Reduced-Complexity Digital Satellite CDMA Systems Robust to Doppler

Reduced-Complexity Digital Satellite CDMA Systems Robust to Doppler

Mehmet R. Yuce¹, Wentai Liu^2,1

1Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695-7914

2Dept. of Electrical and Comp. Eng., University of California at Santa Cruz, Santa Cruz, CA 95064-1077, USA

Abstract-This paper presents two digital satellite CDMA systems which are invariant to Doppler shift. Their performances in a subsampling front-end with a 1-bit A/D converter are evaluated by both theoretical analyses and simulations. The subsampling is done after the received signal is downconverted to an intermediate frequency (i.e IF subsampling). Both systems are based on PSK DS/CDMA communication and do not require either an extra circuit or a pilot transmission for frequency tracking. The use of 1-bit A/D at front-end allows large power consuming circutries such as Automatic Gain Control (AGC), Phase-Lock Loop (PLL) and mixer be eliminated or simply replaced with digital ones. Hardware realizations of each ststem model are also discussed. After the 1-bit A/D converter, the whole sytem inculding the PN code synchronization, the carrier syncronization and timing syncronization can be implemented digitally with the use of application specific integrated circuit (ASIC) technology.

I.INTRODUCTION

In recent years, there has been increased interest in low cost, low size and low power spread spectrum code-division multiple access (DS/CDMA) systems. Due to the advances in VLSI technology, subsampling has recently obtained attention by some IC designers [1]-[3]. The main advantage in subsampling is to extend the digital portion of a receiver circuit, to enable the higher integration level, to decrease the physical size and to reduce the cost. The subsampling behaves like a mixer by sampling the input RF signal at a rate lower than the carrier frequency. Significant power saving is achieved since power consumption is proportional to the operation frequency in a digital CMOS circuit [4].

In this work, two digital mobile satellite CDMA systems invariant to Doppler shift are presented and their performances in a subsampling front-end with a 1-bit A/D converter are evaluated by theoretical analyses and simulations. Those demodulators do not require any pilot signal, PLL or any extra circuit for carrier recovery, therefore resulting in high transmission efficiency and low circuit complexity. Both systems are invariant to frequency offset and local oscillators’

instability, however with approximately 3-4 dB penalties in a single user channel. The paper is organized as follows. The proposed subsampling front-end is presented in section II.

Section III describes two digital CDMA systems invariant to Doppler with overall performance analyses. The first system

This work was supported by NASA Glen Research Center through a grant NAG3-2584

uses dual-channel method [5] while the second system uses the autocorrelation technique [9] to remove the Doppler shift.

In addition, the hardware realizations of each digital CDMA system (i.e. corresponding circuits for the blocks) are also given. Section IV concludes the paper.

II. IF SUBSAMPLING FRONT-END

In the case of subsampling front-end, the carrier frequency is downconverted to a very low frequency by sampling. In this paper we suggest using IF subsampling for digital satellite systems. The received signal is first downconverted to IF (Fig.1), and then subsampling is performed on the IF signal.

The sampling frequency for IF signal is chosen as fs=fIF /N, where N is the subsampling rate and N=(2n+1)/4, n=1,2…

Choosing the subsampling rate at (2n+1)/4 provides the input signal to be sampled at the values of 1,0 and -1, resulting in significant reduction in hardware complexity. Additionally, since the input IF signal will not directly be downconverted to DC, some issues, such as DC offset, 1/f noise and bias in analog circuits, inherent to RF front-end are thus eliminated

Although subsampling is an effective way to reduce circuit complexity and power consumption, it may have two significant impacts on the system performance. One is the overlapped noise due to aliasing from the subsampling process and the other one is the jitter due to sampling aperture jitter [2][3]. Subsampling by factor of N will multiply the noise power by a factor 2N. To prevent the overlapped noise, a bandpass filter (BPF) has to be employed at the front-end to reduce the out-of-band noise such that the overlapped noise does not exceed the input white noise. The jitter on the sampling clock results in phase noise and its spectral density is amplified by the subsampling ratio squared N² [2]. One way to reduce the effect of jitter beside a neat clock is to use a downconversion stage to potentially decrease the subsampling rate in high frequency transmission links. As shown in Fig.1 one stage downconversion including a BPF and an IF amplifier is necessary to meet the subsampling front-end requirements for the satellite communication systems. The effect of subsampling in the frequency domain is shown in Fig. 2. Fig.2-(a) is a circular convolution illustrating the location of images. The images will be located at fi=(k±1/4)fs, where k is an integer number. The input IF signal after BPF is shown in Fig. 2-(b), and the spectral images resulting from subsampling isshowninFig.2-(c).

a) c)

BPF BPF

N0 N0

-f-f IF fIF

N0 f_i=f_s/4 f_i=-f_s/4

fs/2

- fs/2 ^3fs/4

-3f_s/4

…

(k+3/4)f_s

(k+1/2)f_s (k)f_s

(k+1/4)f_s

…

b)

Fig. 1. One stage downconversion for the proposed CDMA systems

Fig. 2. IF signal subsampling: (a) circular convolution, (b) the IF input spectrum after bandpass filtering, (c) signal images after subsampling

Cos (w_ot)

LNA BPF

Low-IF

IF Amp.

IF Signal out

III.DIGITALSATELLITESYSTEMSROBUST TODOPPLER

In this section, two of digital mobile satellite CDMA systems invariant to Doppler shift are described and their performances in a subsampling front-end via a 1-bit A/D converter are evaluated by both theoretical analyses and simulations. Even though subsampling can be used with multi- bit A/D converters, we are mainly interested in 1-bit A/D converter front-end for the reason that a hard-limiter (or 1-bit A/D converter) is a part of many satellite transponders.

A. Digital Satellite CDMA System with Dual-Channel Demodulator Using Subsampling via 1-bit A/D

Kajiwara [5] has recently presented a satellite CDMA system that is based on dual-channel. The transmitted signal is composed of two linear polarized signals that are separated by different PN codes. Only one of the two signals is modulated by data. In this section we analyze Kajiwara’s dual-channel demodulator with a subsampling front-end (Fig.3). The frequency error will be removed at the output of the demodulator because both signals qI(k) and qQ(k) are affected by the same frequency shift.

After 1-bit A/D, the system becomes all-digital and can be implemented with application specific integrated circuit (ASIC) technology. Four mixers in the baseband are replaced with four logic XNOR gates and the low pass filters (matched filters) are implemented as accumulators or by up/down counters. The decimator is used to decreases the accumulators’size. All these have resulted in a highly flexible and low power dual-channel demodulator.

A m p.

d(t) Data

O SC.

C Q^(.) C I^(.)

SS co de genera tor

Circular Polarization

a)

-π/2

Jn 1-Bit A/D

(Hard-Limiter)

L

fs

sgn L

NCO

I_n

Q_n yn

x_I(k) xn

Decimator Accumulator

PN code CQ^(.) Dual-mode

coupler PN code

CI^(.)

Data Out x_Q(k)

RC ^Tclk q_I(k)

q_Q(k) Input IF

Signal r_IF(t)

b) q(k) ^-π/2 I_ref=cos (2π kΝ )

∑

∑

Fig. 3. Modified Kajiwara [5]'s dual-channel model a) transmitter b) receiver with 1-A/D converter and subsampling front-end

At the reiever before the 1-bit A/D converter, the two linear polarized component are expressed as

) 1 ( ) ( ) 2 / cos(

) ( 12 )

( M c t wI t wt n t

i P

t

r_I ∑ _Ii − _{i F} +∆ + _i− + _I

= = τ θ π

) 2 ( ) ( ) cos(

) ( ) ( 12 )

( M c t d t w t wt n t

i P

t

r_Q ∑ _Qi − _{i i IF} +∆ + _i + _Q

= = τ θ

where P is the common signal power, di(t) is the binary data signal of ±1 rectangular shape, M is the number of active users, cIi(t) and cQi(t) are two orthogonal spreading codes assigned to the users, τⁱ is the relative time delays (0≤τⁱ ≤T), w^IF is the radian frequency of the input IF signal,θi is the carrier phase uncertainty caused by channel, ∆w is the frequency uncertainty due to Doppler, and nI and nQ are independent bandpassed additive white Gaussian noise with single-sided power spectral density of No. After A/D and being sampled at t=kTs, the signals at both I and Q branches can be written as

) 3 ( ) ( ) 2 / 2

cos(

) ( 1

2 )

( M c k Nk wkT n k

i P k

r_I ∑ _Ii − _i +∆ _s+ _i− + _I

= = τ π θ π

) 4 ( ) ( ) 2

cos(

) ( ) ( 1

2 )

( M c k d t Nk wkT n k

i P k

r_Q ∑ _Qi − _{i i} +∆ _s+ _i + _Q

= = τ π θ

where N is the subsampling rate defined in section II. Those signals are actually 1-bit quantized (i.e. hard-limiting). The 1- bit quantized output samples at I branch is given by









<

= ≥

= 0 ( ) 0

0 ) ( )) 1

( sgn(

)

( if r k

k r k if

r k

q

I I I

I (5)

The hard-limited samples at Q branch qQ(k) are found similarly. There is no loss assuming in the 1 th signal as the desired signal. After the digital mixers and dispreading with the sampled version of spreading codes cI1(k) and cQ1(k) the samples at each path can be written as [6]

xI(k)=sgn( 2P cos(φ1) +II+NIk) (6) where

) cos(

) ( 2

2 ₁^I_{i i 1 i}

I M R

i P

I ∑ τ −τ φ

= = (7)

NIk = 2 nc(k)cI1(k-τ1)cos(φ1)- 2 ns(k)cI1(k-τ1)sin(φ1) (8) and

xQ(k)=sgn( 2P d1(k)cos(φ1)+IQ+NQk ) (9) where

) cos(

) ( ) (

22 ₁^Q_{i i 1 i i}

Q M R d k

i P

I ∑ τ −τ φ

= = (10)

NQk = 2 nc(k)cQ1(k-τ1)cos(φ1)- 2 ns(k)cQ1(k-τ1)sin(φ1) (11) where φⁱ=∆wkT^s+θⁱ is the phase error, II andIQ are co-channel interferences, R₁^I_iand R₁^Q_i are the cross-correlation functions of c1and ck (k≠1), N^Ik and NQk are Gaussian random variable with zero mean and the variance is

IF 2 0

NQk 2

NIk

2N σ σ N B

σ = = = , where BIF is the bandwidth of

the front-end filter. The noise components nc and ns are the in- phase and quadrature-phase bandpassed noise samples.

Assuming M>>1, the co-channel interferences (7) and (10) can be regarded as independent zero mean Gaussian random variables with variances given by [5], [7] and [8].

p Q p

I I

I G

P M G

P

M ( 1)

2 ) 2 )(

1

2 (

2 =σ = − = −

σ (12)

where Gp, the process gain, is the ratio of the front-end bandwidth to the information rate of the system (Gp=BIF/Rb).

The co-channel interference term together with additive noise (U=I+N) can also be treated as a zero mean Gaussian random variable and the variance is given by

p IF

U N B

G P M

2 ( 1) 0

− +

σ = (13)

The decision algorithm is given as Jn = sgn{xn+ yn}

= sgn{In.Qn} (14) The bit-error probability is Pr (In.Qn=-1|d1=1) when the binary

“1” is transmitted. Total samples at the outputs of the accumulators are given by In and Qn which are the sum of the samples xI and xQ respectively. The accumulator outputs:

⁼

∑

^K

k I

n x k

I ( ) and ⁼

∑

^K

k Q

n x k

Q ( ) (15)

where K =fsT/L= the number of samples after the decimators in a symbol duration T and L is the decimator’s coefficient.

The probability that xQ(k)=1 or xQ(k)=-1 under d1=1 can be written as

P(xQ(k)=1|d1=1)=pQ and P(xQ(k)=-1|d1=1)=1-pQ (16) Defining γ= ^2P/σ^U and

_{= − =} ^∞_∫ ⁻

− cos 1

2

1 )

exp( 2 2 ) 1

cos

( γ φ γ φ π ^{u du}

Q

p_Q (17)

The probability (pI) that xI(k)=1 or xI(k)=-1 can be found similarly. The outputs of the accumulators are described by the

binomial probability mass function. Notice that we will have K/2 samples (K is even) at each I/Q branch and by using the property of binomial mass function; it is simple to obtain the error probability for each binomial function given in (15):

{ }

∑−

= = 1 2

/ /4 /2

K K

k P K errorsamplesoutofK samples P_e

∑ 



 





=

− −

= ^/2

4 /

2

) /

1 2 ( /

K

K k

k K

k p

k p

K (18)

where p=pQ or pI. The probability of error of the receiver given in Fig 3 as indicated in (14) is found by multiplying the two binomial probability mass functions. A computer simulation can accomplish this easily. The index of the new probability mass function will range up to K/2XK/2.

1. Simulation Example

The following parameters are selected for simulations:

• IF frequency, f^IF=150 MHz

• Sampling frequency, f^s=40 MHz

• The BPF bandwidth, B^IF =20 MHz centered at fIF

• The decimator rate, L=1

• The subsampling rate, N=f^IF/ fs=15/4 where n=7.

• The processing gain G^p=BIF/Rb =BIFT=63

• The product of the BPF bandwidth and the PN code period, BIFTc=2

For independent noise sample, the sampling frequency has to be twice the bandwidth of BPF (i.e. fs≥2BIF) and accumulators need to be reset (by RC circuit) every period T. The total number of samples in a symbol duration is K= fsT/L=

2BIFT/L=126. The co-channel interference term is considered as an additional white Gaussian noise and its one-sided power spectral density before the match filter (i.e. accumulator) is equal to (M-1)PBIFTc. Fig.4 shows the simulation results for digital and analog dual-channel receiver for comparison. The relative degradation is due to employing the IF subsampling with the 1-bit A/D converter.

Fig.4. BER performance for multiple access users

B. Digital DDPSK Satellite CDMA System in Subsampling Front-end with 1-bit A/D

The digital double differential PSK (DDPSK) CDMA system is shown in Fig. 5. DDPSK demodulator is invariant to the frequency offset by using double differential technique [9][10]. The baseband includes two-stage differential decoders.

The first stage implements the autocorrelation technique; it converts frequency error into phase error. The second stage eliminates the phase error. This is because of the fact that the adjacent symbols are affected from the same frequency shift and phase error. As a result, the receiver utilizes non-coherent technique and does not require exact phase and frequency.

Jn 1-bit A/D

(Hard-Limiter)

L

fs

r_n

T

sgn

L

T T

90^o

y_n Q_n-1

I_n-1 x_n

DecimatorAccumulator PN code

C(.)

RC Tclk q(k)

Amp.

a(t)

Data

OSC.

a)

T T SS code

generator

b(t) d(t)

b)

Differential Encoders

c(t)

r_n-1

I_n

Q_n x_I(k)

x_Q(k) Input IF

Signal r_IF(t)

∑

∑

Fig.5. Digital DDPSK CDMA system with the 1-bit A/D converter and subsampling front-end a) transmitter b) receiver

In Fig. 6 an example of timing diagram for digital DDPSK satellite system at transmitter and receiver is shown. The binary data an is differentially encoded twice before transmission. After the first and second stage of differential encoding, the data is bn=an⊕bn-1 and dn=bn⊕ d^n-1respectively.

Fig. 6 (c) represents the second order phase difference modulated signal (PDM-2) of the actually transmitted data an

which is obtained from the data dn multiplied by a sinusoidal carrier frequency. In Fig. 6 (d), the received signal has a different frequency (i.e. half cycle loss) from the transmitted signal. This frequency difference might be due to Doppler effect. After the second stage differential decoder, the signal (in Fig. 6 (f)) is the same as an sent by the transmitter and no bit is detected erroneously.

The received and bandpassed signal can be written as ) ( ) ) cos((

) ( ) ( 2 )

(t 1 Pc t d t w wt n t

r ^M _{i i IF i}

IF =i∑ +∆ + +

= θ (19)

where n(t) is the band pass additive white Gaussian with zero mean and single-sided power spectral density equal to N0. The band pass representation of the noise at the output of the BPF is given by

n(t) = nc(t)cos((wIF+∆w )t+θi)-ns(t)sin((wIF+∆w )t+θI) (20) where nc(t) and ns(t) are low-pass Gaussian noise process and the noise variance σⁿ²=N0BIF. After the A/D and sampled at t=kTs, the received signal becomes

) 21 ( ) ( ) 2

cos((

) ( ) ( 2 )

( 1 n k

w wN kN k

k d k c P k

r _i

IF in

i M

n =i∑ + ∆ + +

= π θ

where N is subsampling rate (i.e. N=fIF/fs>1). The samples r(k) are 1-bit quantized. Assuming the 1 th signal as the desired signal, the samples between (n-1)T ≤ t ≤ nT are given by

) ( ) 2

cos(

) ( ) ( 2

2 cos(

) ( ) ( 2 ) (

2 1

1) 1

1

k n kN

k d k c P

kN k

d k c P k r

i s in

i M i

s n

n

wkT wkT

+ + +

+ ∑

+ +

=

∆

∆

= π θ

θ π

= A1+I1+GN1 (22)

The samples and the noise in the interval (n-2)T ≤ t ≤ (n-1)T,

) ( ) 2

cos(

) ( ) ( 2

2 cos(

) ( ) ( 2 ) (

2 )

1 2 (

1) )

1 ( 1 1

1 ( )

k n kN

k d k c P

kN k

d k c P k r

i s n

i i M i

s n

n

wkT T kT w

+ + +

+ ∑

+ +

=

∆

−

∆

= −

−

−

θ π

θ π

= A2+I2+GN2 (23)

Now we have two square waves from two adjacent symbols and the multiplication is done digitally, where xI(k)=rn.rn-1. Since both rn and rn-1 are affected from the same frequency error ∆w, after the digital mixer (or the first stage differential decoder) this frequency error will be removed and thus the samples will include only the phase error which is

φ

⁼∆w. This phase error will also be eliminated after utilizing the second stage differential decoder. Considering the path I in the receiver (Fig.5), after digital mixing rn and rn-1, the new hard limited samples can be written as [6]

) ) cos(

2 sgn(

)

( _{1 I}

I k Pd N

x = φ + ⁽²⁴⁾

Where NI= signal term X (interference term + noise term) + (interference term + noise term) X (interference term + noise term). In order to calculate the probability of (24), we will consider the approach given in [5] and a similar equation is also solved in [9, Eq. 6.42 ]. The total noise term NIcan be treated as a zero mean Gaussian variable. The probability that xI (k)=1 or xI(k)=-1 given d1=1 can be written as

P(xI(k)=1|d1=1)=pI and P(xI(k)=-1|d1=1)=1-pI (25) Following the approach given in [5][6],

) 26 ( cos )

2exp(

1 1 2 ))

cos (2 2exp(

1 1

)) 2 / cos ( 2exp(

1 1 ) ) 0 ( 2exp(

1 1

2 ) 2 exp(

) 1 / cos 2 (

2 2

2 / cos 2

2

U U

P N NI

I

P P

N eff E

u du P

Q p

I

σ φ σ

φ

φ γ σ π

φ φ σ

−

−

=

−

−

=

−

−

=

−

−

≈

∞∫ −

=

−

=

−

The SNR for each sample was defined in (17) as γ= ^2P/σU. The probabilitythatxQ(k)=1is found similarly ,which is

sin ) 2exp(

1 1 ₂

U

Q P

p σ

− φ

−

= (27)

Fig. 6. Timing diagram of autocorrelation technique for digital DDPSK satellite system

+1 +1 +1 -1 +1

+1 +1 +1

+1 +1 +1

-1 -1 -1 -1

-1 -1

PDM-1

PDM-2 (Transmitted Signal)

Binary Data (a_n)

Received Signal (r_k )

After Second Stage (Detected Data)

+1 After One Stage

(I_n)

+1 -1

[X]

Hard-limited Samples (After 1-bit A/D)

[X] ...

Autocorrelation Demodulation

(a)

(b)

(c)

(d)

(e)

(f)

T

[X] ...

Dopp effectedler receiv

ed sig nal

Fig. 7. BER performance of DDPSK for various values of M (K=126, G_p=63)

The signals at the outputs of the second stage differential decoders are xn=In In-1 and yn =Qn.Qn-1, where In and Qn are given in (15). As mentioned previously the outputs of the accumulator are described as binomial mass functions. Their probabilities of error are found by substituting (26) or (27) as the value of p in (18). The error probability at the output of the demodulator can be written as (Pe) = P{zn=xn+ yn≤ 0 | b=1}, when the binary “1” is transmitted. Error occurs when the half of samples at each path is erroneous. In [10], it is suggested to use (2T) delay unit rather than T at the first stage of the receiver given in Fig.5. By doing this high noise correlation is eliminated at the output of the receiver. Simulations show that this scheme improves the performance about 1.8 dB. In Fig.7, BER results of digital DDPSK CDMA receiver are plotted as a function of SNR for different number of users. Simulation parameters of the previous section are used.

C. Further Observations

Dual channel demodulator gives better performance than DDPSK demodulator because large performance degradation is incurred by the two successive multiplications of the signal plus noise at the output of the first and second stage differential decoders in DDPSK. Digital DDPSK is very sensitive to co-channel interference when used in 1-bit A/D front-end. Nevertheless, in a single user channel digital DDPSK approximates digital dual-channel receiver if the received signal includes large frequency error ∆w. This is because of the fact that dual channel receiver removes the frequency error after the match-filtering (i.e. accumulator) where DDPSK removes before the match-filtering.

System performance of the digital receivers can be degraded by quantization error as well. To address the amount of degradation, assume N^q is the quantization noise and Nq=σ^q2/BIF, where σ^q2=2^-2(b-1)/12=1/12 (b=1 for one bit data process). The effect on SNR can easily be calculated. After 1- bit A/D, the new SNR will be E/N^oÆE/(N0+N^q). At high E/N^o the quantization noise becomes effective. However higher sample rate (when K is higher) will exhibit less quantization error. For instance when K=126 and SNR=15, the loss is

~0.18 dB. From simulation results, a worst case loss of 0.2 dB cannot be exceeded as long as the number of K is high enough.

IV.CONCLUSION

We have presented two digital satellite CDMA systems that are invariant to frequency error. The detail of analytical derivations, some simulations and comparisions for multichannels in a subsampling front-end using 1-bit A/D converter are discussed. We have derived equations that represent the error probability of each demodulator. The degradation from the 1-bit A/D converter is 2.2 dB in a single channel user, where about 0.2 dB is from the quantization error. This degradation is not insignificant but 1- bit A/D front-end can be an attractive alternative for applications that are required low power and low complexity at receiver site.

The reason using subsampling via 1-bit A/D in this paper is that many satellite transponders have already been using 1-bit A/D (i.e. hard-limiter) to optimize the power efficiency. The degradation can be avoided if multi-bit A/D converters are used in the subsampling front-end.

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