Signal Constellation of Eight-Phase Shift-Keying Modulation in Noise
The signal constellation for eight-phase shift keying (8-PSK) consists of eight equally spaced signal points arranged on a circle. These points are separated by 45 degrees. In the presence of noise and intersymbol interference, the received signal points deviate from their ideal positions in the constellation. This Demonstration illustrates the dynamical deviation from the ideal positions as the ratio of energy per bit to noise power spectral density changes. An error occurs when the received symbol falls outside its associated decision region. The decision regions are represented by the areas between the black lines. The corresponding theoretical bit error rate (BER) is calculated and displayed.
An 8-PSK signal constellation represents the signal points used in 8-PSK modulation. In 8-PSK, each symbol represents three bits of information. Three bits are transmitted during each symbol time . For one symbol time, the transmitted 8-PSK signal on a carrier frequency of is one of the following eight symbols :
is the energy/symbol and is the energy/bit.
In practical scenarios, noise and intersymbol interference are present, causing random variations in the received signal points. Receiving a large set of symbols results in a "noise cloud"  centered around each ideal signaling point, which is the situation shown in this Demonstration. An error occurs when the received symbol falls outside its associated decision region, which is the area between the black lines. The corresponding theoretical BER is displayed. The presence of noise and intersymbol interference introduces bit errors. Employing an optimum detector [1–3] with additive flat Gaussian noise and no intersymbol interference results in a BER probability of
where is the noise power spectral density and is the probability of in a normal distribution.
 S. Haykin and M. Moher, Introduction to Analog and Digital Communications, 2nd ed., Hoboken, NJ: Wiley, 2007.
 L. E. Couch, Digital and Analog Communication Systems, 7th ed., Upper Saddle River, NJ: Pearson/Prentice Hall, 2007.
 H. Taub and D. L. Schilling, Principles of Communications Systems, 2nd ed., New York: McGraw-Hill Book Company, 1986.
 V. S. Frost. "Introduction to Communication Systems: An Interactive Approach Using the Wolfram Language." University of Kansas Libraries. (Jun 8, 2023) kuscholarworks.ku.edu/bitstream/handle/1808/31779/Introduction-to-Communication-Systems-Deployed-V3.cdf?sequence=5&isAllowed=y.