1

Reconfigurable Intelligent Surfaces for Doppler Effect andMultipath Fading Mitigation

Ertugrul Basar, Senior Member, IEEE and Ian F. Akyildiz, Fellow, IEEE

Abstract—Extensive research has already started on 6G andbeyond wireless technologies due to the envisioned new use-casesand potential new requirements for future wireless networks.Although a plethora of modern physical layer solutions havebeen introduced in the last few decades, it is undeniable that alevel of saturation has been reached in terms of the availablespectrum, adapted modulation/coding solutions and accordinglythe maximum capacity. Within this context, communicationsthrough reconfigurable intelligent surfaces (RISs), which enablenovel and effective functionalities including wave absorption,tuneable anomalous reflection, and reflection phase modifica-tion, appear as a potential candidate to overcome the inherentdrawbacks of legacy wireless systems. The core idea of RISs isthe transformation of the uncontrollable and random wirelesspropagation environment into a reconfigurable communicationsystem entity that plays an active role in forwarding information.In this paper, the well-known multipath fading phenomenon isrevisited in mobile wireless communication systems, and noveland unique solutions are introduced from the perspective ofRISs. The feasibility of eliminating or mitigating the multipathfading effect stemming from the movement of mobile receiversis also investigated by utilizing the RISs. It is shown thatrapid fluctuations in the received signal strength due to theDoppler effect can be effectively reduced by using the real-timetuneable RISs. It is also proven that the multipath fading effectcan be totally eliminated when all reflectors in a propagationenvironment are coated with RISs, while even a few RISs cansignificantly reduce the Doppler spread as well as the deep fadesin the received signal for general propagation environments withseveral interacting objects.

Index Terms—6G, Doppler effect, multipath fading, reconfig-urable intelligent surface (RIS).

I. INTRODUCTION

F IFTH generation (5G) systems have the major threedifferent use-cases with diverse requirements, namely

enhanced mobile broadband (eMBB), ultra-reliable and low-latency communications (URRLC), and massive machine typecommunications (mMTC). Although the 5G standard exploitspromising physical layer (PHY) technologies including mas-sive multiple-input multiple-output (MIMO) systems, millime-ter wave (mmWave) communications, and multiple orthogonalfrequency division multiplexing (OFDM) numerologies, it isdoes not contain revolutionary ideas in terms of PHY layer

Manuscript received December, 2019.E. Basar is with the Communications Research and Innovation Laboratory

(CoreLab), Department of Electrical and Electronics Engineering, Koc Uni-versity, Sariyer 34450, Istanbul, Turkey. e-mail: ebasar@ku.edu.tr

I. F. Akyildiz is with School of Electrical and Computer Engineer-ing, Georgia Institute of Technology, Atlanta, GA 30332, USA. email:ian@ece.gatech.edu

This work was supported in part by the Scientific and TechnologicalResearch Council of Turkey (TUBITAK) under Grant 117E869 and theTurkish Academy of Sciences (TUBA) GEBIP Programme.

Codes available at https://corelab.ku.edu.tr/tools

solutions where the revolutionary ideas are more on soft-warization, namely on software defined networking (SDN) andnetwork functions virtualization (NFV). From this perspective,researchers have already started research on beyond 5G, oreven 6G technologies of 2030 and beyond by exploringcompletely new PHY concepts. Even though future 6G tech-nologies seem to be extensions of their 5G counterparts at thistime [1], potential new user requirements, use-cases, and net-working trends of 6G [2] will bring more challenging problemsin mobile wireless communication, which necessitate radicallynew communication paradigms, particularly at the PHY ofnext-generation radios. The envisioned new communicationsolutions must provide extremely high spectral and energyefficiencies along with ultra-reliability and ultra-security, andmust have highly flexible structures to satisfy the challengingrequirements of diverse users and applications. Although theintensive research efforts of the past two decades, these are stillmissing features in state-of-the-art systems and standards, andslowing down the progress of long-awaited wireless revolution.

Since the invention of modern wireless communications,network operators have been constantly struggling to buildtruly pervasive wireless networks that can provide uninter-rupted connectivity and high quality-of-service (QoS) to mul-tiple users and devices in the presence of harsh propagationenvironments [3]. The main reason of this phenomenon is theuncontrollable and random behavior of wireless propagation,which causes

i) deep fading due to uncontrollable interactions of trans-mitted waves with surrounding objects and their destructiveinterference at the receiver,

ii) severe attenuation due to path loss, shadowing, and non-line-of-sight (LOS) transmissions,

iii) inter-symbol interference due to different runtimes ofmultipath components, and

iv) Doppler effect due to the high mobility of users and/orsurrounding objects.

Although a plethora of modern PHY solutions, includingadaptive modulation and coding, multi-carrier modulation,non-orthogonal multiple access, relaying, beamforming, andreconfigurable antennas, have been considered to overcomethese challenges in the next several decades, the overallprogress has been still relatively slow. The major reason of thisrelatively slow progress is explained by the following so-calledundeniable fact: until the start of modern wireless communi-cations (for decades), the propagation environment has beenperceived as a randomly behaving entity that degrades theoverall received signal quality and the communication QoSdue to uncontrollable interactions of the transmitted radiowaves with the surrounding objects. In other words, communi-

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cation pundits have been mainly focusing on transmitter andreceiver ends of traditional wireless communication systemsfor ages while assuming that the wireless communicationenvironment itself remains an uncontrollable factor and hasusually a negative effect on the communication efficiencyand the QoS. One of the main objectives of this paper isto challenge this view by exploiting the new paradigm ofintelligent communication environments.

In recent years, reconfigurable intelligent surfaces (RISs)have been brought to the attention of the wireless researchcommunity to enable the control of wireless environments [4].RISs are man-made surfaces of electromagnetic (EM) materialthat are electronically controlled with integrated electronicsand aim to modify the current distribution over themselvesin a deliberate manner to enable unique EM functionalities,including wave absorption, anomalous reflection, polarizedreflection, wave splitting, wave focusing, and phase modifi-cation. Recent results have revealed that these unique EMfunctionalities are possible without complex decoding, en-coding, and radio frequency (RF) processing operations andthe communication system performance can be boosted byexploiting the implicit randomness of wireless propagation [5].However, the fundamental issues remain unsolved within thetheoretical and practical understanding as well as modeling ofRIS-aided communication systems.

In contrast to current wireless networks, where the environ-ment is out of control of the operators, RISs have enabled theemerging concept of intelligent communication environments,where the environment is turned into an intelligent entity thatplays an active role in processing signals and accordinglytransferring information. This is a completely new paradigmin wireless communication and has the potential to changethe way the communication takes place. The core technologybehind this promising concept, RISs, is the metasurfaces,which are the 2D equivalent of metamaterials [4]–[6]. Metasur-faces are thin planar artificial structures with sub-wave-lengththickness and enable unnatural EM functionalities for the RF,Terahertz, and optical spectrum. It is worth noting that commu-nications through RISs is different compared with other relatedtechnologies currently employed in wireless networks, suchas relaying, MIMO beamforming, passive reflect-arrays, andbackscatter communications [7], while having the followingmajor distinguishable features:

i) RISs are nearly passive, and, ideally, they do not needany dedicated energy source for RF signal processing;

ii) RISs do not amplify or introduce noise when reflectingthe signals and provide an inherently full-duplex transmission;

iii) RISs can be easily deployed, e.g., on the facades ofbuildings, ceilings of factories, and indoor spaces;

iv) RISs are reconfigurable in order to adapt themselvesaccording to the changes of the wireless environment.

These distinctive characteristics make RIS-assisted commu-nication a unique technology and introduce important commu-nication theoretical as well as system design challenges, someof which will be tackled in this paper.

There has been a growing recent interest in controlling thepropagation environment or exploiting its inherently randomnature to increase the QoS and/or spectral efficiency. For

instance, IM-based [8], [9] emerging schemes such as media-based modulation [10], [11] and spatial scattering modulation[12] use the variations in the signatures of received signalsby exploiting reconfigurable antennas or scatterers to transmitadditional information bits in rich scattering environments. Onthe other hand, RISs are smart devices that intentionally con-trol the propagation environment by exploiting reconfigurablereflectors/scatterers to boost the signal quality at the receiver.Although some early attempts have been reported to controlthe wireless propagation, such as intelligent walls [13], [14],spatial microwave modulators [15], 3D reflectors [16], andcoding metamaterials [17], the surge of intelligent communi-cation environments can be mainly attributed to programmable(digitally-controlled) metasurfaces [18], reconfigurable reflect-arrays [19], [20], software-controlled hypersurfaces [4], [21],and intelligent metasurfaces [22]. For instance, the intelligentmetasurface design of [22], enables tuneable anomalous re-flection as well as perfect absorption by carefully adjustingthe resistance and the capacitance of its unit cells at 5 GHz.

The concept of communications through intelligent surfaceshas received tremendous interest from wireless communicationand signal processing communities very recently due to chal-lenging problems it brings in the context of communication,optimization, and probability theories [5], [6], [23]. Particu-larly, researchers focused on

i) maximization of the achievable rate, minimum signal-to-interference-plus noise ratio (SINR), and energy efficiencyand minimization of the transmit power by joint optimizationof the RIS phases and the transmit beamformer [24]–[29],

ii) maximization of the received signal-to-noise ratio(SNR) to minimize the symbol error probability [30],

iii) efficient channel estimation techniques with passiveRIS elements as well as deep learning tools to reduce thetraining overhead and to reconfigure RISs [31]–[33],

iv) PHY security solutions by joint optimization of thetransmit beamformer and RIS phases [34]–[36],

v) practical issues such as erroneous reflector phases,realistic phase shifts, and discrete phase shifts [37]–[39], and

vi) design of NOMA-based systems for downlink transmitpower minimization and for the minimum decoding SINRmaximization of all users [40], [41].

Furthermore, the first attempts on combining RISs withspace modulation, visible light and free space optical com-munications, unmanned aerial vehicles, wireless informationand power transfer systems, and OFDM systems have beenreported in the past few months (see [5] and referencestherein). Recent improvements have been also reported inhypersurfaces, such as development of a new wave sensingand manipulation system, consideration of neural networks-based solutions for system configuration, and exploration ofnetwork layer modelling issues [42]–[44].

In our paper, we take a step back and revisit the well-knownphenomenons of multipath fading and Doppler effect in mo-bile communications from the perspective of emerging RISs.Although the potential of RISs has been explored from manyaspects as discussed above, to the best of our knowledge, theirpotential in terms of Doppler effect mitigation has not beenfully understood yet. For this purpose, by following a bottom-

3

V

dLOS d1

BSIO

MS

Fig. 1. The basic two-ray propagation model with a mobile receiver and anIO as a reflector.

up approach from simple networks to more sophisticated ones,we explore the potential of RISs to eliminate multipath fadingeffect stemming from Doppler frequency shifts of a mobilereceiver for the first time in the literature. Specifically, weprove that the rapid fluctuations in the received signal strengthdue to the user movement can be effectively eliminated and/ormitigated by utilizing real-time tuneable RISs. We introduce anumber of novel and effective methods that provide interest-ing trade-offs between Doppler effect mitigation and averagereceived signal strength maximization, for the reconfigurationof the available RISs in the system and reveal their potentialfor future mobile networks.

The rest of the paper is organized as follows. In Section II,we consider a simple two-path scenario and revisit multipathand Doppler effects. In Section III, we deal with Dopplereffect elimination with RISs. Section IV deals with moresophisticated networks with multiple RISs and objects. Finally,in Section V we cover practical issues and in Section VI weconclude the paper.

II. REVISITING MULTIPATH AND DOPPLER EFFECTS WITHSIMPLE CASE STUDIES

In this section, we revisit the Doppler and multipath fadingeffects caused by the movement of a mobile receiver under asimple propagation scenario (with and without an RIS). Wefocus our attention to the low-pass equivalent and noise-freereceived signals while a generalization to pass-band signalingis straightforward from the given low-pass equivalent signals.

A. Multipath Fading Due to User Movement and A Reflector

We consider the propagation geometry of Fig. 1 with a basestation (BS), a mobile station (MS) that travels along a straightroute with a speed of V (in m/s), and an interacting object (IO).In this setup, in addition to the LOS signal stemming from theBS, a second copy of the transmitted signal reflected from theIO arrives at the receiver of the MS. For ease of presentation,we consider a reflection coefficient of unity magnitude andphase π, that is R = −1, for the IO. Here, the reflectingsurface is large and smooth enough so that specular reflectionsoccur according to the Snell’s law.

In order to capture the effects of Doppler and multipathfading in the received signal with respect to time, we as-sume the transmission of an unmodulated radio frequency(RF) carrier signal cos(2πfct + Θ0), where fc is the carrierfrequency and Θ0 is the initial phase. Using complex basebandrepresentation, we arrive at the low-pass complex equivalent ofthis signal as x(t) = exp(jΘ0). To illustrate the fade patternand the Doppler spectrum due to the user movement, we focus

on a very short travel distance (a few wavelengths) of the MS,as a result, the received direct and reflected signals have almostconstant amplitudes, while being subject to rapidly varyingphase terms. At the same time, due to the movement of theMS, Doppler shifts are observed at the received signals. Inlight of this information, the received complex envelope isobtained as [45]

r(t) =λ

4π

(e−j

2πdLOS(t)λ

dLOS(t)− e−j

2πdR(t)

λ

dR(t)

)(1)

where we dropped the initial carrier phase Θ0 for clarity.Here dLOS(t) and dR(t) respectively stand for the time-varying radio path distance for the BS-MS and the BS-IO-MS links. For the particular setup considered in Fig. 1, wehave dLOS(t) = dLOS + V t and dR(t) = dLOS + 2d1 − V t,where dLOS and d1 stand for the initial distances betweenthe BS and the MS and the MS and the IO, respectively.Here, we assume that the BS-MS antenna height differenceis sufficiently small so that a horizontal communication linkcan be considered. The same applies from the signal reflectedfrom the IO. In other words, the radio path length variationsfor both rays are directly proportional to the MS travel distancevariations. However, a generalization is straightforward forsignals coming from different angles (see Section IV). Asdiscussed earlier, since we focus into a very short time interval(travel distance of the MS), we may assume that two rays havealmost constant amplitudes at the initial and last positions ofthe MS. Considering this, (1) simplifies to

r(t) =λ

4π

(e−j2πfDt−jφLOS

dLOS− ej2πfDt−jφ1

dLOS + 2d1

)(2)

where fD = V/λ is Doppler shift with respect to the nominalcarrier frequency in the passband or with respect to 0 Hz inlow-pass equivalent representation, φLOS = 2πdLOS/λ, andφ1 = 2π(dLOS + 2d1)/λ. The constant (initial) phase termsof φLOS and φ1 can be readily dropped if they are integermultiples of 2π. In light of this, using the properties of com-plex exponentials1, the magnitude of the complex envelopecan be obtained as follows:

|r(t)| =(λ

4π

)(1

d2LOS

+1

(dLOS + 2d1)2− 2 cos(4πfDt)

dLOS(dLOS + 2d1)

)1/2

.

(3)

In Fig. 2, we plot the magnitude of the complex envelope foran MS travel distance of six wavelengths (corresponding to anobservation time of 0.06 s) considering the following systemparameters2: fc = 3 GHz, V = 10 m/s with varying dLOS andd1 values for a fixed BS-IO total distance of dLOS +d1 = 2000m. As seen from Fig. 2, due to the destructive and constructiveinterference of the arriving two signals, the received signalstrength fluctuates rapidly (with a frequency of 2fD as evident

1For z1 = r1ejξ1 , z2 = r2ejξ2 , and z3 = z1 + z2 = r3ejξ3 , we haver3 = (r21 + r22 + 2r1r2 cos(ξ1 − ξ2))1/2.

2The same simulation parameters (mobile speed, carrier frequency, obser-vation interval, travelled distance, FFT size, sampling distance and time) areused in the following unless specified otherwise.

4

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0 0.01 0.02 0.03 0.04 0.05 0.06

Time (s)

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plex

env

elop

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ag. (

dB)

Time (s)

dLOS=1750 m, d1=250 m

Fig. 2. Variation of the magnitude of the received complex envelope due to MSmovement with V = 10 m/s for varying dLOS and d1 (observation interval:0.06 s, travelled distance: 6λ = 0.6 m).

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dLOS=1750 m, d1=250 m

Fig. 3. Doppler spectrum of the received signal for varying dLOS and d1(FFT size: 256, sampling distance: λ/32 = 0.003125 m, sampling time:λ/(32× V ) = 0.3125 ms).

from (3)) around a mean value, which is determined by thepath loss. This oscillation is also known as the fade pattern ofthe received envelope. It is also worth noting that the variationof the magnitude is more significant for the closer valuesof dLOS(t) and dR(t) (smaller d1). This is also verified bythe Doppler spectrum of the received signal given in Fig. 3for these four cases, which include two sharp componentsat opposite frequencies, i.e., −V/λ = −100 Hz (from theLOS path) and V/λ = 100 Hz (from the IO) with differentnormalized amplitudes due to different travel distances of thetwo rays.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (s)(a)

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=1000 m, d1=1000 m

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=1250 m, d1=750 m

dLOS

=1500 m, d1=500 m

dLOS

=1750 m, d1=250 m

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Doppler shift (Hz)(b)

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Fig. 4. (a) Maximized magnitude of the received signal in the presence of anRIS, (b) Doppler spectrum of the received signal for all scenarios.

B. Eliminating Multipath Fading Due to User Movement withan RIS

In this subsection, we consider the same system model ofSubsection II.A (Fig. 1), however, we assume that a control-lable reflection occurs from the IO through an RIS that ismounted on its facade. In this scenario, intelligent reflection iscaptured by a time-varying and unit-gain reflection coefficientR(t) = ejθ(t). As a result, the received complex envelope isobtained as

r(t) =λ

4π

(e−j2πfDt

dLOS+ej2πfDt+jθ(t)

dLOS + 2d1

). (4)

It is obvious that the magnitude of r(t) is maximized whenthe phases of the direct and reflected signals are aligned,that is, by adjusting the RIS reflection phase as θ(t) =−4πfDt (mod 2π). It is worth noting that this can be onlypossible with an RIS that is able to adjust its reflectioncoefficient dynamically with respect to time (user movement).The practical issues related to this adjustment procedure arediscussed in Section V. With the specified value of θ(t) givenabove, the complex envelope of the received signal becomes

r(t) =λe−j2πfDt

4π

(1

dLOS+

1

dLOS + 2d1

)(5)

whose magnitude is maximized and remain constant withrespect to time during our observation interval and given by

|r(t)|max =λ

4π

(1

dLOS+

1

dLOS + 2d1

). (6)

In light of (5) and (6), we have the following two remarks.Remark 1: Time-varying intelligent reflection of the RIS

eliminates the multipath fading (rapid fluctuations of the re-ceived signal strength) for the scenario of Fig. 1 and enables aconstant magnitude for the received complex envelope, whichis also shown in Fig. 4(a). In other words, it is possible toescape from rapid fluctuations in the received signal due to theuser movement by utilizing an RIS, which has a time-varyingreflection phase.

Remark 2: The received signal is still subject to a Dopplershift of −fD Hz, which is also observed from the Dopplerspectrum of Fig. 4(b). Although the RIS effectively eliminatesfade patterns, due to the direct signal received from the BS,

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dLOS

=1000 m, d1=1000 m

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=1250 m, d1=750 m

dLOS

=1500 m, d1=500 m

dLOS

=1750 m, d1=250 m

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Doppler shift (Hz)(b)

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Fig. 5. (a) Minimized magnitude of the received signal in the presence of anRIS, (b) Doppler spectrum of the received signal for all scenarios.

which is out of the control of the RIS, it is not possibleto eliminate Doppler frequency shifts in this propagationscenario.

C. Increasing Fading and Doppler Effects with An RIS

So far, we focused our attention on the maximization ofthe received signal strength for the scenario of Fig. 1, bycarefully adjusting the RIS reflection phase in real time. Onthe contrary, it might be possible to intentionally degrade thereceived signal strength as well as increase the Doppler spreadfor an unintended mobile receiver or for an eavesdropper.Based on the received signal model of (4), when the receivedtwo signals are in-phase, we obtain the maximum magnitudefor the received signal as in (6). On the other hand, adjustingthe RIS reflection phase as θ(t) = −4πfDt + π (mod 2π),we obtain completely out-of-phase two arriving signals, andthe resulting minimum complex envelope magnitude becomes

|r(t)|min =λ

4π

(1

dLOS− 1

dLOS + 2d1

). (7)

As seen from (7), the degradation in the received signalstrength would be more noticeable for smaller d1. However,the magnitude of the complex envelope becomes constant as in(6), i.e., no fade patterns are observed. In Fig. 5(a), we depictthe minimized complex envelope magnitudes by intentionallyout-phasing the direct and reflected signals for varying dLOSand d1. Comparing Figs. 4(a) and 5(a), we observe up to 9 dBdegradation in magnitude (for dLOS = 1750 m and d1 = 250m), which corresponds to a power variation of 18 dB. Inother words, it is possible to enable up to 18 dB variation inthe received signal power by deliberately co-phasing and out-phasing the multipath components in the considered setup. Itis worth noting that the normalized Doppler spectrum in Fig.5(b) is the same as that of Fig. 4(b).

As another extreme application of an RIS, the Dopplerspread can be increased by intentionally increasing theDoppler shift of the reflected signal by θ(t) = 2π(fD −fD)t (mod 2π), where fD is the desired Doppler shift forthe reflected signal. Here, a maximum desired Doppler shiftof 0.5fs Hz can be observed in simulation, where fs is thesampling frequency for the continuous-wave signal. In Fig. 6,

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Fig. 6. Time varying magnitude and the Doppler spectrum of the receivedsignal with increased Doppler effect for (top) fD = 200 Hz and (bottom)fD = 400 Hz.

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Fig. 7. (a) Magnitude of the received signal in the presence of an RIS withrandom phases, (b) Doppler spectrum of the received signal.

we present the magnitude of the complex envelope as wellas the Doppler spectrum for the case of dLOS = 1500 m andd1 = 500 m by carefully adjusting the reflection phase toincrease the Doppler spread (reduce the coherence time) byfD = 200 Hz and 400 Hz. As seen from Fig. 6, an RIS cancreate new components in the Doppler spectrum, which resultsin a faster fade pattern for the complex envelope.

Going one step further, we consider the concept of randomphase shifts by the RIS, in which the reflection phase isselected at random between 0 and 2π in each time interval.We illustrate the magnitude of the complex envelope and theDoppler spectrum in Fig. 7 for the case random reflectionphases, where the reflection phase is selected at random ineach sampling time for dLOS = 1500 m and d1 = 500 m. Asseen from Figs. 7(a) and (b), although the effect in Dopplerspectrum is not very significant, it would be possible to obtaina very fast fade pattern in time. Specifically, around 5 dBmagnitude variations are observed within a sampling distanceof λ/32 m. It would be possible to obtain an ultra-fast fadepattern by alternating the reflection phase between θ(t) =−4πfDt + π (mod 2π) and θ(t) = −4πfDt (mod 2π) ineach time interval and this is left for interested readers.

6

V

dLOS d1

BSIO

MS

Fig. 8. Communications through an IO with a blocked LOS path.

III. ELIMINATING DOPPLER EFFECTS THROUGHINTELLIGENT REFLECTION

In this section, we focus on a simple scenario in which thedirect link is blocked by an obstacle while the communicationbetween the BS and the MS is established through a reflectionfrom an IO as shown in Fig. 8. We consider the sameassumptions of Section II and investigate the Doppler effecton the received signal in the following two cases.

A. NLOS Transmission without An RIS

Under the assumption of specular reflections from the IOwith a reflection coefficient of R = −1, the received signalcan be expressed as

r(t) = − λ

4π

e−j2πdR(t)

λ

dR(t)(8)

where dR(t) = dLOS +2d1−V t is the time-varying radio pathdistance for a MS moving with a speed of V m/s. Ignoring theconstant phase terms and assuming a very short travel distance,the received signal can be expressed as

r(t) = − λej2πfDt

4π(dLOS + 2d1). (9)

As seen from (9), since only a single reflection occurs withouta LOS signal and other multipath components, the receivedsignal magnitude does not exhibit a fade pattern, that is, fixedwith respect to time and given by |r(t)| = λ/(4π(dLOS +d1)). However, the received signal is still subject to a Dopplerfrequency shift of fD Hz, which is evident from (9), due tothe movement of the MS.

B. NLOS Transmission with An RIS

Here, we focus on the scenario of Fig. 8 while assumingthat the IO is equipped with an RIS that is able to provideadjustable phase shifts, that is, R(t) = ejθ(t), as in Section II.In this case, the received signal can be expressed as

r(t) =λej2πfDt+jθ(t)

4π(dLOS + 2d1). (10)

As seen from (10), the magnitude of the received signal isindependent from the reflection phase and the same as theprevious case (without an RIS). However, it might be possibleto completely eliminate the Doppler effect by adjusting theRIS reflection phase as θ(t) = −2πfDt (mod 2π). We givethe following remark.

Remark 3: When there is no direct transmission betweenthe BS and the MS over which the RIS has no control,intelligent reflection allows one to completely eliminate the

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Fig. 9. Doppler spectrum of the received signal for the scenario of Fig. 8: (a)without an RIS, (b) with an RIS.

Doppler effect, by carefully compensating the Doppler phaseshifts through the RIS.

In Fig. 9, we show the Doppler spectrum of the receivedsignal with and without an RIS for dLOS = d1 = 1000 m. Asseen from Fig. 9, the Doppler effect is eliminated (0 Hz) byadjusting the RIS reflections accordingly.

As discussed earlier, it is not possible the modify themagnitude of the complex envelope with an RIS for thescenario of Fig. 8, however, as done in Section II, the Dopplerspread can be enhanced by θ(t) = 2π(fD − fD)t (mod 2π),where fD is the desired Doppler frequency. The observationof the resulting spectrum is straightforward and left for theinterested readers.

IV. DOPPLER AND MULTIPATH FADING EFFECTS: CASESTUDIES WITH MULTIPLE REFLECTORS

In this section, we extend our system models and analyses inSections II and III into propagation scenarios with multiple IOswith and without intelligent reflection capabilities. We followa bottom-up approach starting with two IOs and illustratethe fading/Doppler effect mitigation capabilities of RISs. Wealso propose a number of effective and novel methods withdifferent functionalities.

A. Direct Signal and Two Reflected Signals without any RISs

In this subsection, by extending our model given in SectionII, we consider the propagation scenario of Fig. 10 with twoIOs. Here, in order to spice up our analyses, we assume thatwhile the BS-MS and BS-IO 1-MS links are parallel to theground, the reflected signal from IO 2 arrives to the MS withan angle of α with respect to the MS route. In this scenario,the initial (horizontal) distances between the BS and the MS,the MS and IO 1, and the MS and IO 2 are shown by dLOS,d1, and d2, respectively. Using a similar analysis as in SectionII, under the assumption of unit gain reflection coefficients forboth IOs, that is R1 = R2 = −1, the time-varying receivedcomplex envelope can be expressed as

r(t) =λ

4π

(e−j2πfDt

dLOS− ej2πfDt

dLOS + 2d1− ej2πfD(cosα)t−jφ2

d2

)(11)

7

V

dLOSd1

BSIO 1

MS

IO 2

d2

Fig. 10. Propagation scenario with two IOs.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (s)(a)

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plex

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0N

orm

aliz

ed fr

eq. r

espo

nse

(dB

)

Fig. 11. The scenario of Fig. 10 without an RIS: (a) Complex envelopemagnitude, (b) Doppler spectrum of the received signal.

where d2 =√d2

2 tan2 α+ (dLOS + d2)2 + d2/(cosα) is theinitial radio path distance for the reflected signal of IO 2,which is obtained after simple trigonometric operations, andφ2 = 2πd2/λ is a fixed phase term. Here, we assume thatthe variations in terms of the large-scale path loss due to themovement of MS are almost negligible (as in Fig. 1) and therays from IO 2 remain parallel for all points of the mobileroute, which corresponds to radio path distance decrementsof V t cosα, with respect to time, for these rays. It is worthnoting that parallel ray assumption is approximately true forshort route lengths [46]. As seen from (11), the received signalhas three Doppler components: −fD Hz, fD Hz, and fD cosαHz due to the rays coming from the BS, IO 1, and IO 2,respectively.

In Fig. 11, we show the magnitude of the complex envelopeas well as the Doppler spectrum for the case of α = 60◦,dLOS = d1 = 1000 m, and d2 = 500 m. As seen from Fig. 11,due to constructive and destructive interference of the directand two reflected signals with different Doppler frequencyshifts (−100 Hz, 100 Hz, and 50 Hz), the magnitude of thecomplex envelope exhibits a more hostile and faster fadingpattern compared to the simpler scenario of Fig. 1 (see Fig.2, top-left subplot).

B. Direct Signal and Two Reflected Signals with One or TwoRISs

In this subsection, we again focus on the scenario of Fig.10, however, under the assumption of one or two RISs that areattached to the existing IOs. Although being more challengingin terms of system optimization and analysis, we focus on the

0 0.02 0.04 0.06Time (s)

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0Doppler shift (Hz)

Method 3

Fig. 12. Magnitude and Doppler spectrum of the received signal with an RISfor scenario of Fig. 10 under three different phase selection methods.

case of a single RIS first, then extend our analysis into thecase of two RISs.

1) One RIS: Let us assume that we have a single RIS thatis mounted on the facade of IO 1 for the scenario of Fig. 10.For this case, the received complex envelope can be expressedas

r(t) =λ

4π

(e−j2πfDt

dLOS+ej2πfDt+jθ1(t)

dLOS + 2d1

−ej2πfD(cosα)t−jφ2

d2

). (12)

Here, we assumed that the intelligent reflection from IO 1is characterized by θ1(t). We investigate the following threemethods for the adjustment of θ1(t), where the correspondingcomplex envelope magnitudes and Doppler spectrums areshown in Fig. 12 for α = 60◦, dLOS = d1 = 1000 m, andd2 = 500 m:• Method 1: θ1(t) = −4πfDt (mod 2π)• Method 2: θ1(t) = 2πfDt(cosα−1)−φ2 +π (mod 2π)• Method 3: θ1(t) = 2πfDt(cosα− 1)− φ2 (mod 2π)

In the first method, we intuitively align the reflected signalfrom the RIS to the LOS signal. As seen from Fig. 12,although this adjustment eliminates the 100 Hz component inthe spectrum and reduces the Doppler spread compared to thecase without RIS (Fig. 11), we still observe two componentsin the spectrum and a noticeable fade pattern for the receivedsignal due to uncontrollable reflection through IO 2. It is worthnoting that this might be the preferred option to obtain a hightime average for the complex envelope magnitude with theprice of a high Doppler spread (faster time variation).

In the second method, we align the reflected signal from theRIS to the one from IO 2, however, this worsens the situationby increasing the relative power of the 50 Hz component in

8

Fig. 13. 3D illustration of the variation of the complex envelope magnitudewith respect to time for all possible RIS reflection angles (scenario of Fig.10).

the Doppler spectrum. As seen from Fig. 12, a more severefade pattern is observed for Method 2 due to destructiveinterference of the reflected signals to the LOS signal. Thiswould be a preferred option in case of an eavesdropper todegrade its signal quality.

In the third method, we follow a clever approach and insteadof aligning our RIS-assisted reflected signal to the existing twosignals, we target to eliminate the uncontrollable reflectionfrom IO 2 by out-phasing the reflected two signals. Thisresults a remarkable improvement in both Doppler spectrumand the received complex envelope by almost mitigatingthe fade pattern. In other words, the RIS scarifies itself inMethod 3 to eliminate the uncontrollable reflection from IO 2,which significantly reduces the multipath effect, while a minorvariation is still observed due to different radio path lengths ofthese two signals. More specifically, for the selection of θ1(t)in Method 3, we obtain

r(t) =λ

4π

(e−j2πfDt

dLOS

+ej2πfD(cosα)t−jφ2

(1

dLOS + 2d1− 1

d2

))(13)

which contains two components. However, the Doppler spreadcan be remarkably reduced when the radio path distances ofthe signals reflected from IO 1 and 2, i.e., dLOS + 2d1 andd2, are close to each other. For instance, for the consideredsystem parameters of dLOS, d1, d2, and α in Fig. 12, we have

1dLOS�(

1dLOS+2d1

− 1d2

), which results almost a single-tone

received signal r(t) ≈ λ4π

(e−j2πfDt

dLOS

). This is also evident

from the Doppler spectrum of the received signal for Method3. However, Method 3 cannot guarantee the highest complexenvelope magnitude, which is also observed from Fig. 12.

To gain further insights, in Fig. 13, we plot the 3D mag-nitude of the complex envelope with respect to time andvarying θ1(t) values between 0 and 2π. As seen from Fig. 13,due to constructive and destructive interference of multipathcomponents (particularly due to the interference of the signalreflected from IO 2), the complex envelope exhibits several

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ad)

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Method 1Optimum

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Fig. 14. Comparison of reflection phases and complex envelope magnitudesfor Method 1 and the optimum method.

deep fades. We also observe that it is not feasible to fix thecomplex envelope magnitude to its maximum value (−48.69dB for this specific setup) as in the case of single reflectionsince the incoming three signals cannot be fully aligned at alltimes. Finally, we note that performing an exhaustive searchfor the determination of the optimum reflection phase thatmaximizes |r(t)| for each time sample might be possible withdifferent system parameters, however, this does not fit withinthe scope of this study, which explores effective solutions forthe RIS configuration. We also verify from Fig. 13 that Method1 achieves approximately the maximum magnitude for thecomplex envelope in the considered experiment. In light ofour discussion above, we give the following remark:

Remark 4: For the case of two reflections with a single RISin Fig. 10, the heuristic choice to maximize the magnitudeof the complex envelope is to align the reflected signal tothe stronger component, that is, the LOS signal (Method 1)under normal circumstances. While this ensures a very highmagnitude for the complex envelope, we still observe a fadepattern in time domain. On the other hand, the RIS can bereversely aligned to the reflected signal from the plain IO(Method 3) to reduce the Doppler spread at the price of aslight degradation in the magnitude of the complex envelope.

Remark 5: For the setup of Fig. 10, the optimal reflectionphase that maximizes the magnitude of the complex envelopeis given by

θ1(t) =π

2(1− sgn(A))− tan−1(−B/A) (14)

where sgn(·) is the sign function and

A =1

dLOScos(4πfDt)−

1

d2

cos(2πfD(1− cosα)t+ φ2)

B =−1

dLOSsin(4πfDt) +

1

d2

sin(2πfD(1− cosα)t+ φ2).

(15)

The proof of (14) is given in Appendix. In Fig. 14, we comparethe reflection phases as well as magnitudes of the complexenvelope for Method 1 and the optimum method for the samesystem parameters. As seen from Fig. 14, Method 1 providesa very close phase behavior compared to the optimal onedue to the stronger LOS path and a very minor degradation

9

can be observed in the magnitude of the complex envelope.Nevertheless, the optimal reflection phase in (14) is validfor all possible system parameters in Fig. 10 and guaranteesthe maximum complex envelope magnitude at all times. Forreference, magnitude values are also shown in the same figurefor Method 3. As seen from Fig. 14, Method 3 reduces theseverity of the fade pattern (Doppler spread) while ensuringthe same minimum magnitude at the price of a lower timeaverage for the complex envelope.

2) Two RISs: Under the assumption of two RISs attachedto the existing two IOs in the system of Fig. 10, the receivedcomplex envelope is obtained as

r(t) =λ

4π

(e−j2πfDt

dLOS+ej2πfDt+jθ1(t)

dLOS + 2d1

+ej2πfD(cosα)t−jφ2+jθ2(t)

d2

)(16)

where the time-varying and intelligent reflection characteristicsof RIS 1 and 2 are captured by θ1(t) and θ2(t), respectively.Here, compared to the previous case, we have more freedomwith two controllable reflections and the magnitude of thereceived signal can be maximized (and the Doppler spreadcan be minimized) by readily aligning the reflected signalsto the LOS signal. This can be done by setting θ1(t) =−4πfDt (mod 2π) and θ2(t) = −2πfDt(1 + cosα) +φ2 (mod 2π), which results

r(t) =λe−j2πfDt

4π

(1

dLOS+

1

dLOS + 2d1+

1

d2

). (17)

Similar to the case with single intelligent reflection (Subsec-tion II.B), we obtain a constant-amplitude complex envelopeand a minimized Doppler spread (with a single component at−fD Hz) due to the clever co-phasing of the multipath com-ponents. Interested readers may easily obtain the magnitudeand the Doppler spectrum of the complex envelope to verifyour findings.

C. Two RISs without a LOS path

Finally, we extend our analysis for the case of non-LOStransmission through two RISs, which yields

r(t) =λ

4π

(ej2πfDt+jθ1(t)

dLOS + 2d1+ej2πfD(cosα)t−jφ2+jθ2(t)

d2

).

(18)

Similar to the case in Section III, by carefully adjustingthe phases of two RISs, the Doppler effect can be totallyeliminated due to the nonexistence of the LOS signal, which isout of control of the RISs. It is evident that this can be done byθ1(t) = −2πfDt (mod 2π) and θ2(t) = −2πfD(cosα)t +φ2 (mod 2π).

D. The General Case with Multiple IOs and the Direct Signal

Against this background, in this subsection, we extend ouranalyses for the general case of Fig. 15, which consists of atotal of R IOs. Here, we assume that N of them are coatedwith RISs, while the remaining M = R−N ones are plain IOs,

V

dLOS

BS

IO 1

MS

IO 2IO R

Fig. 15. The general case of multiple IOs with N RISs and M plain IOs(R = N +M ).

which create uncontrollable specular reflections towards theMS. In this scenario (N RISs and M plain IOs), the receivedcomplex envelope is given by

r(t) =λ

4π

(e−j2πfDt

dLOS+

N∑i=1

ej2πfR,it−jψi+jθi(t)

dR,i

−M∑k=1

ej2πfI,kt−jφk

dI,k

). (19)

Here, we assume that all rays stemming from IOs remainparallel during the movement of the MS for a short periodof time, which is a valid assumption, and without loss ofgenerality, we consider a reflection coefficient of −1 for theplain IOs. Additionally, the corresponding terms in (19) aredefined as follows:• fR,i: Doppler shift for the ith RIS• fI,k: Doppler shift for the kth plain IO• ψi: Constant phase shift for the ith RIS• φk: Constant phase shift for the kth plain IO• dR,i: Initial radio path distance for the ith RIS• dI,k: Initial radio path distance for the kth plain IO• θi(t): Adjustable phase shift of the ith RIS

Here, the Doppler shifts of the RISs and plain IOs are not onlydependent on the speed of the MS, but also on their relativepositions with respect to the MS, i.e., angles of arrival forthe incoming signals: fR,i = fD cosαi and fI,k = fD cosβk,where αi and βk are the angles of arrival for the reflectedsignals of ith RIS and kth plain IO, respectively. In thisgeneralized scenario, we focus on the following two setups:

1) Setup I (N ≤ M): In this setup, we have morenumber of uncontrollable reflectors (plain IOs) than RISs.Consequently, we extend our methods in Subsection IV.Band target either directly aligning N RISs to the LOS path(to improve the received signal strength) or eliminating thereflections stemming from N out of M plain IOs (to reducethe Doppler spread). While the alignment of the reflectedsignals to the LOS signal is straightforward (Method 1), theassignment of N RISs to corresponding IOs in real-timeappears as an interesting design problem. For this purpose,we consider a brute-force search algorithm to determine themost effective set of IOs to be targeted by RISs (Methods 2& 3). More specifically, N out of M IOs can be selectedin C(M,N) different ways, where C(·, ·) is the binomialcoefficient. Since these N RISs can be assigned to N plainIOs in N ! ways, we obtain a total of P (M,N) = C(M,N)N !possibilities (permutations) for the assignment of N RISs toM IOs. Our methodology has been summarized below:

10

• Method 1: We align the existing N RISs to the LOS pathby adjusting their reflection phases as θi(t) = −2πfR,it+ψi − 2πfDt (mod 2π) for i = 1, 2, . . . , N .

• Method 2: For the ith RIS minimizing the effect of thereflection stemming from the kth IO, i.e., ith RIS out-phased with the kth plain IO, we have the followingreflection phase: θi(t) = −2πfR,it + ψi + 2πfI,kt −φk (mod 2π) for i = 1, 2, . . . , N and k = 1, 2, . . . ,M .Considering these given reflection phases, for each timeinstant, we search for all possible N -permutations of Mplain IOs to maximize the absolute value of the complexenvelope. Then, the permutation of IOs that maximizesthe complex envelope magnitude is selected. This methodrequires a search over P (M,N) permutations in eachtime instant, in return, has a higher complexity thanthe first one. Specifically, let us denote the nth per-mutation (the set of IOs) by Pn =

{P1n,P2

n . . . ,PNn}

for n = 1, 2, . . . , P (M,N). For a given time instantt = t0, considering all permutations, we construct thepossible the set of RIS phases as θi(t0) = −2πfR,it0 +ψi + 2πfI,Pint0 − φPin (mod 2π) for i = 1, 2, . . . , Nand the corresponding estimate of the received signalsample rn(t0) is obtained from (19) for the nth per-mutation. Finally, the optimum permutation is obtainedas n = arg maxn |rn(t0)|. Then, the optimal set ofplain IOs to be targeted by RISs are determined as Pnand the RIS reflection phases are adjusted accordingly:θi(t0) = −2πfR,it0 +ψi + 2πfI,Pint0 − φPin (mod 2π)for i = 1, 2, . . . , N . These procedures are repeated for alltime instants. Obviously, this strategy requires the knowl-edge of all Doppler phases at a central processing unit,estimation of the received complex envelope samples, anda dynamic control of all RISs.

• Method 3: This method uses the same exhaustive searchapproach of Method 2, however, instead of maximizingthe the absolute value of the complex envelope, we tryto minimize the variation of it with respect to time byassigning the RISs to IOs with this purpose. Specifically,for a given time instant t = t0, the optimal permutationPn is obtained as n = arg minn

∣∣ |rn(t0)| − |r(t−1)|∣∣,

where r(t−1) is the sample of the received signal at theprevious time instant, while at t = 0, we determine theoptimal permutation as in Method 2. This method directlytargets to eliminate fade patterns of the complex envelopeinstead of focusing on the maximization of the receivedsignal strength by aligning (co-phasing) RISs with certainIOs. In other words, Method 3 eliminates the variationsin the received signal stemming from different Dopplershifts of the incoming signals.

2) Setup II (N > M): In this setup, we have more numberof RISs than the plain IOs, and consequently, have much morefreedom in the system design. Here, we consider the samethree methods discussed above (Setup I) for the adjustmentof RIS reflection phases, however, slight modifications areperformed for Methods 2 and 3 due to fewer number of plainIOs in this setup. In Method 1, we align the existing RISs tothe LOS path as in Setup I. To reduce the Doppler spread by

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300

400

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600

700

800

y-ax

is

IOsBSMS

5

21

3

1086

9

74

Fig. 16. Considered simulation geometry with multiple IOs (the first N ofthem are assumed to have RISs).

Method 2, we search for all possible M -permutations of RISsto target plain IOs, i.e., a total of P (N,M) permutations areconsidered. More specifically, at each time instant, we considerall possible RIS permutations to eliminate the reflections fromM plain IOs, while the remaining N −M RISs are alignedto the LOS path. The permutation of RISs that maximizes theabsolute value of the sample of the received signal is selected.On the other hand, Method 3 aims to minimize the variations inr(t) by assigning M RISs to M plain IOs, while also aligningthe remaining N−M RISs to the LOS path. Our methodologyhas been summarized as follows:

• Method 1: The same as Method 1 for Setup I.• Method 2: Let us denote the nth permutation (the set of

RISs) by Rn ={R1n,R2

n . . . ,RMn}

and the set of RISsthat are not included in the nth permutation by Sn ={S1n,S2

n . . . ,SN−Mn

}, i.e., Pn ∪ Sn = {1, 2, . . . , N}

for n = 1, 2, . . . , P (N,M). For a given time instantt = t0, considering all permutations, we construct thepossible the set of RIS phases to eliminate IO reflec-tions as θRin(t0) = −2πfR,Rint0 + ψRin + 2πfI,it0 −φi (mod 2π) for i = 1, 2, . . . ,M , while aligning theremaining N − M RISs to the LOS path as follows:θSin(t0) = −2πfR,Sint0 + ψSin − 2πfDt0 (mod 2π) fori = 1, 2, . . . , N −M . Then, the corresponding estimateof the received signal sample rn(t0) is obtained from(19) for the nth permutation. Finally, the optimum per-mutation is obtained as n = arg maxn |rn(t0)|. Then,the optimal set of RISs to be paired with IOs andaligned to the LOS path are determined as Rn and Sn,respectively, and the RIS reflection phases are adjustedaccordingly: θRin(t0) = −2πfR,Rint0 +ψRin +2πfI,it0−φi (mod 2π) for i = 1, 2, . . . ,M and θSin(t0) =− 2πfR,Sint0 + ψSin − 2πfDt0 (mod 2π) for i = M +1,M + 2, . . . , N . The above procedures are repeated forall time samples.

• Method 3: This method follows the same proceduresas that of Method 2, except the determination ofthe optimum permutation. This is performed by n =arg minn

∣∣ |rn(t0)| − |r(t−1)|∣∣ considering the current

(estimated corresponding to the nth permutation) andpreviously received signal samples of rn(t0) and r(t−1).

11

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cy r

espo

nse

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N=10,M=0 & Method 1

Fig. 17. Complex envelope and Doppler spectrum for two extreme cases underthe scenario of Fig. 15: (top) N = 0,M = 10 (10 plain IOs without anyRISs) and (bottom) N = 10,M = 0 (10 RISs without any plain IOs).

To illustrate the potential of our methods, we consider the2D geometry of Fig. 16 in our computer simulations, where theMS and the BS are located at (0, 0) and (−1000, 0) in terms oftheir (x, y)-coordinates, respectively. We assume that R = 10IOs are uniformly distributed in a predefined rectangular areaat the right hand side of the origin. We again consider a mobilespeed of V = 10 m/s with fc = 3 GHz and a sampling timeof λ/32, but use the following new simulation parameters: atravel distance of 30λ = 3 m and an FFT size of 1024.

In Fig. 17, we investigate two extreme cases: N = 0,M =10 and N = 10,M = 0. For the case of N = 0,M = 10, i.e.,the case without any RISs, we observe a Doppler spectrumconsisting of many components and in return, a severe deepfading pattern in the time domain. On the contrary, for thecase of N = 10,M = 0, in which all IOs in the system areequipped with RISs, we have a full control of the propagationenvironment by applying Method 1 (aligning the reflectedsignals from all RISs to the LOS path) and observe a constantmagnitude for the complex envelope as in Subsections II.Band IV.B.2. Here, we may readily state that the case ofN = 10,M = 0 with Method 1 provides the maximummagnitude for the complex envelope and can be consideredas a benchmark for all setups/methods with M > 0.

In Figs. 18-20, we consider three different scenarios basedon the number of RISs in the system: N = 3,M = 7 (SetupI), N = M = 5 (Setup I), and N = 7,M = 3 (Setup II)and assess the potential of the introduced Methods 1-3. Asseen from Figs. 18-20, although Method 1 ensures a highcomplex envelope magnitude in average with the price of alarger Doppler spread (faster variation in time), Methods 2 and3 are more effective in reducing the fade patterns observedin the time domain by modifying the Doppler spectrumthrough the elimination of plain IO signals. Particularly, theimprovements provided by Method 3 are more noticeable both

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Fig. 19. Complex envelope magnitude and Doppler spectrum for the generalcase with 10 IOs and N = M = 5 (Setup I).

in time and frequency domains. For instance, for the case ofN = 7,M = 3, Method 3 almost eliminates all Dopplerspectrum components stemming from three plain IOs andensures an approximately constant magnitude for the complexenvelope, as seen from Fig. 20.

To gain further insights, in Table I, we provide a quantitativeanalysis by comparing the peak-to-peak value ∆r of |r(t)|and its time average r (both measured in dB) for all methods,i.e., ∆r = |r(t)|max − |r(t)|min and r = 1

ns

∑ns−1n=0 |r(its)|,

where ns and ts respectively stand for the total numberof time samples and sampling time, which are selected asns = 960 and ts = 0.3125 ms for this specific simulation.As observed from Table I, increasing N noticeably reduces∆r for all methods, while this reduction is more remarkablefor Methods 2 and 3. We also evince that Methods 2 and

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Fig. 20. Complex envelope magnitude and Doppler spectrum for the generalcase with 10 IOs and N = 7,M = 3 (Setup II).

TABLE ICOMPARISON OF METHODS 1-3 IN TERMS OF PEAK-TO-PEAK VARIATION

(∆r IN DB) AND TIME-AVERAGE (r IN DB) OF |r(t)|.

Method 1 Method 2 Method 3

N = 3, M = 7∆r = 11.78r = −47.22

∆r = 5.43r = −47.76

∆r = 3.95r = −50.17

N = M = 5∆r = 7.08r = −45.99

∆r = 3.09r = −49.93

∆r = 2.47r = −50.88

N = 7, M = 3∆r = 2.91r = −44.90

∆r = 1.05r = −46.20

∆r = 0.66r = −46.62

3 cause in a slight degradation in r since they utilize RISsto cancel out reflections from plain IOs. Generalizing ourdiscussion from Subection 4.2.1, we claim that Method 1can be the preferred choice to maximize the (time-averaged)magnitude of the complex envelope due to the stronger LOSpath, however, the complete mathematical proof of this claimis highly intractable. We also observe that Method 2 providesa nice compromise between Methods 1 and 3 by providing amuch lower ∆r with a close r compared to Method 1, whileMethod 3 ensures the minimum ∆r.

E. The General Case with Multiple IOs and without the DirectSignal

In this section, we revisit the general case of the previoussection (Fig. 15), however, without the presence of a LOSpath. For this case, the received signal with N RISs and Mplain IOs can be expressed as follows:

r(t) =λ

4π

(N∑i=1

ej2πfR,it−jψi+jθi(t)

dR,i−

M∑k=1

ej2πfI,kt−jφk

dI,k

).

(20)Here, the three methods introduced in Subsection IV.D can beapplied with slight modifications. For Method 1, since there isno LOS path, the available RISs in the system can be aligned

to the strongest path, which might be from either an RIS or aplain IO and has the shortest radio path distance. For Methods2 and 3, when M ≥ N , we use the same procedures as in theLOS case and assign all N RISs to the plain IOs with differentpurposes. However, when N > M , after applying the samepermutation selection procedures, we determine the RIS withthe strongest path among the remaining N −M RISs in lieuof the LOS path and align the rest of the RISs (N −M − 1ones) to this strongest RIS for each specific permutation. Ourmethodology has been summarized below:

1) Setup I (M ≥ N):• Method 1: We align the existing N RISs to the strongest

path. If the strongest path belongs to a RIS, whose indexis a, we have θi(t) = −2πfR,it + ψi + 2πfR,at −ψa (mod 2π) for i = 1, . . . , a− 1, a+ 1, . . . , N , whileθa(t) = 0. Otherwise, if the strongest path belongs to aplain IO with index a, we have θi(t) = −2πfR,it+ψi+2πfI,at− φa + π (mod 2π) for i = 1, 2, . . . , N . Pleasenote that a = arg mini dR,i if mini dR,i < mink dI,k ora = arg mink dI,k, otherwise.

• Method 2: The same as Method 2 in Subsection IV.D forM ≥ N except that rn(t0) is obtained from (20) for thenth permutation.

• Method 3: The same as Method 3 in Subsection IV.D forM ≥ N except that rn(t0) is obtained from (20) for thenth permutation.

2) Setup II (N > M):• Method 1: The same as Method 1 given above.• Method 2: We follow the same steps for Method 2

in Subsection IV.D for N > M , however, for nthpermutation, the strongest RIS is selected among the setSn (the set of N − M RISs that are not included inthe elimination of IO reflections). Denoting the index ofthis strongest RIS by an, where an = arg mini∈Sn dR,i,we have θSin(t0) = −2πfR,Sint0 + ψSin + 2πfR,ant0 −ψan (mod 2π) for i = 1, 2, . . . , N −M with Sin 6= anand θan(t0) = 0 for this case. The above proceduresare repeated for all permutations and the estimates ofthe received signal samples are obtained as rn(t0) from(20) for n = 1, 2, . . . , P (N,M). After the determinationof the optimal permutation n, we obtain the set of RISstargeting the IOs as Rn while the set of remaining RISsare given by Sn. Finally, RIS angles are determinedas in Method 2 in Subsection IV.D for N > M withthe exception that the phases of the remaining N −MRISs are aligned as θSin(t0) = −2πfR,Sint0 + ψSin +2πfR,ant0 − ψan (mod 2π) for i = 1, 2, . . . , N − Mwith Sin 6= an and θan(t0) = 0. The above proceduresare repeated for all time instants.

• Method 3: This method follows the same procedures asthat of Method 2 given above, except the determinationof the optimum permutation, which is discussed in Sub-section IV.D.

In Figs. 21-22, we investigate the application of Methods 1-3 in two scenarios: N = 3,M = 7 (Setup I) and N = 7,K =3 (Setup II) for the same simulation scenario of Fig. 16 byignoring the LOS path. Compared to Figs. 18-20, we observe

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Fig. 21. Complex envelope magnitude and Doppler spectrum for the generalcase with 10 IOs without a LOS path and N = 3,M = 7 (Setup I).

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Fig. 22. Complex envelope magnitude and Doppler spectrum for the generalcase with 10 IOs without a LOS path and N = 7,M = 3 (Setup II).

that due to the nonexistence of the LOS path, all methodsprovide a similar level of time-average (r) for the complexenvelope while Methods 2 and 3 eliminate deep fades in thereceived signal. In other words, since we do not have a strongerLOS path, Method 1 loses its main advantage in terms of rcompared to the other two methods for both scenarios.

It is worth noting that for the case of N = 0, none of themethods are applicable as in the case of the previous section.However, for M = 0, Doppler effect can be totally eliminateddue to the nonexistence of the LOS path as follows: θi(t) =−2πfR,it+ φi (mod 2π) for i = 1, 2, . . . , N .

As a final note, our aim here is to find heuristic solutionsto mitigate deep fading and Doppler effects under arbitrarynumber of RISs and plain IOs, and the determination of theultimately optimum RIS angles are beyond the scope of this

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Fig. 23. Complex envelope magnitude in the presence of a realistic RIS forthe scenarios of Figs. 1 and 10.

work. Although our methods provide satisfactory results, theremight be a certain permutation of RISs/IOs with specific re-flection phases that may guarantee a maximized received com-plex envelope magnitude and/or the lowest Doppler spread.However, the theoretical derivation of this ultimate optimalsolution seems intractable at this moment.

V. PRACTICAL ISSUES

In this section, we consider a number of practical issuesand investigate the performance of our solutions under certainimperfections in the system.

A. Realistic RISs

Throughout this paper, we assumed that the utilized RISshave a unit-amplitude reflection coefficient with a very highresolution reflection phase θ(t) ∈ [0, 2π) that can be tuned inreal time. However, as reported in recent studies, there can benot only a dependency between the amplitude and the phasebut also a limited range can be supported for the reflectionphase. For this purpose, we consider the realistic RIS designof Tretyakov et al. [22], which has a reflection amplitude of−1 dB with a reflection phase between −150◦ and 140◦. InFig. 23, we compare the complex envelope magnitudes of twoscenarios in the presence of a perfect RIS (P-RIS) and animperfect RIS (I-RIS) with practical constraints: i) the scenarioof Fig. 1 with N = 1,M = 0 and ii) the scenario of Fig. 10with N = M = 1. As seen from Fig. 23, the practical RIS of[22] causes a slight degradation both in magnitude and shapeof the complex envelope, however, its overall effect is notsignificant. A further degradation would be expected in thepresence of discrete phase shifts [39], and this analysis is leftfor interested readers.

B. Imperfect Knowledge of Doppler Frequencies

As discussed in Section IV, in case of multiple RISs, acentral processing unit needs to acquire the knowledge ofDoppler frequencies of all incoming rays to initiate Methods1-3 in coordination with the available RISs. Here, we assumethat due to erroneous estimation of the velocity of the MSand/or relative positions of the IOs, the RISs in the systemare fed back with erroneous Doppler shifts (in Hz), given byfeR,i = fR,i + eR,i and feI,k = fI,k + eI,k, while the dominant

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Fig. 24. Complex envelope magnitude for the general case with 10 IOs witha LOS path and N = 7,M = 3 under erroneous Doppler frequency shifts atRISs (U = 1 and 4) with the perfect case (U = 0).

Doppler shift (fD) stemming from the LOS path is perfectlyknown. Here, eR,i and eI,k respectively stand for the errorsin Doppler shifts for ith RIS and kth plain IO. To illustratethe effect of this imperfection, these estimation error terms aremodelled by independent and identically distributed uniformrandom variables in the range [−U,U ] (in Hz). In Fig. 24, weconsider the scenario of N = 7,M = 3 with U = 0, 1 and 3for the same geometry of Fig. 16. As seen from Fig. 24, whilethe degradation in the complex envelope is not a major concernfor U = 1, a significant distortion has been observed for thecase of U = 4 with respect to time. Here, Methods 2 and3 appear more reliable in the presence of Doppler frequencyestimation errors, however, we observe that the overall systemis highly sensitive to this type of error.

C. High Mobility & Discrete-Time RIS Phases

In this subsection, we will focus on the case of highmobility under the assumption of discrete-time RIS reflectionphases. In this scenario, the RIS reflection phases remainconstant for a certain time duration. It is worth noting thatall methods described earlier are also valid for the case ofhigh mobility if the RIS reflection phases can be tuned inreal-time with a sufficiently high rate. However, in practice,due to limitations in terms of the RIS design and signalingoverhead in the network, the RIS reflection phases can betuned at only (certain) discrete-time instants. Let us denote theRIS reconfiguration interval by tr (in seconds), i.e., the RISphases can be adjusted in every tr seconds only. In our firstcomputer simulation, we consider that the complex envelopeis represented by its samples taken at every ts seconds. Here,we assume that once the RIS reflection phases are adjustedaccording to the LOS path, they remain fixed for Qts seconds.In other words, for Q = 1, we update the RIS reflection

phases at each sampling time and obtain the results giventhroughout the paper. In Fig. 25(a), we perform this simulationfor the high mobility case of V = 100 m/s, fc = 3 GHzand ts = 3.125 µs with N = 1 and M = 0 (for the basicscenario of Fig. 1). Here, ts has been intentionally reduced tocapture the variations in the complex envelope with respect totime due to the higher Doppler spread of the unmodulatedcarrier and a travel distance of 3λ is considered. In thiscase, we assume that RIS reflection phases are modified asθ(t) = −4πfDt (mod 2π) in every Qts seconds, i.e., the RIScannot be reconfigured fast enough compared to the samplingfrequency (variation) of the complex envelope. As seen fromFig. 25(a), a distortion is observed in the complex envelopedue to the delayed reconfiguration of RIS reflection phases.However, we conclude that even if with Q = 50, the variationin the complex envelope is not as significant as in the casewithout an RIS (shown in the figure as a benchmark), whilethe variation is not significant for Q = 20. In what follows, wepresent a theoretical framework to describe this phenomenon.

In mathematical terms, for the considered scenario that isformulated by (4) in terms of its received complex envelope,assuming that the RIS reflection phase is adjusted and fixedat time instant t1 while focusing on the complex envelope attime t2 > t1, we obtain

r(t2) =λ

4π

(e−j2πfDt2

dLOS+ej2πfDt2+jθ(t1)

dLOS + 2d1

)=λe−j2πfDt2

4π

(1

dLOS+

ej4πfD∆t

dLOS + 2d1

)(21)

where ∆t = t2−t1 < tr. Here, we considered the fact that theRIS reflection phase is fixed at time t1 as θ(t1) = −4πfDt1.As a result, we observe a variation in the complex envelopemagnitude, which is a function of both fD and ∆t. It is worthnoting that letting ∆t = 0 in (21), one can obtain (5) for t =t2. After simple manipulations, the magnitude of the complexenvelope is calculated as

|r(t2)|

=

(λ

4π

)(1

d2LOS

+1

(dLOS + 2d1)2+

2 cos(4πfD∆t)

dLOS(dLOS + 2d1)

)1/2.

(22)

It is evident from (22) that the magnitude of the complexenvelope is no longer constant unless 4πfD∆t � 1. In lightof the above analysis, to ensure a constant magnitude for thecomplex envelope, that is, to eliminate the fade pattern due toDoppler spread, we must have tr < 1

40πfDfor the considered

scenario. In other words, the RIS should be tuned fast enoughcompared to fD to capture the variations of the received signal.To illustrate this effect, in Fig. 25(b), for a fixed tr value of12.5 µs, we change the velocity of the MS and observe themagnitude of the complex envelope. As seen from Fig. 25(b),while the smaller Doppler frequency of 500 Hz (V = 50 m/s)can be captured by the RIS since tr <

140πfD

= 15.91 µsfor this scenario, we observe an oscillation in the magnitudefor the higher Doppler frequencies of 2 kHz (V = 200 m/s)and 4 kHz (V = 400 m/s) since the condition of tr < 1

40πfDis no longer satisfied. In light of the above discussion, we

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Fig. 25. Complex envelope magnitude for the scenario of Fig. 1 (N = 1,M =0) a) under high mobility (V = 100 m/s) and fixed reflection phases for aperiod of Qts seconds with Q = 1, 20, and 50, b) under increasing Dopplerfrequencies and a reflection phase update duration of tr = 12.5 µs.

conclude that increasing Doppler frequencies poses a muchbigger challenge for the real-time adjustment of RIS reflectionphases.

Finally, it is worth noting that in case of slow fading(1/fD � Ts), where Ts is the symbol duration, the channelmay be assumed to be static over one or several transmissionintervals and the variations in the magnitude of the complexenvelope from symbol to symbol (in our case, for unmodulatedcosine signals) can be compensated by adjusting RIS reflectionphases at every Ts seconds (with slight variations in magnitudeif Ts > 1

40πfD). On the other hand, in the case of fast fading

(1/fD < Ts), since the channel impulse response changesrapidly within the symbol duration, in order to compensateDoppler and fading effects, i.e., to obtain a fixed magnitude forthe complex envelope during a symbol duration, RIS reflectionphases should be tuned at a much faster rate compared to Ts.As an example, consider the transmission of an unmodulatedcosine signal for a period of 3 ms as in Fig. 25(a). For thiscase, we have fast fading due to the large Doppler spread, andthis can be eliminated by adjusting the RIS reflection phasesat a much faster rate compared to 3 ms, i.e., tr < 7.96 µs.Failure of doing this causes variations in the complex envelopemagnitude as shown in Fig. 25(a).

VI. CONCLUSIONS AND FUTURE WORK

In this paper, we have revisited the multipath fading phe-nomenon of mobile communications and provided uniquesolutions by utilizing the emerging concept of RISs in the pres-ence of Doppler effects. By following a bottom-up approach,first, we have investigated simple propagation scenarios with asingle RIS and/or a plain IO. Then we have developed severalnovel methods for the case of multiple RISs and plain IOsdepending on the their total numbers as well as the presenceof the LOS path. Finally, we have considered a number ofpractical issues, including erroneous estimation of Dopplershifts, practical reflection phases, and discrete-time reflectionphases, for the target setups and evaluated the overall perfor-mance under these imperfections. One of the most importantconclusions of this paper is that the multipath fading effectcaused by the movement of the mobile receiver/transmitter

can be effectively eliminated and/or mitigated by real-timetuneable RISs. A number of interesting trade-offs have beendemonstrated between fade pattern elimination and complexenvelope magnitude maximization. While this work sheds lighton the development of RIS-assisted mobile networks, explo-ration of amplitude/phase modulations and more practical pathloss/propagation models appear as interesting future researchdirections.

APPENDIX

The received complex envelope in (12) can be expressed as

r(t) = rLOSejξLOS(t) + r1e

jξ1(t) + r2ejξ2(t) (23)

where magnitude and phase values of the LOS and tworeflected signals (from IO 1 (RIS) and IO 2) are shown byrLOS, r1, r2 and ξLOS(t), ξ1(t), ξ2(t), respectively. Here, weare interested in the maximization of |r(t)| with respect toξ1(t) = 2πfDt + θ1(t), which captures the reconfigurablereflection phase of the RIS. We use the following trigonometricidentity: For z1 = r1e

jξ1 , z2 = r2ejξ2 , z3 = r3e

jξ3 , andz4 = z1 + z2 + z3 = r4e

jξ4 , we have r4 = (r21 + r2

2 + r23 +

2r1r2 cos(ξ1−ξ2)+2r1r3 cos(ξ1−ξ3)+2r2r2 cos(ξ2−ξ3))1/2.In light of this, the maximization of |r(t)| can be formulatedas

maxθ1(t)

|r(t)|2

maxθ1(t)

rLOSr1 cos(ξLOS(t)− ξ1(t)) + r1r2 cos(ξ1(t)− ξ2(t))

maxθ1(t)

rLOS cos(4πfDt+ θ1(t))

+ r2 cos(2πfDt(1− cosα) + φ2 + θ1(t)) (24)

where the constant magnitude terms and the term does notcontain θ1(t) is dropped. Using the identity cos(x + y) =cosx cos y− sinx sin y and grouping the terms with θ1(t), weobtain

maxθ1(t)

A cos θ1(t) +B sin θ1(t)

maxθ1(t)

sgn(A)√A2 +B2 cos(θ1(t) + tan−1(−B/A)) (25)

where A and B are as defined in (15) and the harmonicaddition theorem [47] is used. Consequently, to maximize thecomplex envelope, we have to ensure

sgn(A) cos(θ1(t) + tan−1(−B/A)) = 1. (26)

This can be satisfied by

θ1(t) =π

2(1− sgn(A))− tan−1(−B/A) (27)

which completes the proof.

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Ertugrul Basar (S’09-M’13-SM’16) received theB.S. degree (Hons.) from Istanbul University,Turkey, in 2007, and the M.S. and Ph.D. degreesfrom Istanbul Technical University, Turkey, in 2009and 2013, respectively. He is currently an AssociateProfessor with the Department of Electrical andElectronics Engineering, Koc University, Istanbul,Turkey and the director of Communications Re-search and Innovation Laboratory (CoreLab). Hisprimary research interests include MIMO systems,index modulation, reconfigurable intelligent sur-

faces, waveform design, visible light communications, and signal processingfor communications.

Dr. Basar currently serves as an Editor of the IEEE TRANSACTIONS ONCOMMUNICATIONS and Physical Communication (Elsevier), and as a SeniorEditor of the IEEE COMMUNICATIONS LETTERS.

Ian F. Akyildiz (F’96) has been a Consulting ChairProfessor with the Department of Information Tech-nology, King Abdulaziz University, Jeddah, SaudiArabia, since 2011. He has been with Computer En-gineering Department, University of Cyprus, sinceJanuary 2017. He has also been a Megagrant Re-search Leader with the Institute for InformationTransmission Problems, Russian Academy of Sci-ences, Moscow, Russia, since May 2018. He is cur-rently the Ken Byers Chair Professor in telecommu-nications with the School of Electrical and Computer

Engineering, the Director of the Broadband Wireless Networking Labora-tory, and the Chair of the Telecommunication Group, Georgia Institute ofTechnology, Atlanta, USA. His current research interests include 5G wirelesssystems, nanonetworks, Terahertz band communications, and wireless sensornetworks in challenged environments. He has been an ACM Fellow, since1997. He received numerous awards from the IEEE and ACM and manyother organizations. His h-index is 116, and the total number of citations isabove 112K as per Google Scholar, as of December 2019.