Continuous-Time IRS-Assisted Communication Under Deterministic Leo Satellite Trajectory Using Closed-Form Scaling Laws and Inverse Aperture Design

Ground-based intelligent reflecting surfaces (IRS) enable passive aperture gain for satellite-to-ground links via coherent superposition y(t)=h_LoS (t)x(t)+?_(n=1)^N?h_(r,n) (t)e^(j?_n (t)) x(t)+n(t) under continuous-time propagation h(t,?)=h_LoS (t)?(?-?_0 (t))+?_(n=1)^N?h_(r,n) (t)?(?-?_n (t)), overcoming discrete-state formulations {s_k }_(k=1)^K ? that preclude closed-form aperture sizing under predictable motion r_s (t), for a deterministic LEO trajectory with orbital velocity v_s=7.67"?" km/s, the separable gain surrogate is posed as G(h,N)=G_h (h)+G_N (N)=-(A_0+?h)+20?log??_10 N with ?G/?h=-?, ?G/(??log??_10 N)=20, where A_0,? are calibrated from first principles and ITU-R P.619-5 slant-path attenuation L_s (h,f_c,?), and continuous-time phase control enforces ?_n (t)=Q_b {-arg?[h_(t,n) (t)h_(n,r) (t)]} to track optimal coherent combining along r_s (t); quantization induces a bounded penalty ?G_b?10?log??_10 ((sin?(?/2^b))/(?/2^b ))^(-2)?O(2^(-2b)) so that for b?2, ?G_b<1"?" dB; at carrier f_c=28"?" GHz, logarithmic aperture scaling G_N (N)=20?log??_10 N (20 dB/decade) is altitude-invariant, while altitude attenuation is linear G_h (h)=-?h with ??0.01"?" dB/km over h?[200,600]"?" km; numerically, (h,N)=(200"?" km,65536)?G=43.2"?" dB relative to the direct path, and Monte Carlo sensitivity over M=1000 atmospheric realizations yields P_95 (?G ?-G?)?1.2"?" dB, validating robustness; the inverse aperture sizing follows in closed form N^* (h,G_req)=10^((G_req-G_h (h))/20)=10^((G_req+A_0+?h)/20), providing a computationally efficient mapping from required gain to minimum IRS element count for non-terrestrial network design under predictable mobility.