EE401: Advanced Communication Theory

Transcription

1 EE401: Advanced Communication Theory Professor A. Manikas Chair of Communications and Array Processing Imperial College London Multi-Antenna Wireless Communications Part-B: SIMO, MISO and MIMO Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 1 / 80

2 Table of Contents 1 Notation 2 Common Symbols 3 The Concept of the Projection Operator 4 Wireless SIMO Channels Space-Selective Fading Small-Scale and Large-Scale fading Scattering Function Wavenumber Spectrum and Angle Spectrum Types (and examples) of Angle Spectrum Correspondence between Frequ., Time & Space Parameters The Relationship between Coherence-Time and Bandwidth The Concept of the "Local Area" Rx Antenna Array Array Manifold Vector Proof Equation 25 (Array Manifold Vector) Multipath SIMO Channel Scatterers in SIMO channels Modelling of the Received Vector-Signal x(t) Multi-user SIMO 5 Wireless MISO Channels Reciprocity Theorem Multipath MISO Channel Scatterers Modelling of the Rx Scalar-Signal x(t) Transmit Diversity Transmit Diversity: "Close Loop" UMTS 3GPP Standard (Close Loop) Transmit Diversity: "Open Loop" (without geometric information) 6 Wireless MIMO Channels Multipath MIMO Channel Scatterers Modelling of the Rx Vector-Signal x(t) MIMO Systems (without geometric information) Capacity - General Expressions Equivalence between MIMO and SIMO/MISO Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 2 / 80

3 Notation Notation a, A denotes a column vector A(or A) denotes a matrix I N N N identity matrix 1 N vector of N ones 0 N vector of N zeros O N,M N M matrix of zeros (.) T transpose (.) H Hermitian transpose A # pseudo-inverse of A, Hadamard product, Hadamard division Kronecker product Khatri-Rao product (column by column Kronecker product) exp(a), exp(a) element by element exponential L[A] linear space/subspace spanned by the columns of A L[A] complement subspace to L[A] P[A] (or P A ) projection operator on to L[A] P[A] (or PA ) projection operator on to L[A] Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 3 / 80

4 Notation The expression "N-dimensional complex (or real) observation space " is denoted by the symbol H and pictorially represented as follows Note that any vector in this space has N elements. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 4 / 80

5 Notation The expression "one-dimensional subspace spanned by the (N 1) vector a" is mathematically denoted by L[a] and pictorially represented by L[a] The expression "M-dimensional subspace (with M 2) spanned by the columns of the (N M) matrix A" is mathematically denoted by L[A] and pictorially represented by L[A] Note that any vector x L[A] can be written as a linear combination of the columns of the matrix A i.e. x = A.a (2) where a is an (M 1) vector with elements that are the coeffi cients of this linear combination. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 5 / 80

6 Common Symbols Common Symbols N φ θ u c F c λ k k number of Rx array elements elevation angle (Direction-of-Arrival) azimuth angle (Direction-of-Arrival) [cos θ cos φ, sin θ cos φ, sin φ] T (3 1) real unit-vector pointing towards the direction (θ, φ) u T u = 1 velocity of light carrier frequency wavelength wavenumber wavevector Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 6 / 80

7 The Concept of the Projection Operator The Concept of the Projection Operator Consider an (N M) matrix A with M N (i.e. the matrix has M columns) Let the columns of A be linearly independent (i.e. a column of A cannot be written as a linear combination of the remaining M 1 columns) Then the columns of A span a subspace L[A] of dimensionality M (i.e. dim{l[a]}=m) lying in an N-dimensional space H (observation space), and this is shown below: Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 7 / 80

8 The Concept of the Projection Operator Any vector x H can be projected on to L[A] by using the concept of the projection operator P[A] (or P A ). That is: P[A] = P A = projection operator on to L[A] (3) = A(A H A) 1 A H (4) Properties of Projection Operator (N N) matrix P A : P A P A = P A P A = P H A (5) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 8 / 80

9 The Concept of the Projection Operator L[A] denotes the complement subspace to L[A] dim (L[A]) = M (6) ( dim L[A] ) = N M (7) P A represents the projection operator of L[A] and is defined as P A = I N P A (8) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 9 / 80

10 The Concept of the Projection Operator Any vector x H can be projected on to L[A] and L[A] as follows Comment: if x L[A] then { PA x = x P A x = 0 N M (9) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 10 / 80

11 Wireless SIMO Channels Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 11 / 80

12 Wireless SIMO Channels Space-Selective Fading Space-Selective Fading A wireless receiver is located (and moves) in our 3D real space. In addition to delay-spread (causing frequency-selective fading) and Doppler-spread (causing time-selective fading) there is also angle-spread Angle Spread causes Space-Selective fading Note that a channel has space-selectivity if it varies (i.e. its transfer function varies) as a function of space and spatial-coherence if its transfer function does not vary as a function of space over a specified distance (D coh ) of interest where D coh = is known as coherence distance. (10) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 12 / 80

13 Wireless SIMO Channels Space-Selective Fading H(r) V 0 for r r 0 D coh 2 (11) In other words a wireless channel has spatial coherence if the envelope of the carrier remains constant over a spatial displacement of the receiver. D coh represents the largest distance that a wireless receiver can move with the channel appearing to be static/constant. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 13 / 80

14 Wireless SIMO Channels Space-Selective Fading Small-Scale and Large-Scale fading If the displacement of the receiver is greater than D coh then the channel experiences small-scale fading (this is due to multipaths added constructively or destructively). If this displacement is very large (i.e. corresponds to a very large number of wavelengths) then the channel experiences large-scale fading (this is due to path loss over large distances and shadowing by large objects - it is typically independent of frequency) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 14 / 80

15 Wireless SIMO Channels Space-Selective Fading Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 15 / 80

16 Wireless SIMO Channels Scattering Function Scattering Function - Wireless Channel Analysis Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 16 / 80

17 Wireless SIMO Channels Wavenumber Spectrum and Angle Spectrum Wavenumber Spectrum and Angle Spectrum If the transfer function of the channel includes "space" then we have: Channel FT H(f, t, r) = Φ }{{} H ( f, t, r) = }{{} S H (τ, f, k) }{{} Transfer function Autocorrelation function (frequency,time,space) Scattering function f frequency t time r location (x,y,z) where τ delay f Doppler frequency k wavenumber vector = k.u(θ, φ) (12) (13) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 17 / 80

18 Wireless SIMO Channels Wavenumber Spectrum and Angle Spectrum Note that if f = 0 and t = 0 then k = = k {}}{ 2π λ [cos θ cos φ, sin θ cos φ, sin φ]t (14) FT Φ H ( r) = }{{} S H (k) }{{} Autocorrelation function Scattering function (space) (wavenumber spectrum) (15) S H (k) = (2π)3 δ( k 2π λ ) ( 2π λ )2 p(θ, φ) }{{} angle spectrum (16) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 18 / 80

19 Wireless SIMO Channels Types (and examples) of Angle Spectrum Types (and examples) of Angle Spectrum 1. Specular Angle-Spectrum: 2. Diffuse Angle-Spectrum: 3. Combination of Specular & Diffused: Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 19 / 80

20 Wireless SIMO Channels Correspondence between Frequ., Time & Space Parameters Correspondence between Frequ, Time & Space Parameters Consider a wireless channel transfer function H(f, t, r) Frequency Time Space Dependency f t r Coherence Spectral domain Spectral width B coh coherence bandwidth τ delay T spread delay spread T coh coherence time f Doppler freq B Dop Doppler spread D coh coherence distance k wavevector k spread wavevector spread Remember the various types of coherence: temporal coherence -coherence time T coh frequency coherence -coherence bandwidth B coh spatial coherence -coherence distance D coh Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 20 / 80

21 Wireless SIMO Channels The Relationship between Coherence-Time and Bandwidth The Relationship between Coherence-Time and Bandwidth We have seen that one of the wireless channels classification is "slow" and "fast fading - as this is shown below: A trade-off between "slow" and "fast" fading is when T coh T cs (or, for CDMA, T coh T c ). This implies T coh B 1 (or, for CDMA: T coh B ss 1) (17) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 21 / 80

22 Wireless SIMO Channels The Relationship between Coherence-Time and Bandwidth An electromagnetic wave of frequency F c (carrier frequency) travelling for the time T coh with velocity c (speed of light, i.e m/s) will cover a distance d, which is given as follows: d = c.t coh = F c λ }{{} c T coh = T coh = =c d F c λ c (18) Using Equations 17 and 18 we have d B 1 = d F c F c λ c B λ c (= c B ) (19) We have seen that in slow fading region T coh > T cs (or, for CDMA, T coh > T c ) and this implies that T coh B > 1 and, finally, d < F c B λ c (= c B ) (20) Thus overall, for slow fading: T coh B 1 = d F c B λ c (21) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 22 / 80

23 Wireless SIMO Channels The Concept of the "Local Area" The Concept of the "Local Area" "Local Area" is the largest volume of free-space (i.e. 4 3 πd 3 ) about a specific point in space r o = [r x0, r y0, r z0 ] T (e.g. the reference point of the Rx-array) in which the wireless channel can be modelled as the summation of homogeneous planewaves, where d F c B λ c (22) Example: Local area for WiFi electromagnetic waves (F c = 2.4GHz,B = 5MHz) d F c B λ c = d = 60m Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 23 / 80

24 Wireless SIMO Channels Rx Antenna Array Antenna Array Consider a single path from Tx single antenna system to an array of N antennas An array system is a collection of N > 1 sensors (transducing elements, receivers, antennas, etc) distributed in the 3-dimensional Cartesian space, with a common reference point. Consider an antenna-array Rx with locations given by the matrix [r 1, r 2,..., r N ] = [ r x, r y, r z ] T (3 N) (23) where r k is a 3 1 real vector denoting the location of the k th sensor k = 1, 2,..., N and r x,r y and r z are N 1 vectors with elements the x, y and z coordinates of the N antennas The region over which the sensors are distributed is called the aperture of the array. In particular the array aperture is defined as follows array aperture. = max ij r i r j (24) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 24 / 80

25 Wireless SIMO Channels Rx Antenna Array Array Manifold Vector It is also know as Array Response Vector. Modelling of Array Manifold Vectors (see Chapter-1 of my book): where S(θ, φ) = exp( j[r 1, r 2,..., r N ] T k(θ, φ)) (25) or = exp( j[r x, r y, r z ]k(θ, φ)) (26) { 2πFc c = (N 1) complex vector k(θ, φ) =.u(θ, φ) = 2π λ c.u(θ, φ) π.u(θ, φ) = wavenumber vector in meters in units of halfwavelength } u(θ, φ) = [cos θ cos φ, sin θ cos φ, sin φ] T (27) = (3 1) real unit-vector pointing towards the direction (θ, φ) u(θ, φ) = 1 (28) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 25 / 80

26 Wireless SIMO Channels Rx Antenna Array In many cases the signals are assumed to be on the (x,y) plane (i.e. φ = 0 ). In this case the manifold vector is simplified to S(θ) = exp( j[r 1, r 2,..., r N ] T k(θ, 0 )) = exp( jπ(r x cos θ + r y sin θ)) (29) A popular class of arrays is that of linear arrays: r y = r z = 0 N in this case, Equation 25 is simplified to S(θ) = exp( jπr x cos θ) (30) Summary: An array maps one or more real directional parameters p or (p, q) to an (N 1) complex vector S(p) or S(p, q), known as array manifold vector, or array response vector, or source position vector. That is p R 1 r S(p) C N (31) or (p, q) R 1 r S(p, q) C N (32) Note: Ideally Equations 31 and 32 should be an one-to-one mapping Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 26 / 80

27 Wireless SIMO Channels Rx Antenna Array Proof Equation 25 (Array Manifold Vector) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 27 / 80

28 Wireless SIMO Channels Rx Antenna Array With reference to the previous figure, consider at the Tx a single uniform planewave δ(t) exp(j2πf c t) that travels with the velocity of light c for time τ and covers a distance d arriving at the Rx., i.e. at the Tx = exp (j2πf c t) δ(t) ( ) 1 α at the Rx s reference point = exp (jφ) exp (j2πf c (t τ)) δ(t τ) d ( ) 1 α ( ) d = exp (jφ) exp j2πf c exp (j2πf c t) δ(t τ) d c }{{} β = β. exp (j2πf c t).δ(t τ) (33) If the k-th antenna of the array is in the "Local Area" about the reference point of the array then at the k-th antenna of the Rx = β. exp (j2πf c (t τ k )).δ(t τ k τ) = β. exp ( j2πf c τ k ). exp (j2πf c t).δ(t τ) That is, the baseband signal is: at the k th antenna of the Rx = β. exp ( j2πf c τ k ).δ(t τ) (35) (34) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 28 / 80

29 Wireless SIMO Channels Rx Antenna Array However, τ k is given as follows: τ k = r T k u(θ, φ) c where r k R 3 1 denotes the Cartesian coordinates of the k-th Rx-antenna in metre. Proof: of Equation 36: With reference to the following figure we have: (36) τ k = r T k u (ut u) 1 u T r k c r T k = r k c (r T k = 2 c = r T k u c (37) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 29 / 80

30 Wireless SIMO Channels Rx Antenna Array Thus, Equation 35 can be expressed as follows: ( at the k-th antenna of the receiver (baseband) = β. exp j 2πF ) c r T k c u(θ, φ).δ(t τ) That is, the Tx planewave exp (j2πf c t) δ(t) arrives at each antenna of the array and produces, at time t = τ, a constant-amplitude voltage-vector as follows: ( ) ( ) 1 st ant. β exp j 2πFc 2 nd c r T 1 u(θ, φ).δ(t τ) exp j 2πFc ( ) c r T 1 ( u(θ, φ) ) ant.. = β exp j 2πFc c r T 2 u(θ, φ).δ(t τ) exp j 2πFc =β c r T 2 u(θ, φ) δ(t τ) N th (. ). ( ) ant. β exp j 2πFc c r T N u(θ, φ).δ(t τ) exp j 2πFc c r T N u(θ, φ) }{{} array manifold vector S (θ,φ) = β.s(θ, φ).δ(t τ) (38) i.e. β Tx,1 β Tx,2. β Tx,N = β.s(θ, φ) (39) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 30 / 80

31 Wireless SIMO Channels Rx Antenna Array where ( ) exp j 2πF c c r T 1 ( u(θ, φ) ) exp j 2πF c c r T 2 u(θ, φ) S(θ, φ) =... ( ) exp j 2πF c c r T k u(θ, φ) (... ) exp j 2πF c c r T N u(θ, φ) ( = exp j 2πF ) c [ ] T r c 1, r 2,... r N u(θ, φ) ( = exp j 2πF ) c [ ] r c x, r y, r z u(θ, φ) (40) (41) (42) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 31 / 80

32 Wireless SIMO Channels Rx Antenna Array Multipath SIMO Channel Let us assume that the transmitted signal arrives at the reference point of an array receiver via L paths (multipaths). Consider that the l th path arrives at the array from direction (θ l, φ l ) with channel propagation parameters β l and τ l representing the complex path gain and path-delay, respectively. Note that θ l and φ l represent the azimuth and elevation angles respectively associated with l-th path. Let us assume that the L paths are arranged such that τ 1 τ 2... τ l... τ L (43) Furthermore, the path coeffi cients β l model the effects of path losses and shadowing, in addition to random phase shifts due to reflection; they also encompass the effects of the phase offset between the modulating carrier at the transmitter and the demodulating carrier at the receiver, as well as differences in the transmitter powers. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 32 / 80

33 Wireless SIMO Channels Rx Antenna Array The vector S l = S(θ l, φ l ) C N, is the array manifold vector of the l-th path. The impulse response (vector) of the SIMO multipath channel is SIMO: h(t) = L S(θ l, φ l ).β l.δ(t τ l ) l=1 (44) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 33 / 80

34 Wireless SIMO Channels Rx Antenna Array i.e. multipath SIMO channel β Tx,1 β Tx,2. β Tx,N L = β l.s(θ l, φ l ) (45) l=1 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 34 / 80

35 Wireless SIMO Channels Rx Antenna Array Scatterers in SIMO channels Consider a scatterer (l-th scatterer, say) which can be seen as a large number of paths (L scat, say) around the direction (θ l, φ l ). That is, the direction-of-arrival of the k-th path satisfies the condition (θ l, φ l ) θ l 2 (θ lk, φ lk ) (θ l, φ l ) + θ l, k (46) 2 In this case the impulse response for this scatterer can be written as follows: l-th scatterer = L scat S(θ lk, φ lk ).β lk.δ(t τ lk ) k=1 (47) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 35 / 80

36 Wireless SIMO Channels Rx Antenna Array If θ l = relatively small then and, thus, (θ l1, φ l1 ) (θ l2, φ l2 )... (θ llscat, φ llscat ) (θ l, φ l ) (48) τ l1 τ lk... τ llscat τ l (49) l-th scatterer = = L scat S(θ lk, φ lk ).β lk.δ(t τ lk ) k=1 L scat S(θ l, φ l ).β lk.δ(t τ l ) k=1 L scat = S(θ l, φ l ).δ(t τ l ) β lk k=1 }{{} β l = S(θ l, φ l ).β l.δ(t τ l ) = S l.β l.δ(t τ l ) (50) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 36 / 80

37 Wireless SIMO Channels Rx Antenna Array In other words a scatterer can be seen as a single path with direction (θ l, φ l ) and fading coeffi cient the term β l that represents the addition/combination of the fading coeffi cients of all paths of this scatterer i.e. L scat β l = β lk (51) k=1. N.B.: Another way to represent a scatterer is using Taylor Series Expansion. This will involve S l (i.e the first derivative of the manifold vector S l ) and θ l. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 37 / 80

38 Wireless SIMO Channels Rx Antenna Array Modelling of the Received Vector-Signal Consider a single Tx transmitting a baseband signal m(t) via an L-path SIVO channel. The received (N 1) vector-signal x(t) can be modelled as follows: where x(t) = h(t) m(t) + n(t) = ( L S(θ l, φ l ).β l.δ(t τ l ) l=1 ) m(t) + n(t) x(t) = L S(θ l, φ l ).β l.m(t τ l ) + n(t) l=1 = Sm(t) + n(t) (52) S = [ S 1, S 2,, S L ], with S l S(θ l, φ l ), l = 1,.., L(53) m(t) = [ β 1 m(t τ 1 ), β 2 m(t τ 2 ),, β L m(t τ L ) ] T (54) n(t) = [ n 1 (t), n 2 (t),, n N (t) ] T (55) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 38 / 80

39 Wireless SIMO Channels Rx Antenna Array Multi-user SIMO Next consider an Rx array of N antennas operating the the presence of M co-channel transmitters/users. In this case we have added the subscript i to refer to the i-th Tx. The received (N 1) vector-signal x(t) from all M transmitters/users (transmitting at the same time on the same frequency band) can be modelled as follows:. with x(t) = M L i=1 l=1 S il.β il.m i (t τ l ) + n(t) = Sm(t) + n(t) (56) S il S(θ il, φ il ) where S, m(t) and n(t) defined as follows: Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 39 / 80

40 Wireless SIMO Channels Rx Antenna Array S S 1 S 2 S M {}}{{}}{{}}{ S 11, S 12,..., S 1L, S 21, S 22,..., S 2L,..., S M 1, S M 2,..., S ML (57) m(t) β 11 m 1 (t τ 11 ) β 12 m 1 (t τ 12 ) m 1 (t). β 1L m 1 (t τ 1L ) β 21 m 2 (t τ 21 ).. β M 1 m M (t τ M 1 ) β M 2 m M (t τ M 2 ) m M (t). β ML m M (t τ ML ) n(t) [ n 1 (t), n 2 (t),..., n N (t) ] T (58) (59) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 40 / 80

41 Wireless MISO Channels Wireless MISO Channels Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 41 / 80

42 Wireless MISO Channels Reciprocity Theorem Reciprocity Theorem Antenna characteristics are independent of the direction of energy flow. The impedance & radiation pattern are the same when the antenna radiates a signal and when it receives it. The Tx and Rx array-patterns are the same. The Tx-array is an array of N elements/sensors/antennas with locations r r = [r 1, r 2,..., r N ] = [ r x, r y,r z ] T (3 N) with r m denoting the location of the m th Tx-sensor m = 1, 2,..., N Notation: the bar at the top of a symbol, i.e. (.), denotes a Tx-parameter. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 42 / 80

43 Wireless MISO Channels Reciprocity Theorem ( ) β exp +j 2π λ c u T i r m δ(t ρ c ) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 43 / 80

44 Wireless MISO Channels Reciprocity Theorem That is, if the Tx is displaced at a specific point r m (within its local area L A ) and the direction of the planewave propagation is described by the vector u i = u(θ, φ) where u i (θ,φ) = [ cos θ cos φ, sin θ cos φ, sin φ ] T If the Tx employs an array of N elements/sensors/antennas with locations r then the channel impulse response (single path) is where k(θ, φ) = h(t) = βs H δ(t ρ c ) (60) S(θ, φ) = exp(+j[r 1, r 2,..., r N ] T k(θ, φ)) (61) or = exp(+j[r x, r y, r z ]k(θ, φ)) (62) = (N 1) complex vector { 2πFc c.u(θ, φ) = 2π λ c.u(θ, φ) π.u(θ, φ) } in meters in units of halfwavelength = wavenumber vector Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 44 / 80

45 Wireless MISO Channels Multipath MISO Channel Multipath MISO Channel Let us assume that the transmitted signal(s) from the Tx-array arrives at the receiver via L resolvable paths (multipaths). Consider that the l th path s direction-of-departure is (θ l, φ l ) and propagates to the Rx with channel propagation parameters β l and τ l representing the complex path gain and path-delay, respectively. Let us assume that the L paths are arranged such that τ 1 τ 2... τ l... τ L (63) remember that the path coeffi cients β l model the effects of path losses and shadowing, in addition to random phase shifts due to reflection; they also encompass the effects of the phase offset between the modulating carrier at the transmitter and the demodulating carrier at the receiver, as well as differences in the transmitter power Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 45 / 80

46 Wireless MISO Channels Multipath MISO Channel The vector S l = S(θ l, φ l ) C N is the array manifold vector of the l-th path. The impulse response of the MISO channel is MISO: h(t) = L β l.s H l δ(t τ l ) l=1 (64) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 46 / 80

47 Wireless MISO Channels Scatterers Scatterers Consider a scatterer (l-th scatterer, say) which can be seen as a large number of paths (L l, say) around the direction (θ l, φ l ). That is, the direction-of-arrival of the k-th path satisfies the condition (θ l, φ l ) θ l 2 (θ lk, φ lk ) (θ l, φ l ) + θ l, k (65) 2 L l = number of paths (l th scatterer) In this case the impulse response for this scatterer can be written as follows: l-th scatterer = L l β lk.s H (θ lk, φ lk ).δ(t τ lk ) k=1 (66) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 47 / 80

48 Wireless MISO Channels Scatterers If θ l = relatively small then and, thus, (θ l1, φ l1 ) (θ l2, φ l2 )... (θ lll, φ lll ) (θ l, φ l ) (67a) τ l1 τ lk... τ lll τ l (67b) l-th scatterer = = L l β lk S H (θ lk, φ lk ).δ(t τ lk ) k=1 L l β lk.s H (θ l, φ l ).δ(t τ l ) k=1 = S H (θ l, φ l ).δ(t τ l ) L l β lk k=1 }{{} β l = S H (θ l, φ l )δ(t τ l ).β l = β l.s H l.δ(t τ l ) (68) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 48 / 80

49 Wireless MISO Channels Scatterers In other words a scatterer can be seen as a single path with direction of departure (θ l, φ l ) and fading coeffi cient the term β l that represents the addition/combination of the fading coeffi cients of all paths of this scatterer i.e. β l = L l β lk. k=1 N.B.: Another way to represent a scatterer is using Taylor Series. Expansion. This will involve S (which is the first derivative of the manifold vector S l ) and θ l Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 49 / 80

50 Wireless MISO Channels Modelling of the Rx Scalar-Signal x(t) Modelling of the Rx Scalar-Signal x(t) Consider a Tx-array of N antennas transmitting a baseband signal m(t) via MISO multipath channel of L resolvable paths (frequency selective MISO). We will consider the following two cases: Case-1: The m(t) is demultiplexed to N different signals (one signal per antenna element) forming the vector m(t). Case-2: All Tx-array elements transmit the same signal m(t). In both cases the transmitted signals may, or may not, be weighted. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 50 / 80

51 Wireless MISO Channels Modelling of the Rx Scalar-Signal x(t) Case-1: The received scalar-signal x(t) can be modelled as follows: x(t) = = L β l S H (θ l, φ l ). (δ(t τ l ) (w m(t)) + n(t) l=1 L β l S H l. (w m(t τ l )) + n(t) (69) l=1 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 51 / 80

52 Wireless MISO Channels Modelling of the Rx Scalar-Signal x(t) Case-2: The signal is "copied" to each antenna and the received scalar-signal x(t) can be modelled as follows: x(t) = = L β l S H (θ l, φ l ). (δ(t τ l ) (wm(t)) + n(t) l=1 L β l S H l w.m(t τ l ) + n(t) (70) l=1 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 52 / 80

53 Wireless MISO Channels Transmit Diversity Transmit Diversity Provides diversity benefits to a mobile using base station antenna array for frequency division duplexing (FDD) schemes. Cost is shared among different users. Order of diversity can be increased when used with other conventional forms of diversity (e.g. multipath diversity). Two main types of diversity combining techniques in 3G: Transmit diversity with feedback from receiver (close loop) Transmit diversity without feedback from receiver (open loop) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 53 / 80

54 Wireless MISO Channels Transmit Diversity Transmit Diversity: "Close Loop" The transmitter transmits some pilot signals The mobile (based on this pilot signals) estimates the Channel State Information (CSI), i.e. channel parameters. The mobile transmits the CSI to the BS (uplink) The base station generates the weights and transmits data to the mobile. That is a beamformer is formed which steers its main lobe towards the Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 54 / 80

55 Wireless MISO Channels Transmit Diversity UMTS 3GPP Standard (Close Loop) N = 2 Tx-antennas where w 1, w 2 are adjusted such as x(t) 2 is maximised, e.g. w 1, w 2 are adjusted based on the feedback information from the receiver Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 55 / 80

56 Wireless MISO Channels Transmit Diversity Transmit Diversity: "Open Loop" without spreader: with spreader: Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 56 / 80

57 Wireless MISO Channels Transmit Diversity Example of STBC Downlink Equations (without spreader): STBC = Space-Time Block Code No geometric/space information is used. Receiver input (L unresolvable multipaths - flat fading): L 1st interval: x 1 = j=1 2nd interval: x 2 = L j=1 ( ) β 1j,Rx m 1 β 2j,Rx m2 + n 1 ( ) β 1j,Rx m 2 + β 2j,Rx m1 + n 2 (71) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 57 / 80

58 Wireless MISO Channels Transmit Diversity where { x1 = β 1,Rx m 1 β 2,Rx m2 + n 1 x 2 = β 1,Rx m 2 + β 2,Rx m1 + n 2 β 1,Rx L β 1j,Rx j=1 and β 2,Rx (72) L β 2j,Rx (73) j=1 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 58 / 80

59 Wireless MISO Channels Transmit Diversity Equivalently, [ ] x1 x 2 [ m1, m = 2 ] [ ] β1,rx m 2, m1 + β 2,Rx }{{} β Rx [ ] n1 n 2 (74) or, in an alternative format: [ ] [ ][ ] [ ] x1 β1,rx, β x = x2 = 2,Rx m1 n1 β2,rx, β 1,Rx m2 + n 2 }{{}}{{}}{{} m n H (75) i.e. where x = Hm + n (76) ( β1,rx H H H = 2 2 ) + β 2,Rx }{{} 2 = β Rx I 2 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 59 / 80

60 Wireless MISO Channels Transmit Diversity Decoder (Rx): This is denoted by the matrix H decoder s o/p : G = H H x (77) = H H Hm + Hn }{{} ñ (78) That is, the decision variables are 2 G = β m Rx + ñ (79) i.e. Note: { 2 G1 = β m1 Rx + ñ 1 2 (80) G 2 = β m Rx 2 + ñ 2 1 the receiver needs to know (estimate) the channel weights β 1,Rx and β 2,Rx but there is no need to send them back to the transmitter (i.e. open loop) 2 β 1,Rx and β 2,Rx can be estimated by transmitting some pilot symbols as m 1 and m 2 and, then, using Equation 74 Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 60 / 80

61 Wireless MIMO Channels Wireless MIMO Channels Consider a single path from a Tx-array of N antennas to an Rx-array of N antennas with locations given by the matrices Tx-array: r = [r 1, r 2,..., r N ] = [ r x, r y, r z ] T (3 N) Rx-array: r = [r 1, r 2,..., r N ] = [ r x, r y, r z ] T (3 N) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 61 / 80

62 Wireless MIMO Channels If the direction-of-departure of this single path planewave propagation is ( θ, φ ) and the direction-of-arrival is (θ, φ) then the impulse response is h(t) = βs(θ, φ).s H (θ, φ).δ(t ρ c ) where Tx: Rx: S = S ( θ, φ ) ( ) = exp +jr T k(θ, φ) ( ) S = S (θ, φ) = exp jr T k(θ, φ) (81) (82) with k(θ, φ) and k(θ, φ) denote the wavenumber vectors of the Tx-array and Rx-array respectively For instance k(θ, φ) is defined as k(θ, φ) = { 2πFc c.u(θ, φ) = 2π λ c.u(θ, φ) in meters π.u(θ, φ) in units of halfwavelength u ( θ, φ ) = [ cos θ cos φ, sin θ cos φ sin φ ] } Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 62 / 80

63 Wireless MIMO Channels Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 63 / 80

64 Wireless MIMO Channels Multipath MIMO Channel Multipath MIMO Channel Let us assume that the transmitted signal(s) from the Tx-array arrives at the Rx-array via L resolvable paths (multipaths). Consider that the l th path s direction-of-departure is (θ l, φ l ) and propagates to the Rx with channel propagation parameters β l and τ l representing the complex path gain and path-delay, respectively. Let us assume that the L paths are arranged such that τ 1 τ 2... τ l... τ L (83) once again, remember that the path coeffi cients β l model the effects of path losses and shadowing, in addition to random phase shifts due to reflection; they also encompass the effects of the phase offset between the modulating carrier at the transmitter and the demodulating carrier at the receiver, as well as differences in the transmitter power. In the following figure the vectors S l = S(θ l, φ l ) C N and S l = S(θ l, φ l ) C N are the Tx- and Rx- array manifold vectors of the l-th path. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 64 / 80

65 Wireless MIMO Channels Multipath MIMO Channel The impulse response of the MIMO channel is MIMO: h(t) = L β l.s l.s H l δ(t τ l ) l=1 (84) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 65 / 80

66 Wireless MIMO Channels Scatterers Scatterers Consider a scatterer (l-th scatterer, say) which can be seen as a large number of paths (L l, say) with directions-of-departure around the direction (θ l, φ l ) and directions-of-arrival around the direction (θ l, φ l ). Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 66 / 80

67 Wireless MIMO Channels Scatterers That is, the direction-of-departure and direction-of-arrival of the k-th path satisfies the condition (θ l, φ l ) θ l 2 (θ l, φ l ) θ l 2 (θ lk, φ lk ) (θ l, φ l ) + θ l, k (85) 2 (θ lk, φ lk ) (θ l, φ l ) + θ l, k (86) 2 In this case the impulse response for this scatterer can be written as follows: l-th scatterer = L l β lk.s(θ lk, φ lk )S H (θ lk, φ lk ).δ(t τ lk ) (87) k=1 If θ l and θ l are relatively small then (θ l1, φ l1 ) (θ l2, φ l2 )... (θ lll, φ lll ) (θ l, φ l ) (88) (θ l1, φ l1 ) (θ l2, φ l2 )... (θ lll, φ lll ) (θ l, φ l ) (89) τ l1 τ lk... τ lll τ l (90) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 67 / 80

68 Wireless MIMO Channels Scatterers Thus, if L l denotes the No. of paths of the l-th scatterer, l-th scatterer = = L l β lk.s(θ lk, φ lk )S H (θ lk, φ lk ).δ(t τ lk ) k=1 L l β lk.s(θ l, φ l )S H (θ l, φ l ).δ(t τ l ) k=1 = S(θ l, φ l )S H (θ l, φ l ).δ(t τ l ) = β l.s(θ l, φ l )S H (θ l, φ l ).δ(t τ l ) L l β lk k=1 }{{} β l (91) = β l.s l.s H l.δ(t τ l ) (92) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 68 / 80

69 Wireless MIMO Channels Scatterers In other words a scatterer can be seen as a single path with direction (θ l, φ l ) and fading coeffi cient the term β l that represents the addition/combination of the fading coeffi cients of all paths of this scatterer i.e. β l = L l β lk. k=1 N.B.: Again, another way to represent a scatterer is using Taylor. Series Expansion. This will involve Ṡ l, S l (i.e the first derivative of the manifold vector S l ), θ l and θ l. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 69 / 80

70 Wireless MIMO Channels Modelling of the Rx Vector-Signal x (t) Modelling of the Rx Vector-Signal x(t) Consider a Tx-array of N antennas transmitting a baseband signal m(t) via VIVO multipath channel of L resolvable paths (frequency selective VIVO). We will consider the following two cases: Case-1: The m(t) is demultiplexed to N different signals (one signal per antenna element) forming the vector m(t). Case-2: All Tx-array elements transmit the same signal m(t). In both cases the transmitted signals may, or may not, be weighted. The Rx is also equipped with an array of N antennas. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 70 / 80

71 Wireless MIMO Channels Modelling of the Rx Vector-Signal x (t) Case-1: The received vector-signal x(t) can be modelled as follows: x(t) = L l=1 β l.s l.s H l. (δ(t τ l ) (w m(t)) + n(t) = L l=1 β l.s l.s H l. (w m(t τ l )) + n(t) (93) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 71 / 80

72 Wireless MIMO Channels Modelling of the Rx Vector-Signal x (t) Case-2: In this case the signal is "copied" to each antenna and may be weighted by a complex weight. The received vector-signal x(t) can be modelled as follows: x(t) = L l=1 β l S ls H l. (δ(t τ l ) (wm(t))) + n(t) = L l=1 β l S ls H l w.m(t τ l ) + n(t) (94) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 72 / 80

73 Wireless MIMO Channels MIMO Systems (without geometric information) MIMO Systems (without geometric information) Let us consider a comm. system with multiple antennas at both the Tx and the Rx. Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 73 / 80

74 Wireless MIMO Channels MIMO Systems (without geometric information) β i,j = gain from the j th Tx-antenna to the i th Rx-antenna β j,tx = gain-vector with its i th element the gain β ij (i.e. from the j-th Tx antenna to all the Rx antennas) If Rx is synchronised to the Tx then for the n th data symbol interval then we have the following received vector-signal: x[n] = Hm[n] + n[n] (N 1) where ] H = [β, β,..., β 1,Tx 2,Tx N,Tx Second order statistics (covariance matrix) of x[n] is as follows: R xx = HR mm H H + R nn }{{} σ 2 ni N (N N) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 74 / 80

75 Wireless MIMO Channels MIMO Systems (without geometric information) In a MIMO system it is assumed that the Matrix H is known Note: The matrix H and the manifold vectors are related according to the following expression H = L l=1 β l S l S H l (95) β 1, 0,..., 0 = S 0, β 2,..., 0...,...,...,... SH (96) 0, 0,..., β L where S = [ S 1, S 2,..., S L ] S = [ S 1, S 2,..., S L ] Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 75 / 80

76 Wireless MIMO Channels Capacity - General Expressions Capacity - of MIMO Channels (without space info) SISO Capacity: General MIMO Capacity expression: C = B log 2 (1 + SNIR in ) bits/s (97) ( ) det (Rxx ) C = B log 2 det (R nn ) bits/s (98) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 76 / 80

77 Wireless MIMO Channels Capacity - General Expressions Based on Equation 98 it can be easily proven that ( C = B log 2 (det I N + 1 )) σ 2 HR mm H H bits/s (99) n Furthermore, for independent parallel channels (i.e. using a multiplexer at Tx, Case-1) : P 1, 0,..., 0 R mm = diagonal = 0, P 2,..., 0...,...,...,... 0, 0,..., P N and thus Equation 99 is simplified to ( C = B log 2 ( N j=1 = B 1 + N log 2 (1 + j=1 2 β Pj j,tx σ 2 n 2 β Pj j,tx σ 2 n ) )) (100) bits/sec (101) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 77 / 80

78 Wireless MIMO Channels Equivalence between MIMO and SIMO/MISO Equivalence between MIMO and SIMO spatial convolution MIMO virtual-simo or MISO Tx antennas:n, Rx antennas:n N N Rx /Tx antennas Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 78 / 80

79 Wireless MIMO Channels Equivalence between MIMO and SIMO/MISO Equivalence between MIMO and SIMO (cont.) Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 79 / 80

80 Wireless MIMO Channels Equivalence between MIMO and SIMO/MISO Equivalence between MIMO and SIMO (cont.) Tx-array: r [r 1, r 2,..., r N ] = [ r x, r y, r z ] T (3 N) Rx-array: r [r 1, r 2,..., r N ] = [ r x, r y, r z ] T (3 N) virtual-array : r virtual r 1 T N + 1 T N r (102) S virtual S S (103) Note : Equation 102 is known as "spatial convolution" Prof. A. Manikas (Imperial College) EE.401: Multi-Antenna Wireless Comms v.16c1 80 / 80