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Computing Theoretical BERs

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Section Overview Example: Using the Theoretical Tab in BERTool Available Sets of Theoretical BER Data

Section Overview

You can use BERTool to generate and analyze theoretical BER data. Theoretical data is useful for comparison with your simulation results. However, closed-form BER expressions exist only for certain kinds of communication systems.

To access the capabilities of BERTool related to theoretical BER data, use the following procedure:

1. Open BERTool, and go to the Theoretical tab.

   

2. Set the parameters to reflect the system whose performance you want to analyze. Some parameters are visible and active only when other parameters have specific values. See Available Sets of Theoretical BER Data for details.

3. Click Plot.

For an example that shows how to generate and analyze theoretical BER data via BERTool, see Example: Using the Theoretical Tab in BERTool.

Also, Available Sets of Theoretical BER Data indicates which combinations of parameters are available on the Theoretical tab and which underlying functions perform computations.

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Example: Using the Theoretical Tab in BERTool

This example illustrates how to use BERTool to generate and plot theoretical BER data. In particular, the example compares the performance of a communication system that uses an AWGN channel and QAM modulation of different orders.

Running the Theoretical Example

1. Open BERTool, and go to the Theoretical tab.

2. Set the parameters as shown in the following figure.

   

3. Click Plot.

   BERTool creates an entry in the data viewer and plots the data in the BER Figure window. Even though the parameters request that E_b/N₀ go up to 18, BERTool plots only those BER values that are at least 10^-8. The following figures illustrate this step.

   

   

4. Change the Modulation order parameter to 16, and click Plot.

   BERTool creates another entry in the data viewer and plots the new data in the same BER Figure window (not pictured).

5. Change the Modulation order parameter to 64, and click Plot.

   BERTool creates another entry in the data viewer and plots the new data in the same BER Figure window, as shown in the following figures.

   

   

6. To recall which value of Modulation order corresponds to a given curve, click the curve. BERTool responds by adjusting the parameters in the Theoretical tab to reflect the values that correspond to that curve.

7. To remove the last curve from the plot (but not from the data viewer), clear the check box in the last entry of the data viewer in the Plot column. To restore the curve to the plot, select the check box again.

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Available Sets of Theoretical BER Data

BERTool can generate a large set of theoretical bit-error rates, but not all combinations of parameters are currently supported. The Theoretical tab adjusts itself to your choices, so that the combination of parameters is always valid. You can set the Modulation order parameter by selecting a choice from the menu or by typing a value in the field. The Normalized timing error must be between 0 and 0.5.

BERTool assumes that Gray coding is used for all modulations.

For QAM, when is odd (M being the modulation order), a rectangular constellation is assumed.

Combinations of Parameters for AWGN Channel Systems

The following table lists the available sets of theoretical BER data for systems that use an AWGN channel.

Modulation	Modulation Order	Other Choices
PSK	2, 4	Differential or nondifferential encoding.
	8, 16, 32, 64, or a higher power of 2
OQPSK	4	Differential or nondifferential encoding.
DPSK	2, 4, 8, 16, 32, 64, or a higher power of 2
PAM	2, 4, 8, 16, 32, 64, or a higher power of 2
QAM	4, 8, 16, 32, 64, 128, 256, 512, 1024, or a higher power of 2
FSK	2	Orthogonal or nonorthogonal; Coherent or Noncoherent demodulation.
	4, 8, 16, 32, or a higher power of 2	Orthogonal; Coherent demodulation.
	4, 8, 16, 32, or 64	Orthogonal; Noncoherent demodulation.
MSK	2	Coherent conventional or precoded MSK; Noncoherent precoded MSK.
CPFSK	2, 4, 8, 16, or a higher power of 2	Modulation index > 0.

BER results are also available for the following:

• block and convolutional coding with hard-decision decoding for all modulations except CPFSK

• block coding with soft-decision decoding for all binary modulations (including 4-PSK and 4-QAM) except CPFSK, noncoherent non-orthogonal FSK, and noncoherent MSK

• convolutional coding with soft-decision decoding for all binary modulations (including 4-PSK and 4-QAM) except CPFSK

• uncoded nondifferentially-encoded 2-PSK with synchronization errors

For more information about specific combinations of parameters, including bibliographic references that contain closed-form expressions, see the reference pages for the following functions:

• berawgn — For systems with no coding and perfect synchronization

• bercoding — For systems with channel coding

• bersync — For systems with BPSK modulation, no coding, and imperfect synchronization

Combinations of Parameters for Rayleigh and Rician Channel Systems

The following table lists the available sets of theoretical BER data for systems that use a Rayleigh or Rician channel.

When diversity is used, the SNR on each diversity branch is derived from the SNR at the input of the channel (EbNo) divided by the diversity order.

Modulation	Modulation Order	Other Choices
PSK	2	Differential or nondifferential encoding Diversity order ≧1 In the case of nondifferential encoding, diversity order being 1, and Rician fading, a value for RMS phase noise (in radians) can be specified.
	4, 8, 16, 32, 64, or a higher power of 2	Diversity order ≧1
OQPSK	4	Diversity order ≧1
DPSK	2, 4, 8, 16, 32, 64, or a higher power of 2	Diversity order ≧1
PAM	2, 4, 8, 16, 32, 64, or a higher power of 2	Diversity order ≧1
QAM	4, 8, 16, 32, 64, 128, 256, 512, 1024, or a higher power of 2	Diversity order ≧1
FSK	2	Correlation coefficient . Coherent or Noncoherent demodulation Diversity order ≧1 In the case of a nonzero correlation coefficient and noncoherent demodulation, the diversity order is 1 only.
	4, 8, 16, 32, or a higher power of 2	Noncoherent demodulation only. Diversity order ≧1

For more information about specific combinations of parameters, including bibliographic references that contain closed-form expressions, see the reference page for the berfading function.

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The BERTool Environment

Using the Semianalytic Technique to Compute BERs

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