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Bit error rate (BER) for imperfect synchronization

`ber = bersync(EbNo,timerr,`

`'timing'`

)

ber = bersync(EbNo,phaserr,`'carrier'`

)

`ber = bersync(EbNo,timerr,`

returns the BER of uncoded coherent binary phase shift keying (BPSK) modulation over an
additive white Gaussian noise (AWGN) channel with imperfect timing. The normalized
timing error is assumed to have a Gaussian distribution. `'timing'`

) `EbNo`

is the
ratio of bit energy to noise power spectral density, in dB. If `EbNo`

is a vector, the output `ber`

is a vector of the same size, whose
elements correspond to the different E_{b}/N_{0}
levels. `timerr`

is the standard deviation of the timing error,
normalized to the symbol interval. `timerr`

must be between 0 and
0.5.

`ber = bersync(EbNo,phaserr,`

returns the BER of uncoded BPSK modulation over an AWGN channel with a noisy phase
reference. The phase error is assumed to have a Gaussian distribution.
`'carrier'`

)`phaserr`

is the standard deviation of the error in the reference
carrier phase, in radians.

The numerical accuracy of this function's output is limited by

Approximations in the analysis leading to the closed-form expressions that the function uses

Approximations related to the numerical implementation of the expressions

You can generally rely on the first couple of significant digits of the function's output.

Inherent limitations in numerical precision force the function
to assume perfect synchronization if the value of `timerr`

or `phaserr`

is
very small. The table below indicates how the function behaves under
these conditions.

Condition | Behavior of Function |
---|---|

`timerr < eps` | `bersync(EbNo,timerr,'timing')` defined
as `berawgn(EbNo,'psk',2)` |

`phaserr < eps` | `bersync(EbNo,phaserr,'carrier')` defined
as `berawgn(EbNo,'psk',2)` |

This function uses formulas from [3].

When the last input is `'timing'`

, the function computes

$$\frac{1}{4\pi \sigma}{\displaystyle {\int}_{-\infty}^{\infty}\mathrm{exp}(-\frac{{\xi}^{2}}{2{\sigma}^{2}})}{\displaystyle {\int}_{\sqrt{2R}(1-2\left|\xi \right|)}^{\infty}\mathrm{exp}(-\frac{{x}^{2}}{2})dxd\xi}+\frac{1}{2\sqrt{2\pi}}{\displaystyle {\int}_{\sqrt{2R}}^{\infty}\mathrm{exp}(-\frac{{x}^{2}}{2})dx}$$

where σ is the `timerr`

input and R is
the value of `EbNo`

converted from dB to a linear
scale.

When the last input is `'carrier'`

, the function computes

$$\frac{1}{\pi \sigma}{\displaystyle {\int}_{0}^{\infty}\mathrm{exp}(-\frac{{\varphi}^{2}}{2{\sigma}^{2}})}{\displaystyle {\int}_{\sqrt{2R}\mathrm{cos}\varphi}^{\infty}\mathrm{exp}(-\frac{{y}^{2}}{2})dyd\varphi}$$

where σ is the `phaserr`

input and R
is the value of `EbNo`

converted from dB to a linear
scale.

As an alternative to the `bersync`

function,
invoke the BERTool GUI (`bertool`

) and use the **Theoretical** tab.

[1] Jeruchim, Michel C., Philip Balaban,
and K. Sam Shanmugan, *Simulation of Communication Systems*,
Second Edition, New York, Kluwer Academic/Plenum, 2000.

[2] Sklar, Bernard, *Digital
Communications: Fundamentals and Applications*, Second
Edition, Upper Saddle River, NJ, Prentice-Hall, 2001.

[3] Stiffler, J. J., *Theory
of Synchronous Communications*, Englewood Cliffs, NJ, Prentice-Hall,
1971.