mohatta2015.pdf

190 SIMULATION NOTES
etc., then the bits in Boolean form (6*, — 0,1) can be detected as follows
z K = z, b K = {z K < 0) (A.3)
z k = m fc+ i(z fc+ i -fhk+i2 k ), b k = (z k < 0), k = K - 1,..., 1. (A.4)
where the modulation form is obtained from the Boolean form using rhk = —2bk + 1.
Also, the expression a < 0 is 1 if true and 0 if false.
While the pulse shape p(t) is continuous in time, it is typically modeled with a
discrete-time, sampled form, using 4 or 8 samples per symbol period. Here we will
use 8 samples per symbol period. To speed simulation up, it is sometimes possible to
avoid modeling p(t) and to work with only one sample per symbol period. Consider
the following assumptions.
• The transmitter uses root-Nyquist pulse shaping.
• The channel consists of symbol-spaced paths.
• The noise is AWGN.
• The receiver initially filters the received signal with a filter that performs a
sliding correlation of the received signal with the pulse shape.
• The output of the receive filter is sampled once a symbol, and the sampling
point is aligned with the point where only one symbol affects the sample.
With these assumptions, the received samples can be modeled as
v(mT s ) = ^2ges(m- t) + ñ(m), (A.5)
t=a
where ñ(m) is the corresponding noise sample after filtering and sampling. Observe
that because of its Nyquist properties, the pulse shape p(t) is not present. By
simulating v{mT s ) directly, we avoid the need to model the pulse shape and to
generate multiple samples per symbol period.
A commonly used "trick" to speed up simulations is to use the same transmitted
signal and same noise realization to evaluate several SNR levels. This is done by
looping over the SNR levels of interest and adding different scaled versions of the
transmitted signal to the noise. (One can also add different scaled versions of the
noise to the transmitted signal, as long as receiver algorithms are insensitive to
scaling.) This can be done after the receive filter, if the transmitted signal and
noise signal are filtered separately.
It is important to "calibrate" a new simulation tool to make sure it is working
properly. One way this is done is by comparing performance results to known results.
To verify M-QAM modulation and demodulation, it is convenient to evaluate
symbol error rate (SER). The decision variable z can be modeled as z = As + n,
where n is real Gaussian noise with variance N a /2. From [Pro89], the SER for
TV-ASK is given by