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.

Note:

- 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.

There are several methods to model the amplitude and phase of each of these components. We look at one method called the “Smiths Fading Simulator” which is based on Clark and Gans model. The simulator can be constructed using the following steps.

1. Define N the number of Gaussian RVs to be generated, fm the Doppler frequency in Hz, fs the sampling frequency in Hz, df the frequency spacing which is calculated as df=(2*fm)/(N-1) and M total number of samples in frequency domain which is calculated as M=(fs/df).

2. Generate two sequences of N/2 complex Gaussian random variables. These correspond to the frequency bins up to fm. Take the complex conjugate of these sequences to generate the N/2 complex Gaussian random variables for the negative frequency bins up to -fm.

3. Multiply the above complex Gaussian sequences g1 and g2 with square root of the Doppler Spectrum S generated from -fm to fm. Calculate the spectrum at -fm and +fm by using linear extrapolation.

4. Extend the above generated spectra from -fs/2 to +fs/2 by stuffing zeros from -fs/2 to -fm and fm to fs/2. Take the IFFT of the resulting spectra X and Y resulting in time domain signals x and y.

5. Add the absolute values of the resulting signals x and y in quadrature. Take the absolute value of this complex signal. This is the desired Rayleigh distributed envelope with the required temporal correlation.

The Matlab code for generating Rayleigh random sequence with a Doppler frequency of fm Hz is given below.

]]>Implementation on Trinket

Implementation on REPL

Windows 8.1 Pro

**System**

Processor Intel(R) Core(TM) i7-5500U CPU @ 2.4GHz

Installed Memory 8.00 GB

System Type 64 Bit Operating System, x64 Based Processor

**Integrated Development Environment (IDE)**

Enthought Canopy

Version 2.1.3.3542 (32 bit)

Operation |
Time in sec (MATLAB) |
Time in sec (PYTHON) |

10 million uniform random variable generation | 0.10 | 0.15 |

10 million normal random variable generation | 0.13 | 0.40 |

for loop counting up to 100 million | 0.40 | 11.60 |

Comparing two vectors of length 10 million each | 0.39 | 0.55 |

Plotting a histogram of 10 million values | 0.89 | 0.76 |

Plotting a scatter plot of 1 million values | 0.30 | 0.23 |

Bit error rate calculation of BPSK for 10 values of SNR | 2.49 | 4.51 |

Although Python is a bit slower than MATLAB for most of the cases but the real difference is in implementation of “for loop” where the speed of MATLAB is 29x that of Python. Another surprising result was that the plot functions for Python were somewhat faster than MATLAB.

]]>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()

]]>

The electromagnetic field in the shadowed region can be calculated by combining vectorially the contributions of all of these secondary sources, which is not an easy task. Secondly, the geometry is usually much more complicated than shown in the above figure. For example consider a telecom tower transmitting electromagnetic waves from a rooftop and a pedestrian using a mobile phone at street level. The EM waves usually reach the receiver at street level after more than one diffraction (not to mention multiple reflections). However, an approximation that works well in most cases is called knife edge diffraction, which assumes a single sharp edge (an edge with a thickness much smaller than the wavelength) separates the transmitter and receiver.

The path loss due to diffraction in the knife edge model is controlled by the Fresnel Diffraction Parameter which measures how deep the receiver is within the shadowed region. A negative value for the parameter shows that the obstruction is below the line of sight and if the value is below -1 there is hardly any loss. A value of 0 (zero) means that the transmitter, receiver and tip of the obstruction are all in line and the Electric Field Strength is reduced by half or the power is reduced to one fourth of the value without the obstruction i.e. a loss of 6dB. As the value of the Fresnel Diffraction Parameter increases on the positive side the path loss rapidly increases reaching a value of 27 dB for a parameter value of 5. Sometimes the exact calculation is not needed and only an approximate calculation, as proposed by Lee in 1985, is sufficient.

Fresnel Diffraction Parameter (v) is defined as:

v=h*sqrt(2*(d1+d2)/(lambda*d1*d2))

where

d1 is the distance between the transmitter and the obstruction along the line of sight

d2 is the distance between the receiver and the obstruction along the line of sight

h is the height of the obstruction above the line of sight

and lambda is the wavelength

The MATLAB codes used to generate the above plots are given below (approximate method followed by the exact method). Feel free to use them in your simulations and if you have a question drop us a comment.

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculation of the path loss based on the value of
% Fresnel Diffraction Parameter as proposed by Lee
% Lee W C Y Mobile Communications Engineering 1985
% Copyright www.raymaps.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
v=-5:0.01:5;
for n=1:length(v)
if v(n) <= -1
G(n)=0;
elseif v(n) <= 0
G(n)=20*log10(0.5-0.62*v(n));
elseif v(n) <= 1
G(n)=20*log10(0.5*exp(-0.95*v(n)));
elseif v(n) <= 2.4
G(n)=20*log10(0.4-sqrt(0.1184-(0.38-0.1*v(n))^2));
else
G(n)=20*log10(0.225/v(n));
end
end
plot(v, G, 'b')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')
```

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exact calculation of the path loss (in dB)
% based on Fresnel Diffraction Parameter (v)
% T S Rappaport Wireless Communications P&P
% Copyright www.raymaps.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
v=-5:0.01:5;
for n=1:length(v)
v_vector=v(n):0.01:v(n)+100;
F(n)=((1+1i)/2)*sum(exp((-1i*pi*(v_vector).^2)/2));
end
F=abs(F)/(abs(F(1)));
plot(v, 20*log10(F),'r')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')
```

An interesting video explaining the phenomenon of diffraction.

]]>For this we consider that in the case of specular reflection the power of the ray is reduced by 3dB (that is R=0.5) and the ray continues as it would if there was no interaction with the object (off course direction would be changed with angle of reflection being equal to angle of incidence).

In the case of scattered ray we assume that the point of interaction between the ray and the object is a point source from where the rays are regenerated. Therefore there is a rapid drop in the E-field strength as the ray propagates away from the point of interaction.

```
clear all
close all
Eo=1;
r1=1:100;
r2=101:200;
r=[r1 r2];
R=0.5;
Er1=Eo./r1;
Er2=R*Eo./r2;
Er=[Er1 Er2];
Es1=Eo./r1;
Es2=Es1(end)./(r2-r1(end));
Es=[Es1 Es2];
plot(r,10*log10(Er),'b')
hold on
plot(r,10*log10(Es),'r:')
hold off
legend('Reflection','Scattering')
xlabel('Distance (m)')
ylabel('E-field (V/m)')
```

The simulation code above considers that a ray travels unobstructed for 100 m and at this point comes in contact with an object and is reflected (reflection coefficient of 0.5 is assumed) or scattered. The ray then again travels for another 100 m without coming in contact with an object.

It is observed that there is a rapid decrease in E-field strength of the scattered ray beyond 100 m. This is because the E-field strength at the point of interaction is much lower than that at the source and when this is considered to be a point source, re-generating the rays, the resulting E-field decays quite rapidly.

We assume that the reflection is specular in most of our simulations as this is closer to reality than assuming a fully scattered ray.

]]>