In this post, we will derive the theoretical equation for bit error rate (BER) with Binary Phase Shift Keying (BPSK) modulation scheme in Additive White Gaussian Noise (AWGN) channel. The BER results obtained using Matlab/Octave simulation scripts show good agreement with the derived theoretical results.

With Binary Phase Shift Keying (BPSK), the binary digits 1 and 0 maybe represented by the analog levels and respectively. The system model is as shown in the Figure below.

**Figure: Simplified block diagram with BPSK transmitter-receiver**

## Channel Model

The transmitted waveform gets corrupted by noise , typically referred to as **Additive White Gaussian Noise** (AWGN).

**Additive** : As the noise gets ‘added’ (and not multiplied) to the received signal

**White** : The spectrum of the noise if flat for all frequencies.

**Gaussian** : The values of the noise follows the Gaussian probability distribution function, with and .

## Computing the probability of error

Using the derivation provided in Section 5.2.1 of [COMM-PROAKIS] as reference:

The received signal,

when bit 1 is transmitted and

when bit 0 is transmitted.

The conditional probability distribution function (PDF) of for the two cases are:

.

** Figure: Conditional probability density function with BPSK modulation**

Assuming that and are equally probable i.e. , the **threshold 0** forms the optimal decision boundary.

- if the received signal is is greater than 0, then the receiver assumes was transmitted.

- if the received signal is is less than or equal to 0, then the receiver assumes was transmitted.

i.e.

and

.

### Probability of error given was transmitted

With this threshold, the probability of error given is transmitted is (the area in blue region):

,

where,

isĀ the complementary error function.

### Probability of error given was transmitted

Similarly the probability of error given is transmitted is (the area in green region):

.

### Total probability of bit error

.

Given that we assumed that and are equally probable i.e. , the **bit error probability** is,

.

**Simulation model**

Matlab/Octave source code for computing the bit error rate with BPSK modulation from theory and simulation. The code performs the following:

(a) Generation of random BPSK modulated symbols +1′s and -1′s

(b) Passing them through Additive White Gaussian Noise channel

(c) Demodulation of the received symbol based on the location in the constellation

(d) Counting the number of errors

(e) Repeating the same for multiple Eb/No value.

Click here to download Matlab/Octave script for simulating BER for BPSK modulation in AWGN chnanel.

**Figure: Bit error rate (BER) curve for BPSK modulation – theory, simulation**

## Reference

[DIGITAL COMMUNICATION: PROAKIS] Digital Communications by John Proakis

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 }