berawgn

BER and SER for uncoded AWGN channels

Syntax

ber = berawgn(EbNo,modtype,M)

ber = berawgn(EbNo,'psk',M,dataenc)

ber = berawgn(EbNo,'oqpsk',dataenc)

ber = berawgn(EbNo,'fsk',M,coherence)

ber = berawgn(EbNo,'fsk',M,coherence,rho)

ber = berawgn(EbNo,'msk',precoding)

ber = berawgn(EbNo,'msk',precoding,coherence)

ber = berawgn(EbNo,'cpfsk',M,modindex,kmin)

[ber,ser] = berawgn(___)

Description

The berawgn function returns the bit error rate (BER) and symbol error rate (SER) in an additive white Gaussian noise (AWGN) channel for uncoded data using various modulation schemes. The first input argument, EbNo, is the ratio of bit energy to noise power spectral density (E_b/N₀) and its units are in dB. Values in the output ber and ser vectors correspond to the theoretical error rate at the specified E_b/N₀ levels for a Gray-coded signal constellation. For more information, see Analytical Expressions Used in berawgn.

ber = berawgn(EbNo,modtype,M) returns the BER at the specified E_b/N₀ levels for the modulation type and modulation order specified by modtype and M, respectively.

ber = berawgn(EbNo,'psk',M,dataenc) specifies the data encoding type as differential or nondifferential for PSK modulation.

ber = berawgn(EbNo,'oqpsk',dataenc) specifies the data encoding type as differential or nondifferential for OQPSK modulation.

ber = berawgn(EbNo,'fsk',M,coherence) specifies the receiver technique as coherent or noncoherent for FSK modulation.

ber = berawgn(EbNo,'fsk',M,coherence,rho) additionally specifies the complex correlation coefficient of the FSK-modulated signal.

ber = berawgn(EbNo,'msk',precoding) specifies whether precoding is applied for MSK modulation.

ber = berawgn(EbNo,'msk',precoding,coherence) additionally specifies the receiver technique as coherent or noncoherent for MSK modulation.

ber = berawgn(EbNo,'cpfsk',M,modindex,kmin) specifies the modulation index, modindex, and the number of paths having the minimum distance, kmin, for CPFSK modulation.

[ber,ser] = berawgn(___) returns the BER and symbol error rate (SER) using any input argument combination from previous syntaxes.

Examples

Return Theoretical BER Data for AWGN Channels

Return theoretical bit error rate data for several modulation schemes in an AWGN channel.

Create a vector of ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ values and specify the modulation order.

EbNo = (0:10)';
M = 4; % Modulation order

Return theoretical BER data for QPSK modulation.

berQ = berawgn(EbNo,'psk',M,'nondiff');

Return equivalent data for DPSK and FSK modulations.

berD = berawgn(EbNo,'dpsk',M);
berF = berawgn(EbNo,'fsk',M,'coherent');

Plot the results.

semilogy(EbNo,[berQ berD berF])
xlabel('Eb/No (dB)')
ylabel('BER')
legend('QPSK','DPSK','FSK')
title("Theoretical Bit Error Rate")
grid

Input Arguments

EbNo — Energy per bit to noise power spectral density ratio

scalar | vector

Energy per bit to noise power spectral density ratio in dB, specified as a scalar or vector.

Data Types: single | double

modtype — Modulation type

'psk' | 'oqpsk' | 'dpsk' | ...

Modulation type, specified as one of these options.

modtype Value	Modulation Scheme	Dependencies
'psk'	Phase shift keying (PSK)	When you set the input dataenc to 'diff', modulation order M must be 2 or 4.
'oqpsk'	Offset quadrature phase shift keying (OQPSK)	None
'dpsk'	Differential phase shift keying (DPSK)	None
'pam'	Pulse amplitude modulation (PAM)	None
'qam'	Quadrature amplitude modulation (QAM)	The modulation order M must be at least 4. When k = log2M is odd, the symbols lie in a rectangular constellation of size M = I × J, where $$I=2^{\frac{k-1}{2}}$$ and $$J=2^{\frac{k+1}{2}}$$. When k is even, the symbols lie in a square constellation of size$$2^{\frac{k}{2}}\times 2^{\frac{k}{2}}$$.
'fsk'	Frequency-shift keying (FSK)	When you set the input coherence to 'noncoherent', modulation order M must be in the range [2, 64].
'msk'	Minimum-shift keying (MSK)	None
'cpfsk'	Continuous-phase frequency-shift keying (CPFSK)	None

Data Types: char | string

M — Modulation order

2^k

Modulation order, specified as an integer equal to 2^k, where k is a positive integer.

Example: 4 or 2^2

Data Types: single | double

dataenc — Data encoding type

'diff' | 'nondiff'

Data encoding type, specified as one of these values.

• 'diff' — For differential data encoding

• 'nondiff' — For nondifferential data encoding

Dependencies

To enable this argument, set the modtype argument to 'psk' or 'oqpsk'.

Data Types: char | string

coherence — Coherent detection type

'conherent' | 'noncoherent'

Coherent detection type, specified as one of these values.

• 'conherent' — For coherent detection

• 'noncoherent' — For noncoherent detection

Dependencies

To enable this argument, set the modtype argument to 'fsk' or 'msk'.

Data Types: char | string

rho — Complex correlation coefficient

complex scalar

Complex correlation coefficient, specified as a complex scalar. For more information about the complex correlation coefficient and how to compute it for nonorthogonal binary frequency-shift keying (BFSK) modulation, see Nonorthogonal 2-FSK with Coherent Detection.

Dependencies

To enable this argument, set the modtype argument to 'fsk' and the M argument to 2.

Data Types: single | double
Complex Number Support: Yes

precoding — Enable precoding

'off' | 'on'

Enable precoding, specified as one of these values.

• 'off' — For conventional MSK

• 'on' — For precoded MSK

Dependencies

To enable this argument, set the modtype argument to 'msk'.

Data Types: char | string

modindex — Modulation index

positive integer

Modulation index, specified as a positive integer.

Dependencies

To enable this argument, set the modtype argument to 'cpfsk'.

Data Types: single | double

kmin — Number of paths having minimum distance

positive integer

Number of paths having the minimum distance, specified as a positive integer. If the number of paths is unknown, specify a value of 1.

Dependencies

To enable this argument, set the modtype argument to 'cpfsk'.

Data Types: single | double

Output Arguments

ber — Bit error rate

scalar | vector

Bit error rate (BER) for uncoded data over an AWGN channel, returned as a scalar or vector. The BER is computed for each E_b/N₀ setting specified by input EbNo according to the modulation type specified by input modtype and related dependencies.

Data Types: double

ser — Symbol error rate

scalar | vector

Symbol error rate (SER) for uncoded data over an AWGN channel, returned as a scalar or vector. The SER is computed for each E_b/N₀ setting specified by input EbNo according to the modulation type specified by input modtype and related dependencies.

Data Types: double

Limitations

The numerical accuracy of the output returned by this function is limited by approximations related to the numerical implementation of the expressions to roughly two significant digits.

Alternatives

As an alternative to the berawgn function, use the BER Analyzer app and configure settings in the Theoretical tab.

References

[1] Anderson, John B., Tor Aulin, and Carl-Erik Sundberg. Digital Phase Modulation. New York: Plenum Press, 1986.

[2] Cho, K., and D. Yoon. “On the General BER Expression of One- and Two-Dimensional Amplitude Modulations.” IEEE Trans. Commun. 50, no. 7, (2002): 1074-1080.

[3] Lee, P. J. “Computation of the Bit Error Rate of Coherent M-ary PSK with Gray Code Bit Mapping.” IEEE Trans. Commun. COM-34, no. 5, (1986): 488-491.

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

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

[6] Simon, M. K. “On the Bit-Error Probability of Differentially Encoded QPSK and Offset QPSK in the Presence of Carrier Synchronization.” IEEE Trans. Commun.. 54, (2006): 806-812.

[7] Lindsey, W. C., and M. K. Simon. Telecommunication Systems Engineering. Englewood Cliffs, N.J: Prentice-Hall, 1973.