoder

Source Decoder

^ X(k)

Source Decoder +

N2(k)

Linear Encoder

Linear Decoder

Figure 1: Block diagram for a nearly robust bandwidth expansion code. as proposed by Mittal and Phamdo (Figure 1). Here, the quantization error from the vector quantizer is sent using the linear coder on the analogue channel, which will improve the quality of the decoded signal when the channel is good (i.e. the system provides graceful improvement).

4.2

Direct Source-Channel Mapping

In two book chapters, Ramstad describes the concept of direct source-channel mappings, and gives practical examples of an image coding system (Ramstad, 1999) and a video coding system (Ramstad, 2000). However, instead of using continuous mappings, these examples use scalar or vector quantization with index assignment and discrete channel symbols. Figure 2 shows the conceptual structure of the systems (excluding the motion compensation in the video coder), where the different subbands in the filterbank are classified and allocated to a mapping according to their importance. The image coder uses scalar quantizers, and the mapping devices either send one subband sample as an 81-PAM symbol, or combine several quantized versions by forming a cartesian product of pdf-optimised quantizers and map it to an 81-PAM symbol. The latter provides 2 : 1 or 4 : 1 compression (Lervik, 1996). Note that this system is memoryless, meaning that errors in the transmission will neither propagate nor cause breakdown in the decoding process. The only effect of transmission errors is increased distortion in the signal components represented by the affected channel symbol. This system shows graceful degradation, and even more interesting: both approximation noise and channel noise are additive, thus they both produce visually equivalent distortion. This implies that if the approximation stage is designed to give visually pleasing artefacts, then the channel noise will also induce visually pleasing artefacts. This is a nice property that in some sense provides robustness in terms of the subjective visual quality. The video coding system is the same as described in Fuldseth’s thesis (Fuldseth, 1997), and is a motion compensated video coder with out-of-band motion estimation. The sourcechannel coding is performed by mapping the output of the different vector quantized subbands directly onto QAM symbols. Compared to a H.263 coder, the proposed coder shows better performance at most channel qualities and provides graceful degradation. However, the gains are not as pronounced as for the image coder. This is due to the fact that the motion compensation produces images (predicted frames) with a smaller

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Figure 2: System model for Ramstad’s image communication system, q(·) denotes quantizing/approximation and T(·) denotes the mapping operation. dynamic range and is therefore mapped to a smaller channel constellation. In his thesis, Fuldseth (1997) also developed a direct source-channel mapping technique called power-constrained channel-optimised vector quantization (PCCOVQ) which is basically the generalised Lloyd-algorithm, modified to take channel noise and a power-constraint into account. Compared to the digital techniques in Section 3 and the quantizer+QAM mentioned above, the PCCOVQ algorithm can be argued to be a more “true” JSCC system since the vector quantizing and channel modulation are performed in one step using a continuous mapping4 , thus avoiding the threshold effect that lowers the robustness of such systems. An interesting point is that for Gaussian sources and Gaussian channel noise, this algorithm creates spiral-like mappings resembling the Archimedes’ spiral shown in Figure 3. This makes sense since a two-dimensional Gaussian distribution has circular symmetry (when the two dimensions have equal variance). This suggests that simpler parametric curves could be used as mapping devices. This will be explored further in section 5.2. The PCCOVQ works best for dimension reduction operations and is not well suited for dimension increase. This is probably because in the case of bandwidth expansion the points in the channel space will have more neighbours than the points in the signal space. This implies that decoding of a noise contaminated channel symbol will not always give lowest possible reconstruction error. Coward and Ramstad (2000) propose a bandwidth expansion mapping very similar to the structure proposed by Mittal and Phamdo (Figure 1); they employ a scalar quantizer and transmit both the quantized value and the quantization error and thereby obtain a memoryless rate 2 system. They denote it hybrid scalar quantizer-linear coder (HSQLC), and show that for a Gaussian source a uniform quantizer is very close to the optimum. Linear solutions of the joint source-channel coding problem are explored by Hjørungnes (2000) where powerconstrained FIR filter banks are optimised with respect to the MSE5 between the input and output vectors. One of the problems with this linear solution is that for bandwidth reduction, the subbands with the smallest coefficients are discarded. This creates transfer matrices that are not of full rank, and perfect reconstruction cannot be obtained for good channels. Except for very bad channels, this system is far from the optimum. 4

The PCCOVQ is actually a discrete set of representation points, but since the points are placed on a (continuous) curve in the source space, it is possible to approximate a continuous mapping by connecting the points with lines. 5 Mean Squared Error.

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5

“Near-Analogue” JSCC Techniques

As seen in the article by Gastpar et al. (2003), the source and channel statistics should be matched in order to reach OPTA. Furthermore, since the channel is usually analogue in wireless communication, the input must also be continuous or analogue if we wish to reach OPTA.

5.1

Dimension Changing Mappings

In his 1949 paper, Shannon (1949) put the communication process in a geometric framework. Signals become points in an N -dimensional space and noise will displace the points in a random manner. To avoid errors, it is important to place the channel points as far apart as possible. This way of looking at the process was revolutionary, since it greatly facilitated the analysis of the system performance. However, another concept introduced in the same paper has not gained the same popularity. The mapping of a signal from the signal space to the channel space is normally done in a one-to-one fashion. However, for an analogue signal it is possible to map the signal to a lowerdimension channel space and thereby achieve bandwidth reduction (compression). This will naturally lead to some approximation noise since exact representation would require infinite energy. For analogue sources, this noise is usually not a problem as long as it is controlled properly. It is also possible to map the signal to a higher dimension providing bandwidth expansion (error protection). Ramstad (2002) elaborates on Shannon’s ideas and gives some examples of mapping devices. He also discusses the noise contributions from both the mapping approximation and the channel. Further, he states some important requirements for a good dimension-reduction mapping (Ramstad, 2002): 1. The mapping should cover the entire source space to lower the approximation noise. 2. The most probable symbols should map to channel symbols with small amplitude in order to lower the average channel power. 3. Signals close in the channel space should remain close when mapped back to the source space. This will ensure that low channel noise leads to small errors in the decoded source signal. (However, it is not required that large noise contributions lead to large errors in the decoded signal.) An example of a mapping fulfilling these requirements is shown in Figure 3. Interestingly, Fuldseth’s PCCOVQ scheme (Fuldseth, 1997) produce a similar spiral mapping for Gaussian sources and channels with CSNR above 20dB.

5.2

Using Space-filling Curves for Dimension Change

There are few papers exploring the possibility to use space-filling curves as mapping devices. Bially (1969) presented algorithms that map points in an N -dimensional cube to a point on a line and vice versa. The use of space-filling curves as a way to obtain compression in a joint coding system was investigated in the thesis by Chung (2000). Space-filling curves, such as Peano and Hilbert curves, are used to obtain 2 : 1 and 3 : 1 compression. In order to approximate a Gaussian distribution matching a Gaussian channel, stretching the space-filling curves is necessary. No practical coding systems were demonstrated, but simulations showed that these mappings are about 1 to 7 dB 7

Positive channel symbol Negative channel symbol Source points Mapped points Received points

3

2

Source 2

1

0

−1

−2

−3

−4

−3

−2

−1

0 Source 1

1

2

3

4

Figure 3: The Archimedes’ spiral as a 2 : 1 mapping, where the x and y axes represent source values, and the length of a spiral from the centre to a point on the spiral arm represents a channel symbol. The combined source values are represented by the point “∗”, the approximated point (channel symbol) is represented by “◦” and the received point, displaced by channel noise, is represented by “+”.

away from the theoretical limits. More interesting is the fact that these mappings tend to be almost parallel to the theoretical upper bounds all the way up to the saturation point. This property implies higher robustness against varying channel SNR.

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Concluding remarks

As mentioned in section 1.2, there are many ways to implement a joint source-channel coding system. Even though only a few different schemes are covered in this article, the principal approaches to the JSSC problem are demonstrated through the examples. The techniques presented are tandem structures with resource control, multiresolution modulation, hybrid digital-analogue coders, and direct source-channel mappings both in the form of numerically optimised systems and parametric space-filling curves. All systems have their strengths and weaknesses, and the choice of approach will depend on the given constraints. Of course, set aside practical considerations, the system that is closest to OPTA will be the best system. However, the direct source-channel mappings (either numerically optimised or space-filling curves) have some intriguing properties that make them interesting for joint source-channel coding. Notably, their lack of the threshold effect for bandwidth reduction, the possibility to balance the contribution of quantizing and channel noise to provide acceptable distortion, and being able to match the channel input distribution to the channel noise in order to approach OPTA.

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References T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. Information Theory, IT-15(6):658–664, Nov. 1969. G. Cheung and A. Zakhor. Bit allocation for joint source/channel coding of scalable video. IEEE Trans. Image Processing, 9(3):340 – 356, March 2000. S.-Y. Chung. On the Construction of Some Capacity-Approaching Coding Schemes. PhD thesis, Massachusetts Institute of Technology, September 2000. http://lids.mit.edu/∼sychung/thesis/index.html. H. Coward and T. A. Ramstad. Robust image communication using bandwidth reducing and expanding mappings. In Conference on Signals, Systems and Computers, volume 2, pages 1384–1388, Asilomar, CA, USA, Oct. 2000. IEEE. A. Fuldseth. Robust subband video compression for noisy channels with multilevel signaling. PhD thesis, Norwegian University of Science and Engineering (NTNU), 1997. M. Gastpar, B. Rimoldi, and M. Vetterli. To code, or not to code: Lossy source-channel communication revisited. IEEE Trans. Information Theory, 49(5):1147–1158, May 2003. A. Hjørungnes. Optimal Bit and Power Constrained Filter Banks. PhD thesis, NTNU Trondheim, 2000. I. Kozintsev and K. Ramchandran. Robust image transmission over energy-constrained time-varying channels using multiresolution joint source-channel coding. IEEE Trans. Signal Processing, 46(4):1012 – 1026, April 1998. J. M. Lervik. Subband Image Communication over Digital Transparent and Analog Waveform Channels. PhD thesis, NTNU, 1996. U. Mittal and N. Phamdo. Hybrid digital-analog (hda) joint source-channel codes for broadcasting and robust communications. IEEE Trans. Information Theory, 48(5): 1082 – 1102, May 2002. T. A. Ramstad. Insights Into Mobile Multimedia Communications, chapter 26: Combined Source Coding and Modulation for Mobile Multimedia Communication, pages 415–430. Academic Press, 1st edition, 1999. T. A. Ramstad. Signal Processing for Multimedia: Robust Image and Video Communication for Mobile Multimedia, volume 174 of Nato Science Series: Computers & Systems Sciences, pages 71–90. IOS Press, 2000. T. A. Ramstad. Shannon mappings for robust communication. Telektronikk, 98(1): 114–128, 2002. Information Theory and its Applications. C. E. Shannon. A mathematical theory of communication. The Bell System technical journal, 27:379–423, 1948. C. E. Shannon. Communication in the presence of noise. Proc. IRE, 37:10–21, jan 1949.

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M. Skoglund, N. Phamdo, and F. Alajaji. Design and performance of vq-based hybrid digital-analog joint source-channel codes. IEEE Trans. Information Theory, 48(3): 708 – 720, March 2002. S. Zahir Azami, P. Duhamel, and O. Rioul. Joint source-channel coding: Panorama of methods. CNES Workshop on Data Compression, Toulouse, France, November 1996. http://citeseer.nj.nec.com/zahirazami96joint.html. H. Zheng and K. J. R. Liu. The subband modulation: A joint power and rate allocation framework for subband image and video transmission. IEEE Trans. Circuits, Syst. for Video Technol., 9(5):823 – 838, August 1999.

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