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Hybrid digital-analog coding with bandwidth expansion for correlated Gaussian sources under Rayleigh fading
- Pradeepa Yahampath^{1}Email author
https://doi.org/10.1186/s13634-017-0474-z
© The Author(s) 2017
Received: 22 November 2016
Accepted: 9 May 2017
Published: 25 May 2017
Abstract
Consider communicating a correlated Gaussian source over a Rayleigh fading channel with no knowledge of the channel signal-to-noise ratio (CSNR) at the transmitter. In this case, a digital system cannot be optimal for a range of CSNRs. Analog transmission however is optimal at all CSNRs, if the source and channel are memoryless and bandwidth matched. This paper presents new hybrid digital-analog (HDA) systems for sources with memory and channels with bandwidth expansion, which outperform both digital-only and analog-only systems over a wide range of CSNRs. The digital part is either a predictive quantizer or a transform code, used to achieve a coding gain. Analog part uses linear encoding to transmit the quantization error which improves the performance under CSNR variations. The hybrid encoder is optimized to achieve the minimum AMMSE (average minimum mean square error) over the CSNR distribution. To this end, analytical expressions are derived for the AMMSE of asymptotically optimal systems. It is shown that the outage CSNR of the channel code and the analog-digital power allocation must be jointly optimized to achieve the minimum AMMSE. In the case of HDA predictive quantization, a simple algorithm is presented to solve the optimization problem. Experimental results are presented for both Gauss-Markov sources and speech signals.
Keywords
- Hybrid digital-analog coding
- Predictive quantization
- Transform coding
- Fading channels
- Speech coding
1 Introduction
In digital communication over a fading channel, the best performance is achieved when both the transmitter and the receiver are adapted to the channel state. If the channel-state information (CSI) is available, the transmitter can adapt coding and modulation to maintain the optimal performance at all times. However, there are common situations in which the transmitter adaptation is not an option. One obvious example is broadcasting where a single transmitter sends information to multiple receivers. Since the channels to different receivers may not be the same, it is not possible to adapt the transmitter to a specific channel state. Another example is when there is no possibility of CSI feedback from a mobile receiver to the transmitter. In either case, the receiver suffers from the “cliff effect” [1]—when channel signal-to-noise ratio (CSNR) decreases, at some point, a less than 1 dB drop in CSNR can take the decoder from perfect operation to complete failure (threshold effect), and when the CSNR increases from this point, the decoder output quality remains fixed regardless of the CSNR (see for example [2] (Fig. 5)). One solution to this problem is multi-resolution coding and modulation [1, 3, 4]. This scheme does not entirely eliminate the cliff effect but improves it to a stair-case effect. For analog sources, a better alternative is hybrid digital-analog (HDA) coding [1, 5, 6] which is the focus of this paper.
It is known that uncoded or analog transmission achieves the optimal performance theoretically attainable (OPTA) in MMSE sense when both the source and the channel are Gaussian and memoryless and have the same bandwidths [7]. Clearly, uncoded transmission cannot be optimal for sources with memory and when the source and channel bandwidths are not matched. For sources with memory, widely used digital source-coding techniques such as predictive quantization (PQ) transform coding (TC) [8] exploit source memory to achieve a coding gain and will outperform uncoded transmission if both the transmitter and the receiver have CSI. However, systems based on these techniques still suffer from the aforementioned cliff effect when the transmitter has no CSI. On the other hand, implementing good analog codes for sources with memory is difficult. A promising approach to benefit from both the robustness of analog transmission against CSNR variations and the source-coding gain due to source correlation is HDA coding. Fundamentally, HDA transmission involves the simultaneous transmission of a source in both digital and analog forms. Most previous work on HDA coding have used a form of layered transmission in which the base layer is digitally coded, and the quantization error of the base layer is transmitted as a refinement layer, using analog pulse amplitude (PAM) modulation [2, 9–12]. While a considerable amount of research has focused on HDA transmission of memoryless sources, much less work has been devoted to developing good HDA codes for sources with memory. In particular, when the source has memory, the optimal HDA coding involves a very different design trade-off compared to coding a memoryless source. The main goal of this paper is to design HDA systems which can simultaneously benefit from high coding gain of PQ or TC and the CSNR-independent optimality of a parallel analog transmission. PQ is the standard technique for moderate to high bit-rate (16–40 kbs) speech coding [13] while TC is a staple in image and video compression.
We consider the transmission of a correlated Gaussian source over a block-fading Gaussian channel whose bandwidth is greater than or equal to the source bandwidth (channel memory is however not considered). In the proposed approach, the source is digitally transmitted using either PQ or TC. The quantization error of the digital encoder is transmitted by linear analog coding over the same channel bandwidth as the digital transmission, by using superposition and power sharing. Given that the transmitter cannot be adapted to the instantaneous CSNR at the receiver, we determine the best analog-digital power allocation by minimizing the average MMSE (AMMSE) with respect to the receiver-CSNR distribution. A closer look at this problem reveals an interesting trade-off between digital and analog transmissions when the source has memory. On the one hand, allocating more power to the digital transmission allows a higher quantization rate and hence a higher predictive or transform coding gain. On the other hand, allocating more power to the analog transmission makes it possible to achieve a greater reduction in distortion as the CSNR increases. The not so obvious variable here that also affects this trade-off is the outage CSNR which is the lowest CSNR at which a receiver can decode the digital signal. For the same power allocation, a higher quantization rate can be chosen at the expense of increased outage CSNR. Therefore, there exists a non-trivial trade-off between the power allocation, quantization rate, and the outage CSNR.
We also address the problem of determining the power allocation and the outage CSNR (or equivalently the quantization rate) in HDA-PQ and HDA-TC systems to achieve optimal (in AMMSE sense) trade-off. To this end, we obtain analytical expressions for the AMMSE of HDA-PQ and HDA-TC systems by relying on the high-rate model of entropy constrained scalar quantizers [14]. Our solutions are therefore asymptotically (in rate) optimal. In general, finding a closed-form solution for the optimal power allocation and outage CSNR appears intractable. However, in the case of HDA-PQ, we identify a simple co-ordinate descent algorithm [15] to determine the optimal solution. This algorithm converges rapidly, typically in 2–3 iterations. We demonstrate that it is quite possible to implement good practical finite-rate HDA-PQ and HDA-TC systems using the asymptotically optimal solutions. Experimental results obtained with Gauss-Markov processes as well as speech signals modeled as a Gaussian auto-regressive (AR) process show that both the system AMMSE and the MMSE of a receiver operating at a given CSNR of practical designs closely match those given by the asymptotic expressions, when the quantization rate is higher than about 1 bit/sample. Our results show that, for highly correlated sources, the HDA systems can substantially outperform both purely digital and purely analog transmission over a wide range of receiver CSNRs.
1.1 Main contribution and related previous work
Compared to previous work on HDA coding of Gaussian sources with memory, the main contribution of this paper is the joint optimization of power allocation and quantization rate of HDA systems based on PQ or TC, with respect to the AMMSE criterion. This optimization problem does not arise when the source is memoryless. We also provide a lower bound to the AMMSE achievable for source with memory, which can be numerically computed for a Gauss-Markov source.
Previously, HDA coding of correlated sources have appeared in [2, 9–12, 16–18]. With the exception of [18], none of these work uses the AMMSE as a criterion for power allocation. While [18] uses the AMMSE, their problem is analog-only transmission of unquantized video DCT coefficients over a fast fading channel. The objective of the power allocation in that case is to benefit from channel-diversity. Therefore, power is allocated among consecutive analog transmissions. As a result, their formulation leads to a mixed discrete and continuous optimization problem which has been solved by a heuristic approach unrelated to ours. The other work cited above does not consider the joint optimization of the power allocation and the quantization rate. Phamdo and Mittal [2] present an implementation of an HDA system for low-bit-rate speech transmission based on the standard FS 1016 CELP codec, by using two independent channels with identical CSNRs for digital and analog transmissions (hence identical power allocations). Yu et al. [9] present similar HDA scheme for video transmission based on H.264/AVC codec but use channel superposition of analog and digital components the power allocation between which is determined by assuming a worst-case CSNR. In [10–12], channel optimized vector quantizers (COVQ) are used as the digital encoder whose quantization error is transmitted in analog form. However, no method for optimizing the power allocation is given. An HDA transform coding scheme is considered in [16], where the analog and digital components are transmitted by time-division multiplexing using equal powers. To the authors’ knowledge, HDA schemes based on linear predictive quantization have not been reported so far.
The rest of this paper is organized as follows. Section 2 describes the HDA system considered in this paper and derives an expression for the decoder MSE. Section 3 finds expressions for the MMSE of asymptotically optimal HDA-PQ and HDA-TC over a Rayleigh fading channel. Section 4 considers the main optimization problem and presents a simple algorithm for solving the problem in the case of HDA-PQ. Section 5 presents some performance bounds for HDA-PQ and HDA-TC systems. Section 6 presents numerical and experimental results, and concluding remarks are given in Section 7.
2 HDA transmission of correlated Gaussian sources over fading channels
and \(\sigma _{\epsilon }^{2}=E\{ \epsilon _{n}^{2}\}\) is the quantization error variance. Therefore, the amplification factor \(\alpha =\sqrt {\frac {\rho {\mathcal {P}}_{T}}{\sigma _{\epsilon }^{2}}}\).
Therefore, the quantization error variance is a function both ρ and θ _{ o }, which we denote by \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\).
Notice that the numerator is independent of the receiver CSNR θ. The denominator is the factor by which the overall distortion at the receiver is reduced due to the analog transmission of the quantization error. Unlike a purely digital system whose MSE will depend only on θ _{ o } independent of the instantaneous CSNR θ, an HDA system will have D(ρ,θ _{ o },θ)→0 as θ→∞. In the following, we first derive expressions for \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) of asymptotically optimal PQ and TC. We then consider determining optimal ρ and θ _{ o } for a Rayleigh fading channel which minimize the AMMSE.
3 Asymptotically optimal quantization in HDA systems
3.1 HDA-PQ
where R is the rate and \(h= \frac {\sqrt {3} \pi }{2}\) for fixed-rate scalar quantization and \(h=\frac {\pi e}{6}\) for entropy constrained scalar quantization [8] (in the latter case, R=H(q) is the entropy of the quantizer output q _{ n }).
where for convenience, we define the constant \(c_{0} \triangleq hA^{2}\). We refer to a PQ which satisfies (12) as an asymptotically optimal PQ. The related work on high-rate analysis of predictive quantizers can be found in [23–25].
In other words, sufficient channel bandwidth must be available to support a high enough quantization rate. With b and θ _{ o } fixed, increasing ρ reduces the allowable quantization rate. Hence, the high-rate model (13) is valid only for “small” ρ. However, as will be seen in Section 6, HDA-PQ provides a useful coding gain only in this regime (typically ρ<30%) anyway, as higher ρ results in low quantization rates at which predictive coding does not yield a considerable gain over pure analog transmission.
Before proceeding, it is worth noting that when prediction is good, the prediction error resembles a white Gaussian process [8]. For the transmission of the latter, analog transmission will be nearly optimal (exactly optimal if the source and channel are bandwidth matched). However, transmitting the open-loop prediction error itself in analog form is not possible in predictive quantization as it would result in channel error propagation in a closed-loop decoder.
3.2 HDA-TC
A detailed description of transform coding (TC) can be found in [8]. For stationary Gaussian sources, it is known that both PQ and TC can asymptotically (in rate) achieve the same MMSE, provided that PQ uses infinite memory linear prediction and TC uses a Karhunen-Loeve transform (KLT) of infinite dimension [8]. However, at low bit-rates, the performance of PQ for Gaussian sources drops below that of TC, due to the degradation of closed-loop predictions based on quantized samples.
where the integer m≤M and hence G _{ m } is a function of ρ and θ _{ o }. While this solution is simple to determine for any given (ρ, θ _{ o }), unlike (13), it does not seem to have a closed-form expression in terms of ρ and θ _{ o }.
4 Robust HDA systems for fading channels
Consider the MSE D(ρ,θ _{ o },θ) in (9), where θ is a random variable (but assumed to remains constant at least for the duration of a single channel codeword), where \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) is given by either (13) or (17). This is the MMSE of an asymptotically optimal HDA-PQ or HDA-TC for a particular (ρ,θ _{ o }). The choice of ρ and θ _{ o } determines how the MMSE varies with the CSNR θ. If θ is known to the transmitter, ρ=0 (purely digital) will achieve the lowest MMSE for any θ, since in this case,\(D(0,\theta _{o},\theta)=\sigma _{\epsilon }^{2}(0,\theta _{o})\) can be minimized by choosing θ _{ o }=θ. In this case, both PQ and TC achieve the maximum possible coding gain. If however the receiver CSNR θ is not available to the transmitter, a purely digital system must be designed for some θ _{ o } which will be different to θ, resulting in a system that is not robust against CSNR variations. On the one hand, the receiver MSE of such a system remains constant even when θ>θ _{ o } despite the increase in the available channel capacity. On the other hand, the channel code and hence the system fail when θ<θ _{ o }, i.e., system goes into outage. We refer to θ _{ o } as the outage CSNR of the digital decoder. When the transmitter cannot be adapted to varying θ, allocating power to the analog transmission (ρ>0) while keeping θ _{ o } fixed will increase the quantization MSE \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) but will make the overall MSE D(ρ,θ _{ o },θ) to decrease with θ. For fixed ρ, increasing θ _{ o } reduces \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) but will increase the outage probability and hence the AMMSE. In order to obtain a robust system which is optimal in some sense over a range of θ, we design the transmitter for ρ and θ _{ o } which minimizes the AMMSE E{D(ρ,θ _{ o },θ)} with respect to the distribution of θ. Such a design is ideal for a system with a single receiver which experiences slow fading or a broadcast environment with a large number of receivers whose empirical CSNR converges to the fading distribution [26].
Figure 2 illustrates the convexity with respect to ρ and quasi-convexity with respect to θ _{ o }. Below we present an efficient method to determine the optimal solution for (ρ,θ _{ o }) in the case of HDA-PQ. Due to the lack of a closed-form expression for the AMMSE, such a simple procedure cannot be devised for HDA-TQ.
4.1 Optimal HDA-PQ
In general, it is difficult to find a closed-form solution to the constrained non-linear minimization problem in (22). In the following, we present a simple coordinate-descent (CD) method [15] to solve this problem. In the CD method, \(\bar {D}(\rho, \theta _{o})\) is minimized alternately with respect to ρ (for fixed θ _{ o }) and θ _{ o } (for fixed ρ _{ o }), until the solution converges. Unlike the joint minimization problem in (22), these two sub-problems are much easier to solve. Since the solution to each problem is conditionally optimal, the CD algorithm is guaranteed to converge to the minimum of \(\bar {D}\). In actual numerical examples, it was found that this method only required 2–3 iterations to converge. In the following, we present the solutions to two sub-problems solved in each CD iteration.
where \(f_{1}(\rho,\theta _{o}) \triangleq c_{1}-\phi (\rho,\theta _{o})\) with c _{1}=ν[h(1+A ^{2})]^{1/b } and ν>1 is a sufficiently large constant chosen to ensure that (14) is not violated at low quantization rates. In our experiments, we have used ν=2. If the constraint (24) becomes active, the solution is not guaranteed to be optimal. However, note that, as the quantization rate decreases, the HDA-PQ performance approaches that of analog-only transmission. Therefore, when HDA-PQ outperforms purely analog transmission, (24) is unlikely to be active. For example, with both Gauss-Markov sources and speech signals, numerical results presented in Section 6 show that when ρ exceeds about 30%, the difference between HDA-PQ and analog systems becomes negligible. It is in this range of ρ that (24) becomes active.
4.1.1 Optimal power allocation for fixed outage CSNR
which can be solved in the interval 0≤ρ<ρ _{max} using a single-variable root-finding method.
4.1.2 Optimum outage CSNR for fixed power allocation
for ρ=0. Given ρ, optimal θ _{ o } can be found by locating the root of (30) or (31) in the interval θ _{ o }∈ [ θ _{ o,min},θ _{ o,max}), where θ _{ o,max} is a suitable value chosen to truncate the pdf p(θ).
5 Comparisons and performance limits
5.1 Analog transmission with block decoding
This analog system achieves no coding gain from source correlation, but it does achieve a gain at the receiver due to linear block decoding. Therefore, (35) is not necessarily worse than (21), though it will be so when source correlation is high. However, since sample-by-sample analog encoding and decoding is a special case of HDA coding, (35) is an upper bound to (21) when the source correlation is ignored in (32), that is when \(\tilde {\lambda }_{i}=1\) in (35).
5.2 HDA vector quantization (HDA-VQ) lower bound
HDA systems considered in this paper can asymptotically achieve performance (AMMSE) that cannot be achieved with either purely analog transmission or purely digital transmission. On an absolute scale, the upper bound to HDA system performance is the optimum performance theoretically attainable (OPTA) when CSI is only available at the receiver. Unfortunately, this bound cannot be determined in any reasonable way, even for a Gaussian source. One obvious upper bound that is easily computed for a Gaussian source is the OPTA when the CSI is available at both transmitter and receiver. This can be found by evaluating the distortion-rate function of the Gaussian process [29] at the rate equal to the capacity of an AWGN channel with the given channel power gain. A more meaningful upper bound for the case when CSI is only available to the receiver can be obtained by replacing the PQ or TC in the HDA coding setup by an optimal (rate-distortion achieving) VQ for the source. The HDA-VQ of a memoryless Gaussian source over a non-fading AWGN channel has previously been studied in [11, 12]. Below, we derive an expression for the AMMSE of HDA-VQ for the G M(a) source and Rayleigh-fading AWGN channel.
Since neither PQ nor TC can outperform optimal VQ, the AMMSE in (18) is bounded below by the minimum value of \(\bar {D}_{HDA-VQ}(\rho,\theta _{o})\). There is no apparent simple way to determine this minimum value since a closed-form expression for δ _{ G }(ρ,θ _{ o }) is not available for all ρ and θ _{ o }. Numerical values of this bound shown in Section 6 have been obtained by performing a grid-search over the (ρ,θ _{ o }) space where 0≤ρ≤1 and 0≤θ _{ o }≤θ _{ o,max} (a suitable upper limit) to determine the minimum of (37).
6 Numerical results and discussion
In this section, we use numerical examples to demonstrate the theoretical performance achievable with asymptotically optimal HDA systems as well as the actual performance of finite-rate HDA systems designed using power allocations and quantizer rates obtained through asymptotic analysis. It is useful to compare the minimum AMMSE of actual HDA-PQ and HDA-TC designs with the HDA-VQ bound for the same source-channel pair. While the latter bound can be difficult to evaluate for a general Gaussian source, it can be numerically evaluated for a Gauss-Markov source (Section 5.2). We also compare the HDA systems with the purely analog system in Section 5.1 and purely digital systems (PQ and TC). We do so for both GM(a) source and speech signals modeled by a Gaussian AR source.
6.1 Performance for Gauss-Markov sources
The power allocations and outage CSNRs of HDA-PQ and HDA-TC systems shown in Fig. 3
HDA-PQ | HDA-TC | ||||
---|---|---|---|---|---|
\(\bar {\theta }\) (dB) | b | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) |
15 | 3 | 27 | –1.9 | 40 | –3.8 |
20 | 3 | 27.5 | –1.3 | 42 | –3.0 |
25 | 3 | 27.5 | –0.8 | 43 | –2.8 |
30 | 3 | 27.5 | –0.4 | 44 | –2.5 |
15 | 6 | 28.5 | –4.6 | 43 | –6.5 |
20 | 6 | 28.5 | –4.1 | 45 | –6.0 |
25 | 6 | 28.5 | –3.8 | 47 | –5.3 |
30 | 6 | 28 | –3.4 | 48 | –5.5 |
Table 1 lists the ρ and θ _{ o } values of HDA-PQ and HDA-TC systems whose AMMSEs are shown in Fig. 3. In general, the power allocated to the analog component of both HDA-PQ and HDA-TC increases with average CSNR \(\bar {\theta }\), but decreases with the increasing bandwidth. The former effect is due to the fact that, when \(\bar {\theta }\) of a Rayleigh fading channel increases, so does the variance of the CSNR. The latter effect can be explained as follows. When more channel bandwidth is made available, the AMMSE can be reduced by increasing the quantization rate and hence the prediction gain.
The power allocations and outage CSNRs of HDA-PQ and HDA-TC systems shown in Fig. 5 (left)
HDA-PQ | HDA-TC | ||||
---|---|---|---|---|---|
a | b | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) |
0.60 | 3 | 100 | – | 61 | –3.3 |
0.80 | 3 | 30 | –1.2 | 48.5 | –3.4 |
0.85 | 3 | 28 | –1.4 | 44.5 | –3.5 |
0.90 | 3 | 27 | –1.9 | 39.5 | –3.8 |
0.95 | 3 | 25.5 | –2.7 | 45 | –5.1 |
0.98 | 3 | 23.5 | –3.7 | 33 | –5.5 |
0.60 | 5 | 100 | – | 62.5 | –5.2 |
0.80 | 5 | 28.5 | –3.1 | 51 | –5.4 |
0.85 | 5 | 28.5 | –3.4 | 47 | –5.6 |
0.90 | 5 | 28 | –3.9 | 42 | –5.8 |
0.95 | 5 | 27.5 | –4.8 | 35.5 | –6.3 |
0.98 | 5 | 25.5 | –5.9 | 36 | –7.6 |
The power allocations and outage CSNRs of HDA-PQ and HDA-TC systems shown in Fig. 5 (right)
HDA-PQ | HDA-TC | ||||
---|---|---|---|---|---|
a | b | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) |
0.6 | 3 | 29.5 | 0.7 | 63 | –2.2 |
0.8 | 3 | 28 | 0.1 | 53 | –2.3 |
0.85 | 3 | 27.5 | –0.3 | 48 | –2.5 |
0.9 | 3 | 27.5 | –0.8 | 43 | –2.8 |
0.95 | 3 | 27.5 | –1.7 | 36.5 | –3.3 |
0.98 | 3 | 26.5 | –2.9 | 38.5 | –4.6 |
0.6 | 5 | 25.5 | –1.2 | 55 | –3.4 |
0.8 | 5 | 27 | –2.0 | 45.5 | –3.8 |
0.85 | 5 | 27.5 | –2.4 | 50.5 | –4.7 |
0.9 | 5 | 28 | –2.9 | 45.5 | –5.0 |
0.95 | 5 | 28.5 | –3.9 | 39 | –5.5 |
0.98 | 5 | 28 | –5.1 | 41 | –6.9 |
6.2 HDA speech transmission
One of the key applications of predictive coding is in moderate-to-high bit-rate speech coding [13]. We designed and simulated HDA-PQ, and for comparison HDA-TC systems, for 4 kHz speech signals sampled at 8 kHz. It is known that speech can be well modeled by a 10th-order auto-regressive process [31]. Therefore, a 10th-order linear predictor was used in predictive coding, while a transform block size of 10 was used for transform coding. In the latter case, the discrete cosine transform (DCT) [8], which is a more practical choice than the KLT for non-Gaussian vectors, was used. The designs were then carried out using a source covariance matrix estimated from an actual training set of 4×10^{5} speech samples. This training set consisted of short sentences spoken by a number of male and female English speakers. As in the case of GM(a) source, the quantization rate (entropy) found by the asymptotic analysis for Gaussian sources were used to design the actual ECQs for HDA-PQ and HDA-TC. For experimentally evaluating the performance of the practical designs, two different test sets (test set 1 and test set 2), each of 4×10^{5} samples, were used. The test set 1 included male and female English speakers, while the test set 2 included male and female French speakers.
The power allocations and outage CSNRs of HDA-PQ and HDA-TC systems shown in Fig. 6
HDA-PQ | HDA-TC | ||||
---|---|---|---|---|---|
\(\bar {\theta }\) (dB) | b | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) | ρ ^{∗}(%) | \(\theta ^{*}_{o} \) (dB) |
15 | 3 | 20 | –0.7 | 38 | –2.2 |
20 | 3 | 21.5 | –0.1 | 39 | –1.7 |
25 | 3 | 22 | 0.3 | 40 | –1.0 |
30 | 3 | 22 | 0.7 | 33 | –0.2 |
15 | 5 | 22 | –2.9 | 39 | –4.2 |
20 | 5 | 23 | –2.4 | 34 | –3.2 |
25 | 5 | 23 | –2.0 | 35 | –3.0 |
30 | 5 | 23 | –1.7 | 35 | –2.5 |
7 Conclusions
This paper presented an approach to designing HDA-PQ and HDA-TC systems for transmitting correlated Gaussian sources over frequency-flat, block Rayleigh fading channels, when CSI is not available to the transmitter. In this case, the encoder is designed to minimize the AMMSE over the receiver-CSNR distribution, so that the system operates well over a range of CSNRs. The main issue addressed in this paper is the joint optimization of the analog-digital power allocation and the outage CSNR (or equivalently the quantization rate) to minimize the AMMSE of HDA-PQ and HDA-TC systems. In particular, a simple algorithm for solving the optimization problem in the case of HDA-PQ was presented. While the power allocations and quantization rates obtained as suggested in this paper can only be asymptotically (in rate) optimal, they were found to be effective in actual HDA systems with finite rates.
Our experimental results showed that, despite the Gaussian assumption, the proposed HDA design approach also worked well with the speech signals. HDA-PQ in particular can be a good approach to adaptive speech coding (e.g., similar to ADPCM [13]) over fading channels and in broadcasting. HDA-PQ is amenable to adaptive quantization in real time due to the simplicity of the system optimization algorithm presented in Section 4.1. A simple approach to adaptive speech coding with HDA-PQ is to use a finite-state model for the source signal, where the state is determined by a segment of consecutive speech samples and each state has a particular set of HDA-PQ parameters (predictor coefficients, quantizer-rate, and power allocation). The method described in this paper can be used to determine optimal parameters for each state. Since unvoiced speech segments resemble white noise, experimental results in this paper suggest that purely analog transmission can likely be as nearly as good as HDA-PQ for such segments. On the other hand, for highly correlated voiced speech segments, a significant amount of total power will get allocated to the digital component.
8 Endnote
^{1} Since the elements of v are linear combinations of M quantization errors, they will be approximately Gaussian if M is sufficiently large.
Declarations
Competing interests
The author declares that he has no competing interests.
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