Bpsk Symbol Error Rate
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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 obtained using Matlab/Octave simulation scripts bit error rate of bpsk show good agreement with the derived theoretical results. With Binary Phase Shift Keying (BPSK), the binary
Symbol Error Rate Vs Bit Error Rate
digits 1 and 0 maybe represented by the analog levels and respectively. The system model is as shown in the Figure below. Figure: symbol error rate qpsk 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
Bit Error Rate Of Qpsk
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 bit error rate matlab code 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) 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 COMMUNICA
theoretical BPSK Bit Error Rate reference curve.
Bpsk Modulation
Parameters Name Type Range Block Diagram System Diagram N/A
Ber Of Bpsk In Awgn Channel Matlab Code
BER/SER Meter System BER/SER Meter N/A Modulation Type List of options N/A Result The bpsk bit error rate matlab code measurement plots a theoretical BPSK bit error probability along the y-axis and the swept variable (typically Eb/N0) along the x-axis. The y-axis should http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/ normally be set to use log scaling. Graph Type This measurement can be displayed on a rectangular graph or tabular grid. Computational Details The measurement generates a reference curve based on the settings of the meter block selected in the BER/SER Meter setting. Values for Pb https://awrcorp.com/download/faq/english/docs/VSS_Measurements/bpsk_berref.htm are calculated for each value specified in the meter's SWPTV parameter. The following BPSK modulation types are supported: COHERENT BPSK: [1] where Q(x) is the Gaussian integral or Q-function: and is approximated numerically, Eb is the average bit energy and N0 is the noise power spectral density. OPTIMUM DIFFERENTIAL BPSK: [2] SUBOPTIMUM DIFFERENTIAL BPSK: [2] where the ideal narrow-band IF filter has bandwidth W=0.57/T, where T is the bit (and symbol) duration. COHERENT DIFFERENTIALLY ENCODED BPSK: [3] References [1] Xiong, F., Digital Modulation Techniques, pg. 127 [2] Xiong, F., Digital Modulation Techniques, pg. 134 [3] Xiong, F., Digital Modulation Techniques, pg. 136 Prev Up Next Home Please send email to awr.support@ni.com if you would like to provide feedback on this article. Please make sure to include the article link in the email. Legal and Trademark Notice
AWGN (7 votes, average: 3.57 out of 5) Loading... In the previous article we saw about how Passband BPSK modulation and demodulation can be done. This concept is extended further to simulate the performance of BPSK modulation technique over an AWGN. Note that this http://www.gaussianwaves.com/2010/04/ber-vs-ebn0-for-bpsk-modulation-over-awgn-2/ is a baseband simulation of BPSK modulation and demodulation. Baseband simulation are faster and yields performace results same as that of pass band simulation. Transmitter: For the BPSK modulation , a series of binary input message bits are generated of which ‘1's are represented by 1v and ‘0's are translated as ‘-1' v (equivalent to NRZ coding as discussed in the previous post). AWGN channel: For BPSK modulation the channel can be modeled as $$ y=ax+n$$ error rate where y is the received signal at the input of the BPSK receiver, x is the 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 σ2. For AWGN the noise variance in terms of noise power spectral density (N0) is given by, $$\sigma^{2}= \frac{N_0}{2}$$ For M-ARY modulation schemes like bit error rate M-PSK including BPSK, the symbol energy is given by, $$ E_s = R_m R_c E_b$$ where Es =Symbol energy per modulated bit (x), Rm = log2(M) , (for BPSK M=2, QPSK M=4, 16 QAM M=16 etc..,). Rc 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 Es=1 for BPSK (Symbol energy normalized to 1) Eb/N0 can be represented as (using above equations), $$ \frac{E_b}{N_0}=\frac{E_s}{R_m R_c N_0}$$ $$ \frac{E_b}{N_0}=\frac{E_s}{R_m R_c N_0}=\frac{E_s}{R_m R_c 2 \sigma^{2}} = \frac{1}{2 R_m R_c \sigma^{2}}$$ From the above equation the noise variance for the given Eb/N0 can be calculated as $$ \sigma^{2} = \left( 2 R_m R_c \frac{E_b}{N_0}\right)^{-1}$$ For the channel model randn function in Matlab is used to generate the noise term. This function generates noise with unit variance and zero mean. In order to generate a noise with sigma σ for the given Eb/N0 ratio , use the above equation , find σ, multiply the ‘randn' generated noise with this sigma , add this final noise term with the transmitted signal to get the received signal. Receiver: BPSK receiver can be a simple threshold detector which categorizes the received signal as ‘0' or ‘1' depending on the threshold that is being set. Calculation of Theoretical BER