Signal Constellation of Quadrature Phase Shift Keying Modulation in Noise

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The signal constellation for quadrature phase shift keying (QPSK) consists of four equally spaced signal points, typically arranged in a square grid. These points are usually located at and with respect to the reference axis. In the presence of noise and intersymbol interference, the received signal points deviate from their ideal positions in the constellation. This Demonstration dynamically illustrates the deviation from the ideal positions as the energy per bit to noise power spectral density ratio changes. An error occurs when the received symbol falls outside its associated decision region. In this Demonstration, the decision regions are represented by colored areas. The corresponding bit error rate (BER) is also displayed.

Contributed by: Victor S. Frost (August 27)
(University of Kansas)
Open content licensed under CC BY-NC-SA


A QPSK signal constellation represents the possible signal points used in a QPSK modulation. In QPSK, each symbol represents two bits of information. Two bits are transmitted each symbol time, .

For one symbol time, the transmitted QPSK signal on a carrier frequency of is one of the following four symbols [1]:

The energy/symbol is and 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" [1] centered around each ideal signaling point as shown in this Demonstration. An error occurs when the received symbol is outside its associated decision region, shown here by a colored area. 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 probability of a bit error (BER) of


where is the noise power spectral density and .

The BER is calculated in the Demonstration.


[1] S. Haykin and M. Moher, Introduction to Analog and Digital Communications, 2nd ed., Hoboken, NJ: Wiley, 2012.

[2] L. W. Couch, Digital and Analog Communications Systems, 7th ed., Upper Saddle River, NJ: Pearson/Prentice Hall, 2007.

[3] H. Taub and D. L. Schilling, Principles of Communication Systems, 2nd ed., New York: McGraw-Hill, 1986.

[4] V. S. Frost. "Introduction to Communication Systems: An Interactive Approach Using the Wolfram Language." University of Kansas Libraries. (Jul 5, 2023)


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