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.

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+W_{1}-W_{2}=λ_{g}/4

It must be remembered that the wavelength here is the guided wavelength which is given as λ_{g}=λ_{o}/√ε_{r}. Here ε_{r} is the relative permittivity of the substrate and λ_{o} 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 W_{1}. In this case the above relationship is reduced to:

L=λ_{g}/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+W_{1}=λ_{g}/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).

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.

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.

We had previously presented a method of generating a temporally correlated Rayleigh fading sequence. This was based on Smith’s fading simulator which was based on Clark and Gan’s fading model. We now present a highly efficient method of generating a correlated Rayleigh fading sequence, which has been adapted from Young and Beaulieu’s technique [1]. The architecture of this fading simulator is shown below.

This method essentially involves five steps.

1. Generate two Gaussian random sequences of length N each.
2. Multiply these sequences by the square root of Doppler Spectrum S=1.5./(pi*fm*sqrt(1-(f/fm).^2).
3. Add the two sequences in quadrature with each other to generate a length N complex sequence (we have added the two sequences before multiplying with the square root of Doppler Spectrum in our simulation).
4. Take the M point complex inverse DFT where M=(fs/Δf)+1.
5. The absolute value of the resulting sequence defines the envelope of the Rayleigh faded signal with the desired temporal correlation (based upon the Doppler frequency fm).

A point to be noted here is that although the Doppler spectrum is defined from -fm to +fm the IDFT has to be taken from -fs/2 to +fs/2. This is achieved by stuffing zeros in the vacant frequency bins from -fs/2 to +fs/2. The MATLAB code for this simulator is given below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RAYLEIGH FADING SIMULATOR BASED UPON YOUNG'S METHOD
% N is the number of paths
% M is the total number of points in the frequency domain
% fm is the Doppler frequency in Hz
% fs is the sampling frequency in Hz
% df is the step size in the frequency domain
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
N=64;
fm=70;
df=(2*fm)/(N-1);
fs=7.68e6;
M=round(fs/df);
T=1/df;
Ts=1/fs;
% Generate 2xN IID zero mean Gaussian variates
g1=randn(1,N);
g2=randn(1,N);
g=g1-j*g2;
% Generate Doppler Spectrum
f=-fm:df:fm;
S=1.5./(pi*fm*sqrt(1-(f/fm).^2));
S(1)=2*S(2)-S(3);
S(end)=2*S(end-1)-S(end-2);
% Multiply square root of Doppler Spectrum with Gaussian random sequence
X=g.*sqrt(S);
% Take IFFT
F_zero=zeros(1, round((M-N)/2));
X=[F_zero, X, F_zero];
x=ifft(X,M);
r=abs(x);
r=r/mean(r);
% Plot the Rayleigh envelope
t=0:Ts:T-Ts;
plot(t,10*log10(r))
xlabel('Time(sec)')
ylabel('Signal Amplitude (dB)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The question now is that how do we verify that the generated Rayleigh fading sequence has the desired statistical properties. This can be verified by looking at the level crossing rate (LCR) and average fade duration (AFD) of the generated sequence as well as the PDF and Autocorrelation function. The LCR and AFD calculated for N=64 and fm=70 Hz and threshold of -10 dB (relative to the average signal power) is given below.

LCR
Simulation: 15.55
Theoretical: 15.46

AFD
Simulation: 453 usec
Theoretical: 508 usec

It is observed that the theoretical and simulation results for the LCR and AFD match reasonably well. We next examine the distribution of the envelope and phase of the resulting sequence x. It is found that the envelope of x is Rayleigh distributed while the phase is uniformly distributed from -pi to pi. This is shown in the figure below. So we are reasonably satisfied that our generated sequence has the desired statistical properties.

Rayleigh Fading Simulator Based on Young’s Filter

In the above simulation of Rayleigh fading sequence we reduced the computation load of Smith’s simulator by reducing the IFFT operations on two branches to a single IFFT operation. However, we still used the Doppler spectrum proposed by Smith. Now we use the filter with spectrum F_{k} defined by Young in [1]. The code for this is given below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RAYLEIGH FADING SIMULATOR BASED UPON YOUNG'S METHOD
% N is the number of points in the frequency domain
% fm is the Doppler frequency in Hz
% fs is the sampling frequency in Hz
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
N=2^20;
fm=300;
fs=7.68e6;
Ts=1/fs;
% Generate 2xN IID zero mean Gaussian variates
g1=randn(1,N);
g2=randn(1,N);
g=g1-j*g2;
% Generate filter F
F = zeros(1,N);
dopplerRatio = fm/fs;
km=floor(dopplerRatio*N);
for k=1:N
if k==1,
F(k)=0;
elseif k>=2 && k<=km,
F(k)=sqrt(1/(2*sqrt(1-((k-1)/(N*dopplerRatio))^2)));
elseif k==km+1,
F(k)=sqrt(km/2*(pi/2-atan((km-1)/sqrt(2*km-1))));
elseif k>=km+2 && k<=N-km,
F(k) = 0;
elseif k==N-km+1,
F(k)=sqrt(km/2*(pi/2-atan((km-1)/sqrt(2*km-1))));
else
F(k)=sqrt(1/(2*sqrt(1-((N-(k-1))/(N*dopplerRatio))^2)));
end
end
% Multiply F with Gaussian random sequence
X=g.*F;
% Take IFFT
x=ifft(X,N);
r=abs(x);
r=r/mean(r);
% Plot the Rayleigh envelope
T=length(r)*Ts;
t=0:Ts:T-Ts;
plot(t,10*log10(r))
xlabel('Time(sec)')
ylabel('Signal Amplitude (dB)')
axis([0 0.05 -15 5])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The above code was used to generate Rayleigh sequences of varying lengths with Doppler frequencies of 5 Hz, 70 Hz and 300 Hz. The sampling frequency was fixed at 7.68 MHz (corresponding to a BW of 5 MHz). It must be noted that in this simulation the length of the Gaussian sequence is equal to the filter length in the frequency domain. It was found that to generate a Rayleigh sequence of reasonable length the length of the Gaussian sequence has to be quite large (2^20 in the above example). As before we calculated the distribution of envelope and phase of the generated sequence as well as the LCR and AFD. These were found to be within reasonable margins.

Note:

1. Level crossing rate (LCR) is defined as number of times per second the signal envelope crosses a given threshold. This could be either in the positive direction or negative direction.
2. Average fade duration (AFD) is the average duration that the signal envelope remains below a given threshold once it crosses that threshold. Simply it is the average duration of a fading event.
3. The LCR and AFD are interconnected and the product of these two quantities is a constant.

[1] David J. Young and Norman C. Beaulieu, "The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform", IEEE Transactions on Communications vol. 48 no. 7 July 2000.

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 h11_{corr} is given as:

h11_{corr}=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:

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.