Fast Fourier Transform or FFT is a powerful tool to visualize a signal in the frequency domain. Shown below is the FFT of a signal (press the play button) composed of four sinusoids at frequencies of 50Hz, 100Hz, 200Hz and 400Hz. The sampling frequency is set at 1000Hz, more than twice the maximum frequency of the composite signal. The amplitude of the sinusoids diminishes with increasing frequency and this is also reflected in the frequency domain. Play with the code, decrease the frequencies, increase the power, visualize the FFT output on logarithmic scale!
We have already seen in previous posts that the BER of BPSK increases significantly when the channel changes from a simple AWGN channel to a fading channel. One solution to this problem, that was proposed by Alamouti, was to use Transmit Diversity i.e. multiple transmit antennas transmit the information over multiple time slots increasing the likelihood of receiving the information. We have considered the simplest case of two transmit antennas and BPSK modulation (QPSK modulation would give the same BER with twice the throughput). Given below is the Python code for this, feel free to modify it and run it from the console given below.
Implementation on Trinket
Implementation on REPL
We have previously calculated the bit error rate of BPSK in an AWGN channel, we now do the same for a Rayleigh fading channel. Remember that we have now shifted our focus from MATLAB to Python since its open and free to use. We are currently using Python-2 but intend to Python-3 once some integration issues with Trinket are sorted out.