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Tarek Hassan
Knowledge Hubwireless communication basics9. Phased Arrays: From Element Phase to RIS Control

wireless communication basics learning note

9. Phased Arrays: From Element Phase to RIS Control

Build phased-array intuition from two- and three-element examples, derive distance and phase compensation, compare architectures, and connect active arrays with RIS beam control.

What Is a Phased Array?

A phased array is a group of antenna elements whose relative excitation phases—and often amplitudes—are controlled so that their electromagnetic fields combine into a desired spatial pattern. The array can strengthen radiation toward a target, suppress selected directions, scan a beam electronically, or focus energy at a near-field point without mechanically rotating the antenna.

For a first mental picture, imagine several people pushing the same swing. If everyone pushes at the correct point in the swing's cycle, the motion becomes stronger. If some push while others oppose the motion, part of the effort cancels. Radio waves behave similarly:

  • phase says where a wave is within its repeating cycle;
  • in phase means peaks and troughs arrive together and reinforce;
  • out of phase means they arrive at different parts of the cycle and add less effectively.

A phased array controls this timing electronically. It cannot change the geometric distance between an element and the target, but it can start or rotate each element's wave so that unequal journeys finish at the target in the same phase.

The word phased refers to controlled relative phase. A collection of antennas is not automatically a useful phased array: the elements need a known geometry, a coherent frequency reference, controllable excitation or combining weights, and adequate calibration.

The complex weight applied to element nn is

wn=anejϕnw_n=a_ne^{j\phi_n}

where ana_n sets amplitude and ϕn\phi_n sets phase. Phase controls where contributions align; amplitude tapering controls the trade-off between main-lobe width and sidelobe level.

References for this section3

The Building Blocks

A practical phased-array system normally contains:

BlockPurpose
Antenna elementsConvert guided RF signals to radiated fields, or the reverse
Reference oscillatorKeeps RF channels coherent at a common carrier frequency
Phase shifters or time-delay unitsSet relative element phase or delay
Gain controlApplies amplitude tapering and corrects channel mismatch
Feed or combining networkDistributes transmit power or combines received signals
RF chains and convertersConnect analogue RF signals to digital baseband
Beam controllerCalculates weights for a direction, user, null, or focal point
Calibration systemMeasures and corrects phase, gain, coupling, and hardware drift

Element spacing is commonly near λ/2\lambda/2 to support wide angular scanning without grating lobes. The final value depends on scan range, element pattern, mutual coupling, bandwidth, packaging, and platform constraints.

References for this section3

First Decide What Kind of Array We Mean

Three related systems are often described with similar beamforming language:

  1. Active transmit phased array: one transmitter or baseband signal is divided into RF branches. Each branch receives a controlled phase and amplitude before its antenna element radiates.
  2. Receive phased array: an external transmitter's wave reaches several receive elements. Their electrical outputs are phase-corrected and added inside the receiver.
  3. Reflecting array or RIS: an external transmitter illuminates controllable cells. Each cell changes the phase of the incident field and re-radiates or reflects it toward a desired point.

If the picture is Tx → array surface → target, the clearest basic model is the third case. The array is acting as a controlled reflector. A conventional passive metallic reflector also re-radiates an incident wave, but an RIS or reflectarray deliberately changes the cell phases to choose where the reflected contributions meet.

Active array: feed → elements → space·Receive array: space → elements → combiner·RIS: Tx → cells → target
References for this section3

Beginner Geometry: Tx, Three Cells, and One Target Point

Let the transmitter be at TT, three controllable cells be at A0A_0, A1A_1, and A2A_2, and the desired point be PP. For cell nn:

  • sn=∣T−An∣s_n=|T-A_n| is the incident distance from Tx to the cell;
  • qn=∣An−P∣q_n=|A_n-P| is the outgoing distance from the cell to the target;
  • Ln=sn+qnL_n=s_n+q_n is the complete Tx–cell–target path.
Original labelled geometry showing a transmitter illuminating three controllable array cells and the three reflected paths meeting at target point P
Original geometry: every cell receives the same transmitted waveform through a different incident distance $s_n$ and sends it through a different outgoing distance $q_n$. The controller compensates the complete path $L_n=s_n+q_n$.

The carrier accumulates propagation phase as it travels. Using the convention ejωte^{j\omega t}, the contribution through cell nn arrives at PP with phase

Φn(P)=−ksn+ϕn−kqn=ϕn−kLn\Phi_n(P) = -k s_n +\phi_n -k q_n = \phi_n-kL_n

where ϕn\phi_n is the controllable phase applied by the cell. Without control, unequal LnL_n values normally produce unequal arrival phases, so the waves add only partly and may even cancel.

Choose a reference path LrefL_{\mathrm{ref}}. Setting

ϕn=k(Ln−Lref)(mod2π)\phi_n = k(L_n-L_{\mathrm{ref}}) \pmod{2\pi}

gives

Φn(P)=−kLref\Phi_n(P) = -kL_{\mathrm{ref}}

for every cell. All contributions then reach PP with the same phase.

References for this section3

Tx-to-Array Focusing, Step by Step

Original four-step diagram showing transmission, unequal phase at three array cells, cell phase compensation, and coherent addition at the target
Original step-by-step diagram: the array first receives unequal incident phases, calculates each complete path to the target, applies a cell-specific correction, and produces aligned arrivals at the chosen point.
References for this section3

Step 1: One Wave Leaves the Tx

The transmitter creates one carrier waveform. It spreads through space and reaches the three cells. Because s0s_0, s1s_1, and s2s_2 are different, the carrier is generally at a different phase when it reaches each cell.

References for this section3

Step 2: Each Cell Has a Different Complete Path

The controller does not compare only Tx-to-cell distance. It includes the remaining cell-to-target distance:

L0=s0+q0,L1=s1+q1,L2=s2+q2L_0=s_0+q_0, \qquad L_1=s_1+q_1, \qquad L_2=s_2+q_2

The cell with the longest complete path accumulates the greatest propagation phase lag before reaching PP.

References for this section3

Step 3: Apply an Individual Phase Correction

The surface adds ϕ0\phi_0, ϕ1\phi_1, and ϕ2\phi_2. These phase shifts do not shorten or lengthen the geometric paths. They rotate the cells' local RF phases so that the unequal propagation phases are compensated at the selected target.

As a small example, let λ=10 mm\lambda=10\ \mathrm{mm} and suppose the complete paths relative to cell 0 are

Ln−L0=[037]mmL_n-L_0 = \begin{bmatrix} 0 & 3 & 7 \end{bmatrix}\mathrm{mm}

The compensating phase settings are

ϕ0=0∘,ϕ1=360∘310=108∘,ϕ2=360∘710=252∘.\begin{aligned} \phi_0&=0^\circ,\\ \phi_1&=360^\circ\frac{3}{10}=108^\circ,\\ \phi_2&=360^\circ\frac{7}{10}=252^\circ. \end{aligned}

The exact signs depend on the time and phase convention. The physical test is unambiguous: after propagation and cell control, the final phases at PP must match.

References for this section3

Step 4: The Waves Meet at the Target

At PP, the aligned electric-field amplitudes add. If three equal contributions each have complex amplitude AA, their coherent sum is

E(P)=A+A+A=3AE(P)=A+A+A=3A

whereas arbitrary phases produce a smaller magnitude

∣E(P)∣=∣AejΦ0+AejΦ1+AejΦ2∣≤3A|E(P)| = \left| A e^{j\Phi_0} +A e^{j\Phi_1} +A e^{j\Phi_2} \right| \leq 3A

The equality occurs only when the three phases are aligned. At other points, the outgoing path lengths differ from the design values, so the same cell settings no longer align perfectly. That spatial change creates the focused beam pattern.

References for this section3

Blocked Tx–UE Link: A Horizontal Three-Element Example

Now consider the exact layout in the following diagram:

  • three controllable elements lie on the horizontal xx-axis;
  • adjacent elements are separated by d=λ/2d=\lambda/2;
  • the Tx is above and to the left;
  • the UE is above and to the right;
  • an obstacle blocks the direct Tx-to-UE path;
  • the lower surface creates a two-hop Tx–array–UE link.

This is a RIS or reflecting phased-array configuration. The ordinary direct signal cannot reach the UE, so the useful field must travel from Tx to the surface and then from the surface to the UE.

Original geometry with three horizontally spaced controllable elements, a transmitter in the upper-left corner, a user in the upper-right corner, an obstacle blocking the direct link, and three reflected paths through the array
Original blocked-link geometry: the direct Tx–UE ray is obstructed. Each horizontal element receives the incident wave through $s_n$, applies phase $\phi_n$, and sends it toward the UE through $q_n$.
References for this section3

Step 1: Define the Coordinates

Measure every coordinate in wavelengths. Choose

A0=(−λ2,0),A1=(0,0),A2=(λ2,0)A_0=\left(-\frac{\lambda}{2},0\right), \qquad A_1=(0,0), \qquad A_2=\left(\frac{\lambda}{2},0\right)

so the adjacent spacing is d=λ/2d=\lambda/2. For an illustrative asymmetric geometry, let

T=(−3λ,4λ),U=(4λ,3λ)T=(-3\lambda,4\lambda), \qquad U=(4\lambda,3\lambda)

The exact numbers are chosen only to make the calculation concrete; the same method works for measured coordinates in metres.

References for this section3

Step 2: Calculate Both Parts of Every Route

For element nn at An=(xn,0)A_n=(x_n,0),

sn=(xn−xT)2+zT2s_n = \sqrt{(x_n-x_{\mathrm{T}})^2+z_{\mathrm{T}}^2}

is the Tx-to-element distance, and

qn=(xU−xn)2+zU2q_n = \sqrt{(x_{\mathrm{U}}-x_n)^2+z_{\mathrm{U}}^2}

is the element-to-UE distance. The complete reflected route is

Ln=sn+qnL_n=s_n+q_n

For the coordinates above:

Elementxnx_nIncident sns_nOutgoing qnq_nTotal LnL_n
A0A_0−0.5λ-0.5\lambda4.7170λ4.7170\lambda5.4083λ5.4083\lambda10.1253λ10.1253\lambda
A1A_1005.0000λ5.0000\lambda5.0000λ5.0000\lambda10.0000λ10.0000\lambda
A2A_2+0.5λ+0.5\lambda5.3151λ5.3151\lambda4.6098λ4.6098\lambda9.9248λ9.9248\lambda

Notice that element A0A_0 is closer to the Tx than A2A_2, but farther from the UE. The controller must use the sum of both distances; choosing the closest element to only the Tx or only the UE is insufficient.

References for this section3

Step 3: See Why the Uncontrolled Reflections Do Not Align

Without programmable phase control, set ϕn=0\phi_n=0. The contribution arriving through element nn has phase

Φn(U)=−kLn\Phi_n(U)=-kL_n

Only the fractional part of Ln/λL_n/\lambda affects phase, because every complete wavelength contributes 360∘360^\circ and returns to the same phase. Relative to the centre-element path L1L_1, the path differences are

[L0−L1L1−L1L2−L1]=[0.12530−0.0752]λ\begin{bmatrix} L_0-L_1 & L_1-L_1 & L_2-L_1 \end{bmatrix} = \begin{bmatrix} 0.1253 & 0 & -0.0752 \end{bmatrix}\lambda

These unequal fractions correspond to unequal propagation phases, so the three reflected waves do not naturally peak together at the UE.

References for this section3

Step 4: Program the Three Element Phases

Use L1L_1 as the reference path and apply

ϕn=k(Ln−L1)(mod2π)\phi_n = k(L_n-L_1) \pmod{2\pi}

The required phase settings are approximately

ElementRelative pathProgrammed phase
A0A_0+0.1253λ+0.1253\lambda+45.1∘+45.1^\circ
A1A_1000∘0^\circ
A2A_2−0.0752λ-0.0752\lambda−27.1∘-27.1^\circ, equivalent to 332.9∘332.9^\circ

After applying these settings,

Φn(U)=ϕn−kLn=k(Ln−L1)−kLn=−kL1\begin{aligned} \Phi_n(U) &=\phi_n-kL_n\\ &=k(L_n-L_1)-kL_n\\ &=-kL_1 \end{aligned}

for all three elements. The element paths remain physically unequal, but their final phases at the UE become equal.

References for this section3

Step 5: Add the Three Contributions at the UE

Including path-dependent amplitude, an illustrative received field is

EUE∝∑n=02βnsnqnej(ϕn−kLn)E_{\mathrm{UE}} \propto \sum_{n=0}^{2} \frac{\beta_n}{s_nq_n} e^{j\left(\phi_n-kL_n\right)}

where βn\beta_n represents the reflection amplitude of element nn. Phase programming aligns the exponential terms. The factors 1/(snqn)1/(s_nq_n) and βn\beta_n need not be equal, so coherent phase does not imply identical amplitude from every route.

The obstacle still blocks the straight line. The array does not remove the obstacle; it creates a controlled alternative path around it. If the UE moves, every qnq_n changes, so the old phase profile is no longer optimal and the surface must calculate or learn a new one.

References for this section3

How Fields from Several Elements Combine

Now return to an active transmit phased array. Here the transmitter feeds each element through a controlled RF branch instead of illuminating a reflecting surface. Assume the nnth element transmits a narrowband signal with weight wn=anejϕnw_n=a_ne^{j\phi_n}. At a target point p\mathbf{p}, its propagation distance is rnr_n. Ignoring polarization and element-pattern factors for the moment, the received complex field is

E(p)∝∑n=0N−1anrnej(ϕn−krn),k=2πλE(\mathbf{p}) \propto \sum_{n=0}^{N-1} \frac{a_n}{r_n} e^{j(\phi_n-kr_n)}, \qquad k=\frac{2\pi}{\lambda}

Maximum coherent addition occurs when every element has the same final phase at the target:

ϕn−krn=ϕcommon(mod2π)\phi_n-kr_n=\phi_{\mathrm{common}}\pmod{2\pi}

One valid transmit-phase choice is therefore

ϕn=k(rn−rref)(mod2π)\phi_n = k(r_n-r_{\mathrm{ref}}) \pmod{2\pi}

where rrefr_{\mathrm{ref}} is any chosen reference distance. The controller does not make the physical paths equal. It introduces phase offsets that compensate for unequal path lengths at the design point.

References for this section3

A Two-Element Array, Step by Step

Consider two identical elements separated by dd.

  1. Equal phase: with ϕ0=ϕ1\phi_0=\phi_1, waves reinforce most strongly where the propagation paths are equal, producing a broadside maximum for a symmetric array.
  2. Off-axis target: toward angle θ0\theta_0, the paths differ approximately by Δr=dsin⁡θ0\Delta r=d\sin\theta_0 in the far field.
  3. Phase compensation: the feed network applies the opposite propagation-phase difference so both signals arrive aligned.

The required progressive phase is

Δϕ=−kdsin⁡θ0=−2πdλsin⁡θ0\Delta\phi = -k d\sin\theta_0 = -\frac{2\pi d}{\lambda}\sin\theta_0
Original three-step diagram showing equal-phase broadside radiation, path difference toward an angled target, and phase compensation using two and three array elements
Original step-by-step diagram: identify the geometric path difference, convert it to phase, then apply a progressive phase profile so all element contributions reach the target coherently.
References for this section3

A Three-Element Numerical Example

Let three elements lie at xn=ndx_n=nd with d=λ/2d=\lambda/2, and steer toward θ0=30∘\theta_0=30^\circ. The adjacent path difference is

Δr=dsin⁡30∘=λ4\Delta r = d\sin30^\circ = \frac{\lambda}{4}

so the progressive phase is

Δϕ=−2πΔrλ=−π2=−90∘\Delta\phi = -2\pi\frac{\Delta r}{\lambda} = -\frac{\pi}{2} = -90^\circ

A suitable phase sequence is

[ϕ0ϕ1ϕ2]=[0∘−90∘−180∘]\begin{bmatrix} \phi_0 & \phi_1 & \phi_2 \end{bmatrix} = \begin{bmatrix} 0^\circ & -90^\circ & -180^\circ \end{bmatrix}

Adding the same constant phase to all three entries does not change the beam direction because only relative phase matters. Reversing the sign convention or the array axis reverses the phase progression, so diagrams and software must define both explicitly.

References for this section3

How Distance and Phase Are Balanced

There are two related design cases.

References for this section3

Far-Field Steering: Balance by Direction

For a linear array, the far-field distance to element nn is approximated by

rn≈r−xnsin⁡θ0r_n \approx r-x_n\sin\theta_0

The common distance rr disappears from relative phase. The weight depends on angle:

ϕn=−kxnsin⁡θ0\phi_n = -kx_n\sin\theta_0

Every point sufficiently far away along the same direction has approximately the same normalized steering vector. The array forms an angular beam whose ideal focus is effectively at infinity.

References for this section3

Near-Field Focusing: Balance by Distance and Direction

For a focal point (xmathrmu,zmathrmu)(x_{mathrm{u}},z_{mathrm{u}}), retain each exact distance:

rn=(xu−xn)2+zu2r_n = \sqrt{(x_{\mathrm{u}}-x_n)^2+z_{\mathrm{u}}^2}

and apply

ϕn=k(rn−rref)(mod2π)\phi_n = k(r_n-r_{\mathrm{ref}}) \pmod{2\pi}

The resulting spherical phase profile can align the waves at one range–angle point. A user at the same angle but a different range sees residual phase error and a smaller coherent sum.

Original comparison of linear phase for far-field steering and curved phase for near-field focusing
Original diagram: far-field steering uses a linear phase ramp determined by angle, while near-field focusing uses element-wise spherical distances determined by both range and angle.

Amplitude can also be balanced. If element-to-target distances differ substantially, ana_n may compensate for relative spreading or implement a desired taper. In practice, strict amplitude inversion can demand excessive power and amplify model error, so constrained optimization is normally preferable.

References for this section3

Transmit and Receive Operation

On transmit, the array applies weights before radiation and the fields combine in space. On receive, each element observes a different phase and amplitude; the receiver applies combining weights so a desired direction or focal point adds coherently.

For a reciprocal calibrated channel, the spatial signature used for receive combining is closely related to the conjugate transmit weight. Hardware reciprocity is imperfect, so transmit and receive RF chains may require separate calibration even when propagation itself is reciprocal.

References for this section3

Analogue, Digital, and Hybrid Architectures

ArchitectureControlMain strengthMain limitation
AnalogueOne or few RF chains feeding phase shiftersLow converter count and powerUsually one/few beams; phase-only and beam-squint constraints
DigitalOne RF chain and converter path per elementFlexible multi-beam, multi-user, and frequency-dependent processingHigh cost, data rate, and power consumption
HybridDigital precoder plus analogue phase networkCompromise between flexibility and RF complexityJoint design, calibration, and hardware constraints

True-time-delay arrays replace frequency-independent phase shifts with physical or synthesized delays. They reduce beam squint over wide bandwidth but increase hardware complexity, insertion loss, and cost.

References for this section3

Phased Array Is Not the Same as MIMO

A phased array describes coherent spatial weighting across antenna elements. MIMO describes a system with multiple signal inputs and outputs. A 64-element analogue phased array driven by one RF chain can produce a narrow beam but carry only one baseband stream. A fully digital or hybrid array with several independent RF dimensions can support both phased-array beamforming and MIMO spatial multiplexing.

References for this section3

How Phased-Array Principles Are Used in RIS

A reconfigurable intelligent surface does not normally generate an independent RF waveform at each cell. Instead, each cell applies a programmable complex reflection or transmission coefficient

Γn=βnejϕn\Gamma_n=\beta_ne^{j\phi_n}

to an incident field. If rB,nr_{\mathrm{B},n} is the distance from the base station to cell nn and rn,Ur_{n,\mathrm{U}} is the distance from that cell to the user, the total propagation phase is associated with rB,n+rn,Ur_{\mathrm{B},n}+r_{n,\mathrm{U}}. A focusing profile satisfies

ϕn=k(rB,n+rn,U−rref)(mod2π)\phi_n = k\left(r_{\mathrm{B},n}+r_{n,\mathrm{U}}-r_{\mathrm{ref}}\right) \pmod{2\pi}

so reflected contributions align at the user.

Original diagram comparing active phased-array transmission with RIS phase-controlled reflection toward a user
Original diagram: an active phased array controls generated element signals, while an RIS controls the phase of an incident wave over the two-hop base-station–surface–user path.

The analogy is useful but not exact:

Active phased arrayRIS
Elements are actively excited or connected to receive chainsCells usually modify an incident wave passively or semi-passively
Transmit power is supplied through RF chains and amplifiersPassive RIS cannot create net RF power
Complex weights may offer detailed amplitude and phase controlPhase states are often quantized and amplitude depends on state
Direct channel sensing may be available per RF chainChannel estimation is harder when cells lack receive chains
One propagation path from array to userCascaded transmitter–RIS and RIS–user paths
References for this section3

Advantages

  • Electronic steering: rapid scanning without mechanical motion.
  • Array gain: coherent combination improves link budget in the selected direction.
  • Spatial selectivity: narrow beams and nulls can reduce interference.
  • Multi-function operation: digital arrays can support communication, sensing, localization, and several users.
  • Low-profile integration: planar arrays fit base stations, vehicles, aircraft, satellites, and compact radar panels.
  • Near-field focusing: large electrical apertures can control energy in both range and angle.
References for this section3

Limitations and Failure Modes

  • Cost and power: many RF chains, converters, amplifiers, or phase shifters increase hardware demand.
  • Calibration: phase and gain errors reduce coherent gain and shift the beam.
  • Mutual coupling: neighbouring elements alter impedance and embedded element patterns.
  • Grating lobes: excessive spacing produces unwanted beams during scanning.
  • Sidelobes: finite apertures radiate energy outside the main beam.
  • Beam squint: fixed phase shifts steer different frequencies toward different angles.
  • Scan loss: element patterns and projected aperture reduce gain at large scan angles.
  • Quantization: limited phase states approximate the desired weight profile.
  • Thermal and manufacturing variation: responses drift across elements and time.
  • Blockage and model error: a correctly calculated line-of-sight beam can fail when the environment changes.
References for this section3

Applications

  • 5G/6G base stations and mmWave access points;
  • satellite terminals and electronically steered antennas;
  • automotive, aviation, weather, and defence radar;
  • radio astronomy and interferometric sensing;
  • indoor positioning, localization, and channel sounding;
  • medical imaging and non-contact monitoring;
  • wireless power transfer and near-field energy focusing;
  • RIS-assisted coverage, localization, and integrated sensing and communication.
References for this section3

Practical Design Checklist

Before calculating phase weights, state:

  1. carrier frequency, bandwidth, and wavelength;
  2. array geometry, element coordinates, patterns, and polarization;
  3. spacing and intended scan sector;
  4. far-field steering or near-field focusing model;
  5. phase convention and transmit/receive direction;
  6. available RF chains, phase resolution, amplitude range, and power constraint;
  7. calibration method and acceptable phase error;
  8. channel-estimation and beam-training overhead;
  9. performance metric: gain, sidelobe level, SINR, localization error, or energy efficiency.
References for this section3

Takeaway

A phased array shapes space by controlling relative element phase and amplitude. Path lengths determine the propagation phase; the array applies compensating weights so fields align at a chosen direction or focal point. Active arrays generate or receive signals through RF hardware, while RIS cells apply the same phase-alignment principle to an incident wave over a cascaded path. Geometry, bandwidth, calibration, hardware architecture, and channel knowledge determine how much theoretical array gain survives in practice.

References for this section3

Complete references and further reading