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README.md

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 # bpsk-ber [![License: MIT](https://img.shields.io/badge/License-MIT-blue.svg)](https://github.com/etfovac/bpsk-ber/blob/master/LICENSE) [![GitHub (pre-)release](https://img.shields.io/badge/release-1.0-yellow.svg)](https://github.com/etfovac/bpsk-ber/releases/tag/v1.0)
- bpsk-ber
+
+### Keywords:
+
+> BPSK,	Binary Phase Shift Keying
+
+> AWGN, Additive white Gaussian noise	 
+
+> SNR, Signal to Noise Ratio
+
+> BER,	Bit Error Rate, POE, Probability Of Error
+
+> Digital Signal processing
+
+
+## Basic Overview
+In digital phase modulation, the bits that need to be transmitted are coded in the carrier phase change.  
+The simplest phase modulation, called Binary Phase Shift Keying (BPSK), uses two phases to encode two binary digits.  
+
+The BPSK modulator is implemented using an input data set (0 and 1) which is sent to the input of the BPSK encoder which has assigned voltages of -1V and + 1V to this bit sequence. The encoder output is multiplied by the carrier cosine signal.
+The integrator works as a low-pass filter and removes harmonics caused by multiplying the received signal by the carrier signal. The output of the integrator is led to a threshold detector which at the output reconstructs through 0 and 1.
+If the signal strength is sufficiently greater than the noise power at the link line, this detected bit string will be identical to the one sent.  
+
+MATLAB simulation that mathematically models the process of determining BER performs the following:
+1. Creating BPSK symbols +1 and -1 from a randomly generated bit sequence (given length)
+2. Adding white (Gaussian) noise (for a given difference between signal level and noise)
+3. Detection of the received signal based on the reception threshold
+4. Counting errors and drawing BER graphics
+
+For the assessment of BER, BPSK coding in the basic frequency range was used, ie. modulation and demodulation were not simulated (moving the signal to a higher frequency, then back to baseband) due to the faster execution of the simulation and because the results are the same in both cases.
+
+
+
+
+### Flowchart
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_system.png" alt="bpsk_system">  
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_signal.png" alt="bpsk_signal">  
+
+bit 1: 	s(t) = A cos(2πfct)    = +A cos(2πfct)   
+bit 0: 	s(t) = A cos(2πfct+ π) = -A cos(2πfct)   
+
+s(t) =  A d(t) cos(2πfct)   
+
+r(t) = s(t) + n(t)   
+
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_snr_formula.png" alt="bpsk_snr_formula" width="150" height="50">  
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_ber_formula.png" alt="bpsk_ber_formula" width="150" height="50">
+
+### Results
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_system_fig1.png" alt="bpsk_system_fig1">  
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_system_fig2.png" alt="bpsk_system_fig2">  
+<img src="https://github.com/etfovac/bpsk-ber/blob/master/graphics/bpsk_ber_fig1.png" alt="bpsk_ber_fig1">  
+
+```  
+BPSK simulation  
+ BPSK - num of data bits to transfer through the channel: 1000000  
+ BER = -0.484644 za SNR = -10  
+ BER = -0.542406 za SNR = -8  
+ BER = -0.622411 za SNR = -6  
+ BER = -0.731174 za SNR = -4  
+ BER = -0.884889 za SNR = -2  
+ BER = -1.105961 za SNR = 0  
+ BER = -1.425159 za SNR = 2  
+ BER = -1.902257 za SNR = 4  
+ BER = -2.618704 za SNR = 6  
+ BER = -3.782516 za SNR = 8  
+ BER = -5.301030 za SNR = 10  
+``` 
+### Conclusion
+The probability of incorrect bit detection (BER) is practically lost for a large SNR and is of the order of 1/N (1 bit in the sequence).  
+All BER curves follow the theoretical BER curve with small deviations.  
+Therefore, the BER does not depend on the number of bits transmitted.  
+The minimum SNR for which BPSK has 1 bit error depends on the total transmitted bits, so that for a larger number of transmitted bits, the SNR must be higher for the signal to be correctly reconstructed.  
+For example. when transmitting 100 bits, it is enough for the SNR to exceed 4 dB so that there is no transmission error, while for the transmission of 100,000 bits, the SNR must exceed 8 dB.