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A Uni?ed Analysis for Direct-Sequence CDMA in the Downlink of OFDM Systems

Anders Persson, Tony Ottosson, and Erik G. Str¨ om

Abstract This paper presents a uni?ed analysis for the downlink BER performance of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA and TFL-CDMA. As a special case, an analysis is obtained also for OFDMA. The intra-cell interference is analyzed under the assumption of Hadamard spreading codes and a correlated frequency-selective Rayleigh-fading channel. The presented results assume error-free channel estimates and a perfectly synchronized receiver. It is concluded that the compared schemes achieve essentially the same performance for low system loads. For high loads, and especially in a near-far scenario, OFDMA is found to outperform the others. Index Terms Code division multiaccess, spread spectrum communication, orthogonal frequency-division multiplexing, fading channels, convolutional codes.

I. I NTRODUCTION Combinations of OFDM and CDMA have been subject to intensive research lately, mainly due to the OFDM modem’s ability to reduce inter-symbol interference (ISI) on frequency-selective channels. Although this property comes at the cost of adding a guard-interval (cyclic pre?x), it has been shown to signi?cantly increase the number of simultaneously supportable users in CDMA systems. Typical OFDM DS-CDMA systems consists of a concatenation of a forward error-correcting code (FEC), a low-rate direct-sequence channelization code, and an OFDM-modem [1]–[3]. Two in principle different approaches can be distinguished in the literature, multi-carrier CDMA (MCCDMA), and multi-carrier direct-sequence CDMA (MCDS-CDMA) [4]–[6]. In MC-CDMA, each direct-sequence codeword is transmitted on adjacent sub-carriers in frequency, within the same

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OFDM symbol. In MCDS-CDMA, the direct-sequence codeword is transmitted on the same carrier but on consecutive OFDM symbols. MC-CDMA can therefore be said to spread the symbols in frequency, while MCDS-CDMA spreads the symbols in time. Schemes that perform spreading in both time and frequency have also been proposed, [7]–[14]. An alternative to using direct-sequence codes for multiple-access is to assign non-overlapping carrier-hopping patterns to the active users within the cell. This method can perfectly preserve orthogonality within the cell and is referred to as OFDMA in the literature, [15, pp.213-]. The system model for the TFL-CDMA scheme presented in [7] will be used in this paper. It will be shown that MC-CDMA, MCDS-CDMA and OFDMA can be ?tted into this model by properly selecting its parameters. On frequency-selective fading channels, the choice between maintaining orthogonality or obtaining diversity through spreading is non-trivial. At low system loads it is clearly bene?cial to seek diversity through spreading. Assuming a single-user detector, it will be shown in this paper that it is already at modest loads preferable to avoid interference by maintaining low cross-correlations. To simplify the analysis, it is commonly assumed in the literature that the fading is uncorrelated among the sub-carriers, see, e.g., [16]. For practical OFDM systems, the fading is, however, correlated in both time and frequency [17]. In this paper, the correlation is taken into account when analyzing the intra-cell interference. Several papers have been devoted to the analysis of the uplink, [18], [19]. In this case, the channel for the desired signal, and the channel for the multiple-access interference (MAI) are modeled as independently fading. The analysis presented in this paper applies to the downlink, where the signal of interest and the intra-cell MAI see the same channel. Practical wireless systems use channel coding and interleaving to achieve coding and diversity gains. Much of the analysis available in the literature is, however, devoted to uncoded systems, [16], [18], [19]. For a reference on coded systems, see [20]. In this paper, convolutional coding is assumed. To summarize, this paper presents a uni?ed analysis for the downlink of convolutionally coded and single-user detected, MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. In the analysis it is assumed that the intra-cell interference and the signal of interest see the same correlated time and frequency-selective fading channel.

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II. S YSTEM MODEL A block diagram of the transmitter for the k th user is shown in ?gure 1. It consists of a direct-sequence transmitter with FEC and interleaving on the coded bits. The interleaved bits are QPSK modulated, and the symbols are thereafter direct-sequence spread by the spreading code ck . The resulting chips are passed through a chip mapping block and, ?nally, transmitted by an OFDM modem. The purpose of the chip mapping is to arrange the order of the chips so that they are positioned in the desired way on the time-frequency grid. As an example, consider a system using an OFDM modem with Mc = 4 sub-carriers and length SF = 4 direct-sequence codewords. Each symbol, sk [l], is now spread to 4 chips, {sk [l]ck [0], sk [l]ck [1], sk [l]ck [2], sk [l]ck [3]}, and several possibilities exists for which sub-carriers, m, and OFDM symbols, n, that should be assigned to each and grid is given in ?gure 2. one of them. An example where each symbol is mapped to a 2 × 2 region of the time-frequency For simplicity, the time-frequency segment over which one symbol is spread will be restricted and time respectively, and the product M N equals the length, SF , of the direct-sequence code. By letting M = SF and N = 1 the MC-CDMA scheme is obtained, and if M = 1 and N = SF the MCDS-CDMA scheme results. Schemes that moves to the extreme in obtaining diversity, by interleaving the direct-sequence chips in time and frequency, have been proposed [20]. For low system loads, these schemes can improve the performance compared to the direct-sequence schemes considered in this paper. To utilize the diversity also at high loads, without causing a severe penalty due to interference, multi-user detectors are used in [20]. Since the focus in this paper is on the downlink, where a low-complexity single-user detector is targeted, this approach is not considered in the comparison. For rectangular regions, the chip mapping is conveniently expressed as the Kronecker product of a data matrix, Sk , containing the k th users symbols and a size M × N spreading matrix G k containing the k th users direct-sequence spreading code. In order to maintain as low as possible cross-correlation between the codewords, a common choice for M and N should be made for to a rectangular M × N region, where M and N denote the amount of spreading in the frequency

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all users within the cell1 . Multiple choices for Gk exist, and in this paper the following is made, Gk [i, j ] = ck [jM + i], i = 0, 1, . . . , M ? 1 j = 0, 1, . . . , N ? 1 . (1)

It is assumed that the binary direct-sequence codewords are normalized, i.e, ±1 , Gk [i, j ] = √ MN

M ?1 N ?1

G2 k [i, j ] = 1.

i=0 j =0

The k th user’s data-matrix, Sk , is chosen as Sk [i, j ] = sk [jMc /M + i] ,

c i = 0, 1, . . . , M ?1 M

j = 0, 1, . . .

.

(2)

For the example in ?gure 2 the following matrices result, ? ? ? ? s [0] sk [2] . . . ck [0] ck [2] ?. ? , Sk = ? k Gk = ? sk [1] sk [3] . . . ck [1] ck [3]

Hadamard codes will be assumed for the direct-sequence spreading in the MC-, MCDS- and

TFL-CDMA schemes. By choosing mutually orthogonal codes, where all but one element in each codeword is zero, OFDMA can also be made to ?t into the model in ?gure 2. That is, for OFDMA the k th users spreading code is chosen as the k th column of the size SF × SF identity matrix. Note that, since only one position per spreading codeword is nonzero for the OFDMA scheme, the summation in the despreading is not necessary to implement. Also, as will be shown later on, the soft-decision maximum-likelihood sequence detector is less computationally complex for the OFDMA scheme compared the direct-sequence schemes. The obvious drawback of not using the direct-sequence component is reduced time and frequency-diversity. It is, however, well known that the ?rst orders of diversity (independently faded contributions) improve the performance dramatically, and that the rate of the improvement thereafter saturates [21, p.785]. A large portion of the available diversity is obtainable by time

1

Different rates for different users is still supported. In this case, M and N are determined by the highest spreading factor

used.

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and frequency-interleaving of the coded bits. Further increasing the diversity, by adding directsequence spreading, can improve the performance somewhat but will introduce multiple-access interference due to the orthogonality loss on the channel. A. Channel model The commonly used tapped delay-line channel model is assumed. The channel is modeled as circular, complex-valued, zero mean Gaussian processes. Furthermore, the taps are assumed to be stationary, mutually independent, and to have autocorrelation ψ l (?), ? ? ψ (?) = ζ J (2πν ?), l = l l l 0 l ? . E {hl (t)hl (t + ?)} = ? 0, l =l Rayleigh-fading, meaning that the channel taps, hl (t), l = 0, 1, . . . , L ? 1, are assumed to be

(3)

Here, J0 (x) is the Bessel function of the ?rst kind and zeroth order, ζ l is the average tap power, and νl is the maximum Doppler frequency of the lth channel tap. This choice of autocorrelation function corresponds to the frequently used model proposed by Clarke [22, pp.116-], [23]. time-varying frequency response can be written as

L?1

For the tapped delay-line channel model, with excess delays τ l , l = 0, 1, . . . , L ? 1, the (4)

H (f, t) =

l=0

hl (t)e?j 2πf τl .

B. Power-control A well functioning power-control scheme is crucial in CDMA systems. In this paper perfect power-control in the average sense is assumed, meaning that the transmit powers are not adjusted according to the short-term fading or interference levels, but kept at a constant level that will result in a given constant long-term average received Eb /N0 . Different transmit powers are considered for different users, i.e., a near-far ratio is included in the analysis. C. Receiver A block-diagram of the receiver is shown in ?gure 3. The ?rst step in the baseband part of the receiver consists of a standard OFDM demodulator, i.e., removal of the cyclic pre?x, and time to frequency-domain conversion using the FFT algorithm. Next, the inverse to the previously described chip mapping operation is performed. The resulting signal is the sequence

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of transmitted chips, attenuated and phase-rotated by the channel, and perturbed by receiver noise, and interference. Prior to direct-sequence despreading, channel-phase compensation is performed. The purpose of this is to accomplish coherent addition in the despreading summation. Multiplying each chip by the conjugate of its corresponding channel-value (SNR maximum-ratio combining) is optimal 2 in the case of white Gaussian receiver noise and zero MAI but, already at moderate system loads, severely increase the codeword-distortion and MAI, [20]. Therefore, in this paper equal-gain chip-combining is used in the despreading summation. Other authors have considered schemes that reduce the intra-cell MAI by applying, e.g., zeroforcing (orthogonality restoring), and minimum-mean-square-error (MMSE) equalizers, see, e.g., [20], [24]. In this paper, no attempt is made to reduce the intra-cell MAI using equalizers. Soft-output demodulation and soft-decision decoding is assumed in the detector. Soft-output demodulation of the QPSK symbols is achieved by simply multiplexing the real and imaginary parts of each received symbol into a real-valued sequence. Prior to soft-decision channeldecoding, the coded bits are multiplied by a post-despreading combining-weight sequence, {g }. Two different combining methods are considered in the next subsection. D. Detector Assuming that the OFDM symbol time is suf?ciently short compared to the coherence time of the channel, and that the cyclic-pre?x exceeds the support of the channel’s impulse-response, the OFDM modem can be said to sample the channel in frequency and time. H [m, n] = H (m?f , n?t ), m = 0, 1, . . . , Mc ? 1 n = 0, 1, . . .

Note that this model is assumed for the analysis only and not for the computer simulations. In section IV, it is argued for that this assumption is valid for the system parameters that will be used in the comparison between the presented analysis and computer simulations. Further reading regarding this matter can be found in [13], [14]. If perfect channel estimates are available at the receiver, the phase of the channel will be completely compensated for and the output signal, rk [i], corresponding to the ith symbol from

2

Optimal in the maximum likelihood sense.

DRAFT

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the k th user’s despreader, can be written as

M ?1 N ?1

r k [i ] = s k [i ]

m=0 n=0 M N ?1

G2 k [m, n]R[i] [m, n]

M ?1 N ?1

+

k =0,k =k

s k [i ]

m=0 n=0

Gk [m, n]Gk [m, n]R[i] [m, n] + w[i].

(5)

and Gk is the k th user’s spreading matrix of size M × N . The last term, w[i], is the additive receiver noise. In (5), all M N codewords are included. If some users (codes) are inactive, the corresponding symbols should be set equal to zero. and let ? denote the set of all valid code-sequences. Assuming Gaussian noise and interference, independent from coded bit to coded bit, the single-user maximum-likelihood sequence detector Let the {q }, where q [j ] ∈ {+1, ?1}, denote a sequence of coded bits for the channel code used,

Here, R[i] [m, n] = |H[i] [m, n]| is the sampled channel envelope that effects the ith symbol,

given the sequence of soft-demodulated coded bits, {zk }, decides in favor of the sequence {q ?k } as {q ?k } = arg max where

2 [j ] σtot m,n j

{q }∈?

G2 k [m, n]R[i[j ]] [m, n] ·z k [j ] · q [j ], 2 σtot [j ]

gk [j ]

(6)

ξk = 2 k =k

2

Gk [m, n]Gk [m, n]R[i[j ]] [m, n]

m,n

+

N0 , 2

and, ξk denotes the QPSK symbol energy for the k th user, and N0 /2 is the power spectral density of the additive receiver noise. The notation i[j ] is used to denote the index rearrangement necessary due to demodulation and deinterleaving. From (6) it is seen that the post-despreading weighting sequence should be chosen as g k [j ] =

m,n

G2 k [m, n]R[i[j ]] [m, n] . 2 σtot [j ]

(7)

This choice will be referred to as SNIR post-despreading weighting since it takes both the power of the noise and the time-varying intra-cell interference into account. A suboptimal choice, but more tractable from an implementation viewpoint, is to neglect the interference and to chose {gk } as g k [j ] =

m,n December 21, 2003 DRAFT

G2 k [m, n]R[i[j ]] [m, n],

(8)

8

where also the constant N0 /2 has been omitted since it no longer effects the result of the maximization in (6). This choice will be referred to as SNR post-despreading weighting since it takes only the signal-to-noise ratio of the coded bits into account. Note that for OFDMA, (7) and (8) become identical since no intra-cell interference exists for this scheme. Again, since only one position per spreading code-word is nonzero for the OFDMA scheme, the summation in (7) is not necessary to implement. In fact, both the phase-compensation and maximum-ratio post-despreading weighting, is for OFDMA achieved by multiplying each symbol by the complex-conjugate of the channel at the corresponding sub-carrier. Clearly, this is a reduction in complexity compared to the schemes that use Hadamard codes, in which each individual chips has to be phase-corrected, and (7) (or (8)) has to be evaluated for each symbol. The detector is implemented using the Viterbi algorithm by which (6) is ef?ciently evaluated and the information-bit sequence estimate, {? bk }, is determined for the convolutional code used. III. P ERFORMANCE A NALYSIS The aim of the analysis is to ?nd an approximation to the bit error rate (BER) on a time- and frequency-selective channel. First a parameterization of the channel is given. This parameterization is thereafter used to derive expressions for the instantaneous power of the intra-cell MAI, the cdf, pdf, and moment-generating function of the signal-to-noise-plus interference ratio (SNIR). Most of the derivations are carried out in [25], and the reader is referred thereto for details. A truncated union bound for soft-decision decoding of the convolutional code is thereafter applied to approximate the BER performance. A. Signal-to-noise plus interference ratio For stationary and mutually independent complex Gaussian channel taps, it is shown in [25] that the statistics of H (f, t) are time and frequency invariant. Therefore, without loss of generality, the despreader output due to the zeroth symbol can be studied. The channel envelope that effects the zeroth symbol is parameterized using the following

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two-dimensional Taylor expansion, ˙ f [ m 0 , n 0 ] · (m ? m 0 ) + ? t R ˙ t [m 0 , n 0 ] · ( n ? n 0 ) , R[m, n] ≈ R[m0 , n0 ] + ?f R m0 = M ?1 , 2 N ?1 n0 = . 2

(9)

That is, for each symbol (spread to M N chips), the channel envelope is modeled as a plane ˙ t , and R ˙ f in time and frequency. with random average value R, and random slopes R At this point it is important to note that (9) is sometimes a crude approximation. This is the case when the spreading in time and/or frequency is high compared to the time and frequencyselectivity of the channel. That is, the accuracy of (9) will depend on the choice of the frequency and time-domain spreading factors relative to the frequency and time-variability of the channel. See [25] for further details. Also, by de?nition the envelope is a positive number but using (9) no longer ensures this. However, this is not a crucial problem in the analysis since the signalto-noise plus interference ratio is sought, where only the squared envelope is of interest. The parametrization made is motivated by its simplicity and it, as will be seen later on, suf?ces for modeling a signi?cant part of the intra-cell interference that results from the orthogonality-loss on the channel. Note that (9) is used in the analysis only, and not in the computer simulations. Using (9) in (5), and noting that G2 k [i, j ] =

M ?1 N ?1

1 , MN

i=0 j =0 M ?1

Gk [i, j ]Gk [i, j ] = δ (k ? k ) ,

m=0 M ?1

(m ? m0 ) = 0, (n ? n0 ) = 0,

n=0

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gives the following approximation to the despreader output

M N ?1 M ?1 N ?1

rk ≈

Rsk

desired signal

˙ f ?f +R

k =0,k =k

sk

m=0 n=0

(m ? m0 )Gk [m, n]Gk [m, n]

˙ f ?f If (k), intra-cell MAI due to frequency-selectivity R M N ?1 M ?1 N ?1

˙ t ?t +R

k =0,k =k

sk

m=0 n=0

(n ? n0 )Gk [m, n]Gk [m, n] +

w

receiver noise

˙ t ?t It (k), intra-cell MAI due to time-selectivity R

˙ f ? f I f (k ) + R ˙ t ?t It (k ) + w, = Rsk + R

(10)

where, for convenience, the symbol index, i, as well as the dependence on m 0 and n0 for R, ˙ t , and R ˙ f have been omitted. R At this point, the following limitation of using (9) should be noted. After using (9) it is seen from (10) that R alone scales the desired symbol. This means that the analysis is unable to model the diversity that is provided by the direct-sequence spreading. This is indeed consistent with (9) being accurate only for relatively small deviations from the center point, [m 0 , n0 ]. Including also higher order terms signi?cantly complicates the remaining steps of the analysis and is left as a topic for future work. Assuming the linearized channel in (9), normalized Hadamard direct-sequence codes, and that the symbols are zero mean and independent among the users, it is shown in [25] that

log2 (M )?1

E | I f (k )| E | I t (k )|

2

=

p=0 log2 (N )?1

4p?1 ξk⊕2p ,

2

=

p=0

4p?1 ξk⊕M 2p , (11)

E {If (k ) (It (k ))? } = 0,

the natural binary representation of the integers a and b. Somewhat surprisingly it is seen that

where ξk = |sk |2 is the symbol energy for the k th user, and a ⊕ b denotes bitwise XOR between

only log2 (M N ) interfering codewords have nonzero cross-correlation with any given codeword, ck , after the linearized channel. It is also seen that among the codewords with nonzero crosscorrelation, some contribute more than others to the interference. For example, assume that a direct-sequence spreading factor of SF = 64 is used and that it is split into the frequency and time-domain spreading factors M = 16 and N = 4 respectively.

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and E |It (2)|2 are now obtained. E |If (2)| E |It (2)|

2

Furthermore, assume that user k = 2 = 000102 is decoded. The following values for E |If (2)|2 1 4p?1 ξ2⊕2p = ξ3 + ξ0 + 4ξ6 + 16ξ10 , 4 p=0 1 4p?1 ξ2⊕16·2p = ξ18 + ξ34 . 4 p=0

1 3

=

2

=

(12)

That is, users 0, 3, 6, 10, 18, and 34 will contribute to the intra-cell MAI seen by user k = 2. If the symbol energies, ξk , are equal it is also seen that users 10 and 34 will dominate the interference part that is due to frequency and time selectivity, respectively. The discussion now returns to the derivation of the signal-to-noise plus interference ratio. For ˙ f , and R ˙ t , it is seen from (10) that the symbol-SNIR, γ , can be written as ?xed values of R, R γ= ˙ 2 ? 2 E | I f (k )| 2 R f f R 2 ξk . 2 2 2 ˙t +R ? t E | I t (k )| 2 + σ w (13)

˙ f , and R ˙ t is derived in [25], and using this expression it is thereafter The joint pdf for R, R shown in [25] that the cdf and pdf for γ can be written as

2 σw P (γ ) =1 ? σ exp ?γ 2 σ 2 2 × σ 2 + 2γσt ?1/2 2 σ 2 + 2γσf ?1/2

,

0 ≤ γ < ∞,

(14)

p(γ ) = exp ?γ

2 σw σ2

2 2 2 2 2 2 2 2 2 4γ 2 σt σf σw + 2γσ 2 σw σf + 2σf σt + σ w σt 2 σ 2 + 2γσf ?3/2 2 σ 2 + 2γσt ?3/2

2 2 2 + σ 4 σw + σf + σt

, (15)

0 ≤ γ < ∞, where σ 2 = E R 2 ξk ,

2 2 ˙2 , = ?2 σf f E R f E | I f (k )| 2 2 ˙2 σt = ?2 , t E R t E | I t (k )| 2 σw = E |w | 2 = N 0 ,

(16)

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and

L?1

E R

2

=

l=0

ζl , ?

L?1 L?1 l=0 τl ζl L?1 l=0 ζl 2

2 ˙f E R

? = 2π 2 ?

L?1 l=0

l=0

τl2 ζl ?

?

? ?, (17)

2 ˙t = π2 E R

νl2 ζl .

Here, ζl , νl , and τl are the average power, maximum Doppler frequency, and delay of the lth channel tap, respectively. It should be stressed that the parameters in (17) and (16), and consequently also P (γ ) and p(γ ) in (14) and (15), are functions of k . At this point, the following intuitively satisfying observations can be made.

?

The average energy gain for the desired signal, quanti?ed by E {R 2 }, is proportional to the average power gain of the channel. ˙ 2 , is proportional to square The frequency-variability of the channel, quanti?ed by E R f of the RMS delay-spread. ˙ 2 , is proportional to the meanThe time-variability of the channel, quanti?ed by E R t square value of the maximum Doppler frequencies.

?

?

B. Approximate bit-error-rate The next step in the analysis is to approximate the bit-error-rate after soft-decision decoding of the convolutional code. A truncated union bound will be used for this. BER ≈ min 1 max , cd P2 (d) 2 d=d

free

d

(18)

Here, P2 (d) is the pair-wise average error probability for two sequences separated by Hamming distance d, and cd is the information-error-weight-spectrum [21, pp.483-]. For low bit-error-rates, say below 10?2 , a reasonable approximation to the true bit-error-rate can be obtained by (18). Note, however, that since the summation in (18) is truncated at d = d max it can no longer be claimed that an upper bound is obtained. At this point in the analysis, it is assumed that the sum of interference and noise is Gaussian. It is also assumed that maximum-ratio combining is used in the decoding, meaning that the

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SNIR of the individual coded bits are added up in the decoding, [26]. In the simulations this corresponds to using the post-despreading weighting sequence in (7). If perfect interleaving of the coded bits is assumed, the pdf of the SNIR for d maximum-ratio combined coded bits equals the d-fold convolution of the pdf given in (15). This corresponds to raising the moment-generating function to d. Using the moment-generating function method presented in [27], P2 (d) may be written as

π/2

1 P2 (d) = π

0

Md

1 2 sin2 (α)

dα,

(19)

where M(s) =

∞

p(γ ) exp (?sγ ) dγ,

0

Re {s} ≥ 0.

Expressions for M(s) are derived in [25] and the reader is referred thereto for details. The ?nal expressions are presented below. It turns out to be convenient to split M(s) into different expressions depending on the values

of σt and σf . In case both frequency and time-domain spreading are used, as in TFL-CDMA is obtained where M ≥ 2 and N ≥ 2, both σt and σf will be nonzero. In this case the following expression σ2s M1 (s) = 1 ? 2πσf σt α (θ ) =

2 σf π

exp (α(θ)β (s)) E1 (α(θ)β (s)) dθ,

0

+

2 σt

2 2 + (σ f ? σt ) cos(θ) , 2 2 4σf σt

2 β (s) = σ 2 s + σw ,

where, E1 (x) is the ?rst order exponential integral [28, p.228, f.5.1.1]. σf will be zero and the following expression is obtained M2 (s) = 1 ? σ 2 s π

2 2σt β (s)

In case time-domain spreading only is used, as in MCDS-CDMA where M = 1 and N ≥ 2, β (s) 2 2σt β (s) 2 2σ t

exp

erfc

,

where erfc(x) is the complementary error function [28, p.297, f.7.1.2].

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σt will be zero and the following expression is obtained [(15) is symmetric in σ f and σt ] M3 (s) = 1 ? σ 2 s π

2 2σf β (s)

In case frequency-domain spreading only is used, as in MC-CDMA where M ≥ 2 and N = 1, β (s) 2 2σf β (s) 2 2σf

exp

erfc

.

In case no direct-sequence spreading is used at all, as in OFDMA, both σ f and σt will be zero. In this case, (15) reduces to the pdf for an exponentially distributed variable. The corresponding moment-generating function is given by [21, pp.773-], M4 (s) =

2 σw . 2 + σ2s σw

The next step in the analysis is to use the appropriate expression for M(s) in (19). At this point M2 (s), and M3 (s), tractable for numerical evaluation, are presented in [25]. IV. N UMERICAL RESULTS

numerical evaluation of the remaining integral(s) seems unavoidable. Expressions for M 1 (s),

In this section, the analytical approximate formulas are compared to Monte-Carlo computer simulations. A. System Parameters A rate

1 2

convolutional code with a memory of six bits is chosen. The generator polynomials,

given in octal format are 1338 and 1778 , [21, p.472, p.493]. The seven ?rst nonzero terms of the information-error-weight spectrum for this code are listed below. {(d, cd )} = (10, 36) , (12, 211) , (14, 1404) , (16, 11633) , (18, 77433) , (20, 502690) , (22, 3322763) , . . . Pseudo-random interleaving over NΠ coded bits is assumed. A more elaborate interleaver design is certainly possible but is left as a topic for future research. The performance will be evaluated for the case of perfect interleaving as well as for a ?nite-depth interleaver. Length 64 spreading codes are assumed. The OFDM modem is assumed to have a bandwidth of 5 MHz, a main-carrier frequency of 2 GHz, and QPSK modulation on all sub-carriers. The number of sub-carriers, is set to 256, and a 26 sample long cyclic-pre?x (≈ 5.3 ?s) is added.

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15

With these parameters the following sub-carrier spacing, ?f , and OFDM-symbol time, ?t , are obtained. ?f = B 5 · 106 ≈ 19.53 kHz = Mc 256 Mc + N g 256 + 26 ?t = = ≈ 56.40 ?s B 5 · 106

For the given parameters, an energy loss of approximately 0.42 dB will result due to the cyclicpre?x. All values of Eb /N0 given in this paper are before removal of the cyclic-pre?x. With the parameters given, each user will transmit at an information bit rate of approximately 71 kbit/s. At full system load (64 active users) this corresponds to approximately 0.9 bits/s/Hz for the cell. B. Channel parameters and system load The standardized six tap channel ”Vehicular Reference Environment A” [29, p.37] is assumed for the average-power-delay pro?le. In the simulations, a time-resolution of 0.2 ?s is used and the delays are rounded upward to the nearest multiple thereof. Se table I. Two different mobile velocities, 5 and 50 km/h, are considered. These velocities corresponds to Doppler frequencies of approximately 9.3 and 93 Hz, respectively. For simplicity, the autocorrelation function in (3) is assumed for all channel taps. For the higher of the two Doppler frequencies, the coherence-time, Tc , for the channel normalized by the OFDM symbol time, ?t , is on the order of Tc /?t ? 1/ (93 · 56.40 · 10?6 ) ≈ 190. This number is high enough to justify the assumption in the analysis of a time-invariant channel for each OFDM symbol. In the simulations, the fading is generated on a sample basis with a 0.2 ?s sampling time, meaning that the variation over one OFDM symbol is included. In an orthogonal scheme, like OFDMA, large differences in the transmit powers for different users can be allowed without causing any intra-cell interference. In nonorthogonal schemes on the other hand, like MC-CDMA, MCDS-CDMA or TFL-CDMA, different transmit powers for different users will lead to a near-far problem that will result in increased background interference in other cells. An accurate model for the transmit powers will depend on the geographical distribution for the users with in the cell of interest as well as in neighboring cells. Also the distributions for the delay and Doppler-spread for the different users will in?uence the result. Clearly, such a model is out of scope in this paper, and the discussion is therefore limited to two simpli?ed scenarios.

December 21, 2003 DRAFT

16

? ?

Scenario I: K active users of equal powers. Scenario II: K active users where the decoded user is assigned a 10 dB lower power compared to each and one of the K ? 1 interferers.

At this point, it is tempting to believe that the users who are assigned the higher power in scenario II will achieve a very low BER. Assuming the same received E b /N0 for all users, it is however realized that the performance for these users in fact will be close to that of scenario I. To see this, consider the interference at the detector for any of the scenario II users that are same power as the desired signal, plus a single interferer with a 10 dB lower power. The resulting BER for these users is therefore upper and lower-bounded by the performance in scenario I with K and K ? 1 active users, respectively. C. Perfect bit-interleaving case In this section the BER-performance is evaluated for the case where SNIR post-despreading weighting (see (7)) and perfect interleaving of the coded bits is assumed. With perfect interleaving is meant that the correlation for the fading between any two coded bits is completely broken up by the interleaver. Note, however, that the interleaving is done on the coded bits and therefore the fading within each M × N block (a QPSK symbol) still has the correlation provided by the found by evaluating (18), using (19) and the appropriate moment-generating function, M(s). channel. This is the case where the presented analysis is applicable and the numerical results are The number of terms to include in the truncated union-bound is, especially for fading channels, assigned the higher power. The total interference is in this case due to K ? 2 interferers of the

a dif?cult choice to motivate. In this paper, the ?rst ?ve nonzero terms in (18) will be included. From the ?gures presented in this paper and in [7], it is seen that this choice results in a reasonably tight approximation for bit-error-rates below 10 ?2 . Figure 4 shows the BER-performance for different values of M , for near-far scenario I and II, respectively. A mobile velocity of 50 km/h, Eb /N0 = 10 dB, and K = 64 users is assumed. It is seen that the analysis is relatively tight for the schemes that use a well localized chipmapping in both near-far scenario I and II. In scenario II, the performance degradation due to interference is seen to be severe for the schemes that use a non-localized chip mapping. Loosely speaking, the interference in this case degrades the performance more compared to what the

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17

additional diversity achieved by the spreading is able to improve on the performance. This is well predicted by the analysis. As explained in section III-A, the additional diversity obtained by the non-localized spreading is not captured in the analysis. This is the reason for the large discrepancy between the simulation and analysis for these schemes in near-far scenario I. Loosely speaking, in this scenario the degradation due to interference is not dominating over the improvement obtained by the additional diversity. Figure 5 assumes the same parameters as in ?gure 4, except that the mobile velocity is reduced to 5 km/h. Comparing ?gures 4 and 5, it is seen that the performance for the schemes that use a high degree of spreading in the time domain is dramatically improved. For these schemes, orthogonality is now better preserved by the almost ?at channel characteristic in time. It can be veri?ed that if the frequency-selectivity of the channel is reduced compared to the channel assumed in this paper, a similar improvement is obtained for the schemes that use a high degree of spreading in the frequency domain. In near-far scenario II, OFDMA is seen to outperform the direct-sequence schemes in the 50 km/h case. In both the 5 and 50 km/h cases for near-far scenario I, as well as for the 5 km/h case for near-far scenario II, the performance difference between OFDMA and the best-performing direct-sequence scheme is small. In ?gure 6, the BER performance for near-far scenarios II and K = 64 users is plotted vs. the Eb /N0 in the receiver. It seen that the schemes with a non-localized spreading in this case are interference limited, and that increasing Eb /N0 therefore not improves the performance much. D. Finite-depth bit-interleaving case The results provided in this subsection assume SNR post-despreading weighting (see (8)) and and computer simulations alone are used to evaluate the performance. In the simulations, a new pseudo-random interleaver sequence is generated for each 6.6 ms block, meaning that the performance is averaged also over the interleaver sequences. Figures 7 and 8 show the simulated performance for near-far scenarios I and II, respectively. A mobile velocity of 50 km/h, and an Eb /N0 of 10 dB is assumed. The BER for K active users is simulated for the different values of the frequency-domain spreading. As the number of active

December 21, 2003 DRAFT

an interleaver depth of 1024 coded bits (≈ 6.6 ms). In this case, the analysis is not applicable

18

users is varied between 1 and SF a set of curves, each and one similar to those in ?gure 4, is obtained. Stacking these curves next to each other, yields the presented three-dimensional plots. Since the users in the OFDMA scheme are orthogonal within the cell, the performance is not at all effected by the number of active users. Also for the case of a ?nite-depth interleaver, the performance difference between the different choices for M and N is negligible for the OFDMA scheme. In ?gures 7 and 8 the bit-error-rate for the OFDMA scheme is shown by the plane spanned by the dashed lines. For low system loads it is seen that the performance does not vary much between the different schemes. Due to the additional diversity obtained by the spreading, the direct-sequence schemes that use a high degree of either time- or frequency-domain spreading (high values of M or N ), achieve slightly better performance for low system loads. For higher loads, the performance for these schemes is degraded due to interference. Compared to in ?gures 4 and 5, all schemes suffer from the limited diversity that is due to the ?nite depth interleaver. In ?gure 8, the relatively small loss of using OFDMA at low loads is indicated. For higher loads, the orthogonality provided by OFDMA is clearly advantageous. From both ?gures, but most clearly from ?gure 8, it is seen that the performance is step-wise degraded as certain users enter the system. That is, certain interferers degrade the performance more than others even though they all are transmitted with equal powers. By recomputing the example in (12) for k = 0 (the zeroth user is decoded in this section) it is seen that, according to the analysis, user 1, 2, 4, 8, 16, and 32 will create interference. The analysis also predicts that user 32 will create the highest interference, followed by user 16, 8, and so on, in descending order. This agrees well with the simulation results. V. F UTURE WORK From the analysis and from ?gure 8, it is seen that the total intra-cell interference power is dominated by a small number of interferers. A reduced complexity multi-user detector, that decodes only a subset of the active users, could therefore possibly be used to improve the performances of the direct-sequence schemes at high system loads. One should also note that for low system loads, several sub-carriers are unused in the OFDMA scheme. By lowering the rate of the convolutional code, and transmitting the additional redundancy interleaved in time and frequency on the free sub-carriers, both an additional coding gain and additional diversity

December 21, 2003 DRAFT

19

can be obtained while still maintaining user orthogonality. Alternatively, the information bit rate per user can in this case be increased without further expanding the bandwidth. Further work is needed to study this. In the analysis, only the ?rst order terms were included in the parametrization of the channel. Indeed, a more accurate analysis would result if also higher order terms could be included. The results and conclusions in this paper assume perfect channel estimates and that the receiver is perfectly synchronized. Further work is needed to study the performance when these parameters are subject to estimation errors. Further work is also needed to study the performance in a multicell scenario. VI. C ONCLUSIONS This paper presents a uni?ed analysis for the BER performance in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Most parts of the derivations are carried out in [25] and the reader is referred thereto for details. The analysis assumes perfect channel estimates and a perfectly synchronized receiver. The channel correlation in time and frequency, that effects each direct-sequence spread symbol, is taken into account in the analysis of the intra-cell interference. In the soft-decision decoding of the convolutional code, it is assumed that the correlated fading is perfectly broken up by the interleaver. For the schemes that transmit the symbols well localized in time and frequency, the analysis is found to reasonably well agree with the results obtained by computer simulations. For low system loads, the performance difference between the studied schemes is found to be relatively small. It is concluded that a large fraction of the available diversity is captured by the interleaving of coded bits, but that the additional diversity obtainable by the direct-sequence spreading can improve the performance at low system loads. Especially in a near-far scenario, it is shown that the improvement at low loads comes at the cost of a severe degradation due to increased interference at high loads. VII. ACKNOWLEDGMENT The authors would like to thank the reviewers and the editor. Their constructive suggestions and comments have signi?cantly improved the quality of the manuscript.

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20

R EFERENCES

[1] D. Lee and L. B. Milstein, “Comparison of multicarrier DS-CDMA broadcast systems in a multipath fading channel,” IEEE Transactions on Communications, vol. 47, no. 12, pp. 1897–1904, Dec. 1999. [2] S. Hara and R. Prasad, “Design and performance of multicarrier CDMA systems in frequency-selective Rayleigh fading channels,” IEEE Transactions on Vehicular Technology, vol. 48, no. 5, pp. 1584–1595, Sept. 1999. [3] R. Prasad and S. Hara, “An overview of multi-carrier CDMA,” Proc. IEEE International Symposium on Spread Spectrum Techniques and Applications, vol. 1, pp. 107–114, 1996. [4] N. Yee, J.-P. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor wireless radio networks,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Yokohama, Japan, Sept. 1993, pp. 109–113. [5] A. Chouly and A. B. S. Jourdan, “Orthogonal multicarrier techniques applied to direct sequence spread spectrum CDMA systems,” in Proc. IEEE Global Telecommunications Conference, Huston USA, Nov. 1993, pp. 1723–1728. [6] K. Fazel and L. Papke, “On the performance of convolutionally coded CDMA/OFDM for mobile communication systems,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Yokohama, Japan, Sept. 1993, pp. 468–472. ¨ m, “Time-frequency localized CDMA for downlink multi-carrier systems,” in Proc. [7] A. Persson, T. Ottosson, and E. G. Stro IEEE International Symposium on Spread Spectrum Techniques and Applications, vol. 1, Prague, Czech Republic, Sept. 2002, pp. 118–122. [8] ——, “Utilizing the channel correlation for MAI reduction in downlink multi-carrier CDMA systems,” in Proc. Radio Science and Communications Conference, RVK, Stockholm, Sweden, June 2002, pp. 749–753. [9] A. Persson, “On convolutional codes and multi-carrier techniques for CDMA systems,” Department of Signals and Systems, ¨ teborg, Sweden, Tech. Rep. 423L, Thesis for the degree of licentiate of engineering, Chalmers University of Technology, Go Jan. 2002. [10] A. Matsumoto, K. Miyoshi, M. Uesugi, and O. Kato, “A study on time domain spreading for OFCDM,” in Proc. IEEE Wireless Personal Multimedia Communications, vol. 2, Hawaii, USA, Oct. 2002, pp. 725–728. [11] H. Atarashi, N. Maeda, S. Abeta, and M. Sawahashi, “Broadband packet wireless access based on VSF-OFCDM and MC/DS-CDMA,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Sept. 2002, pp. 992–997. [12] A. Sumasu, T. Nihei, K. Kitagawa, M. Uesugi, and O.Kato, “An OFDM-CDMA system using combination of time and frequency domain spreading,” IEICE (in Japanese), Tech. Rep. RCS2000-3, Apr. 2000. [13] T. Kadous, K. Liu, and A. Sayeed, “Optimal time-frequency signaling for rapidly time-varying channels,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, May 2001, pp. 2405–2408. [14] G. Leus, Z. Shengli, and G. Giannakis, “Orthogonal multiple access over time- and frequency-selective channels,” IEEE Transactions on Information Theory, vol. 49, no. 8, pp. 1942–1950, Aug. 2003. [15] R. V. Nee and R. Prasad, OFDM For Wireless Multimedia Communications. IEEE communications letters, vol. 6, pp. 276–278, July 2002. [17] P. Frenger, “Multirate codes and multicarrier modulation for future communication systems,” Ph.D. dissertation, School of ¨ teborg, Sweden, Mar. 1999, technical report Electrical and Computer Engineering, Chalmers University of Technology, G o 357. Artech House, 2000. [16] Q. Shi and M. Latva-aho, “Exact bit error rate calculations for synchronous MC-CDMA over a Rayleigh fading channel,”

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[18] B. Smida, C. L. Despins, and G. Y. Delisle, “MC-CDMA performance evaluation over a multipath fading channel using the characteristic function method,” IEEE Transactions on Communications, vol. 49, pp. 1325–1328, Aug. 2001. [19] E. A. Sourour and M. Nakagawa, “Performance of orthogonal multicarrier CDMA in a multipath fading channel,” IEEE Transactions on Communications, vol. 44, pp. 356–367, Mar. 1996. [20] S. Kaiser, “OFDM code-division multiplexing in fading channels,” IEEE Transactions on Communications, vol. 50, no. 8, pp. 1266–1273, Aug. 2002. [21] J. G. Proakis, Digital Communications, 3rd edition. McGraw-Hill, 1995. Pentech Press, 1992. [22] J. D. Parsons, The Mobile Radio Propagation Channel. 1968. [24] S. Kaiser, “On the performance of different detection techniques for OFDM-CDMA in fading channels,” in Proc. IEEE Global Telecommunications Conference, vol. 3, Nov. 1995, pp. 2059–2063. [25] A. Persson, E. G. Str¨ om, and T. Ottosson, “Derivations of a uni?ed analysis for direct-sequence CDMA in the downlink ¨ teborg, Sweden, Tech. of OFDM systems,” Department of Signals and Systems, Chalmers University of Technology, G o Rep. R011, 2003. [26] N. C. Beaulieu, “Introduction to linear diversity combining techniques,” Proceedings of the IEEE, vol. 91, pp. 328–356, Feb. 2003. [27] M. Alouini, “A uni?ed performance analysis of DS-CDMA systems over generalized frequency-selective fading channels,” in Proc. IEEE International Symposium on Information Theory, Aug. 1998, p. 8. [28] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 4th ed. [29] J. P. Castro, The UMTS Network and Radio Access Technology. New York: Dover, 1970. John Wiley & Sons, 2001.

[23] R. H. Clarke, “A statistical theory of mobile radio reception,” Bell System Technical Journal, vol. 47, pp. 1779–1803,

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FIGURES

22

ck

PSfrag replacements

bk FEC Π sk QPSK DS spread tk

chip mapping

Fig. 1.

Transmitter block-scheme for the kth user.

December 21, 2003

OFDM

tk

xk

DRAFT

FIGURES

23

PSfrag replacements

time 0 1 2 3 ... n

0

sk [0]ck [0]

sk [0]ck [2]

sk [2]ck [0]

sk [2]ck [2]

...

1 frequency

sk [0]ck [1]

sk [0]ck [3]

sk [2]ck [1]

sk [2]ck [3]

...

2

sk [1]ck [0]

sk [1]ck [2]

sk [3]ck [0]

sk [3]ck [2]

...

3

sk [1]ck [1]

sk [1]ck [3]

sk [3]ck [1]

sk [3]ck [3]

...

m

Fig. 2.

Spreading for the kth user. The Mc sub-carriers are indexed by m, and the OFDM symbols by n.

December 21, 2003

DRAFT

PSfrag replacements FEC. sk tk FIGURES dk

?[m, n] exp ?j φ (OFDM)?1 ck

24

Π MOD. yk

inv. chip mapping

DS despread

rk

gk

rk (QPSK)

?1

zk Π

?1

(FEC)

? bk

?1

OFDM

Fig. 3. Receiver block-diagram.

December 21, 2003

DRAFT

FIGURES

25

10

0

TFL-CDMA, near-far I TFL-CDMA, near-far II OFDMA, near-far I&II

10

?2

BER

10

?4

10

PSfrag replacements

?6

10

?8

0

1

2

3

log2 (M )

4

5

6

Fig. 4. Simulation (solid) and analysis (dashed) for near-far scenarios I and II, 50 km/h, E b /N0 = 10 dB, K = 64 active users, perfect synchronization, perfect channel estimates, and perfect interleaving.

December 21, 2003

DRAFT

FIGURES

26

10

0

TFL-CDMA, near-far I TFL-CDMA, near-far II OFDMA, near-far I&II

10

?2

BER

10

?4

10

PSfrag replacements

?6

10

?8

0

1

2

3

log2 (M )

4

5

6

Fig. 5. Simulation (solid) and analysis (dashed) for near-far scenarios I and II, 5 km/h, E b /N0 = 10 dB, K = 64 active users, perfect synchronization, perfect channel estimates, and perfect interleaving.

December 21, 2003

DRAFT

FIGURES

27

10

0

10

?2

BER

10

?4

10

?6

PSfrag replacements

10

?8

M=1 M=2 M=4 M=8 M = 16 M = 32 M = 64 OFDMA 6 8 10

Eb /N0

4

12

14

16

18

Fig. 6. Simulation (solid) and analysis (dashed) for near-far scenarios II, 50 km/h, K = 64 active users, perfect synchronization, perfect channel estimates, and perfect interleaving.

December 21, 2003

DRAFT

FIGURES

28

10 10

BER

0

?1

10 10 10

?2

?3

?4

PSfrag replacements

6

4

2

log2 (M )

0

20

40

60

# active users

Fig. 7. Simulation of BER for the zeroth user vs. system load and frequency-domain spreading, for 50 km/h, near-far scenario I, Eb /N0 = 10 dB, a 6.6 ms interleaver, perfect channel estimates, and perfect synchronization. The plane spanned by the dashed lines shows the performance of OFDMA.

December 21, 2003

DRAFT

FIGURES

29

10

0

10

?1

BER

10

?2

32 users 16 users

PSfrag replacements

10

?3

8 users

10

?4

6

4

2

log2 (M )

0

20

40

60

# active users

The maximum loss for OFDMA appears in the single-user case.

Fig. 8. Simulation of BER for the zeroth user vs. system load and frequency-domain spreading, for 50 km/h, near-far scenario II, Eb /N0 = 10 dB, a 6.6 ms interleaver, perfect channel estimates, and perfect synchronization. The plane spanned by the dashed lines shows the performance of OFDMA.

December 21, 2003

DRAFT

TABLES

30

TABLE I AVERAGE - POWER DELAY PROFILE Excess tap delay [?s] Average tap power [dB] τ0 0 ζ0 0 τ1 0.4 ζ1 ?1 τ2 0.8 ζ2 ?9 τ3 1.2 ζ3 ?10 τ4 1.8 ζ4 ?15 τ5 2.6 ζ5 ?20

December 21, 2003

DRAFT

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