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Performance Results via the Semianalytic Technique

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Section Overview When to Use the Semianalytic Technique Procedure for the Semianalytic Technique Example: Using the Semianalytic Technique

Section Overview

The technique described in Performance Results via Simulation works well for a large variety of communication systems, but can be prohibitively time-consuming if the system's error rate is very small (for example, 10^-6 or smaller). This section describes how to use the semianalytic technique as an alternative way to compute error rates. For certain types of systems, the semianalytic technique can produce results much more quickly than a nonanalytic method that uses only simulated data.

The semianalytic technique uses a combination of simulation and analysis to determine the error rate of a communication system. The semianalytic function in Communications Toolbox helps you implement the semianalytic technique by performing some of the analysis.

For more background information on the semianalytic technique, refer to [3].

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When to Use the Semianalytic Technique

The semianalytic technique works well for certain types of communication systems, but not for others. The semianalytic technique is applicable if a system has all of these characteristics:

• Any effects of multipath fading, quantization, and amplifier nonlinearities must precede the effects of noise in the actual channel being modeled.

• The receiver is perfectly synchronized with the carrier, and timing jitter is negligible. Because phase noise and timing jitter are slow processes, they reduce the applicability of the semianalytic technique to a communication system.

• The noiseless simulation has no errors in the received signal constellation. Distortions from sources other than noise should be mild enough to keep each signal point in its correct decision region. If this is not the case, the calculated BER is too low. For instance, if the modeled system has a phase rotation that places the received signal points outside their proper decision regions, the semianalytic technique is not suitable to predict system performance.

Furthermore, the semianalytic function assumes that the noise in the actual channel being modeled is Gaussian. For details on how to adapt the semianalytic technique for non-Gaussian noise, see the discussion of generalized exponential distributions in [3].

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Procedure for the Semianalytic Technique

The procedure below describes how you would typically implement the semianalytic technique using the semianalytic function:

1. Generate a message signal containing at least M^L symbols, where M is the alphabet size of the modulation and L is the length of the impulse response of the channel in symbols. A common approach is to start with an augmented binary pseudonoise (PN) sequence of total length (log₂M)M^L. An augmented PN sequence is a PN sequence with an extra zero appended, which makes the distribution of ones and zeros equal.

2. Modulate a carrier with the message signal using baseband modulation. Supported modulation types are listed on the reference page for semianalytic. Shape the resultant signal with rectangular pulse shaping, using the oversampling factor that you will later use to filter the modulated signal. Store the result of this step as txsig for later use.

3. Filter the modulated signal with a transmit filter. This filter is often a square-root raised cosine filter, but you can also use a Butterworth, Bessel, Chebyshev type 1 or 2, elliptic, or more general FIR or IIR filter. If you use a square-root raised cosine filter, use it on the nonoversampled modulated signal and specify the oversampling factor in the filtering function. If you use another filter type, you can apply it to the rectangularly pulse shaped signal.

4. Run the filtered signal through a noiseless channel. This channel can include multipath fading effects, phase shifts, amplifier nonlinearities, quantization, and additional filtering, but it must not include noise. Store the result of this step as rxsig for later use.

5. Invoke the semianalytic function using the txsig and rxsig data from earlier steps. Specify a receive filter as a pair of input arguments, unless you want to use the function's default filter. The function filters rxsig and then determines the error probability of each received signal point by analytically applying the Gaussian noise distribution to each point. The function averages the error probabilities over the entire received signal to determine the overall error probability. If the error probability calculated in this way is a symbol error probability, the function converts it to a bit error rate, typically by assuming Gray coding. The function returns the bit error rate (or, in the case of DQPSK modulation, an upper bound on the bit error rate).

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Example: Using the Semianalytic Technique

The example below illustrates the procedure described above, using 16-QAM modulation. It also compares the error rates obtained from the semianalytic technique with the theoretical error rates obtained from published formulas and computed using the berawgn function. The resulting plot shows that the error rates obtained using the two methods are nearly identical. The discrepancies between the theoretical and computed error rates are largely due to the phase offset in this example's channel model.

% Step 1. Generate message signal of length >= M^L.
M = 16; % Alphabet size of modulation
L = 1; % Length of impulse response of channel
msg = [0:M-1 0]; % M-ary message sequence of length > M^L

% Step 2. Modulate the message signal using baseband modulation.
modsig = qammod(msg,M); % Use 16-QAM.
Nsamp = 16;
modsig = rectpulse(modsig,Nsamp); % Use rectangular pulse shaping.

% Step 3. Apply a transmit filter.
txsig = modsig; % No filter in this example

% Step 4. Run txsig through a noiseless channel.
rxsig = txsig*exp(1i*pi/180); % Static phase offset of 1 degree
% Step 5. Use the semianalytic function.
% Specify the receive filter as a pair of input arguments.
% In this case, num and den describe an ideal integrator.
num = ones(Nsamp,1)/Nsamp;
den = 1;
EbNo = 0:20; % Range of Eb/No values under study
ber = semianalytic(txsig,rxsig,'qam',M,Nsamp,num,den,EbNo);

% For comparison, calculate theoretical BER.
bertheory = berawgn(EbNo,'qam',M);

% Plot computed BER and theoretical BER.
figure; semilogy(EbNo,ber,'k*');
hold on; semilogy(EbNo,bertheory,'ro');
title('Semianalytic BER Compared with Theoretical BER');
legend('Semianalytic BER with Phase Offset',...
   'Theoretical BER Without Phase Offset','Location','SouthWest');
hold off;

This example creates a figure like the one below.

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Performance Results via Simulation

Theoretical Performance Results

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