Category Archives: LTE

A 4G technology that is now being replaced by 5G worldwide.

M-QAM Bit Error Rate in Rayleigh Fading

BER Performance, Fundamentals, LTE, Modulation, WiMAX, , , ,

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
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]= QAM_fading(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(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;

% Receiver
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

64-QAM Constellation

M-QAM Bit Error Rate in Rayleigh Fading
M-QAM Bit Error Rate in Rayleigh Fading

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

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

BER Performance, Fundamentals, LTE, Modulation, WiMAX, ,

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
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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';

% Receiver
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

64-QAM Constellation in AWGN

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.

M-QAM Bit Error Rate in AWGN
M-QAM Bit Error Rate in AWGN

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).

Inside Qualcomm Snapdragon S4

LTE, WCDMA, ,

We have previously looked at the antennas inside a cell phone. Now we look at another important component of a cell phone; the mobile station modem (MSM). One of the most popular MSM in cell phones today is the Qualcomm Snapdragon S4. The details of this MSM are given in the table below.

Qualcomm Snapdragon S4
Qualcomm Snapdragon S4

As can be seen from the above table this small chipset (can easily fit on a fingertip) packs a punch as far as processing power is concerned. It supports a number of wireless standards from GSM/GPRS to LTE and from CDMA 2000 to TD-SCDMA. One of its close competitors is the NVIDIA Tegra 3 which has four ARM Cortex A9 cores (compared to Snapdragon’s two).

Qualcomm Snapdragon – S4

Antennas on Samsung Galaxy S

Antennas, LTE, WiMAX, , ,

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.

Samsung Galaxy Internal View
Samsung Galaxy Internal View

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)

Antennas, LTE, WiMAX, ,

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

Planar Inverted F Antenna
Planar Inverted F Antenna

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-W2=λg/4

It must be remembered that the wavelength here is the guided wavelength which is given as  λg=λo/√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+W1=λg/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).

Reference
[1] http://www.antenna-theory.com/antennas/patches/pifa.php

Implementing a Non-Uniformly Spaced Tapped Delay Line Channel Model

Channel Modeling, LTE, , ,

Question:
Since you are good on fundamentals I would like to ask you a question that puzzles me. LTE channels models are defined at irregular time intervals as shown in [1].

The EPA, EVA and ETU channel taps can best be described as being sampled at multiples of 10 nsec. However, LTE signal is sampled at multiples of 3.84 MHz (Ts=260.416667 nsec). So how does one perform convolution operation.

Answer:
Empirical multipath channel is usually characterized as a τ-spaced tapped delay line (TDL), whose power delay profile (PDP) is either uniformly spaced, or more frequently, spaced with arbitrary time delay(s). Converting a τ-spaced empirical model to a more tractable T-spaced (sampling time-spaced) statistical model will greatly facilitate the link-level simulation and performance evaluation of the broadband digital wireless system. The practical approaches to accomplish this task could be either approximation or interpolation,

1. Approximation
a. Ceil or Floor a τ-spaced tap to the neighbouring sampling time.
b. Split the energy of a τ-spaced tap to the adjacent sampling times.
2. Interpolation
An Ideal Bandlimited (sinc) Interpolator can fulfil the goal.

It should be noted that although T-spaced statistical model is simpler for analysis, τ-spaced empirical model represents the channel more accurately.

[1] http://www.steepestascent.com/content/mediaassets/html/LTE/Help/PropagationConditions.html