spatial-phase region. In this region the noise vector influences the decision about the
modulated symbols reception.
4.2.
Ser and ber calculation
SER and BER according to the VPM is calculated by the following formula [17-
19]:
(ℎ
2
)=
1
2
∫
−ℎ2
2(−)2
+
1
2
∫
−5ℎ2
18(+
3)
2
+
1
2
∫
−ℎ2
18(−)2
5
4
4
+(3)
(3)
5
4
5
4
, (4)
(ℎ
2
)=
1
2
∫
−ℎ2
18(−)2
−
1
2
∫
−ℎ2
50(−)2
+
1
2
∫
−ℎ2
18(−)2
5
4
−
3
4
5
4
4
5
4
4
2
, (5)
It should be noted that the main feature of these formulas is their authenticity,
because they more accurately calculate SER and BER.
For comparison the VPM formulas with some other well-known formulas, it is
necessary to analyze and compare the data received by these methods. To do this, the
J. Proakis formulas are used to calculate SER and BER.
The input data for the J. Proakis method is the following:
n – error-correcting code block length
ℎ
2
– Energy parameter characterizing the average signal power ratio at the
reception point in the one-sided noise spectral power, in other words – SNR:
ℎ
2
=
0
=
0
,
(6)
where
=
1
;
– energy parameter equal to the ratio of useful signal power and
spectral noise power
0
:
=
0
,
(7)
where
– the symbol rate in the communication channel.
BER in channel can be identified by using these parameters. BER depends on the
method of forming and signal processing.
SER and BER for QPSK and QAM-M can be determined by formulas:
_
=
3
4
−
1
2
Ф(√ℎ
2
)−
1
4
Ф
2
(√ℎ
2
),
(8)
_
=
1
2
[1−Ф(√2ℎ
2
4
)],
(9)
_
=1−(1−
2(1−
1
√
)
√2
∫
(
−
2
2
)
∞
√
3
−1
ℎ
2
)
2
,
(10)
_
=
4(1−
1
√
)
√2
∫
(
−
2
2
)
∞
√
3
−1
ℎ
2
,
(11)
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