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. TETRA Air interface specifications – Overview

IEEE 802.11ac Very High Throughput (for <6GHz band) is an upcoming standard which is development by IEEE standardization committee. The mandate of Task Group AC is supposed to enhance the High Throughput rates achieved by 802.11n.  As described in the document VHT below 6GHz PAR plus 5C’s (802.11-08/0807r4) the group has the following objectives :

a) Maximum Multi station (STA) throughput of atleast 1Gbps and a maximum single link throughput of atleast 500Mbps.

b) Operation below 6GHz while ensuring backward compatibility and coexistence with legacy 802.11 devices in 5GHz unlicensed bands.

Two among some of the presentations discussing the system’s feasibility are :

1) On the feasibility of 1Gbps for various MAC/PHY architectures (IEEE-802.11-08/0307)

In slide3 of the ppt, Intel’s study showed that 160MHz bandwidth with 4×4 MIMO is required to achieve around 1Gbps MAC throughput. However, if we allow simultaneous multiple channel usage, we need only 100MHz bandwidth to hit 1Gbps MAC throughput.

2) PHY and MAC Throughput Analysis with 80 MHz for VHT below 6 GHz (IEEE 802.11-08/0535r0)

The presentation shows that 1Gbps MAC throughput is not reliably achieved with 4×4 MIMO 80MHz bandwidth and 256QAM.

Both the above presentations point to the fact that going to higher throughput cannot be achieved by bandwidth expansion alone. We will see more intelligent channel usage approaches. We will discuss more in the upcoming posts.

Related posts:

1. Trying out PAPR reduction for OFDM by multiplication with j
2. Quiz on IEEE 802.11a specifications
3. LaTex on blogger not working?

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

pi/4 DQPSK

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

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

Table : Phase transitions for DQPSK modulation (Ref Table 5.1 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 DQPSK occupies phase values separated by as shown below in the blue dots. The red lines shows all possible phase transitions.

Figure: Constellation of DQPSK (Ref Figure 5.1 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:

.

Figure: Received delta phase to bit mapping

The theoretical bit error rate defined with the mapping above is (Ref Chapter 5.2.8 of Digital Communications by Proakis, 4th Edition)
,

where

,

,

and

is the modified Bessel function of first kind.

Simulation results

The script performs the following

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

(b) Group two bits together and apply DQPSK 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 simulating BER of pi/4 DQPSK with non-coherent demodulation at the receiver

Figure: BER plot for DQPSK with non-coherent demodulation

Observations

Can see good agreement between simulated and theoretical results.

Reference

Digital Communications by Proakis, 4th Edition

Related posts:

1. Non coherent demodulation of pi/8 D8PSK (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. Quiz on IEEE 802.11a specifications
2. Non coherent demodulation of pi/8 D8PSK (TETRA)
3. Non coherent demodulation of pi/4 DQPSK (TETRA)

In the past, we had discussed BER for BPSK in flat fading Rayleigh channel and BER for BPSK in a frequency selective channel using Zero Forcing Equalization. In this post, lets discuss a frequency selective channel with the use of Minimum Mean Square Error (MMSE) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.

Transmit symbol

Let the transmit symbols be modeled as

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is the symbol index and

is the output waveform.

For simplicity, lets assume that the transmit pulse shaping filter is not present, i.e .

So the transmit symbols can be modeled by the discrete time equivalent

Figure: Transmit symbols

Channel Model

Lets us assume the channel to be a 3 tap multipath channel with spacing i.e.

Figure: Channel model (3 tap multipath)

In addition to the multipath channel, the received signal gets corrupted by noise , typically referred to as Additive White Gaussian Noise (AWGN). The values of the noise follows the Gaussian probability distribution function, with

mean and

variance .

The received signal is

, where

is the convolution operator.

MMSE Equalization

In Minimum Mean Square Error solution, for each sample time we would want to find a set of coefficients which minimizes the error between the desired signal and the equalized signal , i.e.

,

where,

is the error at sample time ,

is column vector of dimension storing the equalization coefficients,

is column vector of dimension storing the received samples,

is the number of taps in the equalizer,

is the cross correlation between received sequence and input sequence ,

is the cross correlation between received sequence and input sequence and

is the auto-correlation of the received sequence.

For solving the Minimum Mean Square Error (MMSE) criterion, we need to find a set of coefficients which minimizes .

Differentiation with respect to and equating to 0,

.

Simplifying,

,

Note :

a) is the variance of the input signal

b) (as there is no correlation between input signal and noise)

Simulation Model

Click here to download: Matlab/Octave script for computing BER for BPSK with 3 tap ISI channel with MMSE Equalization

The attached Matlab/Octave simulation script performs the following:

(a) Generation of random binary sequence

(b) BPSK modulation i.e bit 0 represented as -1 and bit 1 represented as +1

(c) Convolving the symbols with a 3-tap fixed fading channel.

(d) Adding White Gaussian Noise

(e) Computing the MMSE and ZF equalization filter at the receiver (with 7 taps in length)

(f) Demodulation and conversion to bits

(g) Counting the number of bit errors

(h) Repeating for multiple values of Eb/No

The simulation results are as shown in the plot below.

Figure: BER plot for BPSK in a 3 tap ISI channel with MMSE equalizer

Observations

1. Can see around 0.5dB gain with using MMSE equalizer

Reference

Complex to Real : Tutorial 26 – Filters, analog, digital and adaptive equalization

Related posts:

1. BER for BPSK in ISI channel with Zero Forcing equalization
2. MIMO with MMSE SIC and optimal ordering
3. MIMO with MMSE equalizer

Wishing all the readers of dsplog.com a great year 2010 !

Its been a mixed year for dsplog. Some key milestones

a) Crossing 1000 subscribers with 1100+ comments in March 2009
b) Crossing 100 posts with 2200 subscribers and 2600+ comments in October 2009
c) As I write this, we have 102 posts with 2603 subscribers and 3600+ comments.

Towards the last three months, the posting frequency has drastically dropped. Partly attributed to paucity of time following my change of employer and of course, laziness.

Hoping for a great year 2010.

Related posts:

1. dspLog turns two! Happy Birthday!
2. Happy Birthday – dspLog
3. Summary – feedback on [dspLog], July 2008

In the past, we had discussed BER for BPSK in flat fading Rayleigh channel. In this post, lets discuss a frequency selective channel with the use of Zero Forcing (ZF) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.

Transmit symbol

Let the transmit symbols be modeled as

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is the symbol index and

is the output waveform.

For simplicity, lets assume that the transmit pulse shaping filter is not present, i.e .

So the transmit symbols can be modeled by the discrete time equivalent

Figure: Transmit symbols

Channel Model

Lets us assume the channel to be a 3 tap multipath channel with spacing i.e.

Figure: Channel model (3 tap multipath)

In addition to the multipath channel, the received signal gets corrupted by noise , typically referred to as Additive White Gaussian Noise (AWGN). The values of the noise follows the Gaussian probability distribution function, with

mean and

variance .

The received signal is

, where

is the convolution operator.

Zero Forcing Equalization

Objective of Zero Forcing Equalization is to find a set of filter coefficients which can make .

After equalization

.

Note:

The term causes noise amplification resulting poorer bit error rate performance.

Deriving the equalization coefficients

From the post on toeplitz matrix,we know that convolution operation can be represented as matrix multiplication.

% Matlab code for using Toeplitz matrix for convolution
clear all
x = [1:3];
h = [4:6];
xM = toeplitz([x zeros(1,length(h)-1) ], [x(1) zeros(1,length(h)-1) ]);
y1 = xM*h';
y2 = conv(x,h);
diff = y1'-y2
diff = [  0   0   0   0   0 ]

Using similar matrix algebra and assuming that the coefficients has 3 taps, the equation can be equivalently represented as,

Solving for , we have

.

If we assume that has 5 taps,

Solving for , we have

.

Example

% Assuming a 3 tap channel as follows
ht = [0.2 0.9 0.3];
L = length(ht);
kk = 1;
hM = toeplitz([ht([2:end]) zeros(1,2*kk+1-L+1)], [ ht([2:-1:1]) zeros(1,2*kk+1-L+1) ]);
d  = zeros(1,2*kk+1);
d(kk+1) = 1;
c  = [inv(hM)*d.'].';

The frequency response of the channel and the equalizer are shown below:

Figure: Frequency response of the channel and the equalizer

Simulation Model

Click here to download: Matlab/Octave script for computing BER for BPSK with 3 tap ISI channel with Zero Forcing Equalization

The attached Matlab/Octave simulation script performs the following:

(a) Generation of random binary sequence

(b) BPSK modulation i.e bit 0 represented as -1 and bit 1 represented as +1

(c) Convolving the symbols with a 3-tap fixed fading channel.

(d) Adding White Gaussian Noise

(e) Computing the equalization filter at the receiver – the equalization filter is 3, 5, 7, 9 taps in length

(f) Demodulation and conversion to bits

(g) Counting the number of bit errors

(h) Repeating for multiple values of Eb/No

The simulation results are as shown in the plot below.

Figure: BER plot for BPSK in a 3 tap ISI channel with Zero Forcing equalizer

Observations

1. Increasing the equalizer tap length from 3 to 5 showed reasonable performance improvement.

2. Diminishing returns from improving the equalizer tap length above 5.

3. The results are poorer compared to the AWGN no multipath results. This is due to the noise amplification (see the frequency response above) by the zero forcing equalization filter.

Next step is to discuss the zero forcing equalizer in the presence of transmit pulse shaping and then move on to minimum mean square error equalizer.

Related posts:

1. BER for BPSK in ISI channel with MMSE equalization
2. MIMO with Zero Forcing Successive Interference Cancellation equalizer
3. MIMO with Zero Forcing equalizer