# Optimization of Hierarchical Modulation for Use of Scalable Media

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## Abstract

This paper studies the Hierarchical Modulation, a transmission strategy of the approaching scalable multimedia over frequency-selective fading channel for improving the perceptible quality. An optimization strategy for Hierarchical Modulation and convolutional encoding, which can achieve the target bit error rates with minimum global signal-to-noise ratio in a single-user scenario, is suggested. This strategy allows applications to make a free choice of relationship between Higher Priority (HP) and Lower Priority (LP) stream delivery. The similar optimization can be used in multiuser scenario. An image transport task and a transport task of an H.264/MPEG4 AVC video embedding both QVGA and VGA resolutions are simulated as the implementation example of this optimization strategy, and demonstrate savings in SNR and improvement in Peak Signal-to-Noise Ratio (PSNR) for the particular examples shown.

### Keywords

Fading Channel Additive White Gaussian Noise Convolutional Code Scalable Video Space Ratio## 1. Introduction

Recent developments in media source coding have evolved from consideration not only of compression efficiency in terms of rate-distortion curves, but also on methods for providing easy-to-use scalability features. Scalability refers to the ability of the media delivery system to easily provide a range of spatial, temporal, and quality profiles in response to changing system conditions or user demands. For example, a person viewing a sports event on a mobile phone may be content to view a QCIF ( Open image in new window pixels) resolution level at 25 fps, whereas a person with access to an HDTV may wish for a 50 fps, 720 p ( Open image in new window pixels) version of the same media. Such demands can be met using scalable video and audio coding, where lower resolution or lower quality signals can be reconstructed from partial bit streams. This allows simpler delivery of digital media, as networks and terminals can autonomously adapt to issues such as network heterogeneity and error-prone environments (e.g., wireless fading channels) [1]. Scalability allows the removal of parts of the bitstream, while achieving a rate-distortion (R-D) performance with the remaining partial bitstream (at any supported spatial, temporal, or SNR resolution), that is, comparable to a "single-layer" approach [2], that is, nonscalable H.264/MPEG-4 AVC coding (at that particular resolution) [3].

However, in order to take maximum advantage of scalable coding, we need to ensure that scalability is treated at a system level, so that all layers of the communication stack can make intelligent decisions about how to use scalability. For example, in real-time audio-visual traffic, consecutive packets carry data of different importance for the user perceived quality. Header information is of vital importance, whereas texture information (in video coding) can tolerate some errors. So, although data may be lost due to congestion or poor wireless channel conditions, the class of data lost will have the largest impact on user experience [4]. Nevertheless, many current media transmission systems assume all data from higher layers is equal in importance, and rely upon the higher layers to provide the additional redundancy which can help protect more important information. However, it can be agreed that scalable media codecs often have the inherent property that some data is more important than others, and exploiting that knowledge may enhance overall system performance.

One strategy that could be employed is to use time-slicing of data with different priorities; however in [5], Cover proved that if a sender wants to send information simultaneously to several receivers, given specific channel conditions, superimposing high-rate information on low-rate information may achieve higher bandwidth efficiency than the time-sharing strategy.

Scalable coding interacts naturally with hierarchical modulation. Since the packets encoded by scalable codecs can be divided into different classes of priority, a simple scheme would create two classes such as "base information'' and "refinement information'' according to their contribution to the quality/temporal/spatial resolution of the media. The packets belonging to the base level can be allocated to the base bits of the hierarchial constellation, meanwhile the refinement packets can be assigned to the refinement bits of the constellation. The user who is able to decode the base bits of the hierarchical constellation can achieve the lower resolution. Furthermore, if a user is able to decode both the base bits and the refinement bits, a higher resolution is achieved. The enhancement layer cannot reconstruct a higher resolution alone. It has to reuse the information of the lower resolution embedded in the base layer. In order to provide two different resolutions using a nonscalable codec, the media must be encoded twice and the media packets for different resolutions cannot reuse the information from each other. Since the base layer packets encoded by a scalable codec can be reused by the enhancement layer packets, the scalable codec is more efficient than the nonscalable codec in providing multiresolution media simultaneously. In this case, the source packets contributing to the low resolution are allocated to the base bits of the hierarchical constellation and the packets which only contribute to the high resolution are carried by the refinement bits. The users close to the station are able to get all packets decoded and receive a high resolution program. Due to reduced radio signal attenuation, the users far away from the station will probably not be able to decode the refinement bits, but they can still decode the packets for low resolution with acceptable quality.

Note however that the flexibility introduced by hierarchical modulation does not come without a price. In [16], Jiang and Wilford illustrated that a penalty of slightly reduced SNR in base layer bits is introduced by hierarchical modulation. This penalty has equal impact on both scalable and nonscalable codec in a hierarchical system.

However, as we shall see in Section 2, hierarchical modulation also imposes a second performance penalty, namely, for a given choice of hierarchial constellation, and fixed target bit error rates of the two streams, the system will almost certainly be operating at a higher overall SNR than is needed to satisfy the target BERs.

In this paper, we will show how the constellation can be dynamically adapted at the physically layer in order to remove this performance penalty. This adaptation can be done at a session level, or even with finer granularity (e.g., at a one-second interval) in response to the changing dynamics of the transmitted bit-streams.

The paper is presented as follow. In Section 2 we discuss the basic analytical tools for calculating bit error rates in a sample hierarchical system. A simulation of the single-user scenario, a simulation of the multiuser scenario and their results are described in Sections 3 and 4, separately. Section 5 concludes this paper.

## 2. Error Rate Analysis and Optimization in Hierarchical Modulation

As introduced above, hierarchical modulation is a physical layer modulation technique in which the received signal constellations can be treated in two (or more) parts, by first making coarse decisions about the constellation location, followed by a refined decision on the exact location. Figure 1 shows a 16-QAM constellation diagram to illustrate hierarchical modulation. The data carried by this constellation is broken into two classes: a low priority (LP) and high priority (HP) class. The bits from the HP stream are used to select the quadrant of the constellation point, and the LP stream is used to choose the exact constellation point. The notations Open image in new window and Open image in new window represent the intra- and interconstellation group distances, respectively. The ratio Open image in new window is an important parameter, as it defines the achievable error rates of the system in the presence of noise. When Open image in new window is equal to 1, the constellation reverts to a standard 16-QAM constellation. When Open image in new window is larger than 1, the HP stream is more heavily protected against noise than the LP stream. This is compatible with the typical definition of constellation ratio in DVB-T/DVB-H standard [13].

Before assigning the HP and LP streams to Hierarchical Modulation constellation points, we can decrease the bit error probability of the streams by using standard coding techniques such as convolutional coding. A high-rate code is suitable for the LP bit stream because of its lower bit error rate demand. Using different rates of code in the HP and LP bit streams is helpful in achieving arbitrary target bit error rates in the Physical Layer.

Exact (in Open image in new window ) BER expressions for uniform Open image in new window -QAM over an additive white Gaussian noise (AWGN) channel have been developed in [17, 18] based on signal-space concepts and a recursive algorithm, respectively. Exact expressions for the BER of 16-QAM and 64-QAM in nonfading and frequency flat fading channels were derived in [19]. The exact and generic (in Open image in new window ) expression for the BER of uniform square QAM in the presence of AWGN channel was obtained in [20].

For uncoded hierarchical constellation scenarios, an approximate BER expression is described in [9, 10] for 4/16-QAM, 4/64-QAM and in [10] for multicast Open image in new window -PSK. Reference [21] obtains exact and generic expressions in Open image in new window for the BER of the Open image in new window -QAM (square and rectangular) constellations over additive white Gaussian noise (AWGN) and fading channels. Over the AWGN channel, these expressions can be described by a weighted sum of complementary error functions.

In the analysis and simulations which follow, we assume two bit streams, separately fed into convolutional encoders with code rates Open image in new window and Open image in new window , which are then gray-coded and modulated onto a 16-QAM constellation. After the encoding and modulation, the two streams are converged into one symbol sequence and transmitted through an AWGN channel. In the receiver the symbols contaminated by noise are demodulated using a Maximum-Likelihood-Sequence-Estimation technique (Viterbi).

In order to determine the performance of this hierarchical modulated scheme, we carry out an analysis of the error probability for the uncoded case. An exact bit error probability expression has been derived in [21]. In this section, the expression will be further developed into a simpler form. This will allow us to minimize the overall SNR which satisfies the target BERs. For the sake of clarity, we will start the analysis from the original step.

As described in [22], the 16-QAM constellation is equivalent to two 4 PAM signals on quadrature carriers. Since the signals in the phase-quadrature components can be perfectly separated at the demodulator, the probability of error for QAM can be easily determined from the probability of error for PAM. Therefore, the probability of a bit error for the Open image in new window -ary QAM is

where Open image in new window and Open image in new window are the error probabilities of the Open image in new window -ary PAMs with one-half the average power in each quadrature signal of the equivalent QAM system. It should be emphasized here that the error probability discussed here is bit error, which is different from the symbol error in [22].

The error probability for the bits contained in the HP stream is

where Open image in new window is the received symbol contaminated by white Gaussian noise with zero-mean and variance Open image in new window , and Open image in new window is the transmitted symbol (i.e., Open image in new window ). We assume that each symbol is equiprobable. Given this AWGN channel, the error probability can be given generically as

The average bit energy is

where Open image in new window and Open image in new window . Let

Using (1)–(5), we obtain the error probability for the HP bit of 4-PAM as

From the same argument, we can determine the error probability for the LP bit of the 4-PAM constellation as

Assume that the distances between the corresponding signal points in the Imaginary component and the Quadrature component are same:

By substituting the error probabilities for the PAM-system, we can obtain the corresponding QAM-system BERs as a function of Open image in new window :

### 2.1. Optimization of Hierarchical Modulation for AWGN Channel

From (9) we can derive the Signal-to-Noise Ratio (SNR) for low priority bits and high priority bits as a function of space ratio Open image in new window and the target bit error rate for high priority bits and low priority bits:

The overall SNR required by the transmission of both high priority bits and low priority bits is the bigger one of the SNR described by (10). Thus, given target bit error rates for high priority bits and low priority bits,

the optimization of the hierarchical modulation can be described by the following equation:

Since the Open image in new window function in (9) does not have an expression with finite number of coefficients, it is difficult to get an exact expression for (10). There are several approximations proposed in [23, 24, 25, 26]. However, all these approximations are suitable for a specific range of the independent variable. For example, when the independent variable Open image in new window is smaller and far away from Open image in new window ( Open image in new window ), an approximation of Open image in new window function is derived from the Maclaurin series:

The objective of the optimization is to find out an optimum number Open image in new window given Open image in new window , which leads to a minimum overall SNR. Thus the above approximation of Open image in new window function is not suitable. In this section, we first analyze the property of (10) by aid of the BER versus SNR curve. Then, a realistic method is used to calculate the tabulation of the overall SNR versus the space ratio Open image in new window and the target BER for high priority bits and low priority bits.

### 2.2. Optimization of Hierarchical Modulation for Flat Rayleigh Fading Channel

Since an OFDM system is employed in the simulation, the multipath Rayleigh fading channel is converted to a flat Rayleigh fading channel for a specific subcarrier, given that the cyclic prefix length is longer than the number of taps used by the multipath fading channel. In this section, the bit error probability of high priority bits and low priority bits over flat Rayleigh fading channel are deployed and the optimization of the hierarchical modulation over flat Rayleigh fading channel is explained.

In the simulation of this paper we employed a frequency-selective fading channel. That is, we simulated an indoor small scale multiple reflective paths radio environment and there is no line-of-sight component. There is relatively slow motion between the transmitter and the receiver. The mathematical model of the multipath radio channel is expressed by (14):

In the equation above, Open image in new window denotes the sample period and Open image in new window simulates multipath delay components of the fading channel. The coefficient Open image in new window represents the attenuation of the Open image in new window th path. Each Open image in new window can be modeled by

in which the Open image in new window and Open image in new window are independent and identical distributed (i.i.d.) Gaussian random variable with mean Open image in new window and variance Open image in new window . The magnitude Open image in new window has Rayleigh power density function (PDF) described by

In one subcarrier of the OFDM symbol, the multipath Rayleigh Fading channel is converted to a single path channel:

or in normalized continuous version,

The system channel model is described by

in which Open image in new window is the received signal, Open image in new window is the transmitted signal, Open image in new window is the complex flat Rayleigh fading component and n is Additive White Gaussian Noise (AWGN) with mean 0 and variance Open image in new window . When the received signal is equalized in the receiver, the flat Rayleigh fading component is estimated by the receiver and used to divide (19). The following equation is derived from (19):

Equation (20) indicates that by taking into account the flat Rayleigh Fading component Open image in new window the generic error probability over AWGN channel as described by (3) becomes

in which Open image in new window is a Rayleigh distribution random variable and Open image in new window is chi-square random distributed with two degrees of freedom, if the variance of Open image in new window and Open image in new window is 1, which is an assumption without loss of generality. Thus, the following equation is used to calculate the generic error probability over flat Rayleigh fading channel:

in which Open image in new window and Open image in new window is called complementary error function. Complementary function has the following relation with Open image in new window function:

The PDF for chi-square distributed random variable with two degrees of freedom is described by

By introduction of (24) to (22) the generic bit error probability over flat Rayleigh fading channel is derived as

From (25), (1), (4), (5), and (6), we can derive the bit error probability of high priority bits and low priority bits for Hierarchical Modulation over flat Rayleigh fading channel:

From (26) we can derive the Signal-to-Noise Ratio (SNR) for low priority bits and high priority bits as a function of space ratio Open image in new window and the target bit error rate for high priority bits and low priority bits over flat Rayleigh fading channel:

The optimization of the hierarchical modulation over flat Rayleigh fading channel can be described by the following equation:

### 2.3. Analysis of Packet Error Rate over AWGN Channel

The previous analysis is based on bit error rate. In practice, higher layers may be packet-oriented, so that package error rate is the more important parameter. We can make a simple mapping from BER to expected PER, under some simple assumptions. Assuming that the probability of decoding one bit wrongly ( Open image in new window ) is a stationary uncorrelated process, we can consider the decoded bit stream as a Poisson process. This yields the relationship between BER and PER:

in which, Open image in new window is PER, Open image in new window is BER and Open image in new window is the package length.

Using (9)–(29), we find that for a fixed Open image in new window , the required SNR will increase in response to increased packet length.

The packet length is affected by the tradeoff between source coding efficiency and packet error rate. Given a fixed packet length, we can achieve the corresponding optimum space ratio Open image in new window .

### 2.4. Impact of Coding on Performance

Analytical results to date have been based on uncoded bit error rates. In practice, the performance of coded hierarchical modulation systems is of more practical interest. The effect of coding will shift the BER curve to the left by the coding gain.

Coding gain of BER curve according to space ratio Open image in new window at BER of Open image in new window and Open image in new window .

0.5 | 1 | 1.5 | 2 | 2.5 | 3 | |
---|---|---|---|---|---|---|

Difference in Coding Gain (dB) | 8.3 | 2.5 |

## 3. Single-User Scenario Simulation and Results

As a proof-of-concept of the use of the Optimum Hierarchical Modulation scheme for single user in scalable video delivery, we send a still image through an AWGN single carrier channel.

The convolutional code of rate 1/2 and 16-QAM Hierarchical Modulation is employed for transmission. The data bits with higher priority and lower priority are convolutional coded and padded with parity bits. The coded bits with high priority are used to select the base bits in 16-QAM constellation and the coded bits with low priority are used to select the refinement bits in the constellation. The average distances of the base bits and the refinement bits can be tuned in order to give an optimized overall image quality (all save the Open image in new window under the same quality).

We employ a specific example of a scalable still image encoder. The Open image in new window pixels image is processed by a progressive encoder called the Embedded Zero-tree Wavelet-Spatial Orientation Tree (EZW-SOT) [27]. An embedded code represents a sequence of binary decisions that distinguish an image from the all gray image. The embedded coding possesses the property that all the bits are ordered in importance in the bit stream. The importance of the bits can be determined by the precision, magnitude, scale, and spatial location of the wavelet coefficients. For example, there are several real numbers described by 4 digits— Open image in new window . The digit Open image in new window is the most significant digit of each number and the Open image in new window is the least significant digit. Thus, the numbers can be stored by a new order in significance, say, Open image in new window Using embedded coding, a decoder can stop decoding at any position and an optimized quality of the same image will be achieved. A discrete wavelet transform provides a multiresolution presentation of the image. The wavelet coefficients can be embedded coded according to their significance. The zero-tree coding provides a binary map, which can indicate the positions of the significant wavelet coefficients.

## 4. Two Users Scenario Simulation and Results

In the single-user hierarchical modulation scenario, the two or more data channels mapped to the base bits and refinement bits of the constellation points are used to carry the data belonging to different priority levels of one service aiming at one user. As an alternative to the single user case, in the two users case, the two users are assumed to receive the data carried by the hierarchical constellation points and collect the part useful to them. In our simulation, we transmitted an H.264 scalable coded video trailer in which two different resolution sizes are embedded, a VGA ( Open image in new window pixels) size and a QVGA ( Open image in new window pixels) size. The video packets used for decoding the VGA and the QVGA versions are carried by the two different data channels of the hierarchical constellation. All the data packets are encoded using a convolutional code with a rate of Open image in new window before being mapped to the constellations. Assuming the video signal is transmitted in an indoor wireless environment, one user is close to the transmitter and has good average Open image in new window , the other is relatively far from the transmitter and relatively bad average Open image in new window . The user in a good receiving condition is able to decode most of the data packets and is able to watch the VGA version of the trailer. The user in bad receiving condition cannot obtain enough data packets for decoding a VGA trailer due to the wireless loss, but can decode a QVGA size video with acceptable quality.

To evaluate the overall quality performance received by the two users, we calculated the average PSNR performance of the VGA and QVGA versions. In this calculation we assumed that the two users' perceptive quality are equally important. According to the definition of PSNR,

the Mean Squared Error (MSE) is described by

Thus, the average MSE of the VGA and the QVGA version of the video trailer is described by

From (31) and (32) the average PSNR of the VGA and QVGA version of the video is derived to

## 5. Conclusion

This paper proposed an optimized hierarchical modulation strategy directed by cross-layer transport priority information for both single user scenario and two users scenario. In the one user scenario, the hierarchical modulation combined with a convolutional code is designed to achieve the objective bit error rates of two data channels with different priority level by an overall minimum signal to noise ratio. In the two-user scenario, the hierarchical modulation strategy, the video codec scalability in spatial domain is considered and an optimized strategy is proposed for the best perceivable quality of the scalable video transmission. The optimization strategy can be implemented in the timescale of Open image in new window ms to Open image in new window ms. The simulation results show 1.5 dB gain in Open image in new window in the single user scenario and 6 dB gain in PSNR (perceivable quality of the scalable video) in two user scenario. Hierarchical modulation is proved to be a promising candidate for the transmission system for scalable digital media.

## Notes

### Acknowledgment

The authors would like to thank Krishna Sankar because the inspirations about how to calculate the generic bit error probability over flat Rayleigh fading channel is from his web site http://www.dsplog.com/.

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