Module imperfect_csi 

Error rates over Rayleigh fading with an imperfect channel estimate (Chang & Su 2002).

Model

The received sample of a subcarrier is $Y = HX + N$, $H$ zero-mean circular Gaussian of variance $2\sigma_{1}^{2}$, $N$ of variance $2\sigma_{n}^{2}$; the receiver equalises with an estimate $\hat{H}$ jointly Gaussian with $H$, of variance $2\sigma_{2}^{2}$ and cross-correlation $\frac{1}{2}\mathbb{E}[\hat{H} H^{*}] = \mu_{1} + j\mu_{2}$ — every linear pilot-assisted estimate (LS, LMMSE, interpolations) has this form. With $\rho_{i} = \mu_{i}/(\sigma_{1}\sigma_{2})$, $\rho^{2} = \rho_{1}^{2} + \rho_{2}^{2}$ and the mean SNR per bit $\bar{\gamma}{b} = 2\sigma{1}^{2} \mathbb{E}|X|^{2}/(2\sigma_{n}^{2} K)$ ($K$ bits per symbol), the bit error probabilities of M. X. Chang and Y. T. Su (IEEE Trans. Wireless Commun. 1(4), 2002) are, for BPSK (their eq. 31) and Gray QPSK (eq. 32),

$$ P_{b} = \frac{1}{2}\left[1 - \frac{\rho_{1}}{\sqrt{1 + 1/\bar{\gamma}{b} - \rho{2}^{2}}}\right], \qquad P_{b} = \frac{1}{2}\left[1 - \frac{1}{2}\frac{(\rho_{1} + \rho_{2})/\sqrt{2}}{\sqrt{1 + \frac{1}{2\bar{\gamma}{b}} - \frac{(\rho{1} - \rho_{2})^{2}}{2}}} - \frac{1}{2} \frac{(\rho_{1} - \rho_{2})/\sqrt{2}}{\sqrt{1 + \frac{1}{2\bar{\gamma}{b}} - \frac{(\rho{1} + \rho_{2})^{2}}{2}}}\right], $$

for Gray 16-QAM their eq. 53, a sum of 24 terms of the coefficients of their Table I, which reduces with $\hat{H} = H$ to their eq. 69,

$$ P_{b} = \frac{1}{2} - \frac{3}{8}\frac{1}{\sqrt{1 + \frac{5}{2\bar{\gamma}{b}}}} - \frac{1}{4}\frac{1}{\sqrt{1 + \frac{5}{18\bar{\gamma}{b}}}} + \frac{1}{8} \frac{1}{\sqrt{1 + \frac{1}{10\bar{\gamma}_{b}}}}, $$

(whose expansion is $P_{b} \approx 0.4972/\bar{\gamma}{b}$ at high SNR; the paper prints $0.4952/\bar{\gamma}{b}$), and for differential detection over Jakes’ fading of normalised Doppler $f_{D}T_{s}$ (eqs. 40, 41)

$$ P_{b}^{\mathrm{DBPSK}} = \frac{1}{2}\left[1 - \frac{J_{0}(2\pi f_{D}T_{s})}{1 + 1/\bar{\gamma}{b}}\right], \qquad P{b}^{\mathrm{DQPSK}} = \frac{1}{2}\left[1 - \frac{J_{0}(2\pi f_{D}T_{s})}{\sqrt{(2 + 1/\bar{\gamma}{b})(1 + 1/(2\bar{\gamma}{b})) - J_{0}^{2}(2\pi f_{D}T_{s})}}\right]. $$

Examples

use comnum_mapping::imperfect_csi::{ber_bpsk_imperfect_csi, ber_16qam_imperfect_csi};
use comnum_mapping::theory::ber_rayleigh_mrc;
let g = 100.0;                                          // 20 dB
// a perfect estimate: the BER of flat Rayleigh fading
assert!((ber_bpsk_imperfect_csi(g, 1.0, 0.0)? - ber_rayleigh_mrc(g, 1)?).abs() < 1e-15);
// an estimate correlated at 0.999: an error floor
assert!(ber_16qam_imperfect_csi(1e9, 0.999, 0.0, 1.0)? > 1e-3);
// an estimate correlated above 1 is an error, not a NaN
assert!(ber_bpsk_imperfect_csi(g, 1.1, 0.0).is_err());

References

1. M.-X. Chang, Y. T. Su, “Performance analysis of equalized OFDM systems in Rayleigh fading,” IEEE Trans. Wireless Commun. 1(4), 721–732, 2002.

Functions

ber_16qam_imperfect_csi

The BER of Gray 16-QAM equalised by an estimate of correlations $(\rho_{1}, \rho_{2})$ and of standard deviation ratio sigma_ratio $= \sigma_{2}/\sigma_{1}$ (Chang & Su eq. 53 and Table I; $\bar{\gamma}{b} = 5\sigma{1}^{2}/(2\sigma_{n}^{2})$ for the points ${\pm 1, \pm 3}^{2}$).

ber_16qam_rayleigh

The BER of Gray 16-QAM over flat Rayleigh fading with the perfect channel (Chang & Su eq. 69).

ber_bpsk_imperfect_csi

The BER of BPSK equalised by an estimate of correlations $(\rho_{1}, \rho_{2})$ at the mean SNR per bit mean_ebn0_lin (linear; Chang & Su eq. 31).

ber_dbpsk_jakes

The BER of DBPSK over Jakes’ Rayleigh fading of normalised Doppler fd_ts (Chang & Su eq. 40).

ber_dqpsk_jakes

The BER of Gray DQPSK over Jakes’ Rayleigh fading of normalised Doppler fd_ts (Chang & Su eq. 41).

ber_imperfect_csi

The BER of the constellation c equalised by an estimate of correlations $(\rho_{1}, \rho_{2})$ and of standard deviation ratio sigma_ratio at the mean SNR per bit mean_ebn0_lin (linear): ber_bpsk_imperfect_csi for BPSK, ber_qpsk_imperfect_csi for Gray QPSK (4-QAM or 4-PSK) and ber_16qam_imperfect_csi for Gray 16-QAM (sigma_ratio is used by 16-QAM only: the PSK decisions do not depend on $|\hat{H}|$).

ber_qpsk_imperfect_csi

The BER of Gray QPSK equalised by an estimate of correlations $(\rho_{1}, \rho_{2})$ (Chang & Su eq. 32).