exploiting sparsity for wideband mmwave channel...

Exploiting sparsity for wideband mmWave channel estimation in IEEE 802.11ad

Professor Robert W. Heath Jr.

Department of Electrical and Computer EngineeringThe University of Texas at Austin

www.profheath.org

JointworkwithKiranVenugopal, AhmedAlkhateeb andNuriaGonzalez-Prelcic

Thanks to the National Science Foundation Grant No. NSF-CCF-1319556, and the Intel/Verizon 5G program.

WLAN at 60 GHz

u Wireless local area networking (WLAN)ª Gbps peak throughputsª In-room local area networkingª Cable replacement

u Chipsets are available and some products are shippingu Next gen is currently in development (802.11ay)****

ª Will support MIMO spatial multiplexingª Channel bonding for even larger bandwidthsª Targets 100 Gbps data rates

2

Standard Bandwidth Rates Approval Date

IEEE 802.11ad 2.16 GHz 6.76 Gbps Dec. 2012

Wilocity’s chipset**

Tensorcom’s chipset***

• http://nitero.com/ ** http://wilocity.com *** http://www.tensorcom.com/ **** http://www.ieee802.org/11/Reports/ng60_update.htm

Nitero chipset*

3

IEEE 802.11ad PHY structures

Control (required)

OFDM

Low Power SC

Single Carrier(certain modes required)

½-LDPCencoder

𝜋/2-BPSKmodulation

32x spreading

LDPC encoder(1/2,5/8,3/4,13/16)

2x repetition(header & MCS1 only)

Blocking GI insert Modulation

RS(224,208)(N,8) block

code(N = 16,12,9,8)

Blocking GI insert Modulation7x8 block

interleaver

LDPC encoder (1/2,5/8,3/4,13/16)

3x repetition(header only)

IFFT ModulationCarrier mapping

*IEEE Standard for Information technology–Telecommunications and information exchange between systems–Local and metropolitan are networks–Specific requirements-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Am, pp. 1–628, 2012.

used when beams not configured

IEEE 802.11ad PHY header structure

4

Short Training Field (STF) for AGC, time and freq. sync.

Beamforming (BF) Training provides opportunity to

refine or test beams

All preambles are 𝜋/2-BPSK modulated with 32x spreading

STF + Channel Estimation Field (CEF) for identifying PHY format

Header provides decoding information

*T. Rappapport, R. W. Heath, Jr., R. Daniels, J. N. Murdock, Millimeter Wave Wireless Communications, Pearson Education, Inc. 2014

u Complimentary Golay sequences &u Zero-sidelobe property such that:

thereforeu Can also be used for frequency offset estimation

IEEE 802.11ad PHY header features

5

CPHY STF CPHY CEF

-AB -B… BB B -B -AA -B-B -A B -A

48

Other STFSC CEF

OFDM CEF

Orderingindicates the

PHY typeA A … A A -A

16

Correlator A output

Correlator B output

16 A spikes = non-control

-B, -A, B, -A = SC

T. Rappapport, R. W. Heath, Jr., R. Daniels, J. N. Murdock, Millimeter Wave Wireless Communications, Pearson Education, Inc. 2014 * Figure source: Agilent Technologies, “Wireless LAN at 60 GHz – IEEE 802.11ad Explained,” May 2013.

-B -AA -B-B -A B -A

-B -AA -B -B -A B -A

IEEE 802.11ad beamformingu 802.11ad devices my have up to 128 sectors (64 per “antenna”)

ª No codebook: defining sectors/beams is left to implementation

u Two stages of beamformingª Sector sweep

ª Beam refinement• BF Training fields appended to data fields allows eitherTX or RX to test new beam• If TX is testing new beam, RX sends feedback on success or failure of new beam

6

Sector sweep feedbackSector sweep ACK

*T. Rappapport, R. W. Heath, Jr., R. Daniels, J. N. Murdock, Millimeter Wave Wireless Communications, Pearson Education, Inc. 2014

Transmitter tests sectors/beams while receiver listens in omni mode

The roles reverseFeedback occurs

Only single stream, no MIMO

MIMO architecture for <6 GHz WLANs

7

BasebandPrecoding

MIMOCombining

and Equalization

ADC

ADC

ADC

RFChain

RFChain

RFChain

2 to 8 antennas

# antennas = # RF = # pairs ADCs

Bandwidths of 5-100 MHz

MIMOPrecoding

DAC

DAC

DACRF

Chain

RFChain

RFChain

MIMO architectures at mmWave are different

MIMO architectures at mmWave: analog beamforming

8

RFChain

Phase shifters

RFainDAC BasebandBaseband RFChain

RFainADC

beamformer combiner

IEEE 802.11ad uses analog-only beamforming to achieve Gbps throughput

Phase shifters apply for the entire band

Limited to single stream and single user MIMO

* J.Wang, Z. Lan, C. Pyo, T. Baykas, C. Sum, M. Rahman, J. Gao, R. Funada, F. Kojima, H. Harada et al., “Beam codebook based beamforming protocol for multi-Gbpsmillimeterwave WPAN systems,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 8, pp. 1390–1399, 2009.** S. Hur, T. Kim, D. Love, J. Krogmeier, T. Thomas, and A. Ghosh, “Millimeter wave beamforming for wireless backhaul and access in small cell networks,” IEEE Transactions on Communications, vol. 61, no. 10, pp. 4391–4403, 2013.

MIMO architectures at mmWave: hybrid precoding

9

BasebandPrecoding

1-bitADC

DAC

1-bitADCDAC

RFChain

RF Precoding

1-bitADCDAC

1-bitADCDAC

BasebandCombining

Nt NrLt LrNs Ns

RF Combining

FBB FRF WBBWRF

RFChain

RFChain

RFChain

Combine analog and digital precoding

Large antenna arrays at TX/RX for sufficient

received power

IEEE 802.11ay may use hybrid precoding for multi-stream and multi-user MIMO

# of DACs/ADCs << # of antennas

MmWave channel estimation with hybrid architecture

10Channel estimates are an alternative to beam training, work with mulli-stream

Large bandwidth à frequency selective channel

Low link SNR without beamforming

BasebandPrecoding

BasebandCombining

1-bitADCADC

1-bitADCADC

RFChain

RFChain

RF Combining

RF Combining

BasebandPrecoding

BasebandPrecoding

1-bitADCDAC

1-bitADCDAC

RFChain

RFChain

+

+

+

RF Beam-forming

RF Beam-forming

Multistream beam training àlarge training overhead

Prior work

11

J. Wang et al, "Beam codebook based beamforming protocol for multi-Gbps millimeter-wave WPAN systems," in IEEE JSAC, October 2009.A. Alkhateeb, et al, “Channel estimation and hybrid precoding for millimetre wave cellular systems.” IEEE JSTSP, May 2014R. Méndez-Rial et al, "Hybrid MIMO Architectures for Millimeter Wave Communications: Phase Shifters or Switches?," in IEEE Access, 2016.

…

Hierarchical beam training

Sparsity-based channel estimation

BasebandPrecoding

BasebandCombining

1-bitADCADC

1-bitADCADC

RFChain

RFChain

RF Combining

RF Combining

Used for analog

architectures

Works for anyarchitecture

Single stream support only

Works for wideband channel

Prior work considered narrowband channel

Support for multi stream

Avoids explicit channel estimation

Contributions

12

Time domain channel estimation technique for

wideband mmWave systems using hybrid architecture

Exploits sparsity in the channel to reduce beamtraining overhead

Enables multistream comm. in 802.11ad

BasebandPrecoding

BasebandCombining

1-bitADCADC

1-bitADCADC

RFChain

RFChain

RF Combining

RF Combining

BasebandPrecoding

BasebandPrecoding

1-bitADCDAC

1-bitADCDAC

RFChain

RFChain

+

+

+

RF Beam-forming

RF Beam-forming

Wideband mmWave channel model

Obtain for channel estimation

13

Angle of arrival/departure

Antenna array responseComplex path gain Pulse shaping function

Path delay

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

p⇢(S(M) ⌦ fTM ⌦w⇤

M )(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`, ✓`, ↵`, ⌧`}

2

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

p⇢(S(M) ⌦ fTM ⌦w⇤

M )(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`, ✓`, ↵`, ⌧`}

Hd 2 CNr⇥Nt

d = 0, 1, ... Nc � 1

2

Exploit sparsity in the angular and delay domain in the problem formulation

RF Chain

RF Chain

Nt

Ns

RF precoder

RF Chain

RF Chain

NRF

RF combiner

Nr

Fig. 1. Figure illustrating the transmitter and receiver structure assumed in thepaper. The RF precoder and the combiner are implemented using a networkof phase shifters.

II. SYSTEM MODEL

Consider a single-user mmWave MIMO system with a trans-mitter having N

t

antennas and a receiver with Nr

antennas.Both the transmitter and the receiver are assumed to have N

RF

RF chains as shown in Fig. 1. The transmitter uses a hybridprecoder F = F

RF

FBB

2 CNt⇥Ns , Ns

being the numberof data streams that can be transmitted. Denoting the symbolvector at instance n as s

n

2 CNs⇥1, satisfying E[sn

s⇤n

] = 1

NsI,

the signal transmitted at discrete-time n is sn

= Fsn

.The N

r

⇥Nt

channel matrix between the transmitter and thereceiver is assumed to be frequency selective, having a delaytap length N

c

and is denoted as Hd

, d = 0, 2, ..., Nc

�1. With ⇢ denoting the average received power and v

n

⇠N

�0,�2I

�denoting the additive noise vector, the received

signal can be written as

rn

=p⇢Nc�1X

d=0

Hd

Fsn�d

+ vn

. (1)

The receiver applies a hybrid combiner W = WRF

WBB

2CNr⇥NRF so that the post combining signal at the receiver is

yn

=p⇢Nc�1X

d=0

W⇤Hd

Fsn�d

+W⇤vn

. (2)

There are several RF precoder and combiner architecturesthat can be implemented [11]. In this paper, we assume afully connected phase shifting network [11, A1]. We assumea hardware constraint that only quantized angles in

A =

(0,

2⇡

2NQ, · · · ,

�2NQ � 1

�2⇡

2NQ

)(3)

can be realized in the phase shifters. Here NQ

is the numberof angle quantization bits. This implies [F]

i,j

= 1pNt

ej'i,j and[W]

i,j

= 1pNr

ej!i,j , with 'i,j

, !i,j

2 A.

III. CHANNEL ESTIMATION VIA COMPRESSED SENSING

In this section, we present our proposed explicit chan-nel estimation algorithm that leverages the sparse widebandmmWave channel.

A. Frequency Selective Channel Model

Consider a geometric channel model [10], [12] for the fre-quency selective mmWave channel consisting of L scatteringclusters. The dth delay tap of the channel can be expressed as

Hd

=

rN

r

Nt

L

LX

`=1

↵`

prc

(dTs

� ⌧`

)aR

(�`

)a⇤T

(✓`

), (4)

where prc

(⌧) denotes the raised cosine pulse signal evaluatedat ⌧ , ↵

`

2 C is the complex gain of the `th cluster, ⌧`

2 R isthe delay of the `th cluster, �

`

2 [0, 2⇡) and ✓`

2 [0, 2⇡) arethe angles of arrival and departure (AoA/AoD), respectivelyof the `th cluster, and a

R

(�`

) 2 CNr⇥1 and aT

(✓`

) 2 CNt⇥1

denote the antenna array response vectors of the receiver andtransmitter, respectively.

The channel model in (4) can be written compactly as

Hd

= AR

�d

A⇤T

, (5)

where �d

2 CL⇥L is diagonal with non-zero entriesqNtNtL

↵`

prc

(dTs

� ⌧`

), and AR

2 CNr⇥L and AT

2 CNt⇥L

contain the columns aR

(�`

) and aT

(✓`

), respectively. Underthis notation, vectorizing the channel matrix in (5) gives

vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

�

2

6664

↵1

prc

(dTs

� ⌧1

)↵2

prc

(dTs

� ⌧2

)...

↵L

prc

(dTs

� ⌧L

)

3

7775. (6)

Note that the `th column of AT

�AR

is of the form aT

(✓`

)⌦aR

(�`

).

B. Sparse Formulation

We assume block transmission with zero padding (ZP)appended to each transmitted frame and hybrid architectureat the transmitter and the receiver. To formulate the sparserecovery problem, we assume single RF chains are used boththe transmitter and the receiver for the ease of exposition.The formulation extends directly for multiple RF chains at thetransmitter and the receiver. Accordingly, for the mth trainingframe the transmitter uses an RF precoder f

m

that can berealized using quantized angles at the analog phase shifters.

The nth symbol of the mth received frame is

r(m)

n

=p⇢Nc�1X

d=0

Hd

fm

s(m)

n�d

+ v(m)

n

, (7)

where s(m)

n

is the nth non-zero symbol of the mth trainingframe

s(m) = [ 0 · · · 0| {z }Nc�1

s(m)

1

· · · s(m)

N

] . (8)

At the receiver, an RF combiner wm

is used during the mth

* P. Schniter and A. Sayeed, “Channel estimation and precoder design for millimeter wave communications: The sparse way,” in the Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Nov. 2014.

# of tap channel

Clustered in space

AoA anglespread

IEEE 802.11ad beam training - overview

14

Training preamble is sent on each beam pair to select the best analog beam

chip rate 1760 MHz

Golay complementary sequences -- basic

blocks in the preamble

Exploit zero-sidelobeproperty for baseband

chan. estimation

Key idea of channel training

15

Frames

Hybrid precoding for training-frame transmission

Estimate AoA/AoD using sparse recovery

Simultaneously leverage the frame structure & the hybrid architecture

Uniform random phase from

quantized angles

Fixed RFprecoder/combinerfor the whole frame

Channel training stages

16

April 29, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.

(fT ⌦ w⇤)([sN ... sN�Nc+1]⌦ INrNt)

3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nc � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 ... s(m)

N

i

mth

fmwm

Nc � 12

6664

y(m)1

y(m)2

:

y(m)N

3

7775= w⇤

m

⇥H0 H1 ... HNc�1

⇤

2

6664

fms(m)1 fms(m)

2 ... fms(m)N

0 fms(m)1 . .

: . : :

0 0 ... fms(m)N�Nc+1

3

7775+

v(m)T

T

y(m)= S(m) ⌦ (fTm ⌦w⇤

m)

2

664

vec(H0)

.:

vec(HNd�1)

3

775+ v(m)

Hd =

PL`=1 ↵`p(dTs � ⌧`)aR(�`)a⇤T(✓`)

Nc

ZP Golay trainingsequenceLength training data

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

2

April 30, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s

1

0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s

2

s1

... 0]⌦ INrNt)

.

.

.

(fT ⌦ w⇤)([s

N

... sN�Nc+1

]⌦ INrNt)

3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[N

c

� 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1

s(m)

2

... s(m)

N

i

mth

fm

wm

Nc

� 12

6664

y(m)

1

y(m)

2

:

y(m)

N

3

7775= w⇤

m

⇥H

0

H1

... HNc�1

⇤

2

6664

fm

s(m)

1

fm

s(m)

2

... fm

s(m)

N

0 fm

s(m)

1

. .: . : :

0 0 ... fm

s(m)

N�Nc+1

3

7775+

v(m)T

T

y(m)

=

p⇢S(m) ⌦ (fT

m

⌦w⇤m

)

2

664

vec(H0

)

.:

vec(HNc�1

)

3

775+ v(m)

Hd

=

PL

`=1

↵`

p(dTs

� ⌧`

)aR

(�`

)a⇤T

(✓`

)

Nc

precoder

April 30, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s

1

0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s

2

s1

... 0]⌦ INrNt)

.

.

.

(fT ⌦ w⇤)([s

N

... sN�Nc+1

]⌦ INrNt)

3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[N

c

� 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1

s(m)

2

... s(m)

N

i

mth

fm

wm

Nc

� 12

6664

y(m)

1

y(m)

2

:

y(m)

N

3

7775= w⇤

m

⇥H

0

H1

... HNc�1

⇤

2

6664

fm

s(m)

1

fm

s(m)

2

... fm

s(m)

N

0 fm

s(m)

1

. .: . : :

0 0 ... fm

s(m)

N�Nc+1

3

7775+

v(m)T

T

y(m)

=

p⇢S(m) ⌦ (fT

m

⌦w⇤m

)

2

664

vec(H0

)

.:

vec(HNc�1

)

3

775+ v(m)

Hd

=

PL

`=1

↵`

p(dTs

� ⌧`

)aR

(�`

)a⇤T

(✓`

)

Nc

combiner

Discard ZP

April 29, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.

(fT ⌦ w⇤)([sN ... sN�Nd+1]⌦ INrNt)

3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nd � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 0 0... s(m)

N

i

mth

fmwm

Nd � 12

6664

y(m)1

y(m)2

:

y(m)N

3

7775= w⇤

m

⇥H0 H1 ... HNd�1

⇤

2

6664

fms(m)1 fms(m)

2 ... fms(m)N

0 fms(m)1 . .

: . : :

0 0 ... fms(m)N�Nd+1

3

7775+

v(m)T

T

April 29, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.


3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nd � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 0 0... s(m)

N

i

mth

fmwm

Nd � 12

6664

y(m)1

y(m)2

:

y(m)N

3

7775= w⇤

m

⇥H0 H1 ... HNd�1

⇤

2

6664

fms(m)1 fms(m)

2 ... fms(m)N

0 fms(m)1 . .

: . : :

0 0 ... fms(m)N�Nd+1

3

7775+

v(m)T

T

April 29, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.

(fT ⌦ w⇤)([sN ... sN�Nc+1]⌦ INrNt)

3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nc � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 ... s(m)

N

i

mth

fmwm

Nc � 12

6664

y(m)1

y(m)2

:

y(m)N

3

7775= w⇤

m

⇥H0 H1 ... HNc�1

⇤

2

6664

fms(m)1 fms(m)

2 ... fms(m)N

0 fms(m)1 . .

: . : :

0 0 ... fms(m)N�Nc+1

3

7775+

v(m)T

T

y(m)= S(m) ⌦ (fTm ⌦w⇤

m)

2

664

vec(H0)

.:

vec(HNd�1)

3

775+ v(m)

Hd =

PL`=1 ↵`p(dTs � ⌧`)aR(�`)a⇤T(✓`)

Nc

Zero-padding facilitates RF circuit reconfiguration across frames

April 29, 2016

Antenna equations

Let G(�, ✓) denote the antenna gain in the elevation angle ✓ and azimuth �. ForpN⇥

pN square uniform planar antenna array, assuming the peak gain of N is

uniformly constant within the azimuth and elevation half-power beam-widths,

we have

G(�, ✓) = G = N for

⇡

2

� 0.866pN

✓ ⇡

2

+

0.866pN

, and � 0.866pN

� 0.866pN

.

Every where outside the main-lobe, we assume a constant side-lobe gain of g.In order that the total radiated power of the antenna is constant, we need to

satisfy the following equation

Z ✓=⇡

✓=0

Z �=⇡

�=�⇡G(�, ✓) sin(✓)d✓d� = 4⇡. (1)

Since

Z ⇡2 + 0.866p

N

⇡2 � 0.866p

N

Z 0.866pN

� 0.866pN

sin(✓)d✓d� = 2

p3

1pN

sin

✓0.866p

N

◆, (2)

scaling and substituting (2) in (1) leads to

G.2p3

1pN

sin

✓0.866p

N

◆+ g

✓4⇡ � 2

p3

1pN

sin

✓0.866p

N

◆◆= 4⇡. (3)

Solving (3) leads to

g =

pN � 0.276N sin

⇣0.866p

N

⌘

pN � 0.276 sin

⇣0.866p

N

⌘(4)

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.


3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nd � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 0 0... s(m)

N

imth

April 29, 2016

Antenna equations

2

6664

y[1].

.

.

.y[N ]

3

7775=

2

6664

(fT ⌦ w⇤)([s1 0 ... 0]⌦ INrNt)

(fT ⌦ w⇤)([s2 s1 ... 0]⌦ INrNt)

.

.

.


3

7775

2

6664

vec(H[0])

.

.

.

.vec(H[Nd � 1])

3

7775+

2

6664

w ⇤ z[1].

.

.

.w⇤z[N ]

3

7775

h0 0... 0 s(m)

1 s(m)2 ... s(m)

N

i

mth

fmwm

Nd � 12

6664

y(m)1

y(m)2

:

y(m)N

3

7775= w⇤

m

⇥H0 H1 ... HNd�1

⇤

2

6664

fms(m)1 fms(m)

2 ... fms(m)N

0 fms(m)1 . .

: . : :

0 0 ... fms(m)N�Nd+1

3

7775+

v(m)T

T

y(m)= S(m) ⌦ (fTm ⌦w⇤

m)

2

664

vec(H0)

.:

vec(HNd�1)

3

775+ v(m)

training frame

(zeros used for beam switching)

Exploiting sparsity in the angular domain

17

Training data matrix

Noise

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

Channel gain

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

6664

s(m)

1

0 ... 0

s(m)

2

s(m)

1

... 0

: . ... :

s(m)

N

. ... s(m)

N�Nc+1

3

7775

2

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

6664

s(m)

1

0 ... 0

s(m)

2

s(m)

1

... 0

: . ... :

s(m)

N

. ... s(m)

N�Nc+1

3

7775

(Ac

tx

⌦Arx

)x

2

Sparse formulation

Dictionary

Goal: Estimate the non-zeros elements of the vector x

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

6664

s(m)

1

0 ... 0

s(m)

2

s(m)

1

... 0

: . ... :

s(m)

N

. ... s(m)

N�Nc+1

3

7775

(Ac

tx

⌦Arx

)x

Nr

Nt

⇥ L

Nr

Nt

⇥Gr

Gt

2

y(1)

=

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

y(M)

=

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

y = �x+ v

N

vec(Hd

) = (Ac

T

�AR

)

2

664

a1

.:

aL

3

775

2

6664

s(m)

1

0 ... 0

s(m)

2

s(m)

1

... 0

: . ... :

s(m)

N

. ... s(m)

N�Nc+1

3

7775

(Ac

tx

⌦Arx

)x

Nr

Nt

⇥ L

Nr

Nt

⇥Gr

Gt

Gr

Gt

>> L

2

training phase, so that the post combining signal is2

66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)

The use of block transmission with Nc

� 1 zero padding isimportant here, since it would allow for reconfiguring the RFcircuits from one frame to the other and avoids loss of trainingdata during this reconfiguration. This would also avoid interframe interference. Also note that for symboling rate of morethan 1 Gbps (the chip rate used in IEEE 802.11ad preamble,for example, is 1760 MHZ), it is impractical to use differentprecoders and combiners for different symbols. It is morefeasible, however, to change the RF circuitry for differentframes with N ⇠ 64� 512 denoting the frame length in (10).Vectorizing (9) gives

y(m) =p⇢S(m) ⌦ fT

m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)

To formulate the compressive sensing problem we first exploitthe sparse nature of the channel in the angular domain.Accordingly, we assume that the AoAs and AoDs are drawnfrom an angle grid on G

r

and Gt

, respectively. Neglecting thegrid quantization error, we can then express (11) as

y(m)=p⇢⇣S(m)⌦fT

m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx

are the dictionary matrices used for sparserecovery. The N

t

⇥Gt

matrix Atx

consists of columns aT

(✓x

),with ✓

x

drawn from a quantized angle grid of size Gt

, andthe N

r

⇥Gr

matrix Arx

consists of columns aR

(�x

), with �x

drawn from a quantized angle grid of size Gr

. The signal xconsists of the channel gains and pulse shaping filter response,and is of size N

c

Gr

Gt

⇥ 1.Next the band-limited nature of the sampled pulse shaping

filter is used to make the measurement vector more sparse.Define

pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)

Using (13) and (14), (6) can be written as

vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)

Next, we look at the sampled version of the pulse-shapingfilter p

i

having entries pi

(n), for i = 1, 2, · · · , Nc

andn = 1, 2, · · · , G

c

. Then, neglecting the quantization errordue to sampling in the delay domain, we can write (12) as


m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt

⇥ 1 sparse vector containing the complexchannel gains.

Stacking M such measurements obtained from sending Mtraining frames and using different RF precoder and combinerfor each frame, we have

y = � x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)

is the measured signal,

� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3

77752 CNM⇥NcNrNt (18)

is the measurement matrix, and

=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3

77752 CNcNrNt⇥GcGrGt (20)

is the dictionary. The beamforming and combining vectorsfm

, wm

, m = 1, 2, · · · , M used for training have thephase angles chosen uniformly at random from the set A in(3).

AoA/AoD estimation With the sparse formulation of themmWave channel estimation problem in (16), compressedsensing tools can be first used to estimate the AoA and AoD.Note that we can increase or decrease G

r

, Gt

and Gc

to meetthe required level of sparsity. As the sensing matrix is knownat the receiver, sparse recovery algorithms can be used toestimate the AoA and AoD. Following this, the channel gainscan be estimated to minimize the minimum mean squarederror or via least squares by plugging in the columns of thedictionary matrices corresponding to the estimated AoA andAoD.

Clustered in space

AoA anglespreadθ

Exploiting group sparsity due to pulse shaping

18


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


i

having entries pi

(n), for i = 1, 2, · · · , Nc

andn = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y = � x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc


has entries


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


i

having entries pi

(n), for i = 1, 2, · · · , Nc

andn = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y = � x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc


Pulse shaping function


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc



66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc



66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc



66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc


Sampled version


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt


Gc

>> Nc


y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc


Unknown x is GtGrGc X 1, L-sparse vector containing the complex channel gains


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



m

⌦w⇤m

⌘

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3

7775x+ e(m)


y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc

to meetthe required level of sparsity. As the sensing matrix is knownat the receiver, sparse recovery algorithms can be used toestimate the AoA and AoD. Following this, the channel gainscan be estimated to minimize the minimum mean squared

Compressive channel estimation

19

Dictionary with columns

...

Random beamforming matrices

p⇢(S(1) ⌦ fT

1

⌦w⇤1

)(INc ⌦Ac

tx

⌦Arx

)x+ v(1)

p⇢(S(M) ⌦ fT

M

⌦w⇤M

)(INc ⌦Ac

tx

⌦Arx

)x+ v(M)

{�`

, ✓`

, ↵`

, ⌧`

}

Hd

2 Nr⇥Nt

d = 0, 1, ... Nc

� 1

acT

(

˜�x

)⌦ aR

(

˜✓y

)

2

Measurement 1

Measurement MQuantized grid of AoA/AoD

...

Dictionary matrix constructed using antenna array response

Angle grid & delay quantization can be made as fine as required for

sparsity

Extends directly to multiple RF chains during training


66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt



y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc



66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt


y(1)=p⇢⇣S(1)⌦fT

1

⌦w⇤1

⌘�INc⌦A

tx

⌦Arx

��x+ e(1)


y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc



66664

y(m)

1

y(m)

2

...y(m)

N

3

77775

T

=p⇢w⇤

m

⇥H

0

· · · HNc�1

⇤S(m)T⌦ f

m

+ e(m)T , (9)

where S(m) =

2

66664

s(m)

1

0 · · · 0

s(m)

2

s(m)

1

· · · ....

.... . .

...s(m)

N

· · · · · · s(m)

N�Nc+1

3

77775. (10)




m

⌦w⇤m

2

6664

vec(H0

)vec(H

1

)...

vec(HNc�1

)

3

7775+ e(m). (11)


r

and Gt



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

�x+ e(m), (12)

where Atx

and Arx


t

⇥Gt

matrix Atx


(✓x

),with ✓

x


, andthe N

r

⇥Gr

matrix Arx


(�x

), with �x



c

Gr

Gt



pn

(⌧) = prc

(n� ⌧) (13)and �

ps

(n) = diag ([pn

(⌧1

) pn

(⌧2

) · · · pn

(⌧L

)]) . (14)


vec(Hd

) =

rN

t

Nt

L

�A

T

�AR

��

ps

(dTs

)

2

6664

↵1

↵2

...↵L

3

7775. (15)


n

having entries pn

(k), for n = 1, 2, · · · , Nc

andk = 1, 2, · · · , G

c



m

⌦w⇤m

⌘�INc⌦A

tx

⌦Arx

��x+ e(m),

where � =

2

6664

IGrGt ⌦ pT

1

IGrGt ⌦ pT

2

...IGrGt ⌦ pT

Nc

3

7775,

and x is Gc

Gr

Gt


y(M)=p⇢⇣S(M)⌦fT

M

⌦w⇤M

⌘�INc⌦A

tx

⌦Arx

��x+ e(M)


y =p⇢� x+ e, (16)

where y =

2

6664

y(1)

y(2)

...y(M)

3

77752 CNM⇥1 (17)


� =

2

6664

S(1)⌦fT1

⌦w⇤1

S(2)⌦fT2

⌦w⇤2

...S(M)⌦fT

M

⌦w⇤M

3



=�INc ⌦ A

tx

⌦Arx

�� (19)

=

2

6664

�A

tx

⌦Arx

�⌦ pT

1�A

tx

⌦Arx

�⌦ pT

2

...�A

tx

⌦Arx

�⌦ pT

Nc

3



, wm



r

, Gt

and Gc


Measurement matrix

Dictionary matrix

Stack M measurements

Stack M measurements

Simulation results

20

Setup• Tx has 32 antennas, Rx has 16 antennas• Dictionary generated with 64 grids for AoD

and 16 grids for AoA• Frequency selective channel with 4 delay taps• Pulse shaping filter with 0.8 roll-off factor• Frame length = 16• 2 bit quantization for precoder and combiner

phase shifters• Compressive Sampling Matching pursuit

followed by least square estimation

NMSE =

PNcd=0 ||Hd� ˆHd||2FPNc

d=0 ||Hd||2F

3

80-100 training frames are enough to ensure low channel estimation error

With 1 RF chain

21Using multiple RF chains at Tx and Rx gives better channel estimates

Employing hybrid architecture

Effectively increases the precoding and combining

beam patterns

RF Chain

RF Chain

NtNs

RF precoder

RF Chain

RF Chain

NRF

RF combiner

Nr

Reducing the training overhead

22

Increased number of measurements per training

frame

Using multiple RF chains reduces training overhead

Leverage multiple RF chains

Conclusionsu Channel estimation using hybrid architecture

ª Exploiting sparsity enables use of compressive sensing toolsª Efficient wideband mmWave channel estimation with low training overheadª Multiple RF chains at the transceivers reduce the number of training step

u Future workª Investigate different compressive sensing algorithms for the channel estimationª Compare complexity with frequency domain channel estimation techniquesª Comparison of performance between beam training and CS based approaches

23

Questions?

24

exploiting sparsity for wideband mmwave channel...

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