2013 IEEE Wireless Communications and Networking Conference (WCNC): PHY

Effect of Non lineari ties in Wireless Communication

with Digital Receiver

Peng Xuel, 2, 3, Haibin Yangl, 2, Zhan Xu2, 4, Chunhui Zhoul, 2, Shidong Zhoul, 2

I Department of Electronic Engineering, Tsinghua University, Beijing, China 2Tsinghua National Laboratory for Information Science and Technology, Beijing, China

3Naval Aeronautical and Astronautical University, Yantai, China 4School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing, China

E-mail: xuep04@gmail.com

Ahstract-This paper discusses the effect of nonlinearities introduced by Radio Frequency (RF) amplifier in practical digital receivers. Taking downlink as an example, after

considering several main limitations in engineering implementation and simplifying the model of the receiver, the effects of receiver nonlinearities on the performance both in A WGN and Rayleigh fading channel are obtained. In A WGN

channel, the SNR at the input of Am converter has a limit, even though the transmitting SNR is very large with given input power of A/D converter. Combining with the characteristic of quantization, we find trade-off relationship between the SNR at the input and the output of A/D converter. Then, a method is proposed to maximize the SNR at the output of Am converter, and an upper bound of the SNR is also obtained. Considering

nonlinearities and using the optimizing method in this paper, the

SNR at the output of AID converter increase 4 dB at most, as the digital bit increases 1 bit. In Rayleigh fading channel, a threshold of transmitting rate is derived, which is affected by nonlinearities of the receiver.

Index Terms-Nonlinearity; IMD; AWGN; Rayleigh; SNR; capacity; outage; Am; amplifier

I INTRODUCTION

Digital Signal Processing (DSP) is widely used in designing and implementation of communication systems these days. A typical digitized communication system is shown in Fig.l. In most of the analysis and designing, the composite channel with RF transceiver and radio propagation is considered as a linear channel, and many famous conclusions are drawn from this assumption, such as Shannon capacity, diversity-multiplexing tradeoff etc. [1].

However, in a real system, many practical factors have impacts on the performance of the system, including the non­ideal factors of the hardware. There are many different types of hardware non-ideal factors, such as quantization error, sampling clock offset, and the nonlinearities of Radio Frequency (RF) transmitter and receiver.

There has been some research on these non-ideal factors, such as Carrier Frequency Offset (CFO), Sampling Frequency Offset (SFO) and VQ imbalance in OFDM system, which can be eliminated with digital methods introduced in [2]. But there is hardly any digital method to deal with the nonlinearities of

This work is partially supported by National Basic Research Program of China (2012CB3 I 002), China's Major Project (201IZX03004-004), Qualcomm-Tsinghua Joint Research (20093000417) and China's 863 Project (2012AAOI1402).

the RF transmitter and receiver. As a result, many investigations study on it.

The amplifier distortions in RF transmitter and receiver are the major sources of the nonlinearities. Usually, the amplifiers operate in linear region when the input and output power are low enough; otherwise, they become nonlinear and introduce signal distortion as the input power increases, which is discussed in [3] . Therefore, much research focuses on the effect of amplifier distortions and tries to find out ways to eliminate it. A result about the effect of performance evaluation on MIMO system is given in [4]-[5] when the amplifier in transmitter is nonlinear. Furthermore, in [6], a method is presented to compensate for performance degradation of a 16-QAM modulation scheme in Rayleigh fading when the nonlinearity of power amplifier in transmitter exists, and in [7] , the author proposes a method to compensate the nonlinearity of amplifier in MIMO Transmit Diversity Systems with M-QAM modulation.

Fig.l. Architecture of the Digitized Communication System

Currently, most of the investigations focus on the nonlinearities in transmitter and several behavioral models of power amplifier are introduced in [3] , [8], [9], [10], such as power series model, Saleh model, Wiener model, and Volterra model, which can be classify as two types, memoryless model and memory model. But few of the investigations pays attention to the nonlinearities in implementation of receiver, in which there should be some different limitation due to the design of the practical system. In this paper, we discuss the performance variation when considering nonlinearities in the downlink of a typical digitized communication system.

The rest part of this paper is organized as follows: Section II introduces the model of the RF receiver which is simplified from the typical one. In Section III, we discuss the performance variation and the optimizing method in the

978-1-4673-5939-9/13/$31.00 ©2013 IEEE 4037

A WGN channel. Then, the study on performance variation in Rayleigh fading channel is represented in Section IV.

II MODEL OF THE RECEIVER

Fig.2 shows a typical architecture of heterodyne receiver [11]. LNA is the Low Noise Amplifier, and DSA (Digital Step Attenuator) is the Digitally Controlled Attenuator which is used to change the gain of the receiver, both of them hardly bring nonlinearities.

Fig.3 shows the architecture of transmitter, which has the symmetric architecture to the receiver.

<-�-_/' Fig.2. Architecture of RF Receiver

Fig.3. Architecture ofRF Transmitter

The power gains of the amplifiers in the transmitter are gr,] , gr,2 ,gr,RF (from the IF input to the RF output); and the power

gain of amplifiers in the receiver are gR,TNA , gR,RF , gR,] , gR,2 (from the RF input to the IF output).

In this paper, we pay more attention to the receiver, so the transmitter is treated as a relatively simpler one. We define Go = gr,]gr,2gr,RF , which means the amplifiers in the transmitter are regarded as one.

However, in the RF receiver, we define G] = gR,LNAgR,RrgR,] and G2 = gR,2' because the output power of the last stage amplifier is the highest and we want to investigate it separately.

According to the analysis above, the equivalent architecture of composite channel is shown in FigA. It needs to be emphasized that, we assume that the gains of mixers and filters are 1 in the equivalent model.

It is known that there are three types of nonlinearities, which are power compression, harmonic distortions and intermodulation distortion (IMD) [12] . Power compression is well investigated in many papers, some investigations show that reasonable power back-off is an effective method to reduce the effect of power compression [13] [14] [15], and some investigations give the digital compensation methods [7].

Also, the harmonic components can be eliminated by certain filter easily. However, few notice the IMD, while the frequency components caused by IMD is always near the fundamental frequency which is hardly eliminated, although power back-off exists and the amplifier is operated in linear region. Therefore, IMD has great effect on the practical system Mostly, third-order intermodulation distortion (IMD3) is discussed in practice.

TX

RX

FigA. Equivalent Architecture of Composite Channel

P_(dBm) 40 �� ....•.•.. -...•..••• -•... -••••. -.•..••••••....• �:

... ::::::::� .... ! .. ;{.: ..

20 " I

i 0 -10 20 llPJ 3OP.(dBm) [-20 J-40

-80 -80

-100

FigS Definition of Three Order Intercept Point

Generally, we use the third-order intercept point (IP3) and the third-order intermodulation distortion ratio ( IA1R3 ) as indexes. IP3 means the point where the fundamental power (SOUl) equals to the IMD3 power (IoUI)' shown in Fig.5 [16].

IIP3 is input intercept point, and OIP3 is output intercept point. The intersection point of the tangent lines to the fundamental signal and IMD3 is IP3, and the slope of lout is three times to SOUl approximately [16].

IA1R3 is the ratio of output fundamental signal power and IMD3 power, which can be calculated by OIP3 in Eq. (1) when lout is not quite high [16]. And the power (SOl?3) at OIP3 is usually given in the data sheet of the using amplifier [17].

IMR3(dB) = 2(SOl?3 -Sout)(dBm) ( 1)

Now, we define the gain DSA in FigA as Gf)' and we know the gains of AMPO, AMPl, AMP2 are Go' G], G2 respectively. The signal power at point T ( Pr ) is composed of fundamental signal power (Sr ), IMD3 power (Ir ) and noise power ( Nr ), while the signal power at point R ( PH ) is composed of fundamental signal power (S R)' IMD3 power ( I R ) and noise power (NH).

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Furthermore, we define

As Eq. (1) and the definition ofIA1R3 expressed, we get

Thus,

(4)

III ANALYSIS AND OPTIMIZATION OF CAPACITY IN A WGN CHANNEL

Two cases will be discussed in this section. The first is the SNR and the capacity at the output of the receiver which is the input of AID converter, when we assume the output power of the receiver is given. The second is the SNR at the output of AID converter, when we consider the quantization noise in the system.

According to the analysis in the last section, the model is based on the small signal assumption which means the input power of the amplifiers is far from IP3 .

A. SNR and capacity analysis at the input of AID converter In this subsection, we will analyze the performance at the

input of AID converter for given target output power of AMP2, and the channel state.

TABLEL THE OUTPUT AFTER EACH AMPLIFIER

Signal IMD3 Noise T Sr f.1� ST Nr

R SR = GcSJ' IR = f.1�GCST NR

AMPI G]GcSr I] = f.1]2G]PR +GJR G]NR

DSA GDGPcSr In = GnI] GDG]NR

AMP2 G2Gj)G] GcSJ' Iz = f.1�G2�) +G)n G2Gj)G]NR

In A WGN channel, we define the gain of channel as Gc and NT = NR I Gc ' then the fundamental signal power, IMD3

power and noise power at each step can be represented in TABLE I, assuming the DSA and the mixer introduce no nonlinearities.

In TABLE I, we define PR and�) as follow:

(5)

(6)

in which NR is Gaussian noise in Fig.4. Moreover, the output SNR (SNRRo) of the receiver can be denoted as Eq. (7).

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G,G,lJ1 q31 (7) SN�" = ;{,G,G,lJ1GcSj +KG,G,lJI�' +;l;G,G,ll,(1+K)�, +G,G,ll,N"

1) Due to the definition of IA1R3 and the small signal assumption, we can obtain f.1]z «1 , thus, Eq. (7) can be simplified as

2) Since the transmitter of downlink is at the base station, which is of high quality hardware, its nonlinearities can be ignored compared with that in terminal side which is of cheap implementation. So we can assume f.1o = O.

3) In most of the cases, the input of the receiver of downlink is of very low power, which implies that the attenuation value of DSA is low and the output power of AMPI is much lower than AMP2. In addition, in order to obtain quantization of high quality, the AID converter is operating with relatively large input range, e.g. [-IV, +IV], which implies that AMP2 may operate at relatively high output power, and introduce the main part of nonlinearity. So we can assume f.1] = 0 .

Finally, for the downlink, we rewrite Eq. (8) as follow:

{SNRT = Sr I NT

f.1; = a

(9)

(10)

(11)

In A WGN channel, when ST varies slowly, we can change the gain of DSA (Gj)) in time to satisfy the given maximum PRo' thus, Eq. (9) can be rewrite as Eq. (10) in which NJ'=NRIGc·

Then, we define SNRr and 11; as Eq. (11), as a result, we can rewrite Eq. (10) as follow

SNRr SNR =------".--Ro 1 + a + aSN Rr

which has the limit as follow

SNR, lim SN R = ----'-----SNRr .... �

Ro 1 + a+ aSNRr a

Moreover, we can obtain the capacity of the channel.

(12)

(13)

(14)

in which a = f1� can be obtained with the following method. First, according the definition of IMR3 , we get

(16)

in which OIP3 of the last stage amplifier can be found in the data sheet. Considering the input and output of AMP2, we get

(17)

Solving the simultaneous equation ofEq. (16) and Eq. (17), we get

(�)2+� (18) 2S20IP3 27

The analysis above tells that SNR and capacity have upper bound, which is related to the total output power of the receiver, when we consider the distortion of the RF receiver. The result is shown in Fig.6.

80 considering nonlinearity 60 ---b- without considering nonlinearity -

� 40 e 0:: z (f)

-20

25

o 10 20 30 40 50 60 70 SNRt(dB)

I I _ 20- - - - - - - � - - - - - - - - - - � -­N I � 15 e I � I I I .� 10- - - - - - - 1 - - -----1- - -1- - - -

"­'" U 5

10 20 30 40 50 60 70 SNRt(dB)

Fig.6. Variation of SNR&� and capacity when changing SNRT ' when a = 0.01

It shows the difference between considering nonlinearity and without considering nonlinearity, if we change GD in time to ensure PRo unchanged. When considering the nonlinearity,

we obtain the upper bound of SNRRo' which is 1 / a , with SNRT approaching infinite.

In practice, the output of the receiver is the input of AID converter. Fig.7 shows the characteristic of uniform quantization when input is a sine signal or a speech signal, in which 0 is the normalized RMS of amplitude and n is quantization bits [18]. Whatever the input signal of AID converter is, such as sine signal or speech signal, the characteristic of SNR has a linear region. And when AID converter is operating over the linear region, the performance decrease fast. We always make the AID converter operate in the linear region.

It is shown that the SNR at the output of AID converter increases as PRo increases, if the AID converter is operating in linear region. However, we also know the upper bound of SNRRodecreases as PRo increases (shown in Eq. (11), Eq. (12) and Eq. (18)). Obviously, there is trade-off relationship between the SNR at the input and output of AID converter. In next subsection, we will try to optimize the operation point of the receiver by choosing proper GJ) , to maximize the SNR at the output of AID converter.

SNR(dS)

___ L-L--__ --;-_+-. 1 010g TJ(dR) Sine Signal SNR(dR)

-"'-----<'-----+--+-. 1 010g TJ(dR) Speech Signa I

Fig.7. Characteristic of uniform quantization

B. Optimization of SNR at the output of AID converter For the AID converter is operating in linear region, the

quantization noise can be figured out if the signal distribution and the parameters of the AID converter are given. Defining the average power of the quantization noise as N Q ' we rewrite Eq. (9) as follows

m which G = G2GJGC ' and S, ' N" ,S2,011'3 ' G are given,

even N Q is certain, and the calculation method is given in [19].

Considering the convexity of the denominator in Eq. (19), we find that Eq. (19) is upper convex. Thus, we can obtain the maximum of SNRAJ)c in gradient method, when

4040

G = 1

3 N QS�,OIP3

J) G(S[ +N[) 2 (20)

Then, we get the maximum of SNRA1X (SNRAJ)C'.MAX ) as Eq. (2 1 ).

Let

we get

2 2 2 j3 = (N Q /S2,OIP3 )3(23 +i3)

1 + j3 + j3SNRT

(21)

(22)

(23)

in which SNRr = SJ I NJ. It means SNRAJ)c has an upper bound 1 / fJ .

Then, we define Y = S2,0I!'3 I NQ ' we obtain

lim SNRATX' MAX SNRT----7°O '

1 � -� 2 (24) = 10 19-=-10lg(23 +2 3 )+-(Y)dR '" -3.46 +0.67(Y)dR

j3 3

Generally, the SNR at the output of AID converter increase 6 dB, as the digitalizing bit increases 1 bit, if we make the reference level of AID converter remain unchanged and without considering nonlinearity of the receiver [ 19]. However, according to Eq. (24), in which S20IP3 is given, the limit increase of the SNR at the output of AID converter is 4 dB, if the digitalizing bit increases 1 bit, and practically, the increase of the SNR is less than 4 dB if the SNR at the output of AID converter satisfy the optimization in Eq. (23).

III ANALYSIS OF CAPACITY IN RAYLEIGH FADING CHANNEL

We have discussed the case in A WGN channel above. Next, we are going to investigate the case in Rayleigh fading channel.

Consider the model of Rayleigh fading [I].

y[m] = h[m]x[m] + n[m] (25)

where y[m] is the receiving signal, x[m] is the signal we've transmitted, h[m] is the Rayleigh fading coefficient, and n[ m] is the complex Gaussian noise. Further, we define h[ m] as h[m] = h , because of slow fading and assuming h - CN(O,l) . Then the receiving signal power can be denoted as

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1 h 12 ST + NT' where SJ is the transmitting signal power and N[ is the transmitting noise power. We also assume the variation of Sr is so slow, that we can ensure PRo unchanged. Now we can rewrite Eq. (13) as follow

From Eq. (26) we know that SNRRo in Rayleigh fading channel also has an upper bound 1 / a .

Next, considering the case of slow fading, we are going to obtain the outage probability, when the target rate is R bitls/Hz.

PaUl (R) = P{log2 (1 + SNRRo) < R}

= P{lo (1 + 1 h 12 SNRT ) < R} g2 l+a(1 hl2 SNRJ +1)

The Eq. (27) can be further written as

POUI (R)

(27)

= [1-a(2R-I)]SNRT'

(28) {P{I h 1'< (2' -\)(1 + a) } if l-a(2' -1) > 0

I , otherwise

� {,-",PC

SNRt=30dB, pout'

0.9 SNRt=30dB, pout

SNRt=40dB, pout'

0.8 --E-- SNRt=40dB, pout

, 0.7- ---------1 --

0.6

- , 8. 05 - - - - - - - - - � - -

0.4 - - - - - - - - - -' - -I 0.3

,

0.2 - ---------, -,

0.1

R(bitls/Hz)

, otherwise

, _______ 1

,

Fig.S. Comparison between Outage probability considering nonlinearity (pout)

and without considering nonlinearity (pout') SNR, = 30, 40, and a = 0.01

If we don't consider the non1inearities of the receiver, the outage probability can be denoted as

2R -1 P' (R) = 1-exp(- --) aUi

SNRT (29)

Comparing Eq. (28) with Eq. (29), we know that there is a threshold for the target rate R, which depends on a . If R < log2[(1 + a) I a] , it has little difference from the case without considering nonlinearities of the receiver. If R ?: 10gJ(1 + a) I a] , the outage probability is always I, which means the communication interrupts. Fig.8 shows the differences between the two results.

From Fig.8, we know if target rate is low, the outage probability considering nonlinearity doesn't have much difference from without considering nonlinearity. As the target rate increases, the difference becomes obvious. It is more meaningful, if we compare SN Rr between the two cases, when they have the same outage, shown in Fig.9.

i'i' z (f)

30�---

25 � �

5 o 0.1

considering nonlinearities, R=2

�."".- without considering nonlinearities,

-----1:::-- considering nonlinearities, R=4

without considering nonlinearities,

0.2

pout

I 0.3

I 0.4 0.5

Fig.9. Comparison between SNR, considering nonlinearity and without

considering nonlinearity, when R=2, 4 bitlslHz and a = 0.01

IV CONCLUSION

In this paper, we investigate the effect of the amplifier nonlinearity caused by IMD3 in RF receiver and the quantization noise of the AID converter, taking the downlink as an example.

In A WGN channel, the maximum limit of the SNR at the input of the AID converter and the capacity is obtained when considering nonlinearities of the receiver. After combining with the characteristic of quantization, the trade-off relationship between the SNR at the input and output of the AID converter is found.

Naturally, a method is found to maximize the SNR at the output of AID converter, by changing the gain of DSA in the receiver. An equation ofGj) is derived to maximize the SNR at the output of AID converter, and this provides the basis for the power control of the receiver. Using this equation ofGD, we also get an upper bound of the SNR at the output of AID converter. And a definition of the parameter ris given, which is the important index to measure the damage of nonlinearity to the performance of the receiver. After optimization, we found that when the digital bit increase 1 bit, the SNR increases less than 4 dB, which is different from the result without considering nonlinearities of the receiver.

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In Rayleigh fading channel, the degradation of the performance is discussed, and a threshold of the transmitting rate R is obtained. The communication system must run at a rate lower than the threshold

All the conclusions above are reached in downlink, while the situation of uplink is similar.

In this paper, we investigate the performance variation in a single-antenna situation when considering the practical limitation in the wireless communication system with digital receiver. However, the multi-antenna situation is also worthy to discuss, due to the widely used of MIMO, and it will be studied in the future work.

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