1 Introduction

Conventional digital communication systems suffer from the quality saturation effect [10] due to the quantization errors introduced by source coding. Consider digital transmission of a time-discrete source signal sequence in a wireless communication system. The source signal sequence is firstly quantized to a time-discrete digital sequence. It should be noted that the quantization errors are inevitably produced at this step. Furthermore, these errors are unrecoverable at the receiver as they are not transmitted to the receiver at all. In this case, the received quality in terms of end-to-end distortion (i.e., the distortion between the original source signal sequence and the reconstructed sequence at the receiver) will saturate at a certain level as the channel Signal-to-Noise Ratio (SNR) increases. Hybrid Digital-Analog (HDA) transmission [7, 11, 14, 16] has been proposed to combat the quality saturation effect. The basic idea behind HDA transmission is to transmit the quantization errors in an analog mode along with the digital sequence in a digital mode. By doing so, the advantages of both digital and analog transmission are exploited. Particularly, HDA transmission provides graceful improvement for the increasing channel SNR as the reconstruction quality of the quantization errors at the receiver is proportional to the channel SNR.

Wireless Relay Networks (WRNs) have been widely deployed to extend the wireless signal coverage. The quality saturation effect also occurs in WRNs and might even be more serious with relaying transmission. Therefore, we introduce HDA transmission in WRNs to eliminate the quality saturation effect and achieve graceful improvement. Two kinds of relays, i.e., Amplify-and-Forward (AF) relays and Decode-and-Forward (DF) relays, are investigated with HDA transmission respectively. AF relays simply amplify the received signals and retransmit them to the receivers. In this case, HDA operations are transparent to AF relays. On the other hand, DF relays decode the received signals, then re-encode and retransmit them. In this case, HDA operations are also performed by DF relays.

As power allocation has significant impact on the performance of WRNs, many efforts have been made on source-relay power allocation in WRNs [5, 6, 12, 13]. However, they are no longer applicable and should be reconsidered when HDA transmission is involved. The reason lies in two-fold. Firstly, in HDA transmission, digital and analog signals are usually superposed and modulated to improve the bandwidth efficiency. Power allocation should be carefully considered in order to guarantee the separability and decodability of the superposed signals at the receivers as much as possible. Secondly, HDA transmission introduces additional digital-analog power allocation in WRNs. In a WRN with an AF relay, digital-analog power allocation is only performed at the source. In a WRN with a DF relay, digital-analog power allocation is performed at not only the source but also the relay. We note that there also exist a few studies on HDA transmission in WRNs. In [15], Yao and Skoglund proposed two HDA relaying schemes for cooperative transmission and HDA-cast was proposed in [18]. Unfortunately, source-relay power allocation has not been considered in [15] and [18].

As mentioned above, existing studies only consider either source-relay power allocation in WRNs or digital-analog power allocation for HDA transmission. However, both these two kinds of power allocation have significant impact on the distortion performance in WRNs with HDA transmission. Moreover, they are coupled with each other. Therefore, joint power allocation should be considered. In this paper, we study the joint power allocation problem in a three-node WRN (3-WRN) with an AF or a DF relay over Rayleigh fading channels. Our goal is to achieve graceful improvement as well as minimize the end-to-end distortion. To our best knowledge, this is the first work on this problem. Firstly, we introduce superposition modulation in HDA transmission where the analog and digital signals are orthogonally modulated. Then, we formulate and analyze the joint power allocation problem. The main contributions are summarized as follows.

  1. 1)

    We analyze the joint power allocation problem in the case of AF relays and find that this problem is concave. Finally, we derive the close-form expressions of the optimal solution.

  2. 2)

    In the case of DF relays, we formulate the joint power allocation problem as a nonlinear fractional programming problem and then propose an efficient algorithm to search the optimal solution.

  3. 3)

    We perform simulations to evaluate the distortion performance of the proposed joint power allocation schemes in the AF case and that in the DF relay case. Moreover, the proposed joint power allocation schemes are compared with existing schemes under various channel conditions.

The rest of the paper is organized as follows. In Section 2, we discuss some related work briefly. Section 3 gives an overview of the system model of a 3-WRN with HDA transmission. The joint power allocation problems in the AF relay case as well as that in the DF relay case are formulated and analyzed in Sections 4 and 5, respectively. Performance evaluation of the proposed joint power allocation schemes is provided in Section 6. Finally, we give a conclusion of our work in Section 7.

2 Related work

In this paper, we focus the power allocation problem for HDA transmission in WRNs. The problem are related to source-relay power allocation in WRNs and digital-analog power allocation in HDA transmission.

Many efforts have been made to optimize source-relay power allocation in WRNs. [9] and [3] focused on optimizing power allocation to minimize the system distortion while [2] and [6] concerned about minimizing the outage probability. Literature [4] presented a novel power optimization scheme aiming at reducing overheads under the premise of no large performance loss. In [12, 13], Su and Sadek determined optimum power allocation for either the DF or AF relay system based on the asymptotically tight symbol-error-rate approximation.

HDA transmission, integrating advantages of digital and analog coding technique, has attracted much attention. Existing researches on hybrid digital-analog transmission mostly focus on the theoretical analysis and performance optimization of data transmission over Gaussian channels [8, 14]. Moreover, some concrete HDA schemes have been developed for wireless video transmission especially for broadcast/multicast and mobile scenarios [17, 19].

There also exist a few studies on HDA transmission in WRNs. In [15], Yao et al. proposed several HDA relaying protocols for a three-node half-duplex relay network with slow fading channels. They considered both the case that the dimension of the received codeword at the relay was compressed and the case that the dimension was expended. Yu et al. extended their prior work [17] into wireless cooperative communication in [18]. A simple digital-analog power allocation method at the sender and relay was given, which guaranteed the Bit Error Rate (BER) of the digital part (the signal with higher priority, e.g., base layer of video) no higher than a certain value P e , and then the rest of power was allocated to the analog part. However neither [15] nor [18] took source-relay power allocation into consideration.

As mentioned above, existing work only individually considers either source-relay or digital-analog power allocation. The work in conventional WRNs focuses on source-relay power allocation, while the work about HDA transmission consider digital-analog power allocation instead. However, in WRNs with HDA transmission, source-relay and digital-analog power allocation are coupled and should be jointly optimized to achieve the best performance.

3 System model

Consider a 3-WRN consisting of a source S, a relay R and a destination D, as shown in Fig. 1. The source S transmits a discrete-time analog signal to the destination D with the assistance of the relay R. Samples of the source are assumed to be Gaussian independent identically distributed (i.i.d.) with zero mean and unit variance \({\sigma _{s}^{2}}=1\). We assume time-division transmission is applied here, which means that S and R perform transmission in different time slots.

Fig. 1
figure 1

Three-node relay network

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At the first time slot, the sender S transmits the signal to D and R. The received signals y s d and y s r at D and R are respectively expressed as follows.

$$ \begin{array}{l} y_{sd}=h_{sd}X_{s}+n_{sd},\\ y_{sr}=h_{sr}X_{s}+n_{sr}. \end{array} $$
(1)

Similarly, at the second time slot, the received signal y r d at D is

$$ y_{rd}=h_{rd}X_{r}+n_{rd}. $$
(2)

where h s d , h s r and h r d are channel coefficients of the S-D link, S-R link and R-D link, respectively. n s d , n s r and n r d denote the additive white Gaussian noise (AWGN) of three links. We assume all channels are frequency-flat Rayleigh slow-fading channels and have the same noise power N 0. X s and X r denote the signals transmitted from S and R, respectively.

3.1 Overview of HDA transmission

The HDA encoder and decoder are provided in Fig. 2a and b, respectively. The HDA encoder works as the sender and the decoder works as the destination. For a DF relay, the HDA decoder is employed at the first time slot and the HDA encoder is performed in the following time slot. On the contrary, no HDA operation will be involved by an AF relay. The source S is a memoryless Gaussian i.i.d. N-sized sequence with zero mean and unit variance.

Fig. 2
figure 2

HDA system. a HDA encoder; b HDA decoder

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At the encoder, S is firstly quantified, then encoded (including source and channel coding) as a digital signal X d. The analog signal X a is generated by subtracting the quantization result from the source S. Then X d and X a are orthogonally modulated and transmitted with corresponding power.

At the decoder, the received signal is firstly demodulated to obtain the digital signal \(\hat {X^{d}}\) in I component and analog signal \(\hat {X^{a}}\) in Q component. Then \(\hat {X^{d}}\) is decoded by digital channel and source decoders to obtain the reconstructed digital signal \(\hat {S^{d}}\). For \(\hat {X^{a}}\), an analog decoder is employed to obtain the reconstructed quantization errors sequence \(\hat {E}\). The superposition of \(\hat {S^{d}}\) and \(\hat {E}\) is the reconstructed source sequence \(\hat {S}\).

The average end-to-end distortion of the HDA transmission system is expressed as

$$ D=\frac{1}{N}||S-\hat{S}||^{2}. $$
(3)

3.2 HDA transmission in a 3-WRN

In order to improve the bandwidth efficiency, we adopt superposition modulation in HDA transmission in WRNs. The transmitted signal is a complex signal orthogonally modulated by a BPSK-modulated digital signal X d in I component and an analog signal X a in Q component. Then we have

$$ X=X^{d}+jX^{a}. $$
(4)

When superposition modulation is applied at the source, the source signal X s is transmitted in the forms of digital signal \({X_{s}^{d}}\) and analog signal \({X_{s}^{a}}\). In order to make full use of the channel, the coding rates of \({X_{s}^{a}}\) and \({X_{s}^{d}}\) are designed to be as the same as the channel bandwidth. Thus Eq. 1 can be rewritten as

$$ \begin{array}{l} y_{sd}=h_{sd}\left( {X_{s}^{d}}+j{X_{s}^{a}}\right)+n_{sd}\\ y_{sr}=h_{sr}\left( {X_{s}^{d}}+j{X_{s}^{a}}\right)+n_{sr} \end{array} $$
(5)

Similarly, Eq. 2 can be rewritten as

$$ y_{rd}=h_{rd}\left( {X_{r}^{d}}+j{X_{r}^{a}}\right)+n_{rd} $$
(6)

We consider the power allocation problem in a 3-WRN with HDA transmission. Assume the total power budget for the source and relay is limited to P. Let P s and P r denote the transmitting power of S and R, respectively. We have

$$ P_{s}=\alpha P,~P_{r}=(1-\alpha) P. $$
(7)

where α denotes the source-relay power allocation factor ranging between 0 and 1.

We make two assumptions in the rest of the paper. First, Channel State Information (CSI) is always available at the sender. In realistic wireless communications, CSI can be obtained by feedback or channel estimation. Second, for digital-analog power allocation, enough power has been allocated for digital transmission. Such assumption is reasonable for video/audio communication services where the most important data is required to be transmitted in a reliable digital mode. In this case, the receivers are able to correctly decode the digital signal with high probability by requiring a constraint on the BER of the digital signal (P e ). For BPSK, the corresponding digital SNR threshold \(\gamma _{th}^{D}\) satisfies

$$ P_{e}=\frac{1}{2}Q\left( \sqrt{\gamma_{th}^{D}}\right) $$
(8)

where the function Q(x) is defined as

$$Q(x)=\frac{2}{\sqrt{\pi}}{\int}_{x}^{\infty}{e^{-t^{2}}}dt $$

In the following, we will analyze HDA transmission in a 3-WRN with an AF relay as well as that in a 3-WRN with a DF relay.

1) In the Case of AF Relay. The framework of HDA transmission in a 3-WRN with an AF relay is described in Fig. 3. It should be noted that there is no HDA operation at the AF relay. The transmitted signal at S is expressed as

$$ {X_{s}^{d}}=\sqrt{(1-\alpha_{s})P_{s}}{x_{s}^{d}},~{X_{s}^{a}}=\sqrt{\alpha_{s} P_{s}}{x_{s}^{a}}. $$
(9)

where α s denote the analog power allocation factor at S, ranging between 0 and 1. \({X_{s}^{d}}\) and \({X_{s}^{a}}\) are normalized digital and analog signals at S, respectively.

Fig. 3
figure 3

HDA transmission in a 3-WRN with an AF relay

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The relay receives the hybrid signal y s r from S at the first time slot, then amplifies and forwards it with the power budget P r . The transmitted signal at R is

$$ X_{r}=\beta_{r} y_{sr}. $$
(10)

where β r is the amplifying factor, which is inversely proportional to the receiving power. β r can be expressed as

$$ \beta_{r}=\frac{\sqrt{P_{r}}}{\sqrt{{|h_{sr}|}^{2}P_{s}+N_{0}}} $$
(11)

Finally, Eq. 1 can be rewritten as

$$ \begin{array}{l} y_{sd}=h_{sd}(\sqrt{(1-\alpha_{s})\alpha P}{x_{s}^{d}}+j\sqrt{\alpha_{s} \alpha P}{x_{s}^{a}})+n_{sd},\\ y_{sr}=h_{sr}(\sqrt{(1-\alpha_{s})\alpha P}{x_{s}^{d}}+j\sqrt{\alpha_{s} \alpha P}{x_{s}^{a}})+n_{sr}. \end{array} $$
(12)

So does Eq. 2

$$ y_{rd}=h_{rd}\frac{\sqrt{(1-\alpha) P}}{\sqrt{{|h_{sr}|}^{2}\alpha P+N_{0}}} y_{sr}+n_{rd}. $$
(13)

The destination D combines the two received signals y s d and y r d by Maximum Ratio Combination (MRC), and decodes the combined signal by a Linear Minimum Mean Square Error (LLMSE) estimator. Since the digital and analog signals are orthogonally modulated, they can be separately decoded.

2) In the Case of DF Relay. The framework of HDA transmission in a 3-WRN with a DF relay is provided in Fig. 4. Different from the AF relay, the DF relay performs HDA decoding and encoding. Additional digital-analog power allocation is also introduced at the DF relay. The transmitted signal at S is the same as that expressed in Eq. 9. Let α r denote the analog power allocation factor at R, which ranges between 0 and 1. Similarly, the transmitted signal at R is expressed as

$$ {X_{r}^{d}}=\sqrt{(1-\alpha_{r})P_{r}}{x_{r}^{d}},~{X_{r}^{a}}=\sqrt{\alpha_{r} P_{r}}{x_{r}^{a}}. $$
(14)

where \({x_{r}^{d}}\) and \({x_{r}^{a}}\) are normalized digital and analog signals at R, respectively. Then Eq. 13 can be rewritten as

$$ y_{rd}=h_{rd}(\sqrt{(1-\alpha_{r})P_{r}}{x_{r}^{d}}+j\sqrt{\alpha_{r} P_{r}}{x_{r}^{a}})+n_{rd} $$
(15)

It should be noted that Eq. 12 still holds in the case of DF relays.

Fig. 4
figure 4

HDA transmission in a 3-WRN with a DF relay

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4 Joint power allocation of HDA transmission with AF relay

4.1 Problem formulation

In a 3-WRN described in Fig. 1, R and D receive signals y s r and y s d shown as Eq. 12 from S in the first time slot. Since the digital and analog signals are orthogonally modulated, they can be decoded separately. Thus the digital and analog SNRs of y s r and y s d can be expressed as

$$ \begin{array}{l} \gamma_{sd}^{D}=g_{1}(1-\alpha_{s})\eta_{s},~\gamma_{sd}^{A}=g_{1}\alpha_{s} \eta_{s}\\ \gamma_{sr}^{D}=g_{2}(1-\alpha_{s})\eta_{s},~\gamma_{sr}^{A}=g_{2}\alpha_{s} \eta_{s} \end{array} $$
(16)

where η s = P s /N 0 and g 1=||h s d ||2, g 2=||h s r ||2. g 1 and g 2 are power gains of the S-D channel and S-R channel, respectively.

At the second time slot, R amplifies the received signal and then transmits it to D. The received signal at D is y r d shown as Eq. 13. The digital and analog SNRs of y r d can be derived as

$$ \begin{array}{l} \gamma_{rd}^{D}=\frac{{\beta_{r}^{2}}g_{3}g_{2}(1-\alpha_{s}) \eta_{s}}{{\beta_{r}^{2}}g_{3}+1}\\ \gamma_{rd}^{A}=\frac{{\beta_{r}^{2}}g_{3}g_{2}\alpha_{s} \eta_{s}}{{\beta_{r}^{2}}g_{3}+1} \end{array} $$
(17)

where g 3=||h r d ||2 and it represents the power gain of the R-D channel.

The destination D combines the two received signals y s d and y r d by MRC, and then decodes the combined signal by a LLMSE estimator. Therefore, we can obtain the digital and analog SNRs at D as follows.

$$ \begin{array}{l} \gamma^{D}=\gamma_{sd}^{D}+\gamma_{rd}^{D}\\ \gamma^{A}=\gamma_{sd}^{A}+\gamma_{rd}^{A} \end{array} $$
(18)

As we assume that enough power has been allocated for digital transmission and the digital signal can be decoded correctly at the destination D, the end-to-end distortion of the system can be expressed as

$$ D=\frac{{\sigma_{e}^{2}}}{1+\gamma^{A}} $$
(19)

where \({\sigma _{e}^{2}}\) is the variance of quantization errors, constant for a certain coding rate [14].

From Eq. 19, we can see that minimizing D is equivalent to maximizing the analog SNR γ A as \({\sigma _{e}^{2}}\) is constant. Let η r = P r /N 0, then η = η s + η r = P/N 0. The joint power allocation problem in the case of AF relays is formulated as follows.

$$ \begin{array}{lll} \mathcal{P}1:~ &\underset{\alpha_{s},\eta_{s} }{\text{maximize}} & \gamma^{A}=\gamma_{sd}^{A}+\gamma_{rd}^{A}\\ & \text{subject to} & 0\leq\alpha\leq1 \\ && 0\leq\alpha_{s}\leq1 \\ && \gamma^{D}=\gamma_{th}^{D} \end{array} $$
(20)

where α = P s /P = η s /η. The first two constraints are power constraints at S, which means that the transmitting power at S should not be greater than the total power. The second constraint shows that the analog power at S should not be greater than the transmitting power. The equality constraint indicates the digital SNR at D should not be below the threshold \(\gamma _{th}^{D}\), which guarantees the correct decoding of the digital signal.

4.2 Analysis on optimal solution

By substituting Eqs. 11, 16 and 17 in Eq. 18, we have

$$ \begin{array}{l} \gamma^{D}=(1-\alpha_{s})f(\eta_{s}),\\ \gamma^{A}=\alpha_{s}f(\eta_{s}) \end{array} $$
(21)

where f(η s ) is defined as

$$ f(\eta_{s})=g_{1}\eta_{s}+\frac{g_{2}g_{3}\eta_{s}\eta_{r}}{g_{2}\eta_{s}+g_{3}\eta_{r}+1}, 0<\eta_{s}\leq\eta $$
(22)

It’s easy to figure out from the second equation of Eq. 21 that in order to maximize γ A, we should get the maximum values of both α s and f(η s ). Fortunately, from the first equation of Eq. 21, we find that as long as we obtain the optimal solution \(\eta _{s}^{*}\) and the maximum value of \(f(\eta _{s}^{*})\), \(\alpha _{s}^{*}\) can be obtained according to \((1-\alpha _{s}^{*})f(\eta _{s}^{*})=\gamma _{th}^{D}\).

It can be easily proven that f(η s ) is concave. Actually, the first derivative of f(η s ) is

$$ \frac{\partial f}{\partial \eta_{s}}=g_{1} + \frac{g_{2} g_{3}[\eta_{r}(1+g_{3}\eta_{r}) - \eta_{s}(1+g_{2}\eta_{s})]}{(1+g_{2}\eta_{s}+g_{3}\eta_{r})^{2}} $$
(23)

And the second derivative of f(η s ) is

$$ \frac{\partial^{2} f}{\partial {\eta_{s}^{2}}}=\frac{-g_{2}g_{3}[1+(g_{2}+g_{3})\eta+g_{2}g_{3}\eta^{2}]}{(1+g_{2}\eta_{s}+g_{3}\eta_{r})^{3}} $$
(24)

which is obviously a negative number, i.e., f(η s ) is concave for η s . Therefore, we can obtain the optimal solution for \(\eta _{s}^{*}\) by letting \(\frac {{\partial f}}{\partial \eta _{s}}=0\). Then we have

$$ \eta_{s}^{*}\,=\, \left\{\begin{array}{l} \eta,\;\qquad\qquad\qquad\qquad\qquad g_{2}g_{3}\!+g_{1}g_{3}\,-\,g_{1}g_{2}\leq0\\ \min\left\{\eta,\frac{\eta}{2}+\frac{g_{1}\eta}{2g_{3}}+\frac{g_{1}}{2{g_{3}^{2}}}\right\},\ \quad\!\!\! g_{2}=g_{3} \\ \min \left\{ \eta ,\frac{{\sqrt {1 + {g_{3}}\eta } }}{{{g_{3}} - {g_{2}}}}\left( \sqrt {1 + {g_{3}}\eta } - \frac{{\sqrt {{g_{2}}{g_{3}}(1 + {g_{2}}\eta )} }}{{\sqrt {{g_{2}}{g_{3}} + {g_{1}}{g_{3}} - {g_{1}}{g_{2}}} }}\right)\right\}, \\ \;\quad\qquad\qquad\qquad\qquad\qquad otherwise \end{array}\right. $$
(25)

Finally, the optimal solution for joint power allocation can be explicitly expressed as

$$ \left\{ \begin{array}{l} \alpha^{*}=\frac{\eta_{s}^{*}}{\eta}\\ \alpha_{s}^{*}=\left[1-\frac{\gamma_{th}^{D}}{f(\eta_{s}^{*})}\right]^{+} \end{array} \right. $$
(26)

where [x]+ = m a x{x,0}.

Note that \(f(\eta _{s}^{*})\) may be smaller than \(\gamma _{th}^{D}\), which means the channels quality is too bad to decode the digital signal correctly even though all power is allocated to the digital signal. Transmission should stop in this case. Particularly, when \(\eta _{s}^{*}=\eta \), the sender S transmits the hybrid signal directly to the destination D without the assistance of the relay R. The reason is that the quality of S-D channel is much better than that of S-R channel and R-D channel.

5 Joint power allocation of HDA transmission with DF relay

5.1 Problem formulation

In a 3-WRN with a DF relay, the formulation of the joint power allocation problem is similar as that described in Section 4.1, which can be expressed as follows.

$$ \begin{array}{lll} \mathcal{P}2:~ &\underset{\alpha_{s},\alpha_{r},\eta_{s} }{\text{maximize}} & \gamma^{A}=\gamma_{sd}^{A}+\gamma_{rd}^{A}\\ & \text{subject to} & \gamma^{D}=\gamma_{th}^{D} \\ && 0\leq\eta_{s}\leq\eta \\ && 0\leq\alpha_{s}\leq1\\ && 0\leq\alpha_{r}\leq1 \end{array} $$
(27)

However, as digital-analog power allocation is also performed at the DF relay, the problem \(\mathcal {P}2\) is more complicated than the problem \(\mathcal {P}1\). We can see that power constraint at the relay R is also included in the problem \(\mathcal {P}2\) indicated by α r . Moreover, the digital and analog SNRs of y r d cannot be derived from Eq. 17. They should be obtained as

$$ \begin{array}{l} \gamma_{rd}^{D}=(1-\alpha_{r})g_{3}\eta_{r}\\ \gamma_{rd}^{A}=\frac{\alpha_{s}\alpha_{r}g_{2}g_{3}\eta_{s}\eta_{r}}{\alpha_{s}g_{2}\eta_{s}+\alpha_{r}g_{3}\eta_{r}+1} \end{array} $$
(28)

On the other hand, the digital and analog SNRs of y s d and y s r shown as Eq. 16 still hold in the case of DF relays.

5.2 Problem analysis

The problem \(\mathcal {P}2\) is difficult to be solved, as it involves not only source-relay power allocation, but also digital-analog power allocation at the source and DF relay. However, we find the problem can be simplified under some certain channel conditions. Based on this observation, we describe the problem in two separate scenarios in detail as follows.

5.2.1 Quality of S-D channel is better than that of S-R channel

From Eq. 16, we can see that \(\gamma _{sd}^{D} \geq \ \gamma _{sr}^{D}\) as long as g 1≥ g 2, which means if R is able to correctly decode the digital signal, D is certainly able to recover the digital signal correctly. In this case, R should allocate all power to the analog signal. Then we have

$$ \alpha_{r}=1. $$
(29)

To ensure correctly decoding of the digital signal, the digital SNR is set to \(\gamma _{th}^{D}\), which is expressed as

$$ \gamma_{sd}^{D}=(1-\alpha_{s})g_{1}\eta_{s}=\gamma_{th}^{D}. $$
(30)

According to Eq. 30, we can obtain that \(\eta _{s}=\frac {\gamma _{th}^{D}}{(1-\alpha _{s})g_{1}}\geq \frac {\gamma _{th}^{D}}{g_{1}}\). The problem \(\mathcal {P}2\) is reformulated as follows.

$$ \begin{array}{lll} \mathcal{P}3:~ & \underset {\eta_{s} }{\text{maximize}} & \gamma^{A}\\ & \text{subject to} & \frac{\gamma_{th}^{D}}{g_{1}}\leq\eta_{s}\leq\eta \end{array} $$
(31)

Substituting Eqs. 29 and 30 into Eq. 18, we obtain

$$ \gamma^{A}=\frac{{a{\eta_{s}}^{2} + b{\eta_{s}} + c}}{{d\eta_{s} + e}} = \frac{{\chi ({\eta_{s}})}}{{\varphi ({\eta_{s}})}}. $$
(32)

where

$$ \begin{array}{l} a=g_{1}g_{2}-g_{1}g_{3}-g_{2}g_{3}\\ b=\eta g_{3}(g_{1}+g_{2})+g_{1}+g_{2}\gamma_{th}^{D} \left( \frac{g_{3}}{g_{1}}-1\right)\\ c=-\frac{g_{2}g_{3}}{g_{1}}\gamma_{th}^{D}\eta,~d=g_{2}-g_{3}\\ e=g_{3}\eta-\frac{g_{2}}{g_{1}}\gamma_{th}^{D}+1 \end{array} $$
(33)

are all constant factors.

Since \(\mathcal {P}3\) is a non-linear fractional programming, it’s hard to be solved directly. We derive an optimization problem \(\mathcal {P}4\) that is equivalent to \(\mathcal {P}3\) by introducing a non-negative parameter q.

$$ \begin{array}{lll} \mathcal{P}4:~ & \underset {\eta_{s} }{\text{maximize}} & \chi(\eta_{s})-q\varphi(\eta_{s}) \\ & \text{subject to} & \frac{\gamma_{th}^{D}}{g_{1}}\leq\eta_{s}\leq\eta \end{array} $$
(34)

We define the maximum value of the object function of \(\mathcal {P}4\) as

$$ F(q)= \max_{\eta_{s}}~\chi(\eta_{s})-q\varphi(\eta_{s}) $$
(35)

Then a theorem is stated as follows [1].

Theorem 1

Assuming the maximum analog SNR \(q^{*}=\frac {\chi (\eta _{s}^{*})}{\varphi (\eta _{s}^{*})}\) exists, it can be achieved if and only if \(F(q^{*})=\chi (\eta _{s}^{*})-q^{*}\varphi (\eta _{s}^{*})=0\)

We employs Dinkelbach’s method in [1] to solve the problem, which is detailed depicted in Algorithm 1. Then according to Eqs. 29 and 30, we obtain the optimal solution \((\alpha _{s}^{*}, \alpha _{r}^{*}, \alpha ^{*})\).

figure a

5.2.2 Quality of S-D channel is worse than that of S-R channel

Consider the case of g 1<g 2. To correctly decode the signal digital at R and D, digital SNRs at both R and D should satisfy the constraint of the BER, which are expressed as

$$ \begin{array}{l} \gamma_{sr}^{D}=\gamma_{th}^{D},\\ \gamma^{D}=\gamma_{th}^{D}. \end{array} $$
(36)

Substituting Eqs. 16, 17 and 36 into Eq. 19, we can see that the formulation of \(\mathcal {P}3\) still holds, while the constant factors change as follows.

$$ \begin{array}{lll} \mathcal{P}5:~ & \underset {\eta_{s} }{\text{maximize}} & \gamma^{A}=\frac{{a{\eta_{s}}^{2} + b{\eta_{s}} + c}}{{d\eta_{s} + e}} = \frac{{\chi ({\eta_{s}})}}{{\varphi ({\eta_{s}})}} \\ & \text{subject to} & \frac{\gamma_{th}^{D}}{g_{2}}\leq\eta_{s}\leq\eta \end{array} $$
(37)

where

$$ \begin{array}{l} a=g_{1}g_{2}-g_{1}g_{3}-g_{2}g_{3}\\ b=\eta g_{3}(g_{1}+g_{2})+\gamma_{th}^{D}\left( \frac{{g_{1}}^{2}}{g_{2}}-g_{1}-g_{2}+g_{3}\right)+g_{1}\\ c=-\gamma_{th}^{D}\left( \eta g_{3}+\frac{g_{1}}{g_{2}}\gamma_{th}^{D}-\gamma_{th}^{D}\right),~d=g_{2}-g_{3}\\ e=g_{3}\eta+\left( \frac{g_{1}}{g_{2}}-2\right)\gamma_{th}^{D}+1 \end{array} $$
(38)

are constant factors.

The solution of the problem \(\mathcal {P}5\) can be obtained by following the steps of solving the problem \(\mathcal {P}3\).

5.3 Joint power allocation algorithm

Based on the analysis in Section 5.2, we propose a joint power allocation algorithm, which is described in Algorithm 2. Given CSI (i.e., h s d , h s r , h r d , and N 0) and system requirements (i.e., constraints on digital BER P e and total power P), we can obtain the optimal source-relay and analog-digital power allocation (\(\alpha ^{*},~\alpha _{r}^{*},~\alpha _{s}^{*}\)) in the 3-WRN with the DF relay by performing Algorithm 2.

figure b

Obviously the calculation complexity of the algorithm is determined by the number of iteration in Algorithm 1, which is no larger than I m a x . Hence the calculation complexity of the algorithm is O(1).

6 Performance evaluation

We evaluate the performance for the proposed joint power allocation schemes for either AF relay or DF relay over Rayleigh slow-fading channels. All simulations in are implemented and performed in Matlab 2015a. Signal-to-distortion ratio (SDR) [11] is used to measure the transmission performance of a Gaussian i.i.d. source with the size of 1000 samples in a HDA based 3-WRN, which is defined as follows.

$$ SDR\buildrel\Delta\over=10\lg\frac{{\sigma_{s}^{2}}}{D}=10\lg\frac{\mathbb{E}||S||^{2}}{\mathbb{E}||S-\hat{S}||^{2}} $$
(39)

where \(\mathbb {E}||x||\) represents the expectation of x. For \({\sigma _{s}^{2}}=1\), we have

$$ SDR=-10\lg{D} $$
(40)

From Eqs. 39 and 40, we can say that SDR can be used to represent the end-to-end distortion. In order to validate the proposed joint power allocation schemes, we evaluate the SDR performance under various channel conditions. Particularly, two scenarios with different channel conditions described in Section 5.2 are considered for the DF relay. During simulations, the channel variances defined as Eq. 41 are used to adjust the channel quality. The better channel quality is indicated by the greater variances.

$$ \sigma_{sd}^{2}=\mathbb{E}||h_{sd}||^{2},~\sigma_{sr}^{2}=\mathbb{E}||h_{sr}||^{2},~\sigma_{rd}^{2}=\mathbb{E}||h_{rd}||^{2} $$
(41)

The digital BER threshold (P e ) is set to be 10−6, which guarantees the receiver can correctly decode the digital signal with high probability and consequently the distortion is only introduced by analog transmission. Monte Carlo simulation is employed to obtain the average distortion for each channel condition. Each point in the figure is obtained by averaging the results produced by 10000 simulations. In the implementation of Algorithm 1, the maximum iteration number is set to 30 which guarantees the convergence according to our tests, and the tolerance τ is set to be 10−6.

We compare the proposed joint power allocation schemes with the Equal Power Allocation (EPA) scheme in which the total system power is equally allocated for S and R (i.e. \(P_{s} = P_{r} = \frac {1}{2}P\)) and Su’s method proposed in [13].

Figures 5 and 6 describe the SDR performance of HDA transmission in the 3-WRN when channel variance \(\sigma _{sd}^{2} = 1\). Both the AF relay case and DF relay are studied. As expected, the results show that the proposed joint power allocation schemes outperform EPA and Su’s method in terms of SDR under various channel conditions. The gains come from the joint optimization of source-relay power allocation and digital-analog power allocation. Comparing Figs. 5 and 6, we find that joint power allocation in the DF relay case can achieve more gains than that in the AF relay case. The reason is that additional digital-analog power allocation is involved by the DF relay. However, more complex algorithm is also required to search optimal solution for the DF relay case. In addition, we note that with the channel quality of S-R getting better compared with that of R-D, i.e., \(\sigma _{sr}^{2}/\sigma _{rd}^{2}\) getting greater, the performance of EPA and Su’s method come closer, as shown in Figs. 5a, b and c and 6a, b and c. This phenomenon is supported by the research in [13].

Fig. 5
figure 5

SDR of HDA transmission in 3-WRN for \(\sigma _{sd}^{2}=1\): AF case

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Fig. 6
figure 6

SDR of HDA transmission in 3-WRN for \(\sigma _{sd}^{2}=1\): DF case

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In the DF relay case, taking Fig. 5a, b and c for comparison, we can see that more distortion gains are obtained when the quality of the S-D channel is better than that of the S-R channel. In this case, according to Eq. 29, the DF relay allocates all the power for analogy transmission. Therefore, more graceful improvement can be achieved for better channel quality.

We also investigate the optimal solutions for the joint power allocation problem under various channel conditions for the AF relay case and DF relay case, as shown in Figs. 7 and 8. For comparison, the source-relay power allocation scheme in [13] is also drawn, which is only determined by the variance of three channels. Since the power allocation in [13] is only between source and relay, the digital-analog power allocation scheme is the same as our proposed scheme in this paper.

Fig. 7
figure 7

Joint power allocation for \(\sigma _{sd}^{2}=1\): AF case

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Fig. 8
figure 8

Joint power allocation for \(\sigma _{sd}^{2}=1\): DF case

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The results show that α s is small when the SNR is low, which implies that most power is allocated to the digital signal to guarantee it can be correctly decoded. As the SNR increases, more power will be allocated to analog transmission when as requirement for the digital SNR is fixed. In addition, from the curves of P s /P in Fig. 7, we find the trend of P s is mainly determined by the quality of the S-R channel. If the quality of the S-R channel is poor, there will be more power allocated to the source S (see Fig. 7a and c). On the other hand, when the quality of the S-R channel gets better, more power will be allocated to the relay R, as shown in Fig. 7b and d.

As for the DF case, it is shown in Fig. 8 that if the quality of the S-R channel is fixed, α r decreases when the quality of the R-D channel gets worse. This can be explained as follows. When the quality of the R-D channel gets worse, D needs more digital power to ensure the decodability of the digital signal.

7 Conclusion

In this paper, HDA transmission is introduced in WRNs to eliminate the quality saturation effect and achieve graceful improvement for better channel quality. Then joint power allocation is proposed to maximize the end-to-end distortion in the AF relay case as well as the DF relay case, where source-relay power allocation and digital-analog power are jointly considered. In a 3-WRN with a AF relay, we formulate the joint power allocation problem as a concave problem and derive the close-form expressions of the optimal solution. However, the joint power allocation problem is much more complicated in the DF relay case. We formulate it as a nonlinear fractional programming problem and then propose an effective algorithm to search the optimal solution. Simulate results show that the proposed joint power allocation schemes can achieve more distortion gains compared with existing power schemes in WRNs under various channel conditions.

As future work, we will investigate the power allocation problem for HDA transmission in WRNs with partial CSI or without CSI at the sender. Moreover, we will extend our work to distortion-tolerant applications such video streaming services.