Carrier phase or frequency synchronization is a common problem in wireless communication systems. These two problems are interrelated as instantaneous frequency is just the rate of change of phase. The problem of carrier frequency offset might appear due to one of two reasons. Either the oscillators at the transmitter and receiver are not aligned in the frequency domain or there is a Doppler shift introduced by the channel (remember that a moving object in the wireless environment introduces a Doppler shift). In the case of the former the frequency misalignment is given in parts per million (ppm). A typical value for commercially available oscillators is ±20 ppm. Assuming that there is maximum frequency error at both the transmitter and receiver the error increases to ±40 ppm. At 1GHz this translates to 40*1,000,000,000/1,000,000 = 40kHz.Continue reading Modeling Phase and Frequency Synchronization Error
As the data rates supported by wireless networks continue to rise the bandwidth requirements also continue to increase (although spectral efficiency has also improved). Remember GSM technology which supported 125 channels of 200KHz each, which was further divided among eight users using TDMA. Move on to LTE where the channel bandwidth could be as high as 20MHz (1.4MHz, 3MHz, 5MHz, 10MHz, 15MHz and 20MHz are standardized).
This advancement poses a unique challenge referred to as frequency selective fading. This means that different parts of the signal spectrum would see a different channel (different amplitude and different phase offset). Look at this in the time domain where the larger bandwidth means shorter symbol period causing intersymbol interference (as time delayed copies of the signal overlap on arrival at the receiver).
The solution to this problem is OFDM that divides the wideband signal into smaller components each having a bandwidth of a few KHz. Each of these components experiences a flat channel. To make the task of equalization simple a cyclic prefix (CP) is added in the time domain to make the effect of fading channel appear as circular convolution. Thus simplifying the frequency domain equalization to a simple division operation.
Shown below is the Python code that calculates the bit error rate (BER) of BPSK-OFDM which is the same as simple BPSK in a Rayleigh flat fading channel. However there is a caveat. We have inserted a CP which means we are transmitting more energy than simple BPSK. To be exact we are transmitting 1.25 (160/128) times more energy. This means that if this excess energy is accounted for the performance of BPSK-OFDM would be 1dB (10*log10(1.25)) worse than simple BPSK in Rayleigh flat fading channel.
- Although we have shown the channel as a multiplicative effect in the figure above, this is only true for a single tap channel. For a multi-tap channel (such as the one used in the code above) the effect of the channel is that of a filter which performs convolution operation on the transmitted signal.
- We have used a baseband model in our simulation and the accompanying figure. In reality the transmitted signal is upconverted before transmission by the antennas.
- The above model can be easily modified for any modulation scheme such as QPSK or 16-QAM. The main difference would be that the signal would have a both a real part and an imaginary part, much of the simulation would remain the same. This would be the subject of a future post. For a MATLAB implementation of 64-QAM OFDM see the following post (64-QAM OFDM).
- Serial to parallel and parallel to serial conversion shown in the above figure was not required as the simulation was done symbol by symbol (one OFDM symbol in the time domain represented 128 BPSK symbols in the frequency domain).
- The channel model in the above simulation is quasi-static i.e. it remains constant for one OFDM symbol but then rapidly changes for the next, without any memory.
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.
Here is a piece of Python code that calculates Bit Error Rate (BER) of BPSK. The code is a bit slow at the moment, compared to MATLAB implementation, but this is work in progress and further optimizations would be carried out. We would like to point out that the main reason for this slower implementation is that a bit by bit error calculation is done, instead of a vectorial implementation. We already pointed out in our previous post that a “for loop” implemented in Python is not that efficient.
Have you ever thought about how life would be without MATLAB. As it turns out there are free and open source options such as Python. We have so far restricted ourselves to MATLAB in this blog but now we venture out to find out what are the other options. Given below is a most basic Python code that calculates the Bit Error Rate of Binary Phase Shift Keying (BPSK). Compare this to our MATLAB implementation earlier [BPSK BER].
There are various IDEs available for writing your code but I have used Enthought Canopy Editor (32 bit) which is free to download and is also quite easy to use [download here]. So as it turns out that there is life beyond MATLAB. In fact there are several advantages of using Python over MATLAB which we will discuss later in another post. Lastly please note the indentation in the code below as there is no “end” statement in a “for loop” in Python.
from numpy import sqrt from numpy.random import rand, randn import matplotlib.pyplot as plt N = 5000000 EbNodB_range = range(0,11) itr = len(EbNodB_range) ber = [None]*itr for n in range (0, itr): EbNodB = EbNodB_range[n] EbNo=10.0**(EbNodB/10.0) x = 2 * (rand(N) >= 0.5) - 1 noise_std = 1/sqrt(2*EbNo) y = x + noise_std * randn(N) y_d = 2 * (y >= 0) - 1 errors = (x != y_d).sum() ber[n] = 1.0 * errors / N print "EbNodB:", EbNodB print "Error bits:", errors print "Error probability:", ber[n] plt.plot(EbNodB_range, ber, 'bo', EbNodB_range, ber, 'k') plt.axis([0, 10, 1e-6, 0.1]) plt.xscale('linear') plt.yscale('log') plt.xlabel('EbNo(dB)') plt.ylabel('BER') plt.grid(True) plt.title('BPSK Modulation') plt.show()