Given that we have discussed symbol error rate probability for a 4-PAM modulation, let us know focus on finding the symbol error probability for a QPSK (4-QAM) modulation scheme.

**Background**

Consider that the alphabets used for a QPSK (4-QAM) is (Refer example 5-35 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

- Download free e-Book
**discussing theoretical**and**simulated****error rates**for the digital modulation schemes like**BPSK**,**QPSK**,**4**-**PAM**,**16PSK and****16QAM**. Further, Bit Error Rate with**Gray coded**mapping, bit error rate for BPSK over**OFDM**are also discussed. - Interested in
**MIMO (Multiple Input Multiple Output)**communications?**Click here to see the post describing six equalizers with 2×2 V-BLAST.** - Read about using multiple antennas at the transmitter and receiver to improve the diversity of a communication link. Articles include
**Selection diversity, Equal Gain Combining, Maximal Ratio Combining, Alamouti STBC, Transmit Beaforming.**

**Figure: Constellation plot for QPSK (4-QAM) constellation**

The scaling factor of is for normalizing the average energy of the transmitted symbols to 1, assuming that all the constellation points are equally likely.

**Noise model**

Assuming that the additive noise follows the Gaussian probability distribution function,

with and .

## Computing the probability of error

**Consider the symbol **

The conditional probability distribution function (PDF) of given was transmitted is:

.

**Figure: Probability density function for QPSK (4QAM) modulation**

As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the hashed region i.e.

.

Probability of real component of greater than 0, given was transmitted is (i.e. area outside the red region)

, where

the complementary error function, .

Similarly, probability of imaginary component of greater than 0, given was transmitted is (i.e. area outside the blue region).

.

The probability of being decoded correctly is,

.

**Total symbol error probability**

The symbol will be in error, it atleast one of the symbol is decoded incorrectly. The probability of symbol error is,

.

For higher values of , the second term in the equation becomes negligible and the probability of error can be approximated as,

**Simulation Model**

Simple Matlab/Octave script for generating QPSK transmission, adding white Gaussian noise and decoding the received symbol for various values.

Click here to download: Matlab/Octave script for computing the symbol error rate for QPSK modulation

**Figure: Symbol Error Rate for QPSK (4QAM) modulation**

**Observations**

1. Can see good agreement between the simulated and theoretical plots for 4-QAM modulation

2. When compared with 4-PAM modulation, the 4-QAM modulation requires only around 2dB lower for achieving a symbol error rate of .

**Reference**

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Hope this helps.

Krishna

D id you like this article? Make sure that you do not miss a new article
by subscribing to RSS feed OR subscribing to e-mail newsletter.
* Note: Subscribing via e-mail entitles you to download the free e-Book on BER of BPSK/QPSK/16QAM/16PSK in AWGN.*

{ 0 comments… add one now }