Wireless Communication Technologies 16:332:559 (Advanced Topics in Communications) Lecture #17 and #18 (April 1, April 3, 2002)

Transcription

1 Wireless Communication echnologies Lecture 7 & 8 Wireless Communication echnologies 6:33:559 (Advanced opics in Communications) Lecture #7 and #8 (April, April 3, ) Instructor rof. arayan Mandayam Summarized by Sandeepa Muheree (smdatta@ece.rutgers.edu) his lecture note is about multiuser CDMA systems and we study in detail about the performance of such systems. Sandeepa Muheree age of 8

2 Wireless Communication echnologies Lecture 7 & 8.Multiuser BS CDMA System Let s assume that we have a CDMA system with users indexed as, and each user transmits BS modulated signal waveforms simultaneously. hen the transmitted signal of one such user can be represented as, where, ( ) n t c c p ( t n ) c n ( n) c () ( n) c {, + } n ( ) t b b p ( t n ) ( n) b n ( n) {, + } c rocessing Gain he received signal at the receiver can be represented as, where, τ relative time offset φ phase offset o η(t) (, ) In general signal from the various users are received at various propagation delays, τ [, ] and φ [,π ] eceiver in the multiuser scenario can be modeled as figure below: () Sandeepa Muheree age of 8

3 Wireless Communication echnologies Lecture 7 & 8 S (t) τ S (t) τ Σ x x S (t) τ η(t) Figure : eceived signal in multiuser scenario.. Single User eception Focusing on user, we try to loo at the demodulation of bit stream of user. Assume the receiver is synchronized to user. So, without loss of generality we can say, τ and φ. We will consider demodulation of a single bit of user, which is bit b. he matched filter receiver matched to user is illustrated in the following figure, r(t) Z (t) C cosω c t he output of the matched filter corresponding to user is given as, (3) where, z c ( t) c ( t ) b ( t τ )cos( ω ct + φ )cos( ω ct) τ dt + η Sandeepa Muheree age 3 of 8

4 Wireless Communication echnologies Lecture 7 & 8 η η( t) c( t)cos( ω ct) eglecting all double frequency terms, dt ( ) [ b ˆ, ( τ ) + b, ( τ )] φ + η z b + cos (4) We introduce continuous time partial cross-correlation functions, ˆ τ ( τ ) c ( t) c ( t τ dt, ), ( τ ) c ( t) c ( t τ ) τ dt (5) τ hus, in one bit interval of the desired user i.e. user, there would be effectively two bits of every interferer, by assuming that the signals of the interferers are delayed by not more than one bit interval. his is illustrated in the following figure: b (t) b b (t) b (-) b Sandeepa Muheree age 4 of 8

5 Wireless Communication echnologies Lecture 7 & 8 Case of Synchronous users For the case of synchronous system, τ and φ for all. So there is only one term from each interferer i.e. one bit of interferes with one bit of the desired user. hen from eq. (4) we have, z b + b, + η (6) where, c Spreading code vector of length for user c Spreading code vector of length for user η, 4 (a) When code sequence are Orthogonal c c δ herefore the matched filter output is, z b +η (7) he probability of error is given as, b, Q E b, hus performance is same as single user channel (b) When users use random codes he error probabilities are calculated in two steps, noting that c,,. are random binary vectors. Sandeepa Muheree age 5 of 8

6 Wireless Communication echnologies Lecture 7 & 8 Step : Fix the codes and bits of other users. hen calculate the conditional probability of error for synchronous users with fixed data bits and conditioned on, and b. his conditional robability of error is represented as,, b ) (8) b, ({ }{ }, Step : Find the average robability of error b,, by averaging over {, } and {b }. For fixed {, } and {b }, b, ({ }{, b } ) r{ z < b }, + b Q 4 E b, E Q + he average probability of error is, b, b,, (9) b, { }{ } E b, Eb,, E Q + b b, his average probability of error is difficult to calculate...3 Using Gaussian approximation for the Interference Assume for all and the no. of user is very large. he interference term for user is given as, I b, We approximate I to be Gaussian. We proceed to find the mean and variance of I as follows, E [ ] I E b, E[ b ] E[, ] b and, is independent. Also, Sandeepa Muheree age 6 of 8

7 Wireless Communication echnologies Lecture 7 & 8 E [ ] b herefore E [I ] Var [ I] E ( b ) [ ] Var[,] () [ b ] ow, E ( ), c c Var [ ] E ( c c ) [ ], herefore Var [ I ] ( ) he output of the matched filter z can be written as, z b + I herefore For large b, +η Q Q 4 + ( ) ( ) Q ()..4 Case of Asynchronous Users For asynchronous user case, we have non- zero τ and φ. Each bit of the desired user is affected by two bits from the interferer. herefore at the receiver each interferer has two terms. ecall that the matched filter output is given as: Z b + I,( b, τ, φ ) + η where I, is the interference of user to user, and given by Sandeepa Muheree age 7 of 8

8 Wireless Communication echnologies Lecture 7 & 8 I b [ ] cos( φ ) ( ) ( b, τ, φ ) b ( τ ) b ˆ,, ( τ ) ( ) ( b b ), +, Assume that er controlled. he matched filter output can then be written as, Z [ b + I( b, τ, φ )] + η (3) where the resultant interference I ( b, τ, φ ) I, ( b, τ, φ ) ( b b ), τ ( τ τ,... ), φ ( φ φ,... ) b,..., 3 b, 3 τ, 3 hase and ime as assumed to be acquired for user, i.e. the x is synchronized in time and phase w.r.t user. For fixed b, τ, φ the conditional robability of Error when - is transmitted is, { Z > } b, r b r η + ( + I( b, τ, φ ) > Eb Q φ ( I ( b,τ, ) Average probability of error is, b E [ ], b, τ, φ b, Suppose b +, the error probability is given as, Z < b { } b, r + Eb Q φ b E [ ], b, τ, φ b, ( + I ( b,τ, ) φ (4a) (4b) As seen from eq.(4a) & (4b) the error probabilities are not symmetric as the interference is not symmetric. It is not possible to get a closed form for the average probability of error. It is very difficult to compute exact error probability however good bounds and approximations exist. ( Error probability for DS/SS multiple access comm. art-i Upper and lover bound IEEE rancom, May 98 ). Sandeepa Muheree age 8 of 8

9 Wireless Communication echnologies Lecture 7 & 8..5 Error robability for Gaussian Approximation Model interference I ( ) b,τ,φ, as a Gaussian random variable. he other assumptions are: -b m and b n are independent for all i & m n -τ i, τ are independent for all i & uniform over [, ] -φ i, φ are independent for all,π i & uniform over [ ] We now proceed to find the mean and variance of I ( b,τ,φ ). E [ I (,, φ )] τ for all (because bits are independent w.r.t codes), b, [ I, ( b, τ φ )] [ cos φ ] E ( τ ) + ˆ ( τ ) σ Var, E τ, 3 [ ] [, ( τ ) + ˆ, ( τ )] dτ [ ( ) ˆ ( τ )],, τ +, dτ (5) From eq.(3) we have, Z b + I b, herefore for b - we have, E (, τ φ ) + η [ Z b ] + E[ I ( b, τ, φ )] + E[ η ] + E Var [ Z b ] Var I( b, τ, φ) + Var η σ [ Z b ] [ ] [ ], + 4 herefore the average probability of error is given as, Sandeepa Muheree age 9 of 8

10 Wireless Communication echnologies Lecture 7 & 8 Sandeepa Muheree age of 8 + b Q,, 4 σ + b b E E Q, σ (6) As seen, the performance depends critically on σ, which in turn depend on cross correlation properties of chip waveform or code sequence. erformance can be made better if better code sequences are designed with good cross correlation properties σ,. M-sequence and gold sequence are two types of code with fairly desirable cross correlation properties (refer to Error probability for DS/SS multiple access communications, Geranitis and ursley, IEEE rancom, May 98 )

11 Wireless Communication echnologies Lecture 7 & 8. robability of Error for asynchronous users his section contains mathematical analysis, which provides expressions for determining the average bit error rate for users in a single channel, CDMA system. (refer to the paper "erformance evaluation for hase Coded Spread Spectrum Communication - art II, Code Sequence analysis" by ursley, Sarvate & Star, IEEE COM Aug 977)..Gaussian Approximations As seen from eq. (3), the contribution from th interfering user, to user is, b I, b ( t τ ) c ( t τ ) c( t)cos( ω ct + φ ) cosω ctdt (7) It is useful to simplify this equation in order to examine the effects of multiple access interference on the average bit error probability for a single user. he relationship between b t τ ), c t τ ) and c ( ) is illustrated in Fig. ( ( t Integration eriod b c C, i C, i+ C, i+ C, i+- τ γ c C, -γ- C, -γ... C,... b, - b, Figure : iming of the local sequence for user and user he quantities γ and in Fig. are defined from the delay of user relative to user, τ, such that, τ γ c + < < c It is possible to show that, the contribution from th interfering user to user is given as, Sandeepa Muheree age of 8

12 Wireless Communication echnologies Lecture 7 & 8 I, c cos( ϕ ) X + ( ) Y + ( ) U + ( ) V (8) c c c Where X, Y, U, V are random variables having distribution conditioned on A & B which are given as, p X A l + A A l -A, -A+,, A-,A B B py ( l) l + B l-b, -B+,, B-,B U ( l) l -, + p V ( l) l -, + p A and B are defined as follows: A umber of integers in [, - ] for which (c, l+i )(c, l+i+ ) i.e. A, measures for a given code the no. of successive no transitions from + to - in the code c B umber of integers in [, - ] for which (c, l+i )(c, l+i+ ) - i.e. B, measures for a given code the no. of successive transitions from + to - in the code c. ote: A + B - since sets A and B are disoint and span the set of total possible signature sequences of length in which there are a total of - possible chip level transitions he details of these derivations are in the following papers: [] Lehnart & ursley :" Error robability for binary DS-SS communication with andom Signature Sequence" IEEE COM Jan 987 [] Morrow & Lehnart : "Bit to bit error dependence in slotted DS/SSMa pacet systems with andom Signature Sequence" IEEE COM Oct 989 For random signature sequence statistics of A and B are nown. We have seen the variance of the th interference term is given as, σ, Var [ I, (b, τ, φ ) ] Sandeepa Muheree age of 8

13 Wireless Communication echnologies Lecture 7 & 8 [ ] ( ( ) ) E cos E b ( τ ) + b ˆ ( τ ϕ,, (9) ) For random signature sequences it is possible to model, ( ) ( b,( τ ) + b ˆ,( τ )) l X l where{ X } l l are i.i.d random variables with distribution given as, X l Uniform Bernoulli,, with, probability/ with probability/ ow, E [ X l ] [ X ] Var l herefore, [ X ] E l 3 [ ] E cos φ herefore the variance of the th interferer as given in eq.(9) can be written as, σ, 3 3 herefore the Average robability of Error is, Sandeepa Muheree age 3 of 8

14 Wireless Communication echnologies Lecture 7 & 8 b, Q E o b Eb + o σ, Q E o b Eb + o 3 () For large, the average probability of error for asynchronous users is, 3 b, Q (a) For large, the average probability of error for synchronous users is, b, Q (b) hus we see robability of Error is less for asynchronous users than for synchronous users which is due to the better averaging effect due to offset of bits. he expressions in this section are valid only if the number of users is large. Furthermore, depending on the distribution of the power levels for the - interfering users, even when is large, if the interferer power levels are not equal or constant the Central Limit heorem does not hold and the multiple access interference contribution I, cannot be modeled as a Gaussian andom Variable. herefore Gaussian approximation is not appropriate in the following cases () umber of users is not large () Interferers have disparate power levels. In these situations a more in-depth analysis is required which we shall study in the next section... Improved Gaussian Approximation (IGA) his analysis defines the interference terms I, conditioned on the particular operating condition of each user. When this is done, each I becomes Gaussian for large. Define ψ as the variance of the multiple access interference for a specific operating condition i.e. ψ is the conditional variance of I. Sandeepa Muheree age 4 of 8

15 Wireless Communication echnologies Lecture 7 & 8 ψ Var [ I ( b τ, φ, c,..., c, p) φ,, p, c,..., ] Var I (.), τ c [ φ,{ },{ p B] }, Here p ower of th user Offset of th user If the distribution of ψ is nown the bit error rate may be found by averaging over all possible values of ψ. b, E Q b ψ b Q f (.) ψ ψ d ψ It is possible to show that, (3) ψ ( ) + + c B where, cos φ Z (4) c c c Z U V U + cos(φ ) V ( ) + B + c c Closed form expression for f u (.) and f v B (.) can be found so the distribution of Z can be determined. DF of ψ can be obtained by - fold convolution of Z, f ψ (.) f z f z his equation may be used in eq. (3) to determine the average bit error probability. his technique has been shown (refer to paper by Morrow and Lehnert - COM Oct 989) to be accurate for a very small number of interfering users for the case of perfect power control. his results in an Improved Gaussian Approximation. Sandeepa Muheree age 5 of 8

16 Wireless Communication echnologies Lecture 7 & 8..3 Simple Improved Gaussian Approximation (SIGA) he expressions presented in the previous section are complicated and require significant computational time to evaluate. A simplified expression for IGA, is given by Holtzman in the paper "A simple accurate method to calculate SSMA error probabilities", IEEE, COM Mar 99. he simplified bit error probability expressions are based on the fact that a continuous function f(x) may be expressed as, f (x) f ( µ ) + ( x µ ) f ( µ ) + ( x µ ) f ( µ ) + If x is a random variable and µ is the mean of x i.e. E[x] µ herefore, (5) E[ f ( x) ] E[ f ( µ )] + + σ f ( µ ) + f ( µ ) + σ f ( µ ) If derivatives are expressed in terms of finite differences, ( µ + h) f ( µ h) f f f ( x) f ( µ ) + ( x µ ) + ( x µ ) h h eglecting higher order terms, ( µ + h) f ( µ ) + f ( µ h) ( µ + h) f ( µ ) + f ( µ h) σ f E[ f ( x) ] f ( µ ) + h (6) In [Hol9] it is suggested that an appropriate choice for h is 3 σ which yields, E [ f ( x) ] f ( µ ) + f ( µ + 3σ ) + f ( µ 3σ ) (7) If we consider Q (.) f ( x), then using eq. (7), the average probability of error is given by, b, E Q b ψ b Q + b Q 3 µ ψ 6 ( µ + ) + b Q ψ 3σ 6 ψ ( µ ) (8) ψ 3σ ψ Sandeepa Muheree age 6 of 8

17 Wireless Communication echnologies Lecture 7 & 8 µ ψ Mean of variance of Multiple Access Interference (MAI) conditioned on parameters σ ψ Variance of ψ Fig. 3 and Fig.4 illustrate the average bit error rates for single-cell CDMA systems using the different analytical methods discussed in this lecture (i.e. GA, IGA and SIGA). For results shown the processing gain is 3..E+.E- IGA SIGA.E- GA Bit Error ate.e-3.e-4.e-5.e-6.e otal umber of Users, 3 Figure 3: Comparison of BE for a desired user assuming that power level for all users are fixed In this comparison it is assumed that the power levels for all users are fixed. [/] users have a power level of /4. he remaining - - users each have power level equal to that of the desired user,. ote that SIGA provides a very close match to IGA. For small number of users GA falls apart. Sandeepa Muheree age 7 of 8

18 Wireless Communication echnologies Lecture 7 & 8.E- IGA SIGA.E- GA Bit Error ate.e-3.e-4.e otal umber of Users, 3 Figure 4: Comparison GA, IGA and SIGA when interfering users have random but identically distributed power levels he interferer power levels obey a log-normal distribution with a standard deviation of 5dB. he power level of the desired user is constant and equal to the mean value of the power level of each individual interfering user. eferences: [] G. Stuber, rinciples of Mobile Communication [] heodore S. appaport Wireless Communications-rinciples & ractice [3] arayan Mandayam, Lecture notes for 6:33:559, Spring [4] emal araayali, Lectures notes for 6:33:559, Spring Sandeepa Muheree age 8 of 8