DMRS-Based Channel Estimation in 5G NR
1. System Model (OFDM-Based 5G NR)
After CP removal and FFT, the received signal at subcarrier k and OFDM symbol n:
$$ Y(k,n) = H(k,n)X(k,n) + W(k,n) $$
Where:
- X(k,n): transmitted symbol
- H(k,n): channel frequency response
- W(k,n) ~ CN(0, Ī²): AWGN
- Y(k,n): received symbol
2. Channel Representation
Time-Domain Multipath Channel
$$ h(\tau,t) = \sum_{l=0}^{L-1} \alpha_l(t)\delta(\tau - \tau_l) $$
Frequency-Domain Channel
$$ H(k,n) = \sum_{l=0}^{L-1} \alpha_l(n)e^{-j2\pi k \Delta f \tau_l} $$
3. Least Squares (LS) Channel Estimation
$$ Y_{DMRS}(k,n) = H(k,n)X_{DMRS}(k,n) + W(k,n) $$
$$ \hat{H}_{LS}(k,n) = \frac{Y_{DMRS}(k,n)}{X_{DMRS}(k,n)} $$
Vector Form
$$ \mathbf{Y} = \mathbf{X}\mathbf{H} + \mathbf{W} $$
$$ \hat{\mathbf{H}}_{LS} = (\mathbf{X}^H \mathbf{X})^{-1} \mathbf{X}^H \mathbf{Y} $$
LS Estimator Properties
$$ E[\hat{H}_{LS}] = H $$
$$ Var(\hat{H}_{LS}) = \frac{\sigma^2}{|X|^2} $$
4. LMMSE Channel Estimation
$$ \hat{\mathbf{H}}_{LMMSE} = R_{HH} \left( R_{HH} + \sigma^2 (X^H X)^{-1} \right)^{-1} \hat{\mathbf{H}}_{LS} $$
Simplified Scalar Form
$$ \hat{H}_{LMMSE} = \frac{\sigma_H^2}{\sigma_H^2 + \frac{\sigma^2}{|X|^2}} \hat{H}_{LS} $$
5. 2D Time-Frequency Interpolation
Frequency Interpolation
$$ \hat{H}(k) = \frac{k_2-k}{k_2-k_1}H(k_1) + \frac{k-k_1}{k_2-k_1}H(k_2) $$
Time Correlation Function
$$ R(\Delta t) = J_0(2\pi f_D \Delta t) $$
6. MIMO Channel Estimation
$$ \mathbf{Y} = \mathbf{H}\mathbf{X} + \mathbf{W} $$
Orthogonality condition:
$$ X_i X_j^H = 0 \quad (i \neq j) $$
7. Mean Square Error (MSE)
LS MSE
$$ MSE_{LS} = \sigma^2 $$
LMMSE MSE
$$ MSE_{LMMSE} = \sigma_H^2 - \sigma_H^4 \left( \sigma_H^2 + \frac{\sigma^2}{|X|^2} \right)^{-1} $$
8. Doppler Frequency
$$ f_D = \frac{v f_c}{c} $$
9. Equalization
Zero Forcing Equalizer
$$ \hat{X} = \frac{Y}{\hat{H}} $$
MMSE Equalizer
$$ \hat{X} = \frac{\hat{H}^*}{|\hat{H}|^2 + \sigma^2} Y $$
Conclusion
DMRS-based channel estimation in 5G NR uses known reference symbols, LS or LMMSE estimation, followed by time-frequency interpolation and equalization for reliable data detection.