Link-Level Modeling and Performance of CDMA ... - CiteSeerX

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Link-Level Modeling and Performance of CDMA Interference Cancellation Jilei Hou, John E. Smee, Joseph B. Soriaga, Jinghu Chen and Henry D. Pfister Qualcomm Incorporated, 5775 Morehouse Dr., San Diego, CA 92121, USA {jhou,jsmee,jsoriaga,jinghuc}@qualcomm.com Abstract— A general framework is provided to characterize the link level performance of CDMA systems with interference cancellation. This closed-form residual power analysis accounts for the impact of channel estimation errors due to SNR, channel variation, chip asynchronism, and filter mismatch. Simulations further quantify the link level cancellation performance on more realistic sub-chip multipath channels. This work demonstrates that properly designed channel estimation and signal reconstruction techniques achieve high cancellation efficiency over a variety of multipath fading channels. Index Terms— CDMA, cancellation efficiency, channel estimation, interference cancellation, sub-chip multipath channel

study the IC performance on more realistic sub-chip multipath channels, we use a simulation approach which closely approximates the performance of a practical receiver. Simulations compare the performance of IC schemes with the conventional parallel channel estimation (PCE) and with more advanced iterative channel estimation (ICE) of the fingers of each user. It is noted that the system parameters are given based on the CDMA 1xEV-DO RL [5]. However, the proposed analytical and simulation approaches can be easily applied to other CDMA systems on either the RL or the FL.

I. INTRODUCTION

II. CHIP ASYNCHRONOUS MULTIPATH CHANNEL

T is well known that successive interference cancellation (SIC) can achieve the capacity of multiple access channels when the receiver has perfect channel state information [1]. SIC is also of great practical interest since it requires only conventional single-user front-ends and can be readily applied to long-code based commercial CDMA systems without modification to the terminals or the standards [2]. In addition, it is important to note that unlike linear multiuser detectors, error correction coding (ECC) is incorporated into the SIC procedure which is essential to achieve high performance gain when the per-user signal-to-noise-plus-interference ratio (SINR) is low [3]. One practical method of implementing interference cancellation (IC) in a CDMA receiver is to reconstruct the contribution of various transmitted chipx1 streams to the received samples. This involves the estimation of the overall channel between the transmitter chips and the receiver samples. In reality, each user’s channel is time-varying and reliable channel estimation can be a significant challenge. In this paper, we compute the residual power remaining after subtractive cancellation. This quantity provides a measurement of the link-level expected performance gains from IC. Similar subjects have been investigated for the WCDMA forward link (FL) [4]. However, our work is more tailored to the CDMA reverse link (RL) where users typically operate at low SINR regions and it is important to introduce a cancellation scaling factor to minimize the residual power. We provide a more general analytical framework which results in closed-form solutions for the residual power with various realistic constraints such as channel variation, channel estimation errors, filter mismatch, chip asynchronism. To

Consider the continuous time signal transmitted by one user, where the chip sequence is x[n] with unit energy and the transmit pulse shaping filter is p(t )

I

s (t ) = ∑ x[n ] p (t − nTc ) .

Eq. 1

n

Consider the baseband equivalent L -path channel model L

h(t ) = ∑ g l (t )δ (t − τ l ) ,

Eq. 2

l =1

where g l (t ) is the complex path gain for the l -th path at corresponding delay

τ l . With a chip period of

Tc , it is

common to model the multipath channel at a sub-chip resolution of Tc 8 , referred to as chipx8. Therefore, with d l being an integer, we can rewrite the channel delays as τ l = d l ⋅ (Tc / 8) and the overall channel as L T ⎞ ⎛ h(t ) = ∑ g l (t )δ ⎜ t − d l ⋅ c ⎟ . 8⎠ ⎝ l =1

Eq. 3

With the receiver filter given by the low pass filter q(t ) , the composite transmit and receive filter can be written as Eq. 4 φ (t ) = p (t ) ⊗ q (t ) . Without loss of generality, we may take φ (0 ) = 1 . To be consistent with the discrete time sampled channel model, we can write the chipx8 composite filter as φ [m] = φ (mTc / 8) . Summing the contribution from all L paths, the chipx8 baseband samples stored in the receiver can then be written as L ⎛ mT ⎞ r [m] = r ⎜ c ⎟ = ∑ x[n]∑ gl [m]⋅ φ [m − dl − 8n] + z[m] Eq. 5 ⎝ 8 ⎠ n l =1 where z[m ] represents other-user interference and additive thermal noise. In this work, z[m ] is modeled as additive white

2 Gaussian noise (AWGN) and this assumption is validated in the multiuser simulation discussed later. Isolating the contribution of the l -th path signal gives received samples of Eq. 6 r [m] = g l [m] x[n]φ [m − d l − 8n] + zl [m]

∑ n

where zl [m] includes the intra-user multipath interference, inter-user interference, and thermal noise.

the cancellation efficiency as β = 1 − Pr (α ) . As shown in Figure 1, for static channels, the longer the filter length N, the better quality the channel estimates. The gain from increasing N is more pronounced at low Ec / N t where noise is the dominating factor. At high Ec / N t , the benefits from longer finger length diminish.

III. SINGLE PATH ANALYSIS We first consider the IC performance on a path-by-path basis. This single-path approach allows analysis with closed-form solutions which provide insights into the impact on IC performance from various factors. A. Static Channel We start by assuming the channel is fixed over N chips and ignoring the inter-chip interference (e.g., Nyquist pulse is used). If we have the knowledge of the path delay of the l -th path signal, the receiver puts one finger at this delay and the decimated chip-rate received samples can be simplified as, Eq. 7 r [k ] = g l x[k ] + zl [k ] for 0 ≤ k < N . Without loss of generality, we can also assume that the magnitude of g l is one, the magnitude of x[k ] is one, and the average power of zl [k ] is σ 2 . Subtracting the contribution of x[k ] from the received samples requires estimating the complex channel coefficient g l . Note that in the IC procedure, the receiver knows x[k ]

because x[k ] is either a pilot sequence or a successfully decoded data sequence [2]. Under the given assumptions, the minimum variance unbiased channel estimate is given by the simple correlation based method 1 N −1 Eq. 8 gˆ l = ∑ r [k ]x * [k ] N k =0 which has mean g l and variance σ 2 / N . Essentially, this channel estimator is a length-N FIR filter with equal weights. Due to the non-zero variance of the channel estimate, a scaling factor (denoted by α ) should be applied to the cancellation. We can find the optimal cancellation factor (denoted by α ) that minimizes the residual un-cancelled power. The residual power as a function of α is given by ⎡ 1 N −1 2⎤ Pr (α ) = E ⎢ ∑ g l x[k ] − αgˆ l x[k ] ⎥ N ⎣ k =0 ⎦ Eq. 9 2 2 * 2 ˆ = E g l − 2α Re g l g l + α gˆ l

Figure 1 Cancellation factor and residual power B. Time-Varying Channel For a time-varying channel, Eq. 7 becomes r [k ] = g l [k ]x[k ] + z l [k ] .

We assume that g l [k ] is stationary and that its auto-

correlation Rg [τ ] satisfies R g [0] = 1 . We can use the same channel estimation technique as the static channel based on Eq. 8. Now, we attempt to cancel the contribution of the signal x[ k ] in the interval 0 < N1 ≤ k < N 2 < N . In this case, the channel estimate essentially becomes a non-causal sliding window moving average FIR filter with equal weights. It can be shown that the residual power is given by 2 N 2 −1 ⎡ 1 ⎤ 1 N −1 * [ ] [ ] [ ] − α Pr (α ) = E ⎢ g k r i x i ⎥ ∑ ∑ l N i =0 ⎢⎣ N 2 − N1 k = N1 ⎥⎦ Eq. 11

⎛σ 2 ⎞ = 1 − 2αC 2 + α 2 ⎜⎜ + C1 ⎟⎟ , ⎝ N ⎠

*

[

[

]

]

⎛ σ ⎞ ⎟ = 1 − 2α + α ⎜⎜1 + N ⎟⎠ ⎝ We can solve for α * by setting the derivative of Pr (α ) 2

2

equal to zero. We can then show that α * = γ N /(1 + γ N ) and 2 Pr (α * ) = 1/(1 + γ N ) , where γ = 1 / σ = Ec / N t is the chip SINR of x[k ] of the considered path. Alternatively, we define

Eq. 10

where

C1 = C2 =

1 N2

N −1

∑ (N − m )R [m] ,

m = − ( N −1)

g

N 2 −1N −1 1 ∑∑ Re Rg [k − i] . N ( N 2 − N 1 ) k = N1 i = 0

[

]

We can verify that this formulation is a generalization of the static channel case where R g [k ] = 1 for all k. In this case, the constants

C1 and C2 are both unity and the expression

for Pr (α ) is identical. Similar derivations conclude that

α * = C2 /(C1 + σ 2 / N ) and Pr (α * ) = 1 − α *C2 . One popular time-varying wireless channel model is the Jakes model. In this case, the correlation function of the g l [k ]

3 sequence is given by R g [n] = J 0 (2πf d Tc n ) , where f d is the maximum Doppler frequency and Tc is the chip interval, and

J 0 is the Bessel function of the first kind and order zero.

D. Transmit-Receive Filter Mismatch Note that the analysis so far assumes that the IC procedure has perfect knowledge of the composite transmit and receive filter φ (t ) = p (t ) ⊗ q (t ) . If IC uses φ ' (t ) , for the static channel, the residual power from the filter mismatch is Pr (α l ; μ ) = E

g x[n]φ [m − d l − 8n] {l ∑ n a 1 44424443

2

b

− α l gˆ l ∑ x[n]φ ' [m − d l − 8n] { n c 144424443 d

Eq.13

2

= E ∑ x[n]φ [m − d l − 8n] ⋅ E g l − α l gˆ l

2

n

2

+ E α l gˆ l

2

(

)

⋅ E ∑ x[n] φ [m − d l − 8n] − φ [m − d l − 8n] 144444 42444444 3 n '

Δ [m − d l − 8 n ]

Figure 2 IC performance at v=120km/h

Note that the optimal α requires knowledge of Doppler at the receiver. Since we do not assume the receiver knows the Doppler, we compute α as if the channel were static, i.e., α = 1 /(1 + σ 2 / N ) . The residual power is computed accordingly based on Eq. 11. Figure 2 plots the residual power as functions of filter length N, chip SNR Ec / N t at vehicle speed v=120km/h with system parameters chosen to model the EV-DO RL where f c =1.9GHz, Tc =1/(1.2288MHz). Here we choose N1 and N2 to satisfy N2-N1=128 and the interval N1 ≤ k < N 2 sits in the center of the sliding window. The IC performance curves for low to moderate speeds are very close to those of static channels and therefore not shown. Moreover, across typical vehicle speeds, N=2048 achieves a very good tradeoff between noise/interference suppression and channel tracking. C. Chip Synchronous vs. Asynchronous In general, the intra-user and inter-user multi-path delays may be chip asynchronous. It is evident that the residual analysis above applies to the scenario where the multipath delays are chip synchronous. Nevertheless, it can be shown that it applies to chip asynchronous scenario as well. For example, for the static channel, if we define μ as the sub-chip chipx8 offset between any two multipaths, it can be verified that as a function of μ , the residual power becomes ⎛ ⎞ Pr (α ; μ ) = ⎜⎜ ∑ φ 2 [8q + μ ]⎟⎟ ⋅ E g l − αgˆ l ⎝ q ⎠

2

Eq. 12

For typical low-pass transmit and receive filters, it is reasonable to assume that

∑ φ [8q + μ ] ≈ 1 2

for μ =0,1,…,7.

q

Then, the residual analysis degenerates to the chip synchronous case.

⎧ (g l − α l gˆ l )∑ x[n]φ [m − d l − 8n ] ⎪ n + 2 Re ⎨ ˆ , ' α g x n [ ]Δ[m − d l − 8n'] l l∑ ⎪ n' ⎩

[

]

⎫ ⎪ ⎬. ⎪ ⎭

where x, y = E xy * denotes the inner product. Note that expectations related to x[ n] can be separated from those of

g l and gˆ l because it is assumed the chip sequence is independent of the channel coefficients. Furthermore, for an independently distributed chip sequence x[n] we can simplify the norm of the pulse mismatch term to be 2

E ∑ x[n]Δ[m − d l − 8n] = ∑ (Δ[m − d l − 8n]) n

2

Eq. 14

n

Although the general shape of the pulse mismatch may be difficult to model, it is reasonable to assume that the IC procedure can guarantee that the energy in the above mismatch ∑ n (Δ[m − d l − 8n ])2 is below certain threshold ε . Following the same derivation as the previous section, 2

E ∑ x[n]φ [m − d l − 8n] = ∑ φ 2 [8q + μ ] ≈ 1 n

Eq. 15

q

As to the cross term, we have (g l − α l gˆ l )∑ x[n]φ [m − d l − 8n], n

α l gˆ l ∑ x[n']Δ[m − d l − 8n']

Eq. 16

n'

=∑ n

φ [m − d l − 8n]Δ[m − d l − 8n ]

.

⋅ α l* (g l − α l gˆ l ), gˆ l

One important observation is that if

αl

is chosen to

minimize the mean squared error (MSE) E g l − α l gˆ l 2 , it is known

that

(g l − α l gˆ l ), gˆ l

=0

follows

orthogonality principle. Therefore, if we choose

from

αl

the

properly,

the cross term is zero and the residual power then becomes 2 2 Eq. 17 Pr (α l ; μ ) = E g l − α l gˆ l + ε ⋅ E α l gˆ l .

4 This conveniently allows us to evaluate the residual power by separately considering the pulse mismatch and the channel estimation error. For the static channel, we have ⎛ ⎛ ⎛ N ⎞ 1 ⎞⎞ Eq. 18 ⎟ ⎟, E g l − α l gˆ l 2 = ⎜1 − 2α l + α l2 ⎜1 + ⎜⎜ t ⎟⎟ ⎜ ⎟⎟ ⎜ E N c ⎝ ⎠ l ⎝ ⎠⎠ ⎝ ⎛ ⎛N ⎞ 1 ⎞ Eq. 19 ⎟. and E α l gˆ l 2 = α l2 ⎜1 + ⎜⎜ t ⎟⎟ ⎟ ⎜ ⎝ ⎝ Ec ⎠ l N ⎠ The effects of the filter mismatch are illustrated in Figure 3 for static channel with different N. Here the mismatch energy ε is assumed to be 3% which is reasonable given a practical low-pass filter design since it represents close to 20% mismatch in amplitude.

Figure 3 Residual powers with filter mismatch

Similarly, for the time varying channel, if the

αl

is chosen

to minimize the MSE, the residual power is equal to ⎛ ⎛N 2 Pr (α l ) = 1 − 2α l C 2 + (1 + ε )α l ⎜ C1 + ⎜⎜ t ⎜ ⎝ Ec ⎝

⎞ 1⎞ ⎟. ⎟⎟ ⎟ ⎠l N ⎠

Eq. 20

Table 1 Power/Delay Profiles of ITU Channel Models

Five multipath channels from the 3GPP2 evaluation methodology [6] are considered: A (ITU Ped A, 3km/h), B (ITU Ped B, 10km/h), C (ITU Veh A, 30km/h), D (ITU Ped A, 120km/h), and E (1-path Rician with 1.5Hz Doppler). The underlying ITU channels are summarized in Table 1. A. Parallel Channel Estimation (PCE) Typically, the receiver performs IC after the pilot demodulation where the estimated multipath delays are available from the time-tracking loop (TTL). In this simulation, we assume IC places fingers at the offsets from the TTL computation and they are at least one chip apart to minimize the finger correlation. Ped A channel essentially is a single-path channel and the receiver assign one finger at chipx8 offset [0]. Both Ped B and Veh A channels are sub-chip multipath channels and some multipaths are unresolvable at the receiver due to φ (t ) . For these 2 channels, we assume the receiver assigns 3 fingers at chipx8 offsets [0 8 23] and [1 9 17], respectively. The finger offsets are derived based on the assumption that the demodulator usually tracks the multipath long-term peaks. This approach implicitly covers the timing errors since for sub-chip multipath channels, long-term peaks often do not align with the short-term peaks. We first consider IC using PCE. In this method, the channel estimation of one finger does not benefit from the IC of any other fingers of the user and the reconstructed interference of all fingers of the user can be computed in parallel. In the simulation, the system parameters are the same as the ones used for Figure 2 with N=2048. To model the filter mismatch, we assume φ (t ) is the CDMA2000 pulse and

φ ' (t ) is the well-known sinc pulse which is distinctly different from the CDMA2000 pulse. In Figure 4, the PCE IC performance is shown for each of the 5 channels. Here, the channel SNR G is defined as L

ITU Model Pedestrian A Pedestrian B

Chipx8 Delay Profile [0 1 2 4] [0 2 8 12 23 37]

Vehicle

[0 3 7 11 17 25]

A

Power Profile (dB) [-0.5 -10.2 -19.7 -23.3] [-3.9 -4.8 -8.8 -11.9 -11.7 -27.8] [-3.1 -4.1 -12.1 -13.1 -18.1 -23.1]

IV. MULTIPATH CHANNEL SIMULATION In general, the above analysis applies to the multipath channels so long as we assume that the multipaths observed by the receiver are independent. However, the analysis becomes intractable in a sub-chip multipath channels where the multipaths may become unresolvable due to φ (t ) . Therefore, we use simulation to study the IC performance in these channels. The simulation also captures the performance loss due to filter mismatch, channel estimation error, cancellation factor estimation error, and channel variations.

G=

∑ ∑ g l ⋅ φ [m − d l ] m

l =1

E z[m]

2

Eq. 21

2

where g l is the channel coefficient corresponding to the center of the samples over which the channel estimation is based. The cancellation efficiency β is now defined as a peruser quantity rather than a per-path quantity. B. Iterative Channel Estimation (ICE) There are certain constraints associated with PCE IC especially in the sub-chip multipath channels, such as Ped B and Veh A channels. For example, for the IC procedure, since the data symbols are known we have the freedom to put the fingers at locations within one chip. Moreover, if two multipaths are within several chips, the energy of the side lobe of one multipath leaks into the main lobe of the other one where both carry the same data symbol. This multipath correlation phenomenon causes a consistent bias for the estimated channel values if PCE IC is used.

5 To solve the multipath correlation problem and remove the channel estimation bias, we propose IC using iterative channel estimation (ICE). In the first iteration, we perform successive IC from the strongest to the weakest fingers where the ordering is based on estimated per-finger chip SINR. In the following iterations, iterative IC is performed, i.e., for each finger, we remove the other fingers’ interference contribution based on their latest channel estimates, then re-estimate the channel for this finger and reconstruct its interference contribution. This approach also improves the weaker fingers noise/interference suppression since their effective chip SINR’s are higher.

C. Multiuser Packet-Error-Rate (PER) Simulation In this simulation, we have 1 target user and 8 interference users. We assume that the 8 interference users always decode successfully and are therefore removed prior to the decoding of the target user. For the space limits, we do not elaborate on the simulation details. The end results mainly demonstrate that: in terms of cancellation efficiency β , the values obtained for each interference user in this multiuser simulation is virtually the same as the ones obtained in the single-user simulation setup discussed above. This confirms the accuracy of the assumption that z[m ] can be modeled as AWGN. Furthermore, we found that the predicted SNR gains for the target user based on β calculations are consistent with the ones we observed from the PER curves with and without IC. This means that the same link-to-system mapping used for non-IC receivers can be used for system simulation with IC receivers as well. Therefore, using the cancellation efficiency is an accurate and convenient way to evaluate the throughput gains due to IC in the system simulation, as further elaborated in [8]. V. CONCLUSION

Figure 4 IC performance over multipath channels

However, this approach will not be effective if the finger delay estimates are not close to optimal since the TTL estimates may only reflect the long-term average but do not capture the short-term peaks; or sometimes, the TTL may lock on to the side lobes of a very strong finger. Therefore, within each iteration of IC, we also perform iterative TTL updates which allows the finger to move locally ( ± 2 chipx8 offsets) to find the peak through a simple correlation method. This iterative TTL could better fine-tune finger delay estimates; and potentially put more than 1 finger within one chip to handle sub-chip multipaths. In the simulation, we set the finger offsets of the first iteration the same as the PCE IC scheme. The maximum number of iterations is 4. In addition, ICE IC degenerates to 1iteration successive IC if the maximum per-finger estimated chip SNR is less than -18dB. In Figure 4, PCE is compared with ICE. It is observed that ICE gives more visible performance gain for Channels B and C for the reasons discussed above. For Channels A, D, and E, PCE has performance very close to ICE where the gain is from iterative TTL updates which provide better short-term peak delay estimates. Receiver design tradeoffs comparing PCE vs. ICE based on network simulation results are presented in [7]. Note that for typical DO RL systems, due to closed-loop power control [5], the average channel SNR for pilot symbols is roughly -24~-23dB; the average channel SNRs for data symbols could be in the range of (-20,-5) dB depending on the users’ data rates. From Figure 4, it is observed that high cancellation efficiency can be achieved for the EV-DO RL.

In this paper, we provided closed-form analysis of the cancellation efficiency which takes into account various realistic constraints. This analysis can be applied to either single path channels or multipath channels on a path-by-path basis. To consider more realistic performance over sub-chip multipath channels, we resorted to simulation approach where we compared the performance of PCE vs. ICE IC schemes. Insights on when iterative estimation is more effective are elaborated. Finally, via multiuser PER simulation, we verified that the cancellation efficiency accurately predicts the SNR gains observed comparing the non-IC vs. IC PER curves. Overall, we demonstrated that at the link-level, high cancellation efficiency can be achieved on the CDMA RL over a wide range of channels. REFERENCES [1] [2]

[3]

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[8]

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