CRC and Conv. Concatenated Channel Coder. Block. Input. Source Coder. Moldulation. Interleaver. Image. Channel. Block. List Viterbi Channel Decoder

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1 Variable Rate Channel Coding and Enhanced Interleaving for Image Transmission using an Outage Criterion Salim Manji and Narayan. Mandayam WINLA Rutgers, The State University of New Jersey 7 rett Rd. Piscataway, NJ salim, narayan@winlab.rutgers.edu Abstract - We consider an image transmission system consisting of Set Partitioning in Hierarchical Trees (SPIHT) wavelet-based source coding [1] [2], and List Viterbi Algorithm (LVA) channel coding [] using rate compatible punctured convolutional (RCPC) codes [4]. Our goal is to minimize bandwidth usage while maintaining received image quality. We introduce an outage criterion for image transmission. A received image is considered to be in outage if its pixel values are not identical to the image transmitted. In order to reduce correlation in fading, an enhanced block interleaving scheme employing a subblock structure is proposed. Using RCPC codes, we devise a variable rate channel coding system based on the current channel state information. Combining enhanced interleaving and variable rate channel coding, we construct a method to select the highest rate RCPC code that satises the image outage criterion, thus balancing the requirements of high bandwidth eciency and high image quality. I. Introduction The use of an outage criterion for performance of voice communications over wireless networks is a well established idea [5]. Outage is dened as the event that the instantaneous bit error rate is above an allowable threshold, and the requirement is to limit the probability of outage to some small value. We wish to apply an outage criterion towards the transmission of still images over wireless channels. A typical user requests a download of one source compressed image. We dene image outage as the event that the pixel values of the received image are not identical to the image transmitted. We wish to limit the probability of outage to some small fraction,, of all images transmitted. The conventional metric for image quality is peak signal-to noise ratio (PSNR). ecause we consider transmission over error prone wireless channels, average PSNR may be misleading. The SPIHT source decoder is highly sensitive to even a single bit error, hence the variance of the PSNR may be large. Outage gives a simple and clear metric for received image quality. The combination of Set Partitioning in Hierarchical Trees (SPIHT) wavelet-based source coding [1] [2], and channel coding based on the List Viterbi Algorithm (LVA) [] provides bandwidth ecient, error resilient image transmission over wireless channels [6] [7]. The LVA is a concatenated channel coding system that uses an inner code for error correction and an outer code for error detection. The inner code is either a convolutional code or a rate compatible punctured convolutional (RCPC) code [4]. The outer code is typically a cyclic redundancy check (CRC) code. From a mother code of rate R c = 1=r we can develop a family of weaker, higher rate RCPC codes, all of which use the same encoder and decoder. For example, a rate R c = 1=, memory = 4 convolutional code serves as the mother code for a family of RCPC codes of rate 4=n for n = 5; 6; :::; 12 [4]. In order to change the rate of the RCPC code, minimal additional complexity is required at the transmitter and receiver. Given the current channel state conditions, represented by the mean coded symbol energy-to-noise ratio, we can identify the highest rate RCPC code that will satisfy the outage requirement for image transmission. The assumption is that the channel state is relatively constant over the transmission duration of one image. Choosing the highest rate possible minimizes bandwidth usage. Under poor channel conditions, more channel coding is required to satisfy the outage criterion. Assuming that user cost is proportional to the number of bits transmitted over the channel, our scheme minimizes the cost of image transmission while maintaining image quality. The image transmission system model is given in Section II. In Section III, we develop an enhanced block interleaver. Section IV shows simulations results for image transmission over a slow Rayleigh fading channel. Conclusions are oered in Section V. II. System Model The image transmission system model (see Figure 1) that we propose is comprised of SPIHT source coding, LVA channel coding, and interleaving. The raw image to be transmitted is rst compressed using the SPIHT algorithm. The source coding rate is R s bits per pixel (bpp). The rate of compression is chosen such that the recon-

2 structed image is virtually indistinguishable from the original. We call this imperceptibly lossy source compression. Relative to lossless compression, this approach allows for a signicant increase in the source compression ratio. For the purpose of illustration, consider the well known 8-bit grayscale Lena image with a resolution of l = 256 lines and p = 256 pixels per line. Our experimental results show that lossless compression using the SPIHT algorithm requires a source rate of 4.57 bpp. Lossy compression at a rate of R s = 1:0 bpp results in a reconstructed image with PSNR of 7.67 d. The image is virtually indistinguishable from the original yet oers greater than four times the compression of the lossless image. Thus imperceptibly lossy source coding provides an ecient tradeo between high source compression and high image quality. Input Image Image SPIHT Source Coder SPIHT Source Decoder CRC and Conv. Concatenated Channel Coder List Viterbi Channel Decoder lock lock De- PSK Moldulation Coherent PSK Demodulation Figure 1: Image Transmission System Model. Channel The compressed image is then divided into blocks of K c bits and an (N c ; K c ) CRC channel code is applied for error detection. The number of blocks,, required to represent the compressed image is = l p Rs K c : (1) The CRC blocks are further coded using a convolutional or RCPC code with rate R c and memory units. Prior to convolutional coding, a string of zeros is appended to the output of the CRC code. This ensures that the LVA decoder knows the ending state of the convolutional code. The input to the convolutional coder is a block of length N c +. Thus LVA encoding causes a bit stream expansion by a factor of (N c + )=K c over conventional convolutional coding. The overall rate of the concatenated channel coder is R = Kc N c + R c : (2) The length of each coded block is L = K c =R bits. One parameter that we have chosen to x is N c? K c, the number of parity check bits in the CRC code. A value of N c?k c = 16 ensures that there is virtually zero possibility of an undetected block error. This property of perfect error detection is very desirable and useful. As soon as a block error is detected, transmission may be discontinued since it is known that the received image is in outage. Alternatively, if used in conjunction with a retransmission scheme, the CRC code ensures that retransmissions are limited to those blocks known to be in error. The 16 bit CRC polynomial used for all simulation results is g(x) = x 16 + x 15 + x 14 + x 12 + x 11 + x 8 + x 5 + x 4 + x [8]. We wish to limit the bit stream expansion caused by LVA encoding to 10%. Using a convolutional code with = 4 memory units, and a (216, 200) CRC code results in 220 bits fed to convolutional encoder for every 200 information bits resulting in a 10% expansion as required. After channel coding, the next element in the image transmission model is the interleaver, discussed in detail in Section III. We consider coherent PSK modulation over a slow, frequency non-selective Rayleigh fading channel. We assume that the fading is suciently slow to allow for perfect phase estimation. The mean coded symbol energyto-noise ratio, s, is given as s = E( 2 ) E s N 0 ; () where E s is the energy per coded symbol, and is a Rayleigh distributed random variable. We consider a carrier frequency of f c = 2:0 GHz, which is in the neighborhood of the candidate carrier frequencies for third generation wireless networks [9]. With a mobile velocity of v m/s and a channel rate of C = 64 kbits/s, the resulting Doppler frequency is f D = ( 20 )v Hz and the normalized fading rate is f D T s = (1: ?4 )v. A mobile velocity of 1.5 m/s = 5.4 km/h, which is approximately walking speed, results in f D = 10 Hz and f D T s = 1: ?4. A highway speed of 0 m/s = 108 km/h results in faster fading with f D = 200 Hz. The Rayleigh fading channel is modeled using the method described by Jakes [10]. The correlation function of the fading amplitude,, is [11] (t) = J 2 0 2vt c = J 2 0 (2f D t) (4) where J 0 () is a zero order essel function of the rst kind and c is the wavelength of the carrier. III. Enhanced lock Interleaving It is well known that error performance over a fading channel is signicantly improved by the inclusion of interleaving. Optimal interleaving results in independent fading between two adjacent symbols. Analytical work often assumes that the interleaving depth is adequate to provide independent fading. For image transmission, the assumption of innite interleaving can not be made. This is because, in most situations, a typical user request is to download one single image. Thus the maximum interleaving depth is over the bits that comprise one compressed image. This is likely insucient to result in independent fading over a slow fading channel.

3 The simplest implementation of block interleaving for image transmission is illustrated in Figure 2. Interleaving is performed over the blocks that comprise one image; each block consists of L bits. With an interleaving depth of, the delay between the transmission of two adjacent bits within a block is T s, and the resulting correlation in fading amplitude is (T s ) = J 2 0 (2f D T s ). lock it 1 Input 2 L -1 L Figure 2: Simple lock Interleaving. We propose an enhanced version of block interleaving illustrated in Figure. Each block is divided into N subblocks of length L N bits such that N L N = L. Transmission proceeds across the bits within each subblock. The result of enhanced interleaving is that the delay between the transmission of two adjacent bits within a subblock is increased by a factor of N to NT s, and the resulting correlation in fading amplitude is (NT s ) = J0 2 (2f D NT s ). For use in conjunction with variable rate channel coding, we wish to implement enhanced block interleaving in an eective yet computationally simple manner. The set of available RCPC code rates is R c = 4=n for n = 5; 6; :::; 12 and = 4. Using a (216; 200) CRC code, the resulting block length is L = 55n. Therefore, we can choose a subblock length of L N = 55, resulting in N = n subblocks per block. This oers a structure for enhanced block interleaving that requires little additional complexity and easily implemented for all choices of the RCPC code rate. Enhanced block interleaving increases the delay between two adjacent bits within a subblock. At the same time, it decreases the delay between bits in adjacent subblocks. For example, consider bit i in subblock j and bit i in subblock j + 1 of some arbitrary block k. With simple block interleaving, the delay between the transmission of these two bits is L N T s. Enhanced block interleaving reduces the delay to T s, which may result in correlated fading between these two bits. We contend that this reduction in delay will not degrade performance if L N is suciently large. For conventional Viterbi decoding, excessive storage requirements at the receiver are alleviated by truncating path memories after M trellis sections. The choice of M > 5( + 1) is sucient to result in negligible performance degradation [12]. Therefore, choosing L N > 5( + 1)=R c ensures that the correlated fading between bits in adjacent subblocks is of little consequence. We can make an alternative argument using the minimum distance of the convolutional code. Within the trellis structure of the convolutional code, bit i of subblock j and bit i of subblock j + 1 are separated by L N coded bits. In the event of an LVA decoding error, the error path deviates from the transmitted sequence at some point in the trellis and remerges at some later point. If L N is considerably larger than the minimum distance of the code, d min, then the probability that one error path encompasses both bits i of subblocks j and j + 1 is negligibly small. Thus correlated fading between these two bits has no signicant impact on performance. For example, the R c = 1=, = 4 convolutional code has d min = 11 [4]. Choosing L N = 5 d min = 55 ensures that bits i in subblocks j and j + 1 are suciently far apart in the code trellis. This is actually smaller than the previously suggested value of 5( + 1)=R c = 75. lock 1 2 Subblock (1,1) Subblock (2,1) Subblock (,1) Subblock (1,2) Subblock (2,2) Subblock (,2) -1 Subblock (-1,1) Subblock (-1,2) Subblock (,1) Subblock (1,1) (2,1) (,1) (,1) (1,2) (-1,N) (,N) Subblock (,2) Subblock (1,N) Subblock (2,N) Subblock (,N) Subblock (-1,N) Subblock (,N) it Input 1 2 L Figure : Enhanced lock Interleaving. N

4 To see the benets of enhanced block coding, consider image transmission over a narrowband Rayleigh fading channel with f D = 10 Hz and C = 64 kbps. For channel coding, we use a rate R c = 1=, = 4 convolutional code, (216; 200) CRC code, and LVA list length L V =. With a delay of T seconds between the transmission of adjacent bits, the fading correlation is suciently low to yield performance close to uncorrelated if f D T 0:8 [1]. A value of f D T = 0:8 corresponds to the rst zero of the correlation function, (t). Solving (1), the 8-bit grayscale Lena image with l = 256 lines and p = 256 pixels per line compressed at a rate of R s = 1:0 bpp requires = 28 blocks per image. Using simple interleaving, the delay between the transmission of adjacent bits is T = T s, resulting in f D T = 0: The fading correlation between adjacent bits is (T s ) = 0:949. Simple interleaving does little to reduce correlation. With enhanced interleaving, we have N = 12 subblocks per block. The delay between adjacent bits increases by a factor of N to T = NT s, resulting in f D T = 0:615. This is sucient for performance close to uncorrelated. The fading correlation is (NT s ) = 0:162. Figure 4 compares the simulated block error probability with simple and enhanced interleaving for R c = 1= and f D = 10 Hz. A block error is dened as the event that a block is incorrectly decoded by the LVA. We see that enhanced interleaving oers a gain of about 2.5 d for a block error probability of 10?4. If we consider a Rayleigh fading channel with f D = 200 Hz, simple interleaving results in f D T = 1:025 which is sucient for performance close to uncorrelated. In this case, enhanced interleaving will not oer any further performance improvement. In general, f D T s < 0:8 oers a situation where enhanced interleaving is superior. Figure 5 compares the simulated block error probability with R c = 4=5 and f D = 10 Hz. Enhanced interleaving provides a gain of 1.5 d for a block error probability of 10?4. There are N = 5 subblocks per block. The resulting correlation with enhanced interleaving is (NT s ) = 0:202 compared to with simple interleaving. IV. Simulation Results For a given RCPC code, we can determine the block error probability, P, over a range of s, through simulation. We wish to nd the block error probability required to meet the outage criterion, P ;req. Without knowledge of whether block errors are correlated or independent, we can apply a simple worst case scenario to nd P ;req such that the outage criterion is guaranteed to be satised. Assuming that each instance of outage is caused by a single block error, we have P ;req = (5) where is the tolerable outage probability. The value of s that corresponds to a block error probability of P ;req is the minimum value for which this RCPC code is guaranteed to satisfy the outage criterion. For example, a tolerable outage probability of = 5% and = 28 results in P ;req = 1:524 10?4. Figure 6 shows the simulated block error probability for all available RCPC code rates over a Rayleigh fading channel with f D = 10 Hz, corresponding to a velocity of 5.4 km/h. We can generate similar sets of curves for dierent Doppler frequencies corresponding to various mobile velocities. From Figure 6, we can formulate a guideline for choosing the appropriate RCPC code rate, R c, as a function of s. This is shown in Table 1. With this particular mother code, it is not possible to satisfy the outage criterion when s < 2:2 d. If s > 11:0 d, we have a good channel in which case the highest rate RCPC code is used. lock Error Probability 10 Simple Interleaving Enhanced Interleaving ) (d) Figure 4: Simple versus enhanced interleaving for R c = 1= and f D = 10 Hz. lock Error Probability 10 Simple Interleaving Enhanced Interleaving ) (d) Figure 5: Simple versus enhanced interleaving for R c = 4=5 and f D = 10 Hz.

5 lock Error Probability 10 Rc = 4/5 Rc = 2/ Rc = 4/7 Rc = 1/2 Rc = 4/9 Rc = 2/5 Rc = 4/11 Rc = 1/ P,req References [1] J.M. Shapiro, \Embedded Image Coding using Zerotrees of Wavelet Coecients," IEEE Trans. Signal Processing, Vol. 41, pp , Dec [2] A. Said, W.A. Pearlman, \A New Fast and Ecient Image Codec ased on Set Partitioning in Hierarchical Trees," IEEE Trans. Circuits Syst. Video Technol., Vol. 6, pp , June [] N. Seshadri, C.-E. W. Sundberg, \List Viterbi Decoding Algorithms with Applications," IEEE Trans. Comm., Vol. 42, pp. 1-2, Feb/March/Apr ) (d) Figure 6: Simulated P for f D = 10 Hz. Table 1: Guidelines for choosing the RCPC code rate. R c s (d) 1/ / / / / / / /5 > 11.0 V. Conclusions We have considered an image transmission system that uses SPIHT wavelet-based source coding, and LVA concatenated channel coding to provide bandwidth ecient, error resilient image transmission. Consistent with wireless communications, we have applied the notion of outage to image transmission. An image is in outage if the received pixel values are not identical to the ones transmitted. An enhanced block interleaving scheme was introduced using a subblock structure to eectively reduce correlation in fading. Using RCPC codes, we devised a variable rate channel coding system based on the current channel state information. Combining enhanced interleaving and variable rate channel coding, we constructed a method to select the highest rate RCPC code that satises the image outage criterion, thus balancing the requirements of high bandwidth eciency and high image quality. [4] J. Hagenauer, \Rate-Compatible Punctured Convolutional Codes (RCPC Codes) and their Applications," IEEE Trans. Comm., Vol. 6, pp , April [5] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, C. E. Wheatley III, \On the Capacity of a Cellular CDMA System," IEEE Trans. Veh. Technol., Vol. 40, pp , May [6] S. Manji, G. Djuknic, \andwidth Ecient and Error Resilient Image Coding for Rayleigh Fading Channels," Proc. IEEE VTC Spring '99, pp , May [7] P.G. Sherwood, K. Zeger, \Progressive Image Coding for Noisy Channels," IEEE Signal Processing Letters, Vol. 4, pp , July [8] G. Castagnoli, J. Ganz, P. Graber, \Optimum Cyclic Redundancy-Check Codes with 16-it Redundancy," IEEE Trans. Comm, Vol. 8, pp , Jan [9] T. Ojanpera, R. Prasad, \An Overview of Third- Generation Wireless Personal Communications: A European Perspective," IEEE Personal Comm., Vol. 5, pp , Dec [10] W.C. Jakes, Microwave Mobile Communications, IEEE Press, [11] W.C.Y. Lee, Mobile Communications Engineering, McGraw-Hill, [12] J.A. Heller, I.M Jacobs, \Viterbi Decoding for Satellite and Space Communication," IEEE Trans. Comm, Vol. COM-19, pp , Oct [1] R. van Nobelen, D.P. Taylor, \Analysis of the Pairwise Error Probability of Noninterleaved Codes on the Rayleigh-Fading Channel," IEEE Trans. Comm, Vol. 44, pp , Apr Acknowledgments Thanks are due to Carl-Erik W. Sundberg and rian Chen for providing a software implementation of the List Viterbi Algorithm.