# BER for BPSK-OFDM in Frequency Selective Channel

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:

1. 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.
2. 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.
3.  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).
4. 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).
5. 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.

# Sizing Up a Solar System for a Cellular Base Station

Many operators are thinking of moving from the main grid to alternative energy sources such as wind and solar. This is especially true in third world countries where electricity is not available 24/7 and is also very expensive. This has forced operators to switch their base stations to diesel generators (which is also a costly option).

In this article we do a rough estimation of the size a solar system required to run a cellular base station. We start with the assumption that 20 Watts of power are transmitted from a single antenna of base station. For a 3 sector site there are 3 antennas giving us total transmitted power of 60 Watts. Now if 50% of the power is lost in cables and connections we would have to boost up the transmitted power to 120 Watts.

We know that power amplifiers are highly in-efficient (depending upon the load) and a large amount of power is lost in this stage. So we assume an efficiency of 12 % giving us a total input power of 1000 Watts. Another 500 Watts are given to Air Conditioning (200 W), Signal Processing (150 W) and Rectifier (150 W). So the combined AC input to the base station is 1500 Watts. Now we turn our attention to sizing up the solar system.

If we assume that the BS is continuously consuming 1500 Watts over a 24 hour period we have a total energy consumption of  36 kWh. If the solar panels receive peak sun hours of 5 hours/day we would require solar panels rated at 7200 Watts. This could mean 72 solar panels of 100 Watts each or 36 solar panels of 200 Watts each or any other combination. It must be noted that we have not considered any margins for cloudy days when peak sun hours would be reduced. Also, we have not considered any reduction in power consumption when there is no load (or very less load) on the BS.

Next we calculate the amount of batteries required. We assume that the batteries are rated at 200 AH and 12 V. This gives us a total energy storage capacity per battery of 2.4 kWh. So the number of batteries required is calculated as 36 kWh/2.4 kWh = 15. It must be noted that some of the energy would be consumed in real-time and the actual number of batteries required would be lesser. Furthermore we would need an inverter of at least 1500 Watts and charge controller of 125 Amps.

# M-QAM Bit Error Rate in Rayleigh Fading

We have previously discussed the bit error rate (BER) performance of M-QAM in AWGN. We now discuss the BER performance of M-QAM in Rayleigh fading. The one-tap Rayleigh fading channel is generated from two orthogonal Gaussian random variables with variance of 0.5 each. The complex random channel coefficient so generated has an amplitude which is Rayleigh distributed and a phase which is uniformly distributed. As usual the fading channel introduces a multiplicative effect whereas the AWGN is additive.

The function “QAM_fading” has three inputs, ‘n_bits’, ‘M’, ‘EbNodB’ and one output ‘ber’. The inputs are the number of bits to be passed through the channel, the alphabet size and the Energy per Bit to Noise Power Spectral Density in dB respectively whereas the output is the bit error rate (BER).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FUNCTION THAT CALCULATES THE BER OF M-QAM IN RAYLEIGH FADING
% n_bits: Input, number of bits
% M: Input, constellation size
% EbNodB: Input, energy per bit to noise power spectral density
% ber: Output, bit error rate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Transmitter
k=log2(M);
EbNo=10^(EbNodB/10);
x=transpose(round(rand(1,n_bits)));
h1=modem.qammod(M);
h1.inputtype='bit';
h1.symbolorder='gray';
y=modulate(h1,x);

% Channel
Eb=mean((abs(y)).^2)/k;
sigma=sqrt(Eb/(2*EbNo));
w=sigma*(randn(n_bits/k,1)+1i*randn(n_bits/k,1));
h=(1/sqrt(2))*(randn(n_bits/k,1)+1i*randn(n_bits/k,1));
r=h.*y+w;

r=r./h;
h2=modem.qamdemod(M);
h2.outputtype='bit';
h2.symbolorder='gray';
h2.decisiontype='hard decision';
z=demodulate(h2,r);
ber=(n_bits-sum(x==z))/n_bits
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The bit error rates of four modulation schemes 4-QAM, 16-QAM, 64-QAM and 256-QAM are shown in the figure above. All modulation schemes use Gray coding which gives a few dB of margin in the BER performance. As with the AWGN case each additional bit per symbol requires about 1.5-2 dB in signal to ratio to achieve the same BER.

Although not shown here similar behavior is observed for higher order modulation schemes such as 1024-QAM and 4096-QAM (the gap in the signal to noise ratio for the same BER is increased to about 5dB).

Lastly we explain some of the terms used above.

Rayleigh Fading is a commonly used term in simulation of Digital Communication Systems but it tends to differ in meaning in different contexts. The term Rayleigh Fading as used above means a single tap channel that varies from one symbol to the next. It has an amplitude which is Rayleigh distributed and a phase which is Uniformly distributed. A single tap channel means that it does not introduce any Inter Symbol Interference (ISI). Such a channel is also referred to as a Flat Fading Channel. The channel can also be referred to as a Fast Fading Channel since each symbol experiences a new channel state which is independent of its previous state (also termed as uncorrelated).

Gray Coding

When using QAM modulation, each QAM symbol represents 2,3,4 or higher number of bits. That means that when a symbol error occurs a number of bits are reversed. Now a good way to do the bit-to-symbol assignment is to do it in a way such that no neighboring symbols differ by more than one bit e.g. in 16-QAM, a symbol that represents a binary word 1101 is surrounded by four symbols representing 0101, 1100, 1001 and 1111. So if a symbol error is made, only one bit would be in error. However, one must note that this is true only in good signal conditions. When the SNR is low (noise has a higher magnitude) the symbol might be displaced to a location that is not adjacent and we might get higher number of bits in error.

Hard Decision

The concept of hard decision decoding is important when talking about channel coding, which we have not used in the above simulation. However, we will briefly explain it here. Hard decision is based on what is called “Hamming Distance” whereas soft decision is based on what it called “Euclidean Distance”. Hamming Distance is the distance of a code word in binary form, such as 011 differs from 010 and 001 by 1. Whereas the Euclidean distance is the distance before a decision is made that a bit is zero or one.  So if the received sequence is 0.1 0.6 0.7 we get a Euclidean distance of 0.8124 from 010 and 0.6782 from 001. So we cannot make a hard decision about which sequence was transmitted based on the received sequence of 011. But based on the soft metrics we can make a decision that 001 was the most likely sequence that was transmitted (assuming that 010 and 001 were the only possible transmitted sequences).

# M-QAM Bit Error Rate in AWGN

Quadrature Amplitude Modulation has been adopted by most wireless communication standards such as WiMAX and LTE. It provides higher bit rates and consequently higher spectral efficiencies. It is usually used in conjunction with Orthogonal Frequency Division Multiplexing (OFDM) which provides a simple technique to overcome the time varying frequency selective channel.

We have previously discussed the formula for calculating the bit error rate (BER) of QAM in AWGN. We now calculate the same using a simple Monte Carlo Simulation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FUNCTION THAT CALCULATES THE BER OF M-QAM IN AWGN
% n_bits: Input, number of bits
% M: Input, constellation size
% EbNodB: Input, energy per bit to noise power spectral density
% ber: Output, bit error rate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[ber]= QAM_AWGN(n_bits, M, EbNodB)

% Transmitter
k=log2(M);
EbNo=10^(EbNodB/10);
x=transpose(round(rand(1,n_bits)));
h1=modem.qammod(M);
h1.inputtype='bit';
h1.symbolorder='gray';
y=modulate(h1,x);

% Channel
Eb=mean((abs(y)).^2)/k;
sigma=sqrt(Eb/(2*EbNo));
w=sigma*(randn(1,n_bits/k)+1i*randn(1,n_bits/k));
r=y+w';

h2=modem.qamdemod(M);
h2.outputtype='bit';
h2.symbolorder='gray';
h2.decisiontype='hard decision';
z=demodulate(h2,r);
ber=(n_bits-sum(x==z))/n_bits
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The above function basically has three inputs and one output. The inputs are the number of bits to be passed through the channel, the size of the constellation and the signal to noise ratio in dB. The output is the bit error rate (BER). The simulation can be divided into three section namely the transmitter, the channel and the receiver. In this simulation we have used Gray coding which gives us about a dB of improvement at low to medium signal to noise ratio.

As seen above the BER obtained through our simulation matches quite well with the BER obtained through the theoretical formula. Each additional bit per symbol required about 2dB extra in signal to noise ratio to achieve the same bit error rate.

Lastly we explain some of the terms used above.

AWGN

All wireless receivers suffer from thermal noise which is a function of absolute temperature and bandwidth of the receiver. This noise is added to the received signal and makes detection of weak signals a major challenge. Just to given you an idea typical GSM receivers have a noise floor of -113 dBm. Therefore, if the received signal has a power of -100 dBm we get a signal to noise ratio (SNR) of 13 dB. In simulation this noise is usually modeled as a Gaussian Random Process. It is additive, as opposed to channel impairments which are multiplicative and has a flat spectrum (thus called White Noise).

Gray Coding

When using QAM modulation, each QAM symbol represents 2,3,4 or higher number of bits. That means that when a symbol error occurs a number of bits are reversed. Now a good way to do the bit-to-symbol assignment is to do it in a way such that no neighboring symbols differ by more than one bit e.g. in 16-QAM, a symbol that represents a binary word 1101 is surrounded by four symbols representing 0101, 1100, 1001 and 1111. So if a symbol error is made, only one bit would be in error. However, one must note that this is true only in good signal conditions. When the SNR is low (noise has a higher magnitude) the symbol might be displaced to a location that is not adjacent and we might get higher number of bits in error.

Hard Decision

The concept of hard decision decoding is important when talking about channel coding, which we have not used in the above simulation. However, we will briefly explain it here. Hard decision is based on what is called “Hamming Distance” whereas soft decision is based on what it called “Euclidean Distance”. Hamming Distance is the distance of a code word in binary form, such as 011 differs from 010 and 001 by 1. Whereas the Euclidean distance is the distance before a decision is made that a bit is zero or one.  So if the received sequence is 0.1 0.6 0.7 we get a Euclidean distance of 0.8124 from 010 and 0.6782 from 001. So we cannot make a hard decision about which sequence was transmitted based on the received sequence of 011. But based on the soft metrics we can make a decision that 001 was the most likely sequence that was transmitted (assuming that 010 and 001 were the only possible transmitted sequences).

# Antennas on Samsung Galaxy S

We have previously discussed the theory of Planar Inverted F Antennas (PIFA), now let us look at a practical example. Shown below is the rear view of a Samsung Galaxy S phone with six antennas. The description of these antennas is given below.

1. 2.6 GHz WiMAX Tx/Rx Antenna

2. 2.6 GHz WiMAX Antenna Rx Only (as a diversity antenna)

3. WiFi/Bluetooth Tx/Rx Antenna

4. Cell/PCS CDMA/EVDO Tx/Rx Antenna

5. Cell/PCS CDMA/EVDO Rx Only (as a diversity antenna)

6. GPS Antenna Rx Only

The figure above shows the top conducting plane of the PIFAs. The bottom conducting plane (ground plane) is one large plane that extends throughout the length and breadth of the phone.

# Planar Inverted F Antenna (PIFA)

A Planar Inverted F Antenna or PIFA is a very common antenna type being used in cell phones. In fact a cell phone would have multiple PIFAs for LTE, WiMAX, WiFi, GPS etc. Furthermore, there would be multiple PIFAs for diversity reception and transmission. A PIFA is composed of 5 basic elements.

1. A large metallic ground plane

2. A resonating metallic plane

3. A substrate separating the two planes

4. A shorting pin (or plane)

5. A feeding mechanism

The resonant frequency of the PIFA can be calculated from the relationship between the wavelength of the antenna and the dimensions of the antenna. The relationship is given as:

L+W1-W2g/4

It must be remembered that the wavelength here is the guided wavelength which is given as  λgo/√εr. Here εr is the relative permittivity of the substrate and λo is the wavelength in free space. There exist two special cases of the above relationship. First is the case where the shorting plane has width W1. In this case the above relationship is reduced to:

L=λg/4

In the second case the width of the shorting plane is reduced to zero i.e. the shorting plane is actually a shorting pin. In this case the relationship is reduced to:

L+W1g/4

In cell phones with multiple PIFAs the ground plane is actually one large ground plane for all the resonating surfaces and may include the body of the cell phone as well. Lastly, the input impedance of the PIFA is controlled by changing the distance of the feeding pin from the shorting plane. The impedance is zero at the shorting plane and is maximum at the other end (away from the shorting plane).

# QAM Theoretical BER in AWGN

Quadrature Amplitude Modulation (QAM) is an important modulation scheme as it allows for higher data rates and spectral efficiencies. The bit error rate (BER) of QAM can be calculated through Monte Carlo simulations. However this becomes quite complex as the constellation size of the modulation schemes increases. Therefore a theoretical approach is sometimes preferred. The BER for Gray coded QAM, for even number of bits per symbol, is shown below.

Gray coding ensures that a symbol error results in a single bit error. The code for calculating the theoretical QAM BER for k even (even number of bits per symbol) is given below. The formula for calculating the BER for k odd is different, however, the formula given below can be used a first estimate.

EbNodB=-6:2:24
EbNo=10.^(EbNodB/10);
k=8;
M=2^k;
x=sqrt(3*k*EbNo/(M-1));
Pb=(4/k)*(1-1/sqrt(M))*(1/2)*erfc(x/sqrt(2));
semilogy(EbNodB,Pb)

Note:
1. Each additional bit/symbol requires about 2dB extra in SNR to achieve the same BER.
2. 4-QAM is essentially QPSK modulation.

# MIMO Capacity in a Fading Environment

The Shannon Capacity of a channel is the data rate that can be achieved over a given bandwidth (BW) and at a particular signal to noise ratio (SNR) with diminishing bit error rate (BER). This has been discussed in an earlier post for the case of SISO channel and additive white Gaussian noise (AWGN). For a MIMO fading channel the capacity with channel not known to the transmitter is given as (both sides have been normalized by the bandwidth [1]):

Shannon Capacity of a MIMO Channel

where NT is the number of transmit antennas, NR is the number of receive antennas, γ is the signal to interference plus noise ratio (SINR), INR is the NRxNR identity matrix and H is the NRxNT channel matrix. Furthermore, hij, an element of the matrix H defines the complex channel coefficient between the ith receive antenna and jth transmit antenna. It is quite obvious that the channel capacity (in bits/sec/Hz) is highly dependent on the structure of matrix H. Let us explore the effect of H on the channel capacity.

Let us first consider a 4×4 case (NT=4, NR=4) where the channel is a simple AWGN channel and there is no fading. For this case hij=1 for all values of i and j. It is found that channel capacity of this simple channel for an SINR of 10 dB is 5.36bits/sec/Hz. It is further observed that the channel capacity does not change with number of transmit antennas and increases logarithmically with increase in number of receive antennas. Thus it can be concluded that in an AWGN channel no multiplexing gain is obtained by increasing the number of transmit antennas.

We next consider a more realistic scenario where the channel coefficients hij are complex with real and imaginary parts having a Gaussian distribution with zero mean and variance 0.5. Since the channel H is random the capacity is also a random variable with a certain distribution. An important metric to quantify the capacity of such a channel is the Complimentary Cumulative Distribution Function (CCDF). This curve basically gives the probability that the MIMO capacity is above a certain threshold.

It is obvious (see figure above) that there is a very high probability that the capacity obtained for the MIMO channel is significantly higher than that obtained for an AWGN channel e.g. for an SINR of 9 dB there is 90% probability that the capacity is greater than 8 bps/Hz. Similarly for an SINR of 12 dB there is a 90% probability that the capacity is greater than 11 bps/Hz. For a stricter threshold of 99% the above capacities are reduced to 7.2 bps/Hz and 9.6 bps/Hz.

In a practical system the channel coefficients hij would have some correlation which would depend upon the antenna spacing. Lower the antenna spacing higher would be the antenna correlation and lower would be the MIMO system capacity. This would be discussed in a future post.

The MATLAB code for calculating the CCDF of channel capacity of a MIMO channel is given below.

clear all
close all

Nr=4;
Nt=4;
I=eye(Nr);
g=15.8489;

for n=1:10000
H=sqrt(1/2)*randn(Nr,Nt)+j*sqrt(1/2)*randn(Nr,Nt);
C(n)=log2(det(I+(g/Nt)*(H*H')));
end

[a,b]=hist(real(C),100);
a=a/sum(a);
plot(b,1-cumsum(a));
xlabel('Capacity (bps/Hz)')
ylabel('Probability (Capacity > Abcissa)')
grid on

[1] G. J. Foschini and M. J. Gans,”On limits of Wireless Communications in a Fading Environment when Using Multiple Antennas”, Wireless Personal Communications 6, pp 311-335, 1998.

# CDMA vs OFDMA

 Property CDMA OFDMA 1. Channel bandwidth Full system bandwidth Variable system bandwidth to accommodate users with different data rates, 1.25, 2.50, 5.00, 10.00, 15.00 and 20.00 MHz, actual transmission bandwidth is a bit lower than this 2. Frequency-selective scheduling Not possible A key advantage of OFDMA, although it requires accurate real-time feedback of channel conditions from receiver to transmitter 3. Symbol period Very short—inverse of the system bandwidth Very long—defined by subcarrier spacing and independent of system bandwidth 4. Equalization Complicated time domain equalization Simple frequency domain equalization 5. Resistance to mulitpath Rake receiver can combine various multipath components Highly resistant to multipath due to insertion of cyclic prefix (CP) 6. Suitability for MIMO MIMO is not suited to a wideband frequency selective channel MIMO is suited to the independent narrowband flat fading channels that the subcarriers provide 7. Resistance to narrowband interference Resistant to narrow band interference Some subcarriers to be affected by narrowband interference 8. Separation of users Scrambling and orthogonal spreading codes Frequency and time although scrambling and spreading can be added as well

Reference: Agilent 3GPP Long Term Evolution System Overview, Product Development and Test Challenges Application Note.

# A Rayleigh Fading Simulator with Temporal and Spatial Correlation

Just to recap, building an LTE fading simulator with the desired temporal and spatial correlation is a three step procedure.

1. Generate Rayleigh fading sequences using Smith’s method which is based on Clarke and Gan’s fading model.

2. Introduce spatial correlation based upon the spatial correlation matrices defined in 3GPP 36.101.

3. Use these spatially and temporally correlated sequences as the filter taps for the LTE channel models.

We have already discussed step 1 and 3 in our previous posts. We now focus on step 2, generating spatially correlated channels coefficients.

3GPP has defined spatial correlation matrices for the Node-B and the UE. These are defined for 1,2 and 4 transmit and receive antennas. These are reproduced below.

Spatial Correlation Matrix

The parameters ‘alpha’ and ‘beta’ are defined as:

Low Correlation
alpha=0, beta=0

Medium Correlation
alpha=0.3, beta=0.9

High Correlation
alpha=0.9, beta=0.9

The combined effect of antenna correlation at the transmitter and receiver is obtained by taking the Kronecker product of individual correlation matrices e.g. for a 2×2 case the correlation matrix is given as:

Correlation Matrix for 2×2 MIMO

Multiplying the square root of the correlation matrix with the vector of channel coefficients is equivalent to taking a weighted average e.g. for the channel between transmit antenna 1 and receive antenna 1 the correlated channel coefficient h11corr is given as:

h11corr=w1*h11+w2*h12+w3*h21+w4*h22

where w1=1 and w2, w3 and w4 are less than one and greater than zero. For the high correlation case described above the channel coefficient is calculated as:

h11corr=0.7179*h11+0.4500*h12+0.4500*h21+0.2821*h22

From a practical point of antenna correlation is dependent on the antenna separation. Greater the antenna spacing lower is the antenna correlation and better the system performance. However, a base station requires much higher antenna spacing than a UE to achieve the same level of antenna correlation. This is due to the fact the base station antennas are placed much higher than a UE. Therefore the signals arriving at the base station are usually confined to smaller angles and experience similar fading. A UE on the other hand has a lot of obstacles in the surrounding areas which results in higher angle spread and uncorrelated fading between the different paths.