Bpsk Error Function
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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 function of bpsk modulator results obtained using Matlab/Octave simulation scripts show good agreement with the derived theoretical results. bpsk error probability With Binary Phase Shift Keying (BPSK), the binary digits 1 and 0 maybe represented by the analog levels and respectively. The system bpsk error rate 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 bit error rate of bpsk : 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
Bit Error Rate Of Qpsk
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, is 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) Repeatin
votes, average: 3.50 out of 5) Loading... Q functions are often encountered in the theoretical equations for Bit Error Rate (BER) involving AWGN channel. A brief discussion on Q function and its relation to
Bit Error Rate Matlab Code
erfc function is given here. Gaussian process is the underlying model for an ber of bpsk in awgn channel matlab code AWGN channel.The probability density function of a Gaussian Distribution is given by $$p(x) = \frac{1}{ \sigma \sqrt{2 \pi}} e^{ bpsk modulation - \frac{(x-\mu)^2}{2 \sigma^2}} \;\;\;\;\;\;\; (1)$$ Generally, in BER derivations, the probability that a Gaussian Random Variable \( X \sim N( \mu, \sigma^2) \) exceeds x0 is evaluated as the area of the shaded http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/ region as shown in Figure 1. Gaussian PDF and illustration of Q function Mathematically, the area of the shaded region is evaluated as, $$ Pr(X \geq x_0) = \int_{x_0}^{\infty} p(x) dx = \int_{x_0}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} e^{ - \frac{(x-\mu)^2}{2 \sigma^2}} dx \;\;\;\;\;\;\; (2)$$ The above probability density function given inside the above integral cannot be integrated in closed form. So by change of http://www.gaussianwaves.com/2012/07/q-function-and-error-functions/ variables method, we substitute $$ y = \frac{x-\mu}{\sigma} \;\;\;\;\;\;\; (3)$$ Then equation (3) can be re-written as, $$Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = \int_{ \left( \frac{x_{0} -\mu}{\sigma}\right)}^{\infty} \frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy \;\;\;\;\;\;\; (3)$$ Here the function inside the integral is a normalized gaussian probability density function \( Y \sim N( 0, 1)\), normalized to mean=0 and standard deviation=1. The integral on the right side can be termed as Q-function, which is given by, $$Q(z) = \int_{z}^{\infty}\frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy \;\;\;\;\;\;\; (4)$$ Here the Q function is related as, $$ Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = Q\left(\frac{x_0-\mu}{\sigma} \right ) = Q(z)\;\;\;\;\;\;\; (5)$$ Thus Q function gives the area of the shaded curve with the transformation \( y = \frac{x-\mu}{\sigma}\) applied to the Gaussian probability density function. Essentially, Q function evaluates the tail probability of normal distribution (area of shaded area in the above figure). Error functions: The error function represents the probability that the parameter of interest is within a range between \( \sigma \sqrt{2} \) and \( x/ \sigma \sqrt{2} \) and the complementary error function gives the probability that the parameter lies outside that r
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