Downlink processing begins after the UE has completed initial synchronization from the downlink fields and formed the FFT grid Y γ D L \boldsymbol Y_\gamma^\mathrm{DL} Y γ DL
ZC channel estimation → pilot phase tracking → per-symbol channel reconstruction → equalization → QPSK LLRs → LDPC decoding . \text{ZC channel estimation}
\rightarrow
\text{pilot phase tracking}
\rightarrow
\text{per-symbol channel reconstruction}
\rightarrow
\text{equalization}
\rightarrow
\text{QPSK LLRs}
\rightarrow
\text{LDPC decoding}. ZC channel estimation → pilot phase tracking → per-symbol channel reconstruction → equalization → QPSK LLRs → LDPC decoding . TDD processes m ∈ S D L m\in\mathcal S_\mathrm{DL} m ∈ S DL
With sufficient cyclic-prefix length and small intra-symbol Doppler,
Y n , m , γ D L = b n , m , γ D L H n , m , γ D L + Z n , m , γ D L , Y_{n,m,\gamma}^\mathrm{DL}
=b_{n,m,\gamma}^\mathrm{DL}
H_{n,m,\gamma}^\mathrm{DL}
+Z_{n,m,\gamma}^\mathrm{DL}, Y n , m , γ DL = b n , m , γ DL H n , m , γ DL + Z n , m , γ DL , where t m , γ = ( m + γ M ) T O t_{m,\gamma}=(m+\gamma M)T_O t m , γ = ( m + γ M ) T O
H n , m , γ D L = ∑ l = 1 L D L α l D L ( t m , γ ) e j 2 π [ ( f D , l D L + Δ f ˉ c , γ D L ) t m , γ − κ n Δ f ( τ l , p r o p ( t m , γ ) + τ D L R F − τ d U E ( t m , γ ) ) ] . H_{n,m,\gamma}^\mathrm{DL}
=\sum_{l=1}^{L_\mathrm{DL}}
\alpha_l^\mathrm{DL}(t_{m,\gamma})
e^{j2\pi\left[
(f_{D,l}^\mathrm{DL}+\Delta\bar f_{c,\gamma}^\mathrm{DL})
t_{m,\gamma}
-\kappa_n\Delta f
(\tau_{l,\mathrm{prop}}(t_{m,\gamma})
+\tau_\mathrm{DL}^\mathrm{RF}
-\tau_d^\mathrm{UE}(t_{m,\gamma}))
\right]}. H n , m , γ DL = l = 1 ∑ L DL α l DL ( t m , γ ) e j 2 π [ ( f D , l DL + Δ f ˉ c , γ DL ) t m , γ − κ n Δ f ( τ l , prop ( t m , γ ) + τ DL RF − τ d UE ( t m , γ )) ] . Δ f ˉ c , γ D L \Delta\bar f_{c,\gamma}^\mathrm{DL} Δ f ˉ c , γ DL τ d U E ( t ) \tau_d^\mathrm{UE}(t) τ d UE ( t ) Signal Model for the definitions of propagation delay, downlink RF group delay, and TO.
The full-band ZC gives
H ^ n , m s y n c , γ L S = Y n , m s y n c , γ D L z n D L . \hat H_{n,m_\mathrm{sync},\gamma}^\mathrm{LS}
=\frac{Y_{n,m_\mathrm{sync},\gamma}^\mathrm{DL}}
{z_n^\mathrm{DL}}. H ^ n , m sync , γ LS = z n DL Y n , m sync , γ DL . Delay-support limiting and Wiener smoothing produce a lower-noise channel anchor. The dominant peak relative to N l a g N_\mathrm{lag} N lag
For adjacent active downlink symbols that contain the same known pilots, define
R ˉ γ D L [ n ] = 1 ∣ A D L ∣ ∑ m ∈ A D L ( Y n , m , γ D L ) ∗ Y n , m + 1 , γ D L , n ∈ P . \bar R_\gamma^\mathrm{DL}[n]
=\frac{1}{|\mathcal A_\mathrm{DL}|}
\sum_{m\in\mathcal A_\mathrm{DL}}
(Y_{n,m,\gamma}^\mathrm{DL})^*
Y_{n,m+1,\gamma}^\mathrm{DL},
\qquad n\in\mathcal P. R ˉ γ DL [ n ] = ∣ A DL ∣ 1 m ∈ A DL ∑ ( Y n , m , γ DL ) ∗ Y n , m + 1 , γ DL , n ∈ P . A D L \mathcal A_\mathrm{DL} A DL
φ γ D L [ n ] = arg R ˉ γ D L [ n ] ≈ 2 π ( f o , γ D L T O − κ n Δ f N s Δ T s , γ D L ) . \varphi_\gamma^\mathrm{DL}[n]
=\arg\bar R_\gamma^\mathrm{DL}[n]
\approx2\pi\left(
f_{o,\gamma}^\mathrm{DL}T_O
-\kappa_n\Delta fN_s\Delta T_{s,\gamma}^\mathrm{DL}
\right). φ γ DL [ n ] = arg R ˉ γ DL [ n ] ≈ 2 π ( f o , γ DL T O − κ n Δ f N s Δ T s , γ DL ) . A fit weighted by ∣ R ˉ γ D L [ n ] ∣ 2 |\bar R_\gamma^\mathrm{DL}[n]|^2 ∣ R ˉ γ DL [ n ] ∣ 2
( a ^ , b ^ ) = arg min a , b ∑ n ∈ P ∣ R ˉ γ D L [ n ] ∣ 2 ∣ unwrap ( φ γ D L [ n ] ) − a − b κ n ∣ 2 , (\hat a,\hat b)
=\arg\min_{a,b}\sum_{n\in\mathcal P}
|\bar R_\gamma^\mathrm{DL}[n]|^2
\left|
\operatorname{unwrap}(\varphi_\gamma^\mathrm{DL}[n])-a-b\kappa_n
\right|^2, ( a ^ , b ^ ) = arg a , b min n ∈ P ∑ ∣ R ˉ γ DL [ n ] ∣ 2 unwrap ( φ γ DL [ n ]) − a − b κ n 2 , f ^ o , γ D L = a ^ 2 π T O , Δ T ^ s , γ D L = − b ^ 2 π Δ f N s . \hat f_{o,\gamma}^\mathrm{DL}
=\frac{\hat a}{2\pi T_O},
\qquad
\Delta\hat T_{s,\gamma}^\mathrm{DL}
=-\frac{\hat b}{2\pi\Delta fN_s}. f ^ o , γ DL = 2 π T O a ^ , Δ T ^ s , γ DL = − 2 π Δ f N s b ^ . The common phase term estimates residual CFO plus dominant-path Doppler, while the evolving subcarrier slope estimates SFO. Both compensate the intra-frame channel phase.
Optional mid-frame full-band references provide more anchors H ^ n , m a , γ \hat H_{n,m_a,\gamma} H ^ n , m a , γ m a < m < m b m_a<m<m_b m a < m < m b
H ^ n , m , γ b a s e = ( 1 − ξ m ) H ^ n , m a , γ + ξ m H ^ n , m b , γ , ξ m = m − m a m b − m a . \hat H_{n,m,\gamma}^{\mathrm{base}}
=(1-\xi_m)\hat H_{n,m_a,\gamma}
+\xi_m\hat H_{n,m_b,\gamma},
\qquad
\xi_m=\frac{m-m_a}{m_b-m_a}. H ^ n , m , γ base = ( 1 − ξ m ) H ^ n , m a , γ + ξ m H ^ n , m b , γ , ξ m = m b − m a m − m a . The residual CFO/SFO phase from the pilots is then applied at each symbol time. With no additional anchors,
H ^ n , m , γ D L = H ^ n , m s y n c , γ D L e j 2 π ( m − m s y n c ) ( f ^ o , γ D L T O − κ n Δ f N s Δ T ^ s , γ D L ) . \hat H_{n,m,\gamma}^\mathrm{DL}
=\hat H_{n,m_\mathrm{sync},\gamma}^\mathrm{DL}
e^{j2\pi(m-m_\mathrm{sync})
(\hat f_{o,\gamma}^\mathrm{DL}T_O
-\kappa_n\Delta fN_s\Delta\hat T_{s,\gamma}^\mathrm{DL})}. H ^ n , m , γ DL = H ^ n , m sync , γ DL e j 2 π ( m − m sync ) ( f ^ o , γ DL T O − κ n Δ f N s Δ T ^ s , γ DL ) . The uplink uses the same residual-estimation and compensation form on its own references and observations.
The zero-forcing coefficient is
G n , m Z F = ( H ^ n , m D L ) ∗ max ( ∣ H ^ n , m D L ∣ 2 , ϵ ) , G_{n,m}^{\mathrm{ZF}}
=\frac{(\hat H_{n,m}^\mathrm{DL})^*}
{\max(|\hat H_{n,m}^\mathrm{DL}|^2,\epsilon)}, G n , m ZF = max ( ∣ H ^ n , m DL ∣ 2 , ϵ ) ( H ^ n , m DL ) ∗ , while the regularized MMSE coefficient is
G n , m M M S E = ( H ^ n , m D L ) ∗ ∣ H ^ n , m D L ∣ 2 + σ ^ Z 2 / E s . G_{n,m}^{\mathrm{MMSE}}
=\frac{(\hat H_{n,m}^\mathrm{DL})^*}
{|\hat H_{n,m}^\mathrm{DL}|^2+\hat\sigma_Z^2/E_s}. G n , m MMSE = ∣ H ^ n , m DL ∣ 2 + σ ^ Z 2 / E s ( H ^ n , m DL ) ∗ . ZF directly inverts the estimated channel but can amplify noise in deep fades; MMSE regularization limits this noise enhancement. For data resources,
d ^ n , m , γ D L = G n , m Y n , m , γ D L . \hat d_{n,m,\gamma}^\mathrm{DL}
=G_{n,m}Y_{n,m,\gamma}^\mathrm{DL}. d ^ n , m , γ DL = G n , m Y n , m , γ DL . Equalized pilot errors estimate the effective noise variance:
σ ^ e q 2 = 1 ∣ Ω r e f D L ∣ ∑ ( n , m ) ∈ Ω r e f D L ∣ p ^ n , m D L − p n , m D L ∣ 2 . \hat\sigma_\mathrm{eq}^2
=\frac{1}{|\Omega_\mathrm{ref}^\mathrm{DL}|}
\sum_{(n,m)\in\Omega_\mathrm{ref}^\mathrm{DL}}
|\hat p_{n,m}^\mathrm{DL}-p_{n,m}^\mathrm{DL}|^2. σ ^ eq 2 = ∣ Ω ref DL ∣ 1 ( n , m ) ∈ Ω ref DL ∑ ∣ p ^ n , m DL − p n , m DL ∣ 2 . For equalized symbol d ^ \hat d d ^ i i i
L i ( d ^ ) ≈ min a ∈ A i ( 1 ) ∣ d ^ − a ∣ 2 − min a ∈ A i ( 0 ) ∣ d ^ − a ∣ 2 σ ^ e q 2 , L_i(\hat d)
\approx
\frac{
\min\limits_{a\in\mathcal A_i^{(1)}}|\hat d-a|^2
-\min\limits_{a\in\mathcal A_i^{(0)}}|\hat d-a|^2
}{\hat\sigma_\mathrm{eq}^2}, L i ( d ^ ) ≈ σ ^ eq 2 a ∈ A i ( 1 ) min ∣ d ^ − a ∣ 2 − a ∈ A i ( 0 ) min ∣ d ^ − a ∣ 2 , where A i ( b ) \mathcal A_i^{(b)} A i ( b ) i i i b b b
The transmit order is LDPC encoding, scrambling, interleaving, and QPSK mapping. The UE reverses this order with soft deinterleaving, soft descrambling, and LDPC decoding. The resulting H ^ n , m , γ D L \hat H_{n,m,\gamma}^\mathrm{DL} H ^ n , m , γ DL bistatic sensing .
