📄 generalized-hierarchical-psk.htm
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<H4><BOLD>Note</BOLD></H4></CENTER><FONT color=#ff0000>This web page deals with
bit error probability for hierarchical PSK modulations assuming perfect phase
and timing recovery. For performance analysis of under imperfect phase
reference, please click on the following link: </FONT><BR>
<CENTER><A
href="http://www.ece.umn.edu/users/pavan/PSK-phase-error.html">Performance Under
Phase Error</A></TR></TD><BR></CENTER>
<CENTER>
<H2>Performance Analysis of Hierarchical PSK modulations under perfect Phase and
Timing Recovery</H2></CENTER>
<CENTER>
<H4>Gray Coding</H4></CENTER>To Gray code an M-PSK constellation hierarchically,
one can do the following: Number the points on the constellation from 0 to M-1,
and expand them in binary (say integer B has the binary expansion (B1 B2 B3 ...
Bm). The the corresponding gray code (G1 G2 G3 ... Gm) can be written using
these well known equations: <BR>
<CENTER>G1=B1 <BR>Gi=xor(Bi, Bi-1), 1 < i < m, </CENTER><BR>This has been
done interactively by undergraduate students using the following programs.<BR>
<CENTER><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/website/psk_telek.m">psk_telek.m</A></TR></TD><BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/website/graycode.m">graycode.m</A></TR></TD><BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/website/binarycode.m">binarycode.m</A></TR></TD><BR></CENTER><BR>Download
these programs into a directory. Run "psk_telek.m" on the MATLAB prompt. When
prompted enter the theta vector. The program will then plot the constellation
with the gray codes.
<CENTER>
<H4>MATLAB code usage </H4></CENTER>
<P>To use the recursive function, the reader first needs to define the angle
vectors for the constellation being considered. The following examples will
illustrate the definition of priority vector.
<P>Let us consider the generalized 2/4/8-PSK case, as shown in the figure below.
<BR>
<CENTER><IMG src="Generalized-Hierarchical-PSK.files/general_8PSK_change.gif">
</CENTER><BR><BR><BR>The three angles decide the extent of unequal error
protection. If we set the "input_angle_vector" as [pi/2 pi/4 pi/8], then we have
the special case of uniform 8-PSK. If we set the "input_angle_vector" to [pi/2
pi/6 pi/15], then we have a high degree of variation in the performance of the
LSB and the MSB.
<P>Having decided upon the constellation, please use the angle vectors in the
main program below. Please download the following functions, and save them in
one directory. <BR><BR>
<CENTER><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/psk_rec_telek.m">psk_rec_telek.m</A></TR></TD>
<BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/recurse_other.m">recurse_other.m</A></TR></TD>
<BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/lsb_error_new.m">lsb_error_new.m</A></TR></TD>
<BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/F_func.m">F_func.m</A></TR></TD>
<BR><A
href="http://www.ece.umn.edu/users/pavan/PSK/programs/my_integrate.m">my_integrate.m</A></TR></TD>
<BR><BR></CENTER>Please open psk_rec_telek.m, and change the definition of the
angle vector on line 3 accordingly. <BR><BR>input_angle_vector=[pi/2 x y ......
z]' <BR>Please turn "warning off" at the matlab prompt. Then run the program
"psk_rec_telek" at the matlab prompt. <BR><BR>The function first prints out the
angle vector, followed by the BER matrix. <BR><BR>The first column gives the BER
for MSB as a function of CNR from -5dB to 24 dB, in steps of 1 dB each. <BR>The
second column gives the BER for the bit next to MSB, and so on. <BR>The last
column gives the BER for the LSB as a function of CNR. <BR><BR>The program then
plots the BER for all the bits. <BR>As an example, the figure below shows the
BER performance of a generalized 2/4/8/16-PSK constellation. Note the near 20dB
difference between the MSB and LSB performance.
<CENTER><IMG src="Generalized-Hierarchical-PSK.files/four_pri.gif">
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