PRACTICE FINAL EXAM SOLUTION Jing Liang 12/06/2006

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1 PRACICE FIAL EXAM SOLUIO Jing Liang /6/6 PROBLEM (a) (b) (c) (d) (e) he propagation ignal experience power variation due to the contructive and detructive addition of multi-path ignal component, and therefore thee variation occur over very hort ditance, on the order of the ignal wavelength. For Rayleigh fading, there i no light of ight (LOS) component; for Rician fading, there i a fixed LOS component. When the tranmiion bandwidth i much greater than the coherence bandwidth of the channel, or in other word, the tranmitted ignal ymbol period i much maller than the delay pread; the tranmitting channel can provide high frequency diverity. he mot common tatitic model for wirele communication experiencing hadowing effect i the log-normal ditribution model: ξ logψ µ ψ p( ψ) exp, ψ > σ ψ ψ σ ψ where ψ Pt / Pr, ξ / ln, µ ψ i the mean of ψ log in, and ψ σ i the tandard deviation of ψ ψ When the SR i high, the maximum-likelihood detection average error probability for three i.i.d. Rayleigh faded path i: C P γ When γ, C P γ.. 5 When γ, P C/ γ / Scheme : Repetition code cheme. ranmit the ame ymbol over different antenna during ymbol time. At any one time, only one tranmit antenna i turned on and the other are ilent. Scheme : Alamouti cheme. If the channel gain i contant over two ymbol period, the Alamouti cheme can be achieved by tranmitting two different ymbol and at the firt ymbol period at two tranmitting antenna, repectively. And then tranmitting * and * at the econd ymbol period.

2 PROBLEM In thi problem, we know that the complex noie covariant σ n, the ignal energy (covariant) Pt E x. And the channel coefficient j j j, 4 j θ 4 θ h ae e h a e e are contant. he output of variou combiner can be expreed a r tot α i r i. (a) For maximum ratio combiner: he coefficient of the combiner i: (b) (c) jθi αi hx i / σ n e ae i j i j j j j 4 4 rmrc e e x+ w + 4e 4e x+ w j j 4 5x e w 4e w E hl x Pt MRC σn σn + + θ SR hl ( + 4 ) For equal gain combiner: he coefficient of the combiner i: α e i jθi j j j j 4 4 rmrc e e x+ w + e 4e x+ w j j 4 j jθ l l Pt σn σn 7x+ e w + e w E e h x SRegc al For elective combiner: he coefficient of the combiner i: α, α j j 4 rmrc e x+ w + 4e x+ w 4e x+ w E hl x Pt SRSC max max { al } 6.4 σn σn

3 PROBLEM (a) For Orthogonal Property: f (), t f () t f () t f () t dt t co dt Τ t co dt For ormalized Property: f () t E f () t dt t co dt 4 t ( + co ) dt 4 t ( co ) dt + ( + ) f () t E f () t dt dt herefore, both f() t and f() t are normalized. (b) Obviouly, ince f() t () t and f() t () t, uuv () t f() t [ /,]; uuv () t f() t [, ]; For (t): (), t f() t dt ; (), t f() t dt uv [,]

4 (c) Figure. (d) When (t) i tranmitted, it error probability can be calculated a: v v Pc P r fall in the region of S tranmitted c / Q Q σ σ Q Q σ σ P P e c Q Q σ σ (e) From Fig. in ub-problem (c), the minimum ditance i /. And there are two point with thi minimum ditance:,. herefore, by uing the nearet neighbor approximation, we can get: / Pe Q Q σ σ (f) By uing the nearet neighbor approximation, the error probability of thi ytem for a given h can be expre a: h P( h) Q σ he average error probability can be calculated a:

5 P P h p h dh r r r /σh Q e dh σ σ h σ σ 4σ σ h 4 σ h PROBLEM 4 (a) h (), (), Serie to Parallel ranfor m (every ) (), (), IDF d(), d(), d( ) Cyclic Prefix x( L) d( L), x( ) d( ), x() d(), x ( ) d Parallel to Serie ranform RF Modulation Wirele Channel RF Demodulation Remove Prefix Serie to Parallel ranform r(), r(), r( ) Parallel to Serie ranfor m r(), r(), r DF L y() h( l) x( l) + w()... l L y( n) h( l) x( n l) + w( n)... l L y( ) h( l) x( l) + w( ) l (b) From the tranmitting cheme, we know that the IDF output i: dn IDF k ke k and therefore: kn j

6 [ dn] k DF nk j dne n by uing the cyclic prefix: dn, n xn d( + n), n <. From the receiving cheme, we know: rk DF yn n [ ] yne nk j L hl () xn ( l) + wn e n l nk j L ( nl) k lk nk j j j hl () xn ( le ) e + wne n l n L hle () xn ( le ) l n lk nl k j j + L lk l nl k nl k j j j h( l) e x( n l) e + x( n l) e + l n n l Since for the firt term, ( n l) <, x( n l) d( n l + ), therefore the expreion rk L lk j l ( + nl) k nl k j j hle dn ( l+ ) e + d( n l) e + l n n l L lk mk l mk j j j hle dme + dme + l m l m L lk mk j j hle () d( me ) + l m by uing the relationhip between k and dm in the tranmitting cheme, we get: rk L lk j hle { l k } + Wk L lk j hle k + l hk % k + Here, W(k) i the DF of the Gauian variable, if w(n) i AWG, the linear combination of the AWG variable i till AWG, and therefore W(k) i Gauian.

7 he channel coefficient of ub-carrier k i: h% k L h() l e l lk j (c) (d) Since the Rayleigh faded path are i.i.d., the channel coefficient form a tandard complex Gauian random vector (,I), denoted by h [ h, h, h ] L, which each element' real and imagery part ditributed with (,.5). L lk j he OFDM channel coefficient h% k h() l e i the DF of the complex l Gauian random vector. Since the linear combination of each i.i.d. element hl () i till a complex Gauian, which each element real and imagery part ditributed with (,.5 L ), the probability ditribution of h % k i: f ( x) exp( x / L) L he auto-correlation coefficient by it definition i: (% *, % k i) E % % k i R h h h h L lk mi j L j E hle hme l m L L ( lkmi) j * E hlh ( m) e l m * ince different channel hl and hm are i.i.d. with variance, when l m, E hlh ( m) therefore, the ummation i: (% % k i) L m L R h, h h( m) e e m mk ( i) j mk ( i) j herefore, the covariance (ince mean i zero, it i alo the correlation) of h % k and h % i i: (% k, % i) L R h h e m Li ( k) j mi ( k) j e, when ik multiple of, ( ik) j e L, when i k multiple of.,

8 (e) For Gauian variable, the independent propertie mean that their correlation i zero, therefore: Li ( k) j e, when ik multiple of, ( ik) R( h% k, h% i) j e L, when i k multiple of. when i k i the multiple of, h%, h% are alway correlated, ( k i) (% k h% i) Li ( k) when ik multiple of, R h, only if i a integer