Qpsk Bit Error Rate Calculation
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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 results bit error rate calculation using matlab obtained using Matlab/Octave simulation scripts show good agreement with the derived theoretical results. With bit error rate for qpsk matlab code Binary Phase Shift Keying (BPSK), the binary digits 1 and 0 maybe represented by the analog levels and respectively. The system
Ber Calculation Formula
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
Ber Of Qpsk In Awgn Channel Matlab Code
: 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. bit error rate of bpsk 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
6, 2007 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
Ber Of Bpsk In Awgn Channel Matlab Code
a QPSK (4-QAM) modulation scheme. Background Consider that the alphabets used for a symbol error rate definition 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 matlab code for ber vs snr for qpsk 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 http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/ 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 http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/ 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 QPSK modulation over AWGN (2 votes, average: 5.00 out of 5) Loading... This post is a part of the ebook : Simulation of digital communication systems using Matlab - available in both PDF and EPUB format. In the previous article we saw http://www.gaussianwaves.com/2010/10/ber-vs-ebn0-for-qpsk-modulation-over-awgn-2/ about how QPSK modulation and demodulation can be done. This concept is extended further to simulate the performance of QPSK modulation technique over an AWGN. Transmitter: For the QPSK modulation , a series of binary input message bits are generated. In QPSK, a symbol contains 2 bits. The generated binary bits are combined in terms of two bits and QPSK symbols are generated. From the constellation of QPSK modulation the symbol error rate '00' is represented by 1, '01' by j (90 degrees phase rotation), '10' by -1 (180 degrees phase rotation) and '11' by -j (270 degrees phase rotation). In pi/4 QPSK, these phase rotations are offset by 45 degrees. So the effective representation of symbols in pi/4-QPSK is '00'=1+j (45 degrees), '01'=-1+j (135 degrees), '10' = -1-j (225 degrees) and '11'= 1-j (315 degrees). Here we are simulating a pi/4 QPSK system.Once the bit error rate symbols are mapped, the power of the QPSK modulated signal need to be normalized by \(\frac{1}{\sqrt{2}}\). AWGN channel: For QPSK modulation the channel can be modeled as $$ y=ax+n $$ where y is the received signal at the input of the QPSK receiver, x is the complex modulated signal transmitted through the channel , a is a channel amplitude scaling factor for the transmitted signal usually 1. ‘n' is the Additive Gaussian White Noise random random variable with zero mean and variance \(\sigma^2 \). For AWGN the noise variance in terms of noise power spectral density \(N_0\) is given by, $$ \sigma^{2}=\frac{N_{0}}{2} $$ For M-PSK modulation schemes including BPSK, the symbol energy is given by $$ E_s = R_m R_c E_b $$ where \(E_s\) =Symbol energy per modulated bit (\(x\)), \(Rm = log_2(M)\) , (for BPSK M=2, QPSK M=4, 16 QAM M=16 etc..,). \(R_c\) is the code rate of the system if a coding scheme is used. In our case since no coding scheme is used Rc = 1. Eb is the Energy per information bit. Assuming \(E_s=1\) for BPSK (Symbol energy normalized to 1) \(\frac{E_b}{N_0}\) can be represented as (using above equations), $$ \frac{E_b}{N_0} = \frac{E_s}{R_m R_c N_0} $$ $$ \frac{E_b}{N_0} = \frac{E_s}{R_m R_c N_0} = \frac{E_s}{R_m R_c 2 \sigma^{2}} = \f
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