Upload
buidang
View
214
Download
0
Embed Size (px)
Citation preview
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Lecture 6: Modeling of MIMO ChannelsTheoretical Foundations of Wireless Communications1
Lars KildehøjCommTh/EES/KTH
Wednesday, May 11, 20169:00-12:00, Conference Room SIP
1Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication1 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Overview
Lecture 5: Spatial Diversity, MIMO Capacity
• SIMO, MISO, MIMO
• Degrees of freedom
• MIMO capacity
Lecture 6: MIMO Channel Modeling
2 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Overview
Motivation:
• How does the multiplexing capability of MIMO channels depend onthe physical environment?
• When can we gain (much) from MIMO?
• How do we have to design the system?
3 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Line-of-Sight Channels: SIMO• Free space without scattering
and reflections.
• Antenna separation ∆rλc , withcarrier wavelength λc and thenormalized antenna separation∆r ; nr receive antennas.
• Distance between transmitterand i-th receive antenna: di
(D. Tse and P. Viswanath, Fundamentals of Wireless Communi-
cations.)
• Continuous-time impulse between transmitter and i-th receiveantenna:
hi (τ) = a · δ(τ − di/c)
• Base-band model (assuming di/c 1/W , signal BW W ):
hi = a · exp
(−j 2πfcdi
c
)= a · exp
(−j 2πdi
λc
)• SIMO model: y = h · x + w, with w ∼ CN (0,N0I)
→ h: signal direction, spatial signature.
4 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Line-of-Sight Channels: SIMO
• Paths are approx. parallel, i.e.,
di ≈ d + (i − 1)∆rλc cos(φ)
• Directional cosine
Ω = cos(φ) (D. Tse and P. Viswanath, Fundamentals of Wireless Communi-
cations.)
• Spatial signature can be expressed as
h = a · exp
(−j 2πd
λc
)
1exp(−j2π∆rΩ)
exp(−j2π2∆rΩ)...
exp(−j2π(nr − 1)∆rΩ)
→ Phased-array antenna.
• SIMO capacity (with MRC)
C = log
(1 +
P‖h‖2
N0
)= log
(1 +
Pa2nrN0
)→ Only power gain, no degree-of-freedom gain.
5 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Line-of-Sight Channels: MISO
• Similar to the SIMO case:∆t , λc , di , φ, Ω,...
• MISO channel model:
y = h∗x + w ,
with w ∼ CN (0,N0). (D. Tse and P. Viswanath, Fundamentals of Wireless Communi-
cations.)
• Channel vector
h = a · exp
(j
2πd
λc
)
1exp(−j2π∆tΩ)
exp(−j2π2∆tΩ)...
exp(−j2π(nt − 1)∆tΩ)
• Unit spatial signature in the directional cosine Ω:
e(Ω) = 1/√n · [1, exp(−j2π∆Ω), . . . , exp(−j2π(n − 1)∆Ω)]T
→ et(Ωt) and er(Ωr ) with nt ,∆t and nr ,∆r , respectively.
6 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Line-of-Sight Channels: MIMO
• Linear transmit and receive array with nt ,∆t and nr ,∆r .
• Gain between transmit antenna k and receive antenna i
hik = a · exp (−j2πdik/λc)
• Distance between transmit antenna k and receive antenna i
dik = d + (i − 1)∆rλc cos(φr )− (k − 1)∆tλc cos(φt)
• MIMO channel matrix (with Ωt = cos(φt) and Ωr = cos(φr ))
H = a√ntnr exp
(−j 2πd
λc
)er(Ωr )et(Ωt)
∗
→ H is a rank-1 matrix with singular value λ1 = a√ntnr
→ Compare with SVD decomposition in Lecture 5:
H =k∑
i=1
λiuiv∗i
7 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Line-of-Sight Channels: MIMO
• MIMO capacity
C = log
(1 +
Pa2nrntN0
)→ Only power gain, no degree-of-freedom gain.
• nt = 1: power gain equals nr → receive beamforming.
• nr = 1: power gain equals nt → transmit beamforming.
• General nt , nr : power gain equals nr · nt→ Transmit and receive beamforming.
• Conclusion: In LOS environment, MIMO provides only a power gainbut no degree-of-freedom gain.
8 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Geographically Separated Antennas at the Transmitter
Example/special case
(D. Tse and P. Viswanath, Fundamentals of
Wireless Communications.)
• 2 distributed transmit antennas,attenuations a1, a2, angles of incidenceφr 1, φr 2, negligible delay spread.
• Spatial signature (nr receive antennas)
hk = ak√nr exp
(−j 2πd1k
λc
)er(Ωrk)
• Channel matrix H = [h1, h2]
• H has independent columns as long as (Ωr i = cos(φr i ))
Ωr = Ωr 2 − Ωr 1 6= 0 mod1
∆r
→ Two non-zero singular values λ21, λ
22; i.e., two degrees of freedom.
→ But H can still be ill-conditioned!
9 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Geographically Separated Antennas at the Transmitter
• Conditioning of H is determined by how the spatial signatures arealigned (with Lr = nr∆r ):
| cos(θ)| = |fr (Ωr2−Ωr1)| = | er(Ωr1)∗er(Ωr2)︸ ︷︷ ︸=fr (Ωr2−Ωr1)
| =
∣∣∣∣ sin(πLrΩr )
nr sin(πLrΩr/nr )
∣∣∣∣• Example (a1 = a2 = a)
λ21 = a2nr (1 + | cos(θ)|)λ2
2 = a2nr (1− | cos(θ)|) ⇒λ1
λ2=
√1 + | cos(θ)|1− | cos(θ)|
(D. Tse and P. Viswanath, Fundamentals of Wireless Commu-
nications.)
• fr (Ωr ) is periodic with nr/Lr .
• Maximum at Ωr = 0; fr (0) = 1.
• fr (Ωr ) = 0 at Ωr = k/Lr withk = 1, . . . , nr − 1.
• Resolvability 1/Lr ,if Ωr 1/Lr , then the signalsfrom the two antennas cannotbe resolved.
10 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Geographically Separated Antennas at the Transmitter
Beamforming pattern
(D. Tse and P. Viswanath, Fundamentals of Wireless
Communications.)
• Assumption: signal arrives withangle φ0; receive beamformingvector er(cos(φ0)).
• A signal form any other direction φwill be attenuated by a factor
|er(cos(φ0))∗er(cos(φ))|= |fr (cos(φ)− cos(φ0))|
• Beamforming pattern
( φ, |fr (cos(φ)− cos(φ0))| )
• Main lobes around φ0 and any angle φ for which cos(φ) = cos(φ0).
→ In a similar way, separated receive antennas can be treated.
11 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– LOS Plus One Reflected Path
• Direct path:φt1, Ωr1, d (1), and a1.
• Reflected path:φt2, Ωr2, d (2), and a2.
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
• Channel model follows from signal superposition
H = ab1er(Ωr1)et(Ωt1)∗ + ab2er(Ωr2)et(Ωt2)∗,
with
abi = ai√ntnr exp
(−j 2πd (i)
λc
).
→ H has rank 2 as long as
Ωt1 6= Ωt2 mod1
∆tand Ωr1 6= Ωr2 mod
1
∆r.
→ H is well conditioned if the angular separations |Ωt |, |Ωr | at thetransmit/receive array are of the same order or larger than 1/Lt,r .
12 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– LOS Plus One Reflected Path
• Direct path:φt1, Ωr1, d (1), and a1.
• Reflected path:φt2, Ωr2, d (2), and a2.
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
• H can be rewritten as H = H′′H′, with
H′′ = [ab1er(Ωr1), ab2er(Ωr2)] and H′ =
[et∗(Ωt1)
et∗(Ωt2)
]→ Two imaginary receivers at points A and B (virtual relays).
• Since the points A and B are geographically widely separated, H′
and H′′ have rank 2 and hence H has rank 2 as well.
• Furthermore, if H′ and H′′ are well-conditioned, H will bewell-conditioned as well.
→ Multipath fading can be viewed as an advantage which can beexploited!
13 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– LOS Plus One Reflected Path
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
• Significant angular separation is required at both the transmitterand the receiver to obtain a well-conditioned matrix H.
• If the reflectors are close to the receiver (downlink), we have a smallangular separation ⇒ not very well-conditioned matrix H.
• Similar, if the reflectors are close to the transmitter (uplink).
→ Size of an antenna array at a base station will have to be manywavelengths to be able to exploit the spatial multiplexing effect.
14 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Physical Modeling– Summary
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
15 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– General Concept
• Antenna lengths Lt , Lr limit the resolvability2 of the transmit andreceive antenna in the angular domain.
→ Sample the angular domain at fixed angular spacings of 1/Lt at thetransmitter and 1/Lr at the receiver.
→ Represent the channel (the multiple paths) in terms of these inputand output coordinates.
• The (k, l)-th channel gainfollows as the aggregation of allpaths whose transmit andreceive directional cosines lie ina (1/Lt × 1/Lr ) bin around thepoint (l/Lt , k/Lr ).
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
2Note, if Ωr,t 1/Lr,t , the paths cannot be separated.16 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Angular Domain Representation (ADR)
• Orthonormal basis for the received signal space (nr basis vectors)
Sr =
er(0), er(
1
Lr), . . . , er(
nr − 1
Lr)
→ Orthogonality follows directly from the properties of fr (Ω).
• Orthonormal basis for the transmitted signal space (nt basis vectors)
St =
et(0), et(
1
Lt), . . . , et(
nt − 1
Lt)
→ Orthogonality follows directly from the properties of ft(Ω).
• Orthonormal bases provide a very simple (but approximate)decomposition of the total received/transmitted signal up to aresolution 1/Lr , 1/Lt .
17 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Angular Domain Representation
Examples: Receive beamform patterns of the angular basis vectors in Sr
• (a) Critically spaced (∆r = 1/2),each basis vector has a single pairof main lobes.
• (b) Sparsely spaced (∆r > 1/2),some of the basis vectors havemore than one pair of main lobes.
• (c) Densely spaced (∆r < 1/2),some of the basis vectors have nopair of main lobes.
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
18 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– ADR of MIMO Channels
(Assumption: critically spaced antennas)
• Observation: The vectors in St and Sr form unitary matrices Ut andUr with dimensions (nt × nt) and (nr × nr ), respectively. (→ IDFTmatrices!)
• With3 xa = U∗t x and ya = U∗r y we get
ya = U∗r HUtxa + U∗r w = Haxa + wa, with wa ∼ CN (0,N0Inr ).
• Furthermore, with H =∑
i abi er(Ωri )et(Ωti )
∗, we get
hakl = er(k/Lr )∗Het(l/Lt)
=∑i
abi [er(k/Lr )∗er(Ωri )]︸ ︷︷ ︸(1)
· [et(Ωti )∗et(l/Lt)]︸ ︷︷ ︸(2)
• The terms (1) and (2) are significant for the i − th path if∣∣∣∣Ωri −k
Lr
∣∣∣∣ < 1
Lrand
∣∣∣∣Ωti −k
Lt
∣∣∣∣ < 1
Lt.
(→ Projections on the basis vectors in Sr , St .)
3The superscript “a” denotes angular domain quantities.19 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Statistical Modeling in the Angular Domain
• Let Tl and Rk be the sets of physical paths which have most energyin directions of et(l/Lt) and er(k/Lr ).
• hakl corresponds to the aggregated gains abi of paths which lie inRk ∩ Tl .
• Independence and time variation• Gains of the physical paths abi [m] are independent.
⇒ The path gains hakl [m] are independent across m.
• The angles φri [m]m and φti [m]m evolve slower than abi [m].
⇒ The physical paths do not move from one angular bin to another.
⇒ The path gains hkl [m] are independent across k and l .
• If there are many paths in an angular bin ⇒ Central Limit Theorem
⇒ hakl [m] can be modeled as complex circular symmetric Gaussian.
• If there are no paths in an angular bin ⇒ hakl [m] ≈ 0.
• Since Ut and Ur are unitary matrices, the matrix H has the samei.i.d. Gaussian distribution as Ha.
20 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Statistical Modeling in the Angular Domain
Example for Ha
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
• (a) Small angular spread at the transmitter.
• (b) Small angular spread at the receiver.
• (c) Small angular spread at both transmitter and receiver.
• (d) Full angular spread at both transmitter and receiver.
21 / 1
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Degrees of Freedom and Diversity
Degrees of freedom
• Based on the derived statistical model, we get the following result:with probability 1, the rank of the random matrix Ha is given by
rank(Ha) = min number of non-zero rows,
number of non-zero columns .
• The number of non-zero rows and columns depends on two factors:• Amount of scattering and reflection; the more scattering and
reflection, the larger the number of non-zero entries in Ha.
• Lengths Lr and Lt ; for small Lr , Lt many physical paths are mappedinto the same angular bin; with higher resolution, more paths can berepresented.
Diversity: The diversity is given by the number of non-zero entries in Ha.Example: Same number of degrees of freedom but different diversity.
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)
22 / 1
Notes
Notes
Lecture 6Modeling of MIMO
Channels
Lars KildehøjCommTh/EES/KTH
Modeling of MIMO Fading Channels– Antenna Spacing
So far: critically spaced antennas with ∆r = 1/2.
• One-to-one correspondence between the angular windows and theresolvable bins.
Setup 1: vary the number of antennas for a fixed array length Lr,t .• Sparsely spaced case (∆r > 0.5)
• Beamforming patterns of some basis vectors have multiple mainlobes.
• Different paths with different directions are mapped onto the samebasis vector.
→ Resolution of the antenna array, number of degrees of freedom, anddiversity are reduced.
• Densely spaced case (∆r < 0.5)• There are basis vectors with no main lobes which do not contribute
to the resolvability.
• Adds zero rows and columns to Ha and creates correlation in H.
Setup 2: vary the antenna separation for a fixed number of antennas.
• Rich scattering: number of non-zero rows in Ha is already nr ; i.e.,no improvement possible.
• Clustered scattering: scattered signal can be received in more bins;i.e., increasing number of degrees of freedom.
23 / 1
Notes
Notes