This work has met the research ethics requirement and received the university Research Ethics Committee’s approval, with the number EA1772021.

A sender transmits signal x( t), a continuous wave (CW) that has been modulated by symbol s( t), on M antennas at time t. The transmit signal on antenna m is varied by phase φ _m ( t), i.e. x m ( t ) = 2 P M ℜ { s ( t ) e j [ ω 0 t + φ m ( t ) ] } (1)

ω 0 is the carrier frequency. Ε [ | s ( t ) | 2 ] = 1 . After propagating over the wireless channel, the signal picked up at single receive antenna is y ( t ) = 2 P M Λ ℜ { h ( t ) s ( t ) e j ω 0 t }

Λ is the path loss, h( t) represents the overall time-varying channel gain. h ( t ) = ∑ m = 1 M h m e j φ m ( t ) (2)

h _m is the amplitude gain from antenna m due to channel fading. Let r( t) = h( t) s( t). y ( t ) = 2 P M Λ ℜ { r ( t ) e j ω 0 t } (3)

The effective RF power level at the input of an energy harvester is P RF = Ε [ y 2 ( t ) ] (4)

Assume that { φ m ( t ) } ∀ m are uniformly distributed over 2π and are independent, and h m = 1   ∀ m . It can be shown that P RF = P ¯ RF | r ( t ) | 2 M (5)

| r( t)| ² is the received signal envelope. P ¯ RF = Ε [ P RF ] = P Λ (6)

Given certain expression for s( t), we can determine| r( t)|and thus P _RF. In the following, we consider a combination of multiple sinewaves of which the center frequencies are constantly separated apart. Suppose the multisine waveforms are given by s ( t ) = ∑ n = 0 N − 1 1 N exp ( j n Δ w t ) ,   Δ w = 2 π Δ f (7)

Δ _w is the inter-carrier frequency spacing. Assume that { φ m ( t ) } ∀ m are uniformly distributed over 2π and are independent, and h m = 1   ∀ m . r ( t ) = ∑ m = 1 M ∑ n = 0 N − 1 1 N e j n Δ w t e j φ m ( t ) (8)

Consider a special case of M = 1. r ( t ) = 1 N ∑ n = 0 N − 1 e j [ n Δ w t + φ ( t ) ] (9)

To calculate | r( t)|, let us first consider a simple case of N = 2 sinewaves where n = 1, 2. r ( t ) = 1 2 ( e j [ Δ 1 t + φ ( t ) ] + e j [ Δ 2 t + φ ( t ) ] ) ,     Δ 2 = 2 Δ 1 (10)

It can be shown that | r ( t ) | 2 = 1 + cos θ (11)

where θ = θ ( t ) = Δ 1 t ,   0 ≤ θ < 2 π (12)

When M = 1 and N = 3, we find that | r ( t ) | 2 = 1 + 2 3 ( 2 cos θ + cos 2 θ ) (13)

By induction, it can be shown that for M = 1, | r ( t ) | 2 = 1 + 2 N [ ( N − 1 ) cos θ + ( N − 2 ) cos 2 θ + ... + 2 cos ( N − 2 ) θ + cos ( N − 1 ) θ ] (14)

To the best of our knowledge, (14) is a novel expression.

Based on a simulation study on a rectenna circuit and curve fitting on the collected data, Clerckx and Kim ⁵ have shown that the RF-to-DC energy conversion can be represented by a polynomial as follows: ln P DC = u ( ln P RF ) 2 + v ln P RF + w (15)

Based on the results from the circuit simulations conducted by Clerckx and Kim, ⁵ when the input was a CW, the coefficients u, v, and w of the polynomial took the following values: u = − 0.0669 ,   v = − 0.1317 ,   w = − 6.3801 for  − 40  dBm ≤ P RF ≤ − 5  dBm (16)

Since P RF = P ¯ RF | r ( t ) | 2 M , ln P DC = u [ ln ( P ¯ RF | r ( t ) | 2 M ) ] 2 + v ln ( P ¯ RF | r ( t ) | 2 M ) + w

After some workings, we have the following expression: ln P DC = ln P DC, 0 + ( ln | r ( t ) | 2 M ) [ u ( ln | r ( t ) | 2 M + 2 ln P ¯ RF ) + v ] (17)

where ln P DC, 0 = u ( ln P ¯ RF ) 2 + v ln P ¯ RF + w (18)

P _{DC, 0} is the DC power harvested from P ¯ RF . Eq. (17) can also be expressed in the following form: P DC = P DC, 0 ⋅ e td (19)

e _td is the gain from transmit diversity, e td = ( | r ( t ) | 2 M ) u ( ln | r ( t ) | 2 M + 2 ln P ¯ RF ) + v (20)

The average harvested DC power, P ¯ DC = Ε [ P DC ] = P DC, 0 ⋅ e ¯ td (21)

where e ¯ td = Ε [ e td ] = Ε [ ( | r ( t ) | 2 M ) u ( ln | r ( t ) | 2 M + 2 ln P ¯ RF ) + v ] (22)

The resultant RF-to-DC conversion efficiency is given by ε 0 = P DC, 0 P ¯ RF (23)

Substituting (23) into (21), P ¯ DC = P ¯ RF ε 0 e ¯ td (24)

The average RF-to-DC conversion efficiency is thus given by ε avg = P ¯ DC P ¯ RF = ε 0 e ¯ td (25)