Bit error rate (BER) for Rayleigh and Rician fading channels

`ber = berfading(EbNo,'pam',M,divorder)`

ber = berfading(EbNo,'qam',M,divorder)

ber = berfading(EbNo,'psk',M,divorder)

ber = berfading(EbNo,'depsk',M,divorder)

ber = berfading(EbNo,'oqpsk',divorder)

ber = berfading(EbNo,'dpsk',M,divorder)

ber = berfading(EbNo,'fsk',M,divorder,coherence)

ber = berfading(EbNo,'fsk',2,divorder,coherence,rho)

ber = berfading(EbNo,...,K)

ber = berfading(EbNo,'psk',2,1,K,phaserr)

[BER,SER] = berfading(EbNo, ...)

The first input argument, `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.

Most syntaxes also have an `M`

input that specifies
the alphabet size for the modulation. `M`

must have
the form 2^{k} for some positive integer `k`

.

`berfading`

uses expressions that assume
Gray coding. If you use binary coding, the results may differ.

For cases where diversity is used, the E_{b}/N_{0} on
each diversity branch is `EbNo/divorder`

, where `divorder`

is the diversity order (the number of diversity branches) and is a
positive integer.

`ber = berfading(EbNo,'pam',M,divorder)`

returns
the BER for PAM over an uncoded Rayleigh fading channel with coherent
demodulation.

`ber = berfading(EbNo,'qam',M,divorder)`

returns
the BER for QAM over an uncoded Rayleigh fading channel with coherent
demodulation. The alphabet size, `M`

, must be at
least 4. When $$k={\mathrm{log}}_{2}M$$ is odd, a rectangular
constellation of size $$M=I\times J$$ is used, where $$I={2}^{\frac{k-1}{2}}$$ and $$J={2}^{\frac{k+1}{2}}$$.

`ber = berfading(EbNo,'psk',M,divorder)`

returns
the BER for coherently detected PSK over an uncoded Rayleigh fading
channel.

`ber = berfading(EbNo,'depsk',M,divorder)`

returns
the BER for coherently detected PSK with differential data encoding
over an uncoded Rayleigh fading channel. Only *M* =
2 is currently supported.

`ber = berfading(EbNo,'oqpsk',divorder)`

returns
the BER of coherently detected offset-QPSK over an uncoded Rayleigh
fading channel.

`ber = berfading(EbNo,'dpsk',M,divorder)`

returns
the BER for DPSK over an uncoded Rayleigh fading channel. For DPSK,
it is assumed that the fading is slow enough that two consecutive
symbols are affected by the same fading coefficient.

`ber = berfading(EbNo,'fsk',M,divorder,coherence)`

returns
the BER for orthogonal FSK over an uncoded Rayleigh fading channel. `coherence`

should
be `'coherent'`

for coherent detection, or `'noncoherent'`

for
noncoherent detection.

`ber = berfading(EbNo,'fsk',2,divorder,coherence,rho)`

returns
the BER for binary nonorthogonal FSK over an uncoded Rayleigh fading
channel. `rho`

is the complex correlation coefficient.
See Nonorthogonal 2-FSK with Coherent Detection for the definition
of the complex correlation coefficient and how to compute it for nonorthogonal
BFSK.

`ber = berfading(EbNo,...,K)`

returns the
BER over an uncoded Rician fading channel, where `K`

is
the ratio of specular to diffuse energy in linear scale. For the case
of `'fsk'`

, `rho`

must be specified
before `K`

.

`ber = berfading(EbNo,'psk',2,1,K,phaserr)`

returns
the BER of BPSK over an uncoded Rician fading channel with imperfect
phase synchronization. `phaserr`

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

`[BER,SER] = berfading(EbNo, ...)`

returns
both the BER and SER.

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

As an alternative to the `berfading`

function,
invoke the BERTool GUI (`bertool`

), and use the **Theoretical** tab.

[1] Proakis, John G., *Digital
Communications*, 4th ed., New York, McGraw-Hill, 2001.

[2] Modestino, James W., and Mui, Shou Y., *Convolutional
code performance in the Rician fading channel*, IEEE Trans.
Commun., 1976.

[3] Cho, K., and Yoon, D., “On the
general BER expression of one- and two-dimensional amplitude modulations”, *IEEE
Trans. Commun.*, Vol. 50, Number 7, pp. 1074-1080, 2002.

[4] Lee, P. J., “Computation of the
bit error rate of coherent M-ary PSK with Gray code bit mapping”, *IEEE
Trans. Commun.*, Vol. COM-34, Number 5, pp. 488-491, 1986.

[5] Lindsey, W. C., “Error probabilities
for Rician fading multichannel reception of binary and N-ary signals”, *IEEE
Trans. Inform. Theory*, Vol. IT-10, pp. 339-350, 1964.

[6] Simon, M. K , Hinedi, S. M., and Lindsey,
W. C., *Digital Communication Techniques – Signal
Design and Detection*, Prentice-Hall, 1995.

[7] Simon, M. K., and Alouini, M. S., *Digital
Communication over Fading Channels – A Unified Approach to
Performance Analysis*, 1st ed., Wiley, 2000.

[8] Simon, M. K , “On the bit-error
probability of differentially encoded QPSK and offset QPSK in the
presence of carrier synchronization”, *IEEE Trans.
Commun.*, Vol. 54, pp. 806-812, 2006.