In TETRA specifications, one of the modulation technique used is Differential 8 Phase Shift Keying (D8PSK). We will discuss the bit error rate with non-coherent demodulation of D8PSK in Additive White Gaussian Noise (AWGN) channel.

pi/8 D8PSK

The incoming bit sequence is grouped into three bits and is mapped into differential phase as follows:

B(3k-2)	B(3k-1)	B(3k)
0	0	0
0	0	1
1	0	1
1	0	0
0	1	0
0	1	1
1	1	1
1	1	0

Table : Phase transitions for D8PSK modulation (Ref Table 5.2 of ETSI 301-893 V3.2.1)

The modulation symbol is formed by applying a phase offset to previous symbol and is defined as follows:

and

.

Alternately, the phase transitions can be represented as

.

The constellation of the D8PSK occupies phase values separated by as shown below in the blue dots. The red lines shows all possible phase transitions.

Figure: Constellation of D8PSK (Ref Figure 5.2 of ETSI 301-893 V3.2.1)

Channel Model

The transmitted waveform gets corrupted by noise , typically referred to as Additive White Gaussian Noise (AWGN).

Additive : As the noise gets ‘added’ (and not multiplied) to the received signal

White : The spectrum of the noise if flat for all frequencies.

Gaussian : The values of the noise follows the Gaussian probability distribution function,

with mean and

variance .

The received symbol is,

Non Coherent Receiver

A non-coherent receiver relies on the phase transitions between consecutive symbols to form an estimate of the transmitted bits. The sequence of operation is as follows:

a) On the received symbols estimate the phase

b) Find the delta of the estimated phase between consecutive symbols

c) Quantize the estimated delta phase values lying from as follows and convert to bits per the following encoding:

.

Simulation results

The script performs the following

(a) Generate random binary sequence of +1′s and 0′s.

(b) Group three bits together and apply D8PSK modulation

(c) Add white Gaussian noise.

(d) At the receiver, estimate the phase of the received symbols. Based on the delta of the received phase, estimate the transmitted bits

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download the Script for computing BER for non coherent demodulation of pi/8 D8PSK in AWGN

Figure: BER plot for D8PSK with non-coherent demodulation

Comments

As I did not find the theoretical BER equations for D8PSK, was unable to compare it with the simulated results.

Reference

Digital Communications by Proakis, 4th Edition

Related posts:

1. Non coherent demodulation of pi/4 DQPSK (TETRA)
2. Coherent demodulation of DBPSK
3. Bit Error Rate (BER) for frequency shift keying with coherent demodulation

TETRA (TErrestrial Trunked RAdio) is a wireless specification intended to be used by government agencies, public health services, rail transport, military etc. TETRA specification comes from ETSI (European Telecommunication Standards Institute). We plan to start a post series on the building blocks of TETRA physical layer and radio specifications and this is the first post towards that step.

Documents

a) All the standardization documents from ETSI are available online here

b) From the above list, the key document for us to study the physical layer and RF specification is
Doc. Nb. EN 300 392-2 Ver. 3.2.1, Terrestrial Trunked Radio (TETRA); Voice plus Data (V+D); Part 2: Air Interface (AI). Ref. REN/TETRA-03152

Physical Layer Overview

TETRA specification allows for two modulation types

a) Phase Modulation

The overview of the building blocks in phase modulation is as shown below

Figure: Overview of TETRA phase modulation (reference Fig 4.1 ETSI EN 300 392-2 Ver 3.2.1)

For phase modulation, based on the data type, the following channel coding schemes are employed

a) Reed Muller (30,14) code or Block Code

b) Rate Compatible Punctured Convolutional code (RCPC)

c) Block Interleaver

d) Scrambler

e) Modulation of Differential Quarternary Phase Shift Keying ( DQPSK) or Differential 8PSK ( D8PSK)

Figure: Overview of TETRA QAM modulation (reference Fig 4.2 ETSI EN 300 392-2 Ver 3.2.1)

For QAM modulation, based on the data type, the following channel coding schemes are employed

a)  Reed Muller (16,5) code or Block code

b) Parallel Concatenated Convolutional Code (PCCC)

c) Interleaver

d) Scrambler

e) Modulation of 4-QAM, 16-QAM or 64–QAM

In the upcoming posts in this series we will discuss each of these building blocks in more detail.

Related posts:

1. Non coherent demodulation of pi/8 D8PSK (TETRA)
2. Non coherent demodulation of pi/4 DQPSK (TETRA)
3. Cylcic prefix in Orthogonal Frequency Division Multiplexing

In this post lets discuss a closed-loop transmit diversity scheme, where the transmitter has the knowledge of the channel. As there is a feedback path required from the receiver, to communicate the channel seen by the receiver to the transmitter, the scheme is called closed-loop transmit diversity scheme. Recall that the transmit diversity using Space Time Coding (Alamouti STBC) does not require the knowledge of the channel. In this post, we will restrict our discussion to a 2 transmit, 1 receive case. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Channel Model

1. We have 1 receive antennas and two transmit antenna.

2. The channel is flat fading – In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]

3. The channel experienced by each receive antenna is randomly varying in time. For the receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of are Gaussian distributed having mean and variance .

4. The channel experience by each transmit antenna to receive antenna is independent from the channel experienced by other transmit antennas.

5. On each receive antenna, the noise has the Gaussian probability density function with

with and .

6. At each transmit antenna, the channel is known.

Figure: 2 transmit 1 receive beam steering

Transmit Beamforming

On the receive antenna, the received signal is,

where,

is the received symbol,
is the channel on the transmit antenna,
is the transmitted symbol and
is the noise on the receive antenna.

When transmit beamforming is applied, we multiply the symbol from each transmit antenna with a complex number corresponding to the inverse of the phase of the channel so as to ensure that the signals add constructively at the receiver. In this scenario, the received signal is,

,

where,

and

.

In this case, the signal at the receiver is,

.

For equalization, we need to divide the received symbol with the new effective channel, i.e,

.

BER Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1′s and -1′s.

(b) Multiply the symbols with the beam steering matrics – corresponding to the phase of the channel

(c) Perform equalization at the receiver

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulting BER for BPSK in flat fading Rayleigh channel with and without beamforming

Figure: BER plot for 2 transmit 1 receive beamforming for BPSK in Rayleigh channel

Observations

1. Sending the same information on multiple transmit antenna does not provide diversity gain. Intuituvely, this is due to the fact that the effective channel in a 2 transmit antenna case is again a Rayleigh channel; hence the BER performance is identical to 1 transmit 1 receive Rayleigh channel case.

2. If the transmit symbols are multiplied by a complex phase to ensure that the phases align at the receiver, there is diversity gain. However, the BER performance seems to be slighly poorer than the 1 transmit 2 receive MRC case. I guess, the noise is scaled by in the case of transmit beamforming, whereas the noise scaling is different in the case of Maximal Ratio Combining. I need to study bit more for a precise answer.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Related posts:

1. MIMO with MMSE SIC and optimal ordering
2. Alamouti STBC with 2 receive antenna
3. Maximal Ratio Combining (MRC)

Typical communication systems use I-Q modulation and we had discussed the need for I-Q modulation in the past. In this post, let us understand I-Q imbalance and its effect on transmit signal.

I-Q modulation and demodulation

The typical I-Q modulation and demodulation is as shown in the figure below. Consider that the information to be transmitted is a complex signal,

.

At the output of I-Q modulation transmitter is,

Figure: I-Q modulation Transmitter and receiver

At the transmitter, the information is sent on and is sent on .

At the receiver, we multiply the signal with and followed by low pass filtering (LPF) to extract, and respectively.

From trigonmetric identies,
,

,

.

The math for extracting the information at the receiver is as follows:

.

Similarly for the Q-arm,

.

Ignoring the scaling factor of 1/2, we are able to recover both and .

Phase imbalance in IQ modulation

In an ideal I-Q modulator, the phase difference between the signals used for modulating the I arm and Q arm is 90degrees, resulting in using and for sending and .

When there is phase imbalance, the phase difference might not be exactly 90 degrees. We can consider that is used for sending and for sending .

Amplitude imbalance in IQ modulation

When there is amplitude imbalance, there is small variation in the amplitude of thse sine and cosine arms in the modulator. This can be modelled as using for sending and using for sending , where is constant lying between 0 and 1.

The transmit signal including the effect of phase and amplitude imbalance is,

.

Received signal with IQ imbalance in transmitter

Assuming that we have ideal IQ demodulator ie. at the receiver, we multiply the signal with and followed by low pass filtering (LPF) to extract, and respectively.

.

.

As can be observed from the above equations, the desired signal is distorted due to the presence of IQ imbalance.

Effect on spectrum due to I-Q imbalance

To understand the effect of I-Q imbalance on the transmit signal, let us consider that the transmitted signal is i.e,

and

From the post on negative frequency, we know that such signal has a frequency component at and NO frequency component at .

In the presence of I-Q imbalance at the transmitter, the received signal is,

and

.

Click here to download Matlab/Octave script for plotting receive spectrum with transmit IQ imbalance

Figure: Spectrum of received signal in the presence of IQ imbalance at the transmitter

It is reasonably intuituve to see that the received signal has frequency components at and also at . The component at was introduced due to I-Q imbalance.

Related posts:

1. Simulating Minimum Shift Keying Transmitter
2. MSK transmitter and receiver
3. Cylcic prefix in Orthogonal Frequency Division Multiplexing

Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the communication. In this article, lets discuss a simple binary convolutional coding scheme at the transmitter and the associated Viterbi (maximum likelihood) decoding scheme at the receiver.

Update: For some reason, the blog is unable to display the article which discuss both Convolutional coding and Viterbi decoding. As a work around, the article was broken upto into two posts. This post descrbes a simple Binary Convolutional Coding scheme. For details on the Viterbi decoding algorithm, please refer to the post – Viterbi decoder.

Chapter 8, Table 8.2-1 of Digital Communications by John Proakis lists the various rate 1/2 convolutional coding schemes. The simplest among them has constraint length with generator polynomial . There are three parameters which define the convolotional code:

(a) Rate : Ratio of the number of input bits to the number of output bits. In this example, rate is 1/2 which means there are two output bits for each input bit.

(b) Constraint length : The number of delay elements in the convolutional coding. In this example, with there are two delay elements.

(c) Generator polynomial : Wiring of the input sequence with the delay elements to form the output. In this example, generator polynomial is . The output from the arm uses the XOR of the current input, previous input and the previous to previous input. The output from the uses the XOR of the current input and the previous to previous input.

Figure 1: Convolutional code with Rate 1/2, K=3, Generator Polynomial [7,5] octal

From the Figure 1, it can be seen that the operation on each arm is like a FIR filtering (aka convolution) with modulo-2 sum at the end (instead of a normal sum). Hence the name Convolutional code.

State transition

For understanding the Viterbi way of decoding the convolutional coded sequence, lets understand the relation between the input and output bits and the state transition.

Table 1: State transition table and the output values

For details on the Viterbi decoding algorithm, please refer to the post – Viterbi decoder.

References

Tutorial on Convolutional Coding with Viterbi Decoding – Mr. Chip Fleming

Digital Communications by John Proakis

Related posts:

1. Viterbi with finite survivor state memory
2. Viterbi decoder
3. Matlab or C for Viterbi Decoder?

In this post, we will explore a probable way of reducing PAPR (peak to average power ratio) in OFDM by changing the phase of some of the subcarriers. This is in response to the comment to post on Peak to Average power ratio for OFDM, where Mr. Elibom suggested to reduce the PAPR by cyclically rotate some of the subcarriers and using.

Further, the presentation in the IEEE TGN, PAPR in HT-LTF (11-06/1595r1), mentions that in 40MHz mode where a 128pt FFT is used, PAPR of HT-LTF (High Throughput Long Training Field) can be reduced by multiplying the upper 20MHz subcarriers by j. Using quick Matlab simulations, we will try to validate that claim for HT-LTF and further check the PAPR for a general random BPSK and QPSK modulation.

PAPR in HT-LTF

The peak to average power ratio for a signal is defined as
, where
corresponds to the conjugate operator.

Expressing in deciBels,
.

Matlab/Octave script for computing PAPR in HT-LTF
close all; clear all;
nFFT = 128; nDSC = 114;
% The 128pt HT-LTF sequence for 40MHz as defined in 11-06/1595r4
htLTF = [ zeros(1,6) 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 ...
1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 0 ...
0 0 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 ...
1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 zeros(1,5)];



x1F = htLTF; % no rotation
x2F = [htLTF(1:64) j*htLTF(65:128)]; % with 90degree rotation



% Taking iFFT, time domain
x1t = (nFFT/sqrt(nDSC))*ifft(fftshift(x1F.')).'; % no rotation
x2t = (nFFT/sqrt(nDSC))*ifft(fftshift(x2F.')).'; % with rotation



% computing the peak to average power ratio for symbol without rotation
meanSquareValue1 = sum(x1t.*conj(x1t),2)/nFFT;
peakValue1 = max(x1t.*conj(x1t),[],2);
paprSymbol1 = peakValue1./meanSquareValue1;
paprSymbol1dB = 10*log10(paprSymbol1)
% computing the peak to average power ratio for symbol with 90 degree rotation
meanSquareValue2 = sum(x2t.*conj(x2t),2)/nFFT;
peakValue2 = max(x2t.*conj(x2t),[],2);
paprSymbol2 = peakValue2./meanSquareValue2;
paprSymbol2dB = 10*log10(paprSymbol2)

As claimed in slide7 of power point presentation PAPR In HT-LTF (11-06/1595r1), the PAPR with

(a) No rotation is 5.6317dB and

(b) With 90 degree rotation is 3.4066dB.

So, around 2.2dB reduction in PAPR is achieved.

PAPR for random BPSK and QPSK

Armed with the above result, I tried to simulate the PAPR for with random BPSK (and QPSK) modulation on a 128pt FFT with and without rotating the upper 20MHz subcarriers by j.

Click here to download the Matlab/Octave script for simulating PAPR for BPSK/QPSK modulation OFDM with j rotation

The results from the simulations is shown below.

Figure: OFDM PAPR for BPSK/QPSK with j rotation

Observations

1. For random BPSK sequence, the multiplication by j resulted in increase of PAPR by around 0.5dB.

2. For random QPSK sequence, the PAPR with and without multiplication by j is almost the same.

Based on the above simulation results, may I conclude that there is no noticable reduction in PAPR by multiplying upper 20MHz subcarriers by j. There might be some special cases (like HT-LTF sequence), where the multiplication by j reduces PAPR. However for random BPSK/QPSK sequence, there is no improvement.

Related posts:

1. Peak to Average Power Ratio for OFDM
2. BPSK BER with OFDM modulation
3. BER for BPSK in OFDM with Rayleigh multipath channel

In the previous post on Binary to Gray code conversion for PSK, I had claimed that “for a general M-QAM modulation the binary to Gray code conversion is bit more complicated“. However following a closer look, I realize that this is not so complicated.

The QAM scenario can be treated as independent PAM modulation on I arm and Q-arm respectively. For example, let us consider 16-QAM scenario.

Each constellation point can represent bits, with two bits on the I axis and two on the Q axis. For 16-QAM, the values taken by the I and Q axes are {-3, -1, +1, +3}. The two bits on the I and Q arm can be Gray coded as shown in the table below

Table: Gray coded constellation mapping for 16-QAM

The constellation diagram with the bit mapping is shown below.

Figure: 16QAM constellation with Gray coded mapping

As can be observed from the figure above, the adjacent constellation symbols differ by only one bit. As simple as that.

Simulation model for Binary to Gray coded mapping for 16-QAM


% Matlab/Octave code for converting bits into 16-QAM constellation

clear all
M = 16; % number of constellation points
k = log2(M); % number of bits in each constellation
% defining the real and imaginary PAM constellation
% for 16-QAM
alphaRe = [-(2*sqrt(M)/2-1):2:-1 1:2:2*sqrt(M)/2-1];
alphaIm = [-(2*sqrt(M)/2-1):2:-1 1:2:2*sqrt(M)/2-1];
% input - decimal equivalent of all combinations with b0b1b2b3
ip = [0:15];
ipBin = dec2bin(ip.'); % decimal to binary
% taking b0b1 for real
ipDecRe = bin2dec(ipBin(:,[1:k/2]));
ipGrayDecRe = bitxor(ipDecRe,floor(ipDecRe/2));
% taking b2b3 for imaginary
ipDecIm = bin2dec(ipBin(:,[k/2+1:k]));
ipGrayDecIm = bitxor(ipDecIm,floor(ipDecIm/2));
% mapping the Gray coded symbols into constellation
modRe = alphaRe(ipGrayDecRe+1);
modIm = alphaIm(ipGrayDecIm+1);
% complex constellation
mod = modRe + j*modIm;

Note:

The above code snippet works only for 4QAM and 16QAM.

Related posts:

1. 16QAM Bit Error Rate (BER) with Gray mapping
2. Binary to Gray code conversion for PSK and PAM
3. Bit error rate for 16PSK modulation using Gray mapping