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Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Lecture 6: Modeling of MIMO Channels Theoretical Foundations of Wireless Communications1

Lars Kildehøj CommTh/EES/KTH Wednesday, May 11, 2016 9:00-12:00, Conference Room SIP

1Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Overview

Lecture 5: Spatial Diversity, MIMO Capacity

• SIMO, MISO, MIMO
• Degrees of freedom
• MIMO capacity

Lecture 6: MIMO Channel Modeling

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Overview

Motivation:

• How does the multiplexing capability of MIMO channels depend on

the physical environment?

• When can we gain (much) from MIMO?
• How do we have to design the system?

3 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Physical Modeling

– Line-of-Sight Channels: SIMO

• Free space without scattering

and reflections.

• Antenna separation ∆rλc, with

carrier wavelength λc and the normalized antenna separation ∆r; nr receive antennas.

• Distance between transmitter

and i-th receive antenna: di

(D. Tse and P. Viswanath, Fundamentals of Wireless Communi- cations.)

• Continuous-time impulse between transmitter and i-th receive

antenna: 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.

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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

• 

      1 exp(−j2π∆rΩ) exp(−j2π2∆rΩ) . . . exp(−j2π(nr − 1)∆rΩ)        → Phased-array antenna.

• SIMO capacity (with MRC)

C = log

• 1 + Ph2

N0

• = log
• 1 + Pa2nr

N0

• → Only power gain, no degree-of-freedom gain.

5 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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

• 

      1 exp(−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.

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Physical Modeling

– Line-of-Sight Channels: MIMO

• MIMO capacity

C = log

• 1 + Pa2nrnt

N0

• → 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 gain

but no degree-of-freedom gain.

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 incidence φr 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 = 0 mod 1 ∆r → Two non-zero singular values λ2

1, λ2 2; i.e., two degrees of freedom.

→ But H can still be ill-conditioned!

9 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Physical Modeling

– Geographically Separated Antennas at the Transmitter

• Conditioning of H is determined by how the spatial signatures are

aligned (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)

λ2

1

= 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 with

k = 1, . . . , nr − 1.

• Resolvability 1/Lr,

if Ωr ≪ 1/Lr, then the signals from the two antennas cannot be resolved.

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Physical Modeling

– Geographically Separated Antennas at the Transmitter

Beamforming pattern

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

• Assumption: signal arrives with

angle φ0; receive beamforming vector er(cos(φ0)).

• A signal form any other direction φ

will 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 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 = ab

1er(Ωr1)et(Ωt1)∗ + ab 2er(Ωr2)et(Ωt2)∗,

with ab

i = ai

√ntnr exp

• −j 2πd(i)

λc

• .

→ H has rank 2 as long as Ωt1 = Ωt2 mod 1 ∆t and Ωr1 = Ωr2 mod 1 ∆r . → H is well conditioned if the angular separations |Ωt|, |Ωr| at the transmit/receive array are of the same order or larger than 1/Lt,r.

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Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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′′ = [ab

1er(Ωr1), ab 2er(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 be

well-conditioned as well. → Multipath fading can be viewed as an advantage which can be exploited!

13 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 transmitter

and the receiver to obtain a well-conditioned matrix H.

• If the reflectors are close to the receiver (downlink), we have a small

angular 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 many wavelengths to be able to exploit the spatial multiplexing effect.

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Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Physical Modeling

– Summary

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 15 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH

Modeling of MIMO Fading Channels

– General Concept

• Antenna lengths Lt, Lr limit the resolvability2 of the transmit and

receive antenna in the angular domain. → Sample the angular domain at fixed angular spacings of 1/Lt at the transmitter and 1/Lr at the receiver. → Represent the channel (the multiple paths) in terms of these input and output coordinates.

• The (k, l)-th channel gain

follows as the aggregation of all paths whose transmit and receive directional cosines lie in a (1/Lt × 1/Lr) bin around the point (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 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 a resolution 1/Lr, 1/Lt.

17 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 pair

• f main lobes.
• (b) Sparsely spaced (∆r > 1/2),

some of the basis vectors have more than one pair of main lobes.

• (c) Densely spaced (∆r < 1/2),

some of the basis vectors have no pair of main lobes.

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 18 / 1

Notes Notes

Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 and

Ur with dimensions (nt × nt) and (nr × nr), respectively. (→ IDFT matrices!)

• 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 ab i er(Ωri)et(Ωti)∗, we get

ha

kl

= er(k/Lr)∗H et(l/Lt) =

• i

ab

i [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

Lr and

• Ωti − k

Lt

• < 1

Lt .

(→ Projections on the basis vectors in Sr, St.)

3The superscript “a” denotes angular domain quantities. 19 / 1 Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 energy

in directions of et(l/Lt) and er(k/Lr).

• ha

kl corresponds to the aggregated gains ab i of paths which lie in

Rk ∩ Tl.

• Independence and time variation
• Gains of the physical paths ab

i [m] are independent.

⇒ The path gains ha

kl[m] are independent across m.

• The angles {φri[m]}m and {φti[m]}m evolve slower than ab

i [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

⇒ ha

kl[m] can be modeled as complex circular symmetric Gaussian.

• If there are no paths in an angular bin ⇒ ha

kl[m] ≈ 0.

• Since Ut and Ur are unitary matrices, the matrix H has the same

i.i.d. Gaussian distribution as Ha.

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Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 mapped

into the same angular bin; with higher resolution, more paths can be represented.

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 6 Modeling of MIMO Channels Lars Kildehøj CommTh/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 the

resolvable 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 main

lobes.

• Different paths with different directions are mapped onto the same

basis vector. → Resolution of the antenna array, number of degrees of freedom, and diversity 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.

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Notes Notes