Basics of Information Theory • If g(x) is a function defined on a disc discrete RV X , we have, � � ( ( ) ) ( ) = E g x p x g ( x ) X X ∈ X x ( ) ( ) ( ( ( ) ) ) ( ) 2 2 2 = µ = = σ + µ = − = = E x 3 . 5 ; E x 15 . 17 1 ; E log p x 2 . 58 H X 2 X Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 18 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • For a Bernoulli RV, • the possible outcomes are X={0,1} with corresponding probabilities ( ) { ( ) { { { } } = = 1 − 1 − p X p X x x 1 1 p , p , p p ( ) ( ( ( ) ) ) ( ) ( ) ( ) = − = − − − − = log 2 log l 1 log 1 H X E p x p p p p H p X 2 2 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 19 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • H(p) is purely a concave function • It is maximum when p=1/2 (supr supreme uncertainty) • and it is zero for p=1 or 0 (uncert certainty is minimum) • Entropy has a key role in information • Entropy has a key role in information ation theory ation theory • Differential entropy: ∞ = � ( ) ( ) ( ( ) ) ) ( ( ( ) ) ) − = − h X f x log f x dx E log f x X 2 X 2 X − ∞ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 20 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Find the entropy of complex multiva ltivariate Gaussian distribution • A zero mean multivariate complex G ex Gaussian distribution with covariance matrix R has the followin owing pdf 1 ( ) ( ) − 1 ( ) − − H 1 H 1 ϕ = − = = π − x exp x R x R exp x R x N π R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 21 Communications, Cambridg ridge University Press, 2017
( ) ( ( ) ( ) ( ) ( ) ( ) − H 1 = − ϕ = − − π − h x E log x log e E ln R x R x f f 2 2 f � � � � � ( ) ( ) ( ) ( ) ( ) ) − � − � H 1 � 1 � = π + = π + log e ln R E x R x log e e ln R E x R x 2 f 2 f i j � � � � ij � � i j , � � � � � � � � � � ( ) ( ) ( ) ( ) ( ) � − � � − � � 1 � � 1 � = π + = = π + log e ln R E x x R log e ln R E x x R 2 f i j 2 f j i � � � � � � � � ij ij � � � � i j , i j , � � � � � � � � � � � � � � � � � � � � � ( ) ( ) ( ) + � ( ) ( ( ) � − � � − � 1 � � 1 � = π = π + log e ln R R R log l e ln R RR 2 2 � ji � � � � � � ij jj � � � � i j , i j , � � � � � � � � � � ) ( ) ( ( ) ) ( ) ( ) ( ( � � � � = π + = π + = π log e ln R I log e ln R N log e ln e R � � � � � 2 2 2 jj � � � � � � i j , � � ( ) = � π log e R � 2 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 22 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory ( ) ( ) ( ) = − I X ; Y Y H X H X | Y • Mutual Information: ( ( ( ) )) ( ( ( ) ) ) = − − − E log l p X E log p X / Y 2 2 • the decrement in the � � � ( ) � p X , Y � � � � ( ( ( ) )) • uncertainty (entropy) of X = − − − E log l p X E log � � � � � � 2 2 ( ) ( ) � � � � � � P P Y Y � � • because of knowledge of Y because of knowledge of Y � − � � � ( ) ( ) P X P Y � � � � = E log 2 � � � � ( ) � � p X , Y � � ( ) ( ) ( ) = + − H X X H Y H X , Y ( ) = I Y , X ( Y ) ( ) = − H Y H Y / X Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 23 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Note that conditioning cut downs en ns entropy ( ) ( ) ( ) ≤ = − 0 I X ; Y H Y H Y / X � ( ( ) ) ( ) ( ) ≤ H Y / X H Y • In the above expression, • H(Y) is the differential entropy of y of random variable (RV) Y and • H(Y/X) is the conditional differen rential entropy • The equality is possible for independ pendent Y and X Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 24 Communications, Cambridg ridge University Press, 2017
Basics of Information n Theory • Convex and Concave functions: • Definition: f(x) is strictly convex ove over (a,b) if • ( ( ( ( ) ) ) ) ( ) ( 1 ( ) ( 1 ) ( ) ) ( ) ( ( ) ) λ λ + + − − λ λ < < λ λ + + − − λ λ ∀ ∀ ≠ ≠ ∈ ∈ < < λ λ < < f f u u 1 1 v v f f u u f f v v u u v v a a , , b b , , 0 0 1 1 • In other words, each chord in f(x) lie lies above f(x) • For convex function f(x) , -f(x) is a co a concave Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 25 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • x 2 , x 4 , e x and xlog(x) ( x≥0 ) are strictly rictly convex function • log(x) , √x are strictly concave functi nction • x is a concave and convex function on • Jensen’s inequality: Jensen’s inequality: • For an arbitrary convex function f( � � ) and any random variable X, ( ( ) ) ( ( )) ≥ E f X f E X X Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 26 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • For strictly convex function f(x) ( ( ) ) ( ( ) ) > E f X f E X • For example, f(x)=x 2 is a strictly conv convex function • Let us say the possible outcomes are Let us say the possible outcomes are s are X={-1,+1} with equal s are X={-1,+1} with equal probabilities of p={1/2,1/2} • Then E(X)=0 , f(E(X))=0 , but, E(f(X))=1 ))=1 • Hence E(f(X))>f(E(X)) Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 27 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Kullback-Leibler distance: • Relative differential entropy of two wo pdf f and g is expressed as � ( ) � � � f x � � ( ) ( ) ( ) ( ( ( ) ) ) = = − − || log log D f g f x dx h X E g X � � 2 ( ) ( ) 2 f f � � � � g g x x • Information inequality: ( ) ≥ D f || g 0 { ( ) } = > S x : f x 0 • Proof: • Define Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 28 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory � � � � � ( ) � � ( ) � � ( ) � � g x g X g X � � � � ( ) ( ) � � � � � � − = − = ≤ D f || g f x log dx E E log log E � � � � � � � � � ( ) � 2 ( ) f 2 ( ) 2 f � � � � � � f x f X f X � � � � • In the above, we have used Jensen’s en’s inequality and log being a concave function concave function � ( ) ( ) � g x � � � � ( ) ( ) ( ) − ≤ = = = = D f || g log f x dx log g x dx log 1 0 � � 2 ( ) 2 2 � � f x S S Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 29 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Find the entropy maximizing distribu tribution over the interval (a,b) • Assume f(x) is a distribution over th r the interval (a,b) • We have, ( ( ) ) ( ) ( ) ( ( ( ( ( ))) ( ( ( ))) ( ) ( ) ( ( ) ) ≤ = − − = − + − 0 D f || u h X E log u X h X log b a f f 2 f 2 � ( ) ( ) ≤ − h f X log b a 2 1 ( ) = • A uniform distribution ( u x ) − b a • maximizes entropy over the interval rval Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 30 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • For a given covariance matrix K, find find the zero mean entropy maximizing distribution over the inf infinite interval ( ) n − ∞ ∞ , • Answer: • A real multivariate Gaussian distribu • A real multivariate Gaussian distribu tribution with the pdf tribution with the pdf � − � 1 1 − x − ( ) � T 1 � φ = 2 π x K exp x x K 2 � � 2 • Proof: ( ) ( ) ( ( ( ) ) ) ≤ = − − φ 0 D f || u h X E log X f f 2 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 31 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory ( ) ) ) ( ( ) ( � log ≤ ϕ X X X h E f f 2 � � 1 1 ( ( ) ) log log ln ln − T 1 � � = − − π π − e E 2 K K x K x 2 f � � � � 2 2 2 2 1 ( ) ( ) log log ln ln − T 1 = π + + e E 2 K x K x 2 f 2 � � � � 1 � ( ) ) ( ) ( ) log ln � − � � 1 � = π + K K e E 2 E x x � �� 2 f f i j � 2 � �� � ij , � � i j � � 1 � ) ( ) ( ) ( log � ln � − � 1 � = π + e 2 K E E x x K � �� 2 f i j � 2 � �� � ij , i j Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 32 Communications, Cambridg ridge University Press, 2017
� � � � 1 � ) ( ) ) ( ) ( ( log � ln � − � 1 � = π + e 2 K E E x x K � �� 2 f j i � 2 � �� � ij , � � i j � � 1 ) ( � ) ) ( ) ( ( log ln � � � − � 1 = π + K K K K e 2 � �� 2 � 2 � �� ji � ij , � � i j � � 1 � ) ( ) ( ) ( log log ln ln � � � − 1 � = π + e 2 K KK K � �� 2 � 2 � �� � jj , , � � i j � � 1 ) ( ) ( � � ) ) ( ( ( ( ) ) jj � log log ln ln � � � � � � � = = π π + + e e 2 2 K K I I I I � � � �� 2 � � �� 2 � , , i j 1 � � ) ( ) ( log log ln ln = π + e 2 K n � � � � 2 2 1 � � ) ( ) ( log ln = π e 2 e K � � � � 2 2 1 � � ( ) log = π 2 K e � � � � 2 2 ( ) = h X ϕ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 33 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Capacity of a parallel Gaussian chan hannel • Let us consider n independent Gaus aussian channel with I-O relation for the i th channel as = = + + Y Y X X N N i i i i i i ( ) 2 • where are zero mean Gaussian ssian i.i.d. σ N i ~ N 0 , N i • with the power constraint n � ( ) − 1 2 ≤ n E E X P i = i 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 34 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory city of the i th Gaussian channel first • Let us analyze and find the capacity • Assumption: X i and N i are independ endent RV with zero mean ( ) ( ) 2 2 2 2 = + + = P + σ E Y E X N 2 X N i N i i i i • We may express information capacit • We may express information capacit pacity as pacity as max ( ) ( ) = • ; C f I X Y X i i i ( ) 2 ≤ E X P i • Maximization of mutual information tion is w.r.t. pdf of X i subject to the power constraint Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 35 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) � = − = − + = − = − I X ; Y h Y h Y | X h Y h X N | X h Y h N | X h Y h N i i i i i i i i i i i i i i i • We know that optimal input ( X i ) is G ) is Gaussian distributed, hence output ( Y i ) is also Gaussian distributed and and the noise ( N i ) is also Gaussian distributed � + � + � � ( ( ( ( ) ) ) ) ) ) ( ( ) ) 1 1 1 1 1 1 P P • Hence, • Hence, ( ( ) ) � � � � 2 2 2 2 ≤ ≤ π P π + + σ σ − − π π σ σ = = I X ; Y log 2 2 e log 2 e log 1 i i 2 2 � � 2 2 2 2 σ • In the above case, we have conside sidered X i , Y i and N i are real RV, hence, there is a half factor in the capacity city expression • If we consider complex RV, the capa capacity for real and imaginary part will get added up and the half facto actor becomes 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 36 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • The I-O relation can be represented ted in the vector form as • y = x + n • Let us find the capacity for this case ase ( ( ) ) ( ) ( ) ( ( ) ) ( ) ( ) ( + ( x ) ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) = = − − = = − − + = = − − = = − − I x ; y h y h y | x h y h n | x h y h n | x h y h n • where the real random vectors are are � � � � � � Y X N 1 1 1 � � � � � � Y X N � � � � � � 2 2 2 = = = y ; x ; n � � � � � � � � � � � � � � � � � � � � � Y X N n n n Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 37 Communications, Cambridg ridge University Press, 2017
Basics of Information Theory • Note that mutual information is max maximum when N i and X i are i.i.d. Gaussian with zero mean • Hence, for optimal input X i , we have have � � ) � n n � P 1 � � ( ) ( ) ( i ≤ = − = + X ; Y I C h Y h N N log 2 1 � � i i 2 2 σ � � = = i 1 i 1 i + T. M. Cover and J. A. Thomas, Elements of Inform ormation Theory , Wiley, 2006. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 38 Communications, Cambridg ridge University Press, 2017
Capacity of random SIMO ch hannel ignal + can be written as • For a time slot m, the received signa ( ) ( ) = + y m h m x ( m ) n ( m ) ( 0 ) ( ) ( ) ~ N N 2 h ( ) ~ m N 0, I 0, σ I N n m R R C C • Dropping the time index m , we can can rewrite the I-O relation of SIMO system as • y = h x+ n + S. Barbarossa, Multiantenna Wireless Communic unication Systems , Artech House, 2003. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 39 Communications, Cambridg ridge University Press, 2017
Capacity of random SIMO ch hannel � � � � � � y n h • where 1 1 1 [ ] � � � � � � 2 y n h � � � � � � = E x P 2 2 2 = = = y ; n ; h � � � � � � � � � � � � � � � � � � � � � � � � � � � � y n h � � � � � � � � � � � � N N N R R R R R R • The covariance of the received signa ignal vector can be calculated as [ ] [ ] ( ) ( ) ( )( ) H H H H H H 2 = = + + = + = + σ R yy E yy E h x n h x n E E xx hh E nn P hh I N R • Note that we are assuming the chan channel is deterministic at a particular instant of time Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 40 Communications, Cambridg ridge University Press, 2017
Capacity of random SIMO ch hannel • Note that we have assumed that x and n are independent • Then, mutual information ( ) ( ) ( ) ( ) ( ) = − = − − I x ; y h y h y | x h y h n • due to translation invariance of the the entropy ( h ) and independence ( ) ( ( ) ) ( ) ( ) � = + = = y | x h x n / n / n h h x h x h Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 41 Communications, Cambridg ridge University Press, 2017
Capacity of random SIMO ch hannel • Since jointly proper Gaussian random ndom vectors maximize the differential entropy • Hence, N ( ) ( ) ( ) R 2 2 = π = π = π σ = π σ y log R ; n log R R log I log h e h e e e 2 yy 2 nn 2 N 2 R • We next use the upper bound on th n the mutual information by rewriting the capacity of the channel as Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 42 Communications, Cambridg ridge University Press, 2017
Capacity of random SIMO ch hannel � � ( ) ( ( )) N ( ) ( ) ( ) 2 R = − ≤ π − π σ y y n R I x ; h h log det e e log e � � � � 2 yy 2 ( ( ) ) � � � H 2 π + σ � � det e P hh I � � � � � � P N � � � � H R = = = = + + I I hh hh log log log log 2 det det � � � � � � 2 ( ) N R � � N 2 σ 2 R � � � π σ e • For any two matrices M×N matrix A A and N×M matrix B , we have, ( ) ( ) + = + det I AB det I BA M N Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 43 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Hence, for SIMO system, using the a he above identity, we have, � + � P 2 ( ) � � ≤ I x ; y log det 1 h 2 � � 2 � � σ • Therefore, instantaneous capacity Therefore, instantaneous capacity is is � + � P 2 � � = C W log 2 1 h � � 2 σ = � N R uare of Frobenius or Euclidean or L 2 - • where is the square 2 2 H = h h h h i norm of vector h = i 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 44 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • The average capacity of this channe nnel is given by � � � � + � P � 2 � � � � = C E log 2 1 h � � � � � 2 � � � σ • Assume iid Rayleigh fading SIMO ch Assume iid Rayleigh fading SIMO ch channel channel = � N R is a sum of the squ square of independent Gaussian RVs • 2 2 H = h h h h i = i 1 • hence it is Chi-square distributed wi d with 2N R degrees of freedom Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 45 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Its pdf is x − − 1 ( ) − N 1 = 2 f x x e R 2 ( ) N h − 2 N 1 ! R R • Therefore, the average capacity of t Therefore, the average capacity of t of this channel is given by of this channel is given by ∞ x � + � − 1 � P P � � − N R 1 = 2 C log 2 1 x x e dx ( ) � � N R 2 2 σ − 2 N 1 ! R 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 46 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Average capacity of random SIMO c O channel � � � � N P � � 2 2 � � R = + C E log 2 1 h � � � � 2 � σ � N � � R • For high SNR case For high SNR case � � � � � � 2 � � � � � � � � � � h � N P � � P � � � 2 � � � � R ≈ = + C E log h E log N E log � � � � � � � � � � � 2 2 R 2 � � � � 2 � 2 � � � N σ σ � � � N � � � � � R � R � � � � � 2 � � � � � h N P � � � � � � R ≈ + log � � log C E � � 2 2 � 2 � σ � � N � � � � � � R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 47 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Note that in high SNR region, • the average capacity of the i.i.d .i.d. Rayleigh channel is N R P • equal to that of AWGN having an g an effective SNR of 2 σ σ • with an additional term which reduc • with an additional term which reduc educes capacity educes capacity • The second tends to zero as → → ∞ N R 2 • since the PDF of h N R • approaches a Dirac delta function ce n centered at 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 48 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Approximate Outage probability: • For a threshold or target rate of R (b (bits/s/Hz), • outage probability is given by � � � � � � � � � 2 � � R P h − 2 1 � � � < 2 � ( ) � � � � = + = < Pr Pr log 1 Pr h ob R ob R ob � � � 2 2 P σ � � � � � � � � � � � � � 2 σ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 49 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel R − 2 1 2 • Hence the corresponding threshold old on is h 2 P σ / • Let us compute the outage probabil ability as follows R − 2 1 P 2 x σ − � 1 ( ) − N 1 2 = P R x e dx R out ( ) R N N − 2 1 ! R R 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 50 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel x y = • Substituting , we have, 2 R − 1 2 1 P 2 2 σ � � 1 1 ( ) ( ) − 1 − 1 − − N N y y dy = P R y e R out ( ) − N 1 ! R 0 • For a high SNR, y tends to zero R − − y ≈ e 1 2 1 y < P 2 2 2 σ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 51 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel R − 1 2 1 R P 2 − 1 2 1 ( ) 2 N σ � R R P − � � 1 1 1 2 1 2 ( ) − N 1 N ≈ = � = P R y dx y R R 2 σ out ( ( ) ) ( ( ) ) ) ) N N N N − − � � N N 1 1 ! ! N N ! ! ! ! � � � � 2 2 R R R R P P R R 0 ( ) � � 0 N ! R � � 2 σ • Hence there is diversity gain of N R R with respect to (w.r.t.) SISO case. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 52 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Exact Outage Probability: � � γ P ( ) � � H H + < = + γ < γ = Pr ob log I hh R Pr r o ob log 1 h h R ; � � � � 2 N 2 2 σ σ R N N � � � � T T � R − 2 1 ( ) ( ) ( ) H H R R H R H � + γ < = + γ < = γ < − = < r ob log 1 h h R Pr ob 1 h h 2 2 Pr ob h h 2 1 Pr ob h h 2 γ � Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 53 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Hence R − 2 1 P 2 x σ − � 1 ( ) − N R 1 = 2 P R x x e dx out ( ( ) ) N − − 2 2 N N 1 1 ! ! R R R 0 0 � � R − 2 1 1 � � γ N , � � R γ � � ( ) ∴ = P R out ( ) Γ N R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 54 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel ( ) , • where is the incomplete ga e gamma function γ z a z � � ( ( ) ) ( ( ) ) , , ; ; ; ; − − − low a 1 t 1 a a γ = = + − 1 a z t e dt a z z F a a z inc 1 1 0 • and R − 2 1 x , , = = = a N z t R γ 2 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 55 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • Hypergeometric functions • Pochhammer symbol defined as ( ( ) ) Γ + a n ( ) ( ) ( ( ) ( ) ( ) ) � � = = = = + + + + + + − − a a a a a a a a 1 1 a a n n 1 1 ( ) n Γ a ( ) ( ) ( ) ) ( ) 2 � = = = + = + a 1 , a a , a a a 1 a a , 0 1 2 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 56 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • The Hypergeometric function is defi defined for two complex vectors { } { 1 } , , , , , , , , � = � a a a a = b b b b b 1 2 p 2 q q • and single variable z and single variable z • Note that a is a vector of p elements ents and • b is a vector of q elements p F • that’s why the Hypergeometric func function is denoted as q Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 57 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • It is defined as ( ) ( ) ( ) ( ) ( ) � ∞ a a a a a k � � z − 1 2 3 p 1 p k k k k ( ) k k = a ; b ; F z ( ) ( ) p q ( ) ( ) ) ( ) ( ) ) � � b b b b b b k k ! ! 1 1 2 2 q q k k k k = k 0 k Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 58 Communications, Cambridg ridge University Press, 2017
Capacity of random S SIMO channel • For integer a � � − k a 1 z � ( ) ( ) , ! ! � − � low z γ = − − − 1 1 a z a e e � � ! inc � k � = k 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 59 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • For a time slot m , the received signa ignal can be written as ( ) ( ) ( ) ( ( ) = + y m h m x m n m n � � � � P ( ) ( ) ( ) ~ N N N � � ( ) h m ~ N 0, I x m N 0, I ( ) T T T 2 σ n m ~ N 0, C C � N � C T ( ) ( ) 2 2 ≤ E x m P Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 60 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • If we assume that channel is not kno t known at the transmitter, • we can have equal power allocation tion and therefore, P • each transmitting antenna will send end signal with power of N N T T • The instantaneous capacity for unifo niform power allocation is given as � � P P 2 � � = + C log 1 h 2 uniform 2 � N σ σ � T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 61 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • The average capacity of this channe nnel is given by � � P 2 � � = + C E log 2 1 h h 2 � � � � � � σ � � N T T = � N T 2 2 • the PDF of this RV is h h j = j 1 x − 1 ( ) − 1 N 2 = f x x e T 2 ( ) N h − 2 N 1 ! T T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 62 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • Therefore, the approximate average rage capacity of this channel • for high SNR case is given by � � � � � � �� �� 2 � � � � � � h h P P � � � �� � ≈ + C log � � E log g � �� 2 2 2 � 2 � σ � N � �� � T • One point to be noticed is that there here is no power or array gain in the first term of the average capacity Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 63 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • Outage probability: � � ( ) � � � � � � R � � � � − N 2 1 P 2 2 ( ) T � � � � � � = + < = < P out R Pr ob log 2 1 h R R Pr ob h � � 2 2 � � � � � � P P � � � � � � σ σ N T � � N � � � � � � � � 2 σ � � � � � � R − 2 1 � 2 ( ) � � = < P R h out P � � � � 2 σ � � N T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 64 Communications, Cambridg ridge University Press, 2017
Capacity of random M MISO channel • Hence, at high SNR (similar analysis ysis with SIMO case above), we get, N ( ) ) T R − 2 1 1 � � ( ) ( ) ≈ ≈ P P R R out out N N � � � � � � T P ( ) N � � � 2 N ! T T 2 σ σ � � N � T • Hence there is diversity gain of w.r.t w.r.t. SISO case Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 65 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Average capacity • For equal power allocation: � � � � R � � � � λ H i P P � � � � � � � � = = + + C C E E W W log log 2 1 2 1 � � � � � 2 � σ σ � � N � � = i 1 T • where are the singular values o es of the channel matrix λ i Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 66 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Alternatively we could also write the e the mean MIMO capacity for ergodic fading channels in terms of determi rminant of matrices as � � � � � � P Q � � � � = + C E W log 2 det I � � � � N � � 2 2 � � R R R R σ σ � � � � N N � � � � T � � H < HH , N N • where Q is the Wishart matrix defin efined as � R T = Q � H ≥ � H H , N N R T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 67 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Assume W=1 for brevity of the analy nalysis � � � � � � P Q � � � � = + C E log 2 det I � � N � � 2 � � R σ σ � � � � N N � � � � T T otation of P/σ 2 as γ • We will use a more convenient nota { } = m min N T N , • Assuming the channel matrix is full r full rank, then, R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 68 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • It is equivalent to m times (m is the the rank of the full rank matrix H ) bitrary and unordered eigenvalue � of • finding the expectation of an arbitra the Q matrix � � � � � � � � � � � � + � � � + � � � m � � � � � � γ γ γ γ � � � � � � � � ( ( ) ) ( ( ) ) � � � � � = λ = λ C log e E ln 1 m log 2 e E ln 1 � � � � 2 k � � � � � � � � � N N � � � � � T T = k 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 69 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels unordered � is given by • we have the marginal PDF of an uno l ( ) ( ) � �� � − 2 j − + − m − 1 i 1 2 j ! 2 i 2 j 2 j 2 n 2 m 1 ��� + − l n m ( ) ( ) − λ � �� � λ = λ p e ( ( ) ) − 2 i l − − + + � � − − �� �� − − � � i i j j 2 2 j j l l m m 0 2 2 j l j l ! ! ! ! n n m m j j ! ! ! ! = = = = = = i i 0 0 j j 0 0 l l 0 { N } = n max N , T N R { N } = m min N , T N R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 70 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels � � ∞ γ λ � ( ) ( ) � � = + λ λ C m log e ln 1 p d 2 � � N 0 T l � � ( ) ( ) � �� � ∞ − − + − − 2 j 1 2 j ! m 1 i 2 i 2 2 j 2 j 2 n 2 m γ λ ��� ��� � � + − l n m ( ( ) ) ( ) ( ) − λ � � � � � � �� �� � � = = + + N λ λ λ λ λ log log e e ln 1 ln 1 e e d d 2 2 ( ( ) ) − − 2 2 i l i l − − − − − − + + � � �� �� � � 2 ! ! ! i i j j j j 2 2 j j l l � � � � j l n m j = = = i 0 j 0 l 0 0 T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 71 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels ∞ � + � γ � λ d � � ( ) + − l n m − • In order to calculate = λ λ λ ln 1 I e � � � � N T T 0 • The complementary incomplete gam gamma function is given by ∞ = � k dx − − + − − ( ) x q k 1 Γ − + υ − + > q k , e x ; q k 0 υ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 72 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels ∞ � + � γ � λ d � � ( ) + − l n m − = λ λ λ I ln 1 e � � � � N γ γ N T 0 T x = λ − = + − υ = ; q 1 l n m ; γ N T T N N d λ λ d � T λ λ = = = ν = x x dx γ ν ∞ q ( ) � Γ − + υ � q k , ( ) ( ) ( ) − − υ υ q 1 x � ν = + = − I ln 1 x x e dx q 1 1 ! e q k υ = 1 k 0 + M.-S. Alouini and A. J. Goldsmith, “Capacity of Ray Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” s,” IEEE Trans. Veh. Technol., vol. 48, pp. 1165–1181, Jul 1999. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 73 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels ( ) − q 2 � = ν ν I I q ∞ q ( ) � � Γ − + υ q k , � � − − − q 2 q 1 q 2 − υ e υ ( ) ( )( ) x ( ( ) ( ) ∴ = υ + = = − υ I ln 1 x x e dx q 1 ! k k υ υ = k 1 0 − q q 1 � � υ − + − υ − + − ( ) ( ) ( ) ( ) k q 2 k q 1 = − υ Γ − + υ = − υ Γ − + + υ q 1 ! e q k , q q 1 ! e q k 1, = = k 1 k 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 74 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • The exponential integral function of n of order r could be expressed as ∞ � − υ − ( ) y r � υ = υ > r = E e y dy ; 0, , r 0,1, r 1 1 • The exponential integral function is n is a particular case of the complementary incomplete gamma ma function ∞ � − − + − ( ) x q k 1 Γ − + υ = − + > , ; 0 q k e x dx q k ∞ � − − ( ) x r Γ − υ = − > > 1 , ;1 0 0 r e x dx r υ υ Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 75 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels ∞ � • Substituting x= ν y, dx= ν dy, we hav have, − − ( ) x r Γ − υ = − > 1 r , e x dx ;1 r 0 υ ∞ ∞ ∞ � � � − r − ν − + − ν − ( ) y ( ) r 1 y r Γ − υ = ν ν = ν 1 r , e y dy e y dy 1 1 1 1 1 1 ∞ � − υ − ( ) y r • Therefore, � υ = υ > = E e y dy ; 0, r 0,1, r r 1 − ( ) r 1 ( ) υ = υ Γ − υ E 1 r , r − q 1 � ( ) ( ) 1 ! ! , ν − + − k q 1 = − ν Γ − + + ν I q e e q k 1 = k 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 76 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • If we assume that r-1=q-k-1, then 1 n 1-r=-q+k+1 • Hence, − q 1 � � ( ) ( ) υ ∴ = − υ I q 1 ! e E r = 0 k N T • Putting back, = − − = + − υ = r q k ; q 1 l n n m ; γ N + − � � l n m m T � � N ( ) � � γ T = + − I l n m ! e E � � + − + − l n m 1 k � � γ = k 0 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 77 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Note that k=0, l+n-m+1-k= l+n-m+1 +1 • and • k=l+n-m, l+n-m+1-k=1 • Hence it similar to k going from 1 to Hence it similar to k going from 1 to 1 to l+n-m+1 1 to l+n-m+1 • Or k+1 going from 0 to l+n-m • Hence, it can be further expressed a ed as N + − � � l n m m T � � N ( ) � � γ T = + − I l n m ! e E � � + k 1 � γ � = k 0 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 78 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Hence the average capacity of i.i.d. R . Rayleigh fading MIMO channel � � ∞ γ λ � ( ) ( ) � � = + λ λ C m log e ln 1 p d 2 � � N T 0 l l ( ( ) ( ) ( ) ) � � � � � � �� �� � � ∞ ∞ − − 2 2 j j − − + + − − m m − − 1 1 i i 1 1 2 2 j j ! ! 2 2 i i 2 2 j j j j 2 2 j j 2 2 n n 2 2 m m γ λ γ ��� � + − l n m ( ) ( ) − λ � � � �� � = + λ λ log e ln 1 e d 2 ( ) 2 − i l � − �� − � − + i j j 2 j l � � 2 j l ! ! n m j ! N = = = i 0 j 0 l 0 T 0 N � � − 2 j − + m 1 i ( ) ( ) ( ) � � � � n m l � � T l � � ��� − + − � − − − + 2 i 2 j 2 j 2 n 2 m N 1 2 j ! n m l ! � � � � � � ( ) γ T � � = C e log e E � � � � � � + 2 k 1 ( ) � − � N , N 2 i l � � � − � � − � γ i j 2 j l − + � 2 j ! l ! n m j ! � T R = = = = i 0 j 0 l 0 k 0 + H. Shin and J. H. Lee, “Capacity of multiple-antenna nna fading channels: Spatial fading correlation, Double scattering and Keyhole,” IEEE Trans. Information The Theory , 2003, pp. 2636-2647. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 79 Communications, Cambridg ridge University Press, 2017
Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 80 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Fig. Average capacity vs SNR (dB) of ) of open loop MIMO system Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 81 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Example • The average capacity for i.i.d. Raylei yleigh fading MIMO channel can be calculated as N N � � � � − 2 j ( ) ( ) ( ) � � � � − + � � T m 1 i l n m l � � ��� − + − � − − − + 2 i 2 j 2 j 2 n 2 m 1 2 j ! n m l ! N ( ) � � � � � � γ T � � = C e log e E � � � � � � + 2 k 1 � ( ) � N , N − 2 i l � − � � − � � γ � i j 2 j l − + � � 2 j ! l ! n m j ! T R = = = = i 0 j 0 l 0 k 0 ∞ ( ) � − − xy k • where = E x e y dy k 1 • (exponential integral function of ord f order k), and Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 82 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • find the average capacity of • (a) MIMO channel with antennas at the transmitter = = = N N N R T and receiver • (b) MISO channel with antenna • (b) MISO channel with antenna nnas at the transmitter and 1 antenna nnas at the transmitter and 1 antenna N N T at the receiver • (c) SIMO channel with one antenna nna at the transmitter and N R antennas at the receiver Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 83 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • (a) Given that = = N N N R T • and hence, { } = = m min N , N N T T R R { } = = n max N , N N T R = n m Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 84 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels N � − � − 2 j ( ) ( ) ( ) � � � � � � N 1 i l l T � � ��� − � 2 i 2 j 2 j 1 2 j j ! l ! N ( ) � � � � � � γ T � � = C e log e E � � � � � � + 2 k 1 ( ) � − � N , N 2 i l � − � � − � � � γ i j 2 j l � 2 j ! l l ! j ! � = = = = i 0 j 0 l 0 k 0 ) � N � − � − 2 j 1 ( ( ) ) ( ( ) ( ) ) ( ) � � ( ( ) � � T N i l l � � � � − ��� ��� � 2 i 2 j N 1 2 j ! l ! 2 j ! � � � � � � � � � � ( ) ( ) γ γ T T � � � � = = C C e e log log e e E E � � � � � � � � + 2 ( ) k 1 ( ) N , N � − � 2 i l l � � � − � − γ i j 2 j l ! l ! � 2 j ! l ! j ! � = = = = i 0 j 0 l 0 k 0 N � − � − 2 j � � � � N 1 i ( ) ( ) l � � T l l � � ��� − � 2 i 2 j 2 j 1 2 ! N j ( ) � � � � � � � γ T � � = C e log e E � � � � � � + 2 k 1 � ( ) � N , N − 2 i l � − � � � � � γ i j l � � 2 j ! j ! = = = = i 0 j 0 l 0 k 0 N � − � 2 − j � � � � � � � � T N 1 i ( ) l � l � ��� − � 2 j 2 i 2 j 2 j 1 N � � � � � � � ( ) � � γ T � � = C e log e E � � � � � � � � + 2 k 1 � − � N , N 2 i l � � � − � � � � � γ j i j l � 2 � = = = = i 0 j 0 l 0 k 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 85 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels (b) Given that = N 1 R { } { } • and hence, = = n max N , 1 N = = m m min N , 1 1 T T T • Therefore, i=j=l=0, we have, N − N 1 � � T ( ) � T N � � γ T = C e log e E � � + + 2 k 1 N , 1 � γ � T = 0 k Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 86 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels (c) = • Given that 1 N T { { } } = { { } } = = = • and hence, and hence, m min 1 , N 1 = = n max 1 , N N R R R R • Therefore, i=j=l=0, we have, 1 − N 1 � � ( ) � R 1 � � γ = C e log e E E � � + 2 k 1 1 , N � γ � R = k 0 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 87 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Example • Show that a simple upper bound on d on the average capacity of Rayleigh fading MIMO channel is given as � � � � � + � � � � � � � N γ γ N � � ( ) ) R � � ≤ + γ C min N log 1 , N log 1 � � 2 2 R T � � � � N � � T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 88 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Note that the log-det function is con concave over the set of nonnegative matrices • Therefore, applying Jensen’s inequa quality, we have � � � � γ γ ( ) ( ) H H � � = + ≤ + = + γ log I HH log I HH log 2 1 C E E N 2 N 2 N R � � R R N N � � T T • In the above we have used the relat elation ( ) H = E HH N I T N N R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 89 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels � � � * � � � � � � h h h h h h h � � 11 12 1 N 11 21 21 N 1 � � � � T R � � � � � � � � h h h h h h h ( ) 21 22 2 12 22 22 2 � � N N H = E HH E T R � � � � � � � � � � � � � � � � � � � � � � � � � � � � � h h h h h h � � � � � � � � � � � N N 1 1 N N 2 2 N N N N 1 1 N N 2 2 N N N N N N N N R R R R R R T T T T T T R R T T � ( ) � 2 2 2 � � + + + 0 0 E h h h � � 11 12 1 N T � � ( ) � � 2 2 2 � � � + + + + 0 E h h h 0 � � 21 22 2 N = T � � � � � � � � � � ( ) 2 2 2 � � + + + 0 0 E h h h � � � � N 1 N 2 N N R R R T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 90 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • In retrospect, the matrices HH H and and H H H have identical nonzero eigenvalues, therefore � � � + � � � γ γ γ ( ) N � � H H R � � = + ≤ + = C E log I H H log I E H H N log 2 1 � � � � 2 N 2 N T � � � � � � � � T N N T N N N N � � � � T T T T T T ( ) H • In the above we have used the relat elation = E H H N I R N T • By combining the above two cases, es, we can obtain the upper bound as � � � + � � � N γ � � ( ) R � � ≤ + γ , C min N log 1 N log 2 1 � � R 2 T � � � � N � � T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 91 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Outage capacity of iid Rayleigh fadin ading MIMO channel � � � � � � P � � ( ) H � � = + + < Pr ob R Pr ob W log 2 det I HH R � � N R � � 2 � � N T σ σ � � � � N � � � � T • It has been shown + also that the ins instantaneous capacity C inst leads to a Gaussian RV for all values of N T an and N R + P. J. Smith and M. Shafi, “On a Gaussian approximat imation to the capacity of wireless MIMO systems,” IEEE ICC , 2002, New York, April 2002. Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 92 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels • Therefore the outage probability ma y may be nearly approximated for all combination of N T and N R antenna nnas as � � µ − R � � � � ( ) ( ) C ≈ ≈ P P R R Q Q � � � � out out � σ � C • where R is the target data rate, • μ C =E[C inst ], • σ c is the square root of the variance nce of the C inst 2 ∞ z 1 − � ( ) • Q-function is tail integral of a Gauss ussian pdf = 2 Q x e dz π 2 x Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 93 Communications, Cambridg ridge University Press, 2017
Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 94 Communications, Cambridg ridge University Press, 2017
Capacity of i.i.d. Rayleigh fad ading MIMO channels Fig. CDF of open loop N T ×N R MIMO c O channel capacity for SNR=5dB • For a 5×5 MIMO channel, • the 0.2 outage capacity is approx proximately • 7.5 bits/sec/Hz for SNR of 5 d • 7.5 bits/sec/Hz for SNR of 5 d 5 dB 5 dB • Whereas, for a 7 ×7 MIMO channel, nel, • the 0.2 outage capacity is approx proximately • 10.5 bits/sec/Hz for SNR of 5 f 5 dB Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 95 Communications, Cambridg ridge University Press, 2017
Capacity of separately corr rrelated Rayleigh fading MIMO channel • Instantaneous capacity of separately ately correlated Rayleigh fading MIMO channel � � � � � � � � � � � � H P Q P HH � � � � � � � � = + = + C W log 2 det I W W log 2 det I � � � � N N � � 2 � � � � 2 � � R R σ σ σ σ � � � � � � � � N N N N � � � � � � � � T T T T • For separately correlated MIMO cha channel, 1/2 1/2 = H R H R R w T X X � � � �� � C P � 1/2 1/ H H /2 � � �� = + log det I R H R H R 2 N R R w T w R 2 � � N σ W � �� R X X X X � T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 96 Communications, Cambridg ridge University Press, 2017
Capacity of separately corr rrelated Rayleigh fading MIMO channel • For N T =N R =N , and assuming that the t the matrices and and are full rank, we have have for high SNR case, • R R R T X X ( ( ) ) ( ( ) ) + + = = + + det det I I AB AB det det I I BA BA � � � �� � C P � � � H 1/2 H H /2 �� = log det H H R R R 2 w w R R R T 2 � σ � � �� W N X X X � T � � � � � � P { ) } { } ( ( ) � � H �� = + + log det H H log de et R log det R 2 w w 2 R 2 T 2 � � σ � N � X X � � T Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 97 Communications, Cambridg ridge University Press, 2017
Capacity of separately corr rrelated Rayleigh fading MIMO channel • Hence the MIMO channel capacity h ity has been reduced (why reduced?) • and the amount of reduction in the the capacity is given by { { } } { { } } ( ( ) ) ( ( ) ) + log det R log 2 det R 2 2 R R 2 2 2 T T X X X X • Example { } { ) } ( ) ( • Show that is always negative. + log det R log de det R 2 R 2 T X X Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 98 Communications, Cambridg ridge University Press, 2017
Capacity of separately corr rrelated Rayleigh fading MIMO channel • Note that [ ] T H = R E H H T X [ ] H = R E HH R X • The diagonal elements are 1 and The diagonal elements are 1 and • off-diagonal elements hold a valu value between 0 and 1 • Hence ( ) X = trace R N T T ( ) X = trace R N R R Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 99 Communications, Cambridg ridge University Press, 2017
Capacity of separately corr rrelated Rayleigh fading MIMO channel • The geometric mean is bounded by by the arithmetic mean 1 � � N N N 1 1 � R R ( ) R ∏ � � λ ≤ λ = = = R trace 1 i i R � � � � N N N N N N X = = = = i i 1 1 i i 1 1 R R R R • Note that product of all eigenvalues lues of a matrix is equal to the determinant of the matrix N R ( ) ∏ • Therefore = λ ≤ det R 1 R i X = i 1 Rakhesh Singh Kshetrimayum, Fu , Fundamentals of MIMO Wireless 1/19/2018 100 Communications, Cambridg ridge University Press, 2017
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