In high performance communications systems it is common for both the
transmitter (Tx) and receiver (Rx) to add noise to the signal. Unless
you have a perfect example of one or the other, it is impossible to
measure the source: both the Tx and Rx will contribute. This has the
effect of having a minimum bit error rate (BER) even when no noise is
added to the signal, called the Back-to-Back (B2B) BER, and is often
expressed in Decibels as the signal-to-noise ratio (SNR) which would the
B2B BER in a perfect Rx.
The other common effect is to add an offset to the SNR. That is, the
system may always perform as though the SNR were a few Decibels
less.
We can take a BER function in Decibels and add the offset and B2B BER
to it as follows. The linear SNR is \(\gamma =
E_b/N_0\), but in the B2B case the input noise \(N\) is zero, yet there is still a finite
error rate observed. This is modeled as \(\gamma = E_b/{(N_0+\beta)}\) where \(\beta\) is \(1/B2B\).
Creating BER Functions with B2B & Offset
If we have a function \(P_{dB}(
s_{dB})\) of the SNR in Decibels, we can express the B2B and
offset in decibels too.
\[
P_e( s_{dB}, B_{dB}, O_{dB}) = BER( -dB( undB( -s_{dB}) + undB( -B_{dB})
- O_{dB})
\]
The B2BQ package provides a function factory to convert
any such function \(P_{dB}\) into a
function with B2B and Offset in Decibels B2BConvert.
QPSKdB.B2B <- B2BConvert( QPSKdB)
O1 <- 3
B1 <- 16
s <- 0:20
plot( s, y=QPSKdB( s), typ="l", log="y", ylim=c( 1e-10, 0.5),
panel.first = grid(), main="QPSK Performance Limits", xlab="SNR in dB",
ylab="BER")
lines( s, y=QPSKdB.B2B( s, Inf, O1), col="blue", lwd=2)
lines( s, y=QPSKdB.B2B( s, B1, O1), col='red')
lines( s, y=QPSKdB.B2B( rep( B1, length( s)), Inf, O1), col='black', lty=2)
legend( "bottomleft", legend=c( "Theory", "Offset", "Offset + B2B", "B2B"),
lty=c( 1, 1, 1, 2), lwd=c(1, 2, 1, 1),
col=c( 'black', 'blue', 'red', 'black'))
Using nls
Let’s generate some example data and use nls to fit the
curve to the data.
N <- 1000000
(r <- rbinom( length( s), N, QPSKdB.B2B( s, B1, O1)))
#> [1] 161193 134359 107870 83728 61830 43368 28246 17266 9817 5083
#> [11] 2195 1024 393 134 44 11 7 1 0 0
#> [21] 0
bplot1 <- function( s, r, N, O, B, ylim=c( 1e-10, 0.5)) {
plot( s, y=QPSKdB.B2B( s, B, O), col='red', log='y', type='l', ylim=ylim,
main="QPSK Performance Limits", xlab="SNR in dB",
ylab="BER", panel.first = grid())
points( s, r/N)
lines( s, y=QPSKdB( s))
lines( s, y=QPSKdB.B2B( s, Inf, O), col="blue", lwd=2)
lines( s, y=QPSKdB.B2B( rep( B1, length( s)), Inf, O1), col='black', lty=2)
legend( "bottomleft",
legend=c( "Data", "Theory", "Offset", "Offset + B2B", "B2B"),
lty=c( NA, 1, 1, 1, 2), lwd=c(NA, 1, 2, 1, 1), pch=c( 1, NA, NA, NA, NA),
col=c( 'black', 'black', 'blue', 'red', 'black'))
}
bplot1( s, r, N, O1, B1)
nls creates a non-linear least squares fit to estimate
the parameters.
df1 <- data.frame( Errors=r, Trials=rep( N, length( s)), SNR=s)
nls.fit1 <- nls( Errors/Trials ~ QPSKdB.B2B( SNR, b, o), data=df1,
start=list( b=10, o=0))
summary( nls.fit1)
#>
#> Formula: Errors/Trials ~ QPSKdB.B2B(SNR, b, o)
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> b 15.948539 0.052466 304 <2e-16 ***
#> o 2.992966 0.002874 1041 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 8.646e-05 on 19 degrees of freedom
#>
#> Number of iterations to convergence: 7
#> Achieved convergence tolerance: 5.708e-08
bplot1(s, r, N, O1, B1)
c <- coef( nls.fit1)
lines( s, y=QPSKdB.B2B( s, c['b'], c['o']), col='green')
Those are remarkably close estimates. However, we have fit the error
rate to the function, not the error counts. There are zeros in the data,
and this doesn’t match the function at high SNRs very well. The
following will perform the fit without the zero points.
nls.fit2 <- nls( Errors/Trials ~ QPSKdB.B2B( SNR, b, o),
data=subset( df1, Errors > 0), start=list( b=10, o=0))
summary( nls.fit2)
#>
#> Formula: Errors/Trials ~ QPSKdB.B2B(SNR, b, o)
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> b 15.948539 0.057174 278.9 <2e-16 ***
#> o 2.992966 0.003132 955.5 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 9.422e-05 on 16 degrees of freedom
#>
#> Number of iterations to convergence: 7
#> Achieved convergence tolerance: 1.248e-07
bplot1(s, r, N, O1, B1)
c <- coef( nls.fit2)
lines( s, y=QPSKdB.B2B( s, c['b'], c['o']), col='green')
That was unsatisfying; the answer is almost identical to the version
including the zeros. Plotting the data on a linear scale indicates
why.
plot( s, y=QPSKdB.B2B( s, B1, O1), col='red', type='l', panel.first = grid(),
main="QPSK Performance Limits", xlab="SNR in dB", ylab="BER")
points( s, r/N, panel.first = grid())
lines( s, y=QPSKdB( s))
lines( s, y=QPSKdB.B2B( s, Inf, O1), col="blue", lwd=2)
lines( s, y=QPSKdB.B2B( s, c['b'], c['o']), col='green')
legend( "bottomleft",
legend=c( "Data", "Theory", "3 dB Offset", "Offset + B2B", "Fit"),
lty=c( NA, 1, 1, 1, 1), lwd=c(NA, 1, 2, 1, 1), pch=c( 1, NA, NA, NA, NA),
col=c( 'black', 'black', 'blue', 'red', 'green'))
The error rate is so close to zero past 10 dB that a least squares
error minimization is not affected by any fit errors. A better technique
might be to fit the “Q” function of both sides of the equation. The Q
function is Markum’s Q function and is given by
\[
Q(s) = \frac{1}{2} \mathrm{Erfc} \left( s / \sqrt{2}\right)
\]
In R, this is simply pnorm( x, lower.tail=FALSE),
although the package provides the function Q_(x) for
convenience. Since the Q function of zero is infinite, we have to
exclude zero results again.
nls.fit3 <- nls( Q_( Errors/Trials) ~ Q_( QPSKdB.B2B( SNR, b, o)),
data=subset( df1, Errors > 0), start=list( b=10, o=0))
summary( nls.fit3)
#>
#> Formula: Q_(Errors/Trials) ~ Q_(QPSKdB.B2B(SNR, b, o))
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> b 15.948434 0.057152 279.1 <2e-16 ***
#> o 2.992959 0.003142 952.4 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.746e-05 on 16 degrees of freedom
#>
#> Number of iterations to convergence: 7
#> Achieved convergence tolerance: 5.456e-07
bplot1(s, r, N, O1, B1)
c <- coef( nls.fit3)
lines( s, y=QPSKdB.B2B( s, c['b'], c['o']), col='green')
