石墨烯在太赫兹频段实现的无线片上网络（WiNoC）（五）

Appendix

A. Proof of Theorem 4

As the signal-to-noise ratio (SNR) is required for evaluating the achievable capacity of a communication system, we first derive the total noise power of the nanocommunication between two GNAs. At a frequency 

[Hz], the total noise temperature at 

 located at  [m] from  (i.e.,  [K]) consists of system electronic noise temperature (i.e.,  [K]), molecular absorption noise temperature (i.e.,  [K]), and other noise source temperature (i.e.,  [K]); that is,Assuming that , we haveHere,  is caused by the molecules within the transmission medium and thus can be expressed via the transmittance of the medium asSubstituting (A.3) into (A.2), we obtainThe total noise power at  given transmission bandwidth  is therefore given byNote that the THz channel is highly frequency-selective and the molecular absorption noise is nonwhite. Therefore, we can divide the total bandwidth  into  narrow subbands and regard them as parallel channels. Subject to the transmission power constraint at  (i.e., ), the channel capacity, in bits/s, of the nanocommunications between the GNAs can be expressed bywhere  [Hz] is the width of subband,   [Hz] is the centre frequency of the th subband,  is the power allocated for the th subband,  is the total path loss, and  is the total noise temperature at 

. Substituting (17) and (A.4) into (A.6), we obtain (19). The theorem is proved.

B. Proof of Proposition 5

Let 

 be the Lagrange multiplier associated with the power constraint 

; the Lagrangian of (19) can be formed asBy denoting  as in (21), we can rewrite (B.1) asDifferentiating  with respect to , , we haveSolving , we obtainwhere 

.

Since 

, , the solution in (B.4) iswhere  and  can be solved by utilising the power constraint at  as in water-filling approach. That is,

This completes the proof.

Notations

[m]:	Distance between two GNAs

[m]:	Height of the chip package

, 

[m]:	Height of the GNAs at

 and 

[Hz]:	Transmission frequency and channel bandwidth

[atm]:	Ambient pressure applied on chip

, 

, and 

[K]:	System electronic, molecular absorption, and other noise source temperature, respectively

, 

, and 

:	DPL, MAA, and total path loss, respectively

and 

[W]:	Transmitted power and received power

and 

:	Transmitter antenna gain and receiver antenna gain

[m/s]:	Phase velocity

:	Relative permittivity of material

:	Transmittance of medium

:	Medium absorption coefficient

:	Isotopologue

 of gas 

:	Individual absorption coefficient of

[mol/m3]:

Molecular volumetric density of

[m2/mol]:

Absorption cross section of

[%]:	Mixing ratio of

[m2Hz/mol]:

Line density for the absorption of

[Hz−1]:

Spectral line shape of

, 

[Hz]:

Resonant frequency of

 and resonant frequency at 

atm

[Hz−1]:	Van Vleck-Weisskopf asymmetric line shape [36]

[Hz]:

Linear pressure shift of

[Hz]:

Lorentz half width of

[36]

, 

[Hz]:

Broadening coefficient of air and

respectively

:	Temperature broadening coefficient.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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