e Book DspLog Error Rates Awgn

[dspLog] Signal Processing for Communication

[dspLog[dspLog ] ] Signal processing for Communication

Krishna Pillai

All Rights Reserved Krishna Pillai, [dspLog] Copyright : 20072008

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Change LogChange Log

Version Date Author Changes

1.00 28Sep2008 Krishna Pillai Initial Draft

All Rights Reserved Copyright : 20072008 Krishna Pillai, [dspLog]

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Table of ContentsTable of Contents1. Background................................................................................................................................................4

TARGET AUDIENCE..........................................................................................................................................4PROBABLE CONTENTS......................................................................................................................................4AUTHOR.......................................................................................................................................................4

2. Error Rates in AWGN...............................................................................................................................5BER FOR BPSK IN AWGN..........................................................................................................................5

Channel Model.......................................................................................................................................................5Computing the probability of error........................................................................................................................5Simulation model...................................................................................................................................................7

SYMBOL ERROR RATE FOR 4PAM..................................................................................................................8Channel Model.......................................................................................................................................................8Computing the symbol error rate...........................................................................................................................8Simulation Model.................................................................................................................................................10

SYMBOL ERROR RATE FOR QPSK (4QAM).................................................................................................11Noise model.........................................................................................................................................................12Computing the probability of error......................................................................................................................12Simulation Model.................................................................................................................................................13

SYMBOL ERROR RATE FOR 16QAM MODULATION...........................................................................................14Noise model.........................................................................................................................................................15Computing the probability of error......................................................................................................................15Simulation model.................................................................................................................................................17

SYMBOL ERROR RATE FOR 16PSK MODULATION............................................................................................18Deriving the symbol error rate.............................................................................................................................19Simulation model.................................................................................................................................................21

16QAM BIT ERROR RATE (BER) WITH GRAY MAPPING.................................................................................22Gray coded bit mapping in 16QAM modulation................................................................................................22Symbol Error and Bit Error probability..............................................................................................................23Bit energy and symbol energy.............................................................................................................................2316QAM BER........................................................................................................................................................24Simulation model.................................................................................................................................................24

BPSK BER WITH OFDM MODULATION.......................................................................................................25OFDM modulation...............................................................................................................................................25Cyclic prefix.........................................................................................................................................................26Frequency spread.................................................................................................................................................26Relation between Eb/No and Es/No in OFDM....................................................................................................26Simulation model.................................................................................................................................................26

3. References.................................................................................................................................................28


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1. 1. BackgroundBackground The blog [dspLog] started some in February 2007 on the Blogger platform and was hosted at http://dsplog.blogspot.com with the following objective:

Explain and discuss the basic text book concepts in digital signal processing and digital communication using simple Octave scripts.

Seeing consistent increase in traffic over the 67 months from inception, it was decided to move to an independent platform hosted at www.dsplog.com.

Target audience● Students taking courses in digital signal processing/digital communication and requiring help to

understand some of the concepts.

● Practicing engineers who are fresh to the domain of digital signal processing and requiring help.

● Experienced engineers can share tips and tricks associated with the digital signal processing trade.

Probable contents● Understanding basic concepts pertaining to FIR/IIR filtering

● Sample rate conversion

● Bit and Symbol error probabilities for typical digital communication

● Orthogonal Frequency Division Multiplexing

● more…

AuthorKrishna Pillai is a Signal Processing Engineer at an Indian firm based out of Bangalore, India. His typical activities on a working day involve identifying and modeling digital signal processing algorithms for wireless receivers.


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2. 2. Error Rates in AWGNError Rates in AWGNOver a one year period, error rates for most of the modulation schemes like

BPSK, QPSK, PAM, QAM are in Additive White Gaussian Noise (AWGN) environment have been discussed in the [dspLog]. This eBook summarizes the

BER for BPSK in AWGNLet us derive the equation for bit error probability wit BPSK modulation scheme. With Binary Phase Shift Keying (BPSK), the binary digits 1 and 0 maybe represented by the analog levels and respectively.

Figure: Simplified block diagram with BPSK transmitterreceiver

Channel ModelThe 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 and .

Computing the probability of errorUsing the derivation provided in Section 5.2.1 of [COMMPROAKIS] as reference.

The received signal, OR corresponding to transmitted bit 1 OR 0


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respectively.

The conditional probability distribution function (PDF) of for the two cases are:

.

Figure: Conditional probability density function with BPSK modulation

For decoding, a decision rule with threshold as 0 might be optimal i.e.

for received signal and .

With this threshold, the probability of error given is transmitted is (the area in blue region):

, where

the complementary error function, .

Similarly, the probability of error given is transmitted is (the area in green region):


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.

The total probability of bit error,

.

Assuming, and are equally probable i.e. , the bit error probability is,

.

Simulation modelOctave/Matlab source code for computing the bit error probability with BPSK modulation from theory

and simulation. The code performs the following:

(a) Generation of random BPSK modulated symbols +1's and 1's

(b) Passing them through Additive White Gaussian Noise channel

(c) Demodulation of the received symbol based on the location in the constellation

(d) Counting the number of errors

(e) Repeating the same for multiple Eb/No value.

Click here to download the Matlab/Octave script for simulating BPSK bit error rate.


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http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/#Simulation Model




Figure: Bit error curve for BPSK modulation theory, simulation

Symbol Error rate for 4PAMConsider that the alphabets used for a 4PAM is (Refer example 5

34 in [DIGCOMMBARRYLEEMESSERSCHMITT]).

The average energy of the constellation assuming all the alphabets are equally likely is,

.

The constellation plot for a 4PAM signal after normalization can be as shown below.


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Figure: Constellation plot for 4 PAM modulation

Channel ModelNow comes the interesting task analysis of symbol error probability for the modulation scheme

in additive white Gaussian noise condition. The noise follows the Gaussian probability distribution

function, with and .

Computing the symbol error rateUsing the derivation provided in Section 5.2.1 of [COMMPROAKIS] as reference:

The received signal can be

or or or .

Let us first consider the case was transmitted.

The conditional probability distribution function (PDF) of given was transmitted is :

.

Figure: Probability distribution function when the alphabet +3 is sent

Using midway point between +1 and +3 as the detection threshold, i.e. for received

signal

.

With this threshold, the probability of error given is transmitted is (the area in blue region):


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.

Note: The complementary error function, .

Given that the constellation of +3 and 3 is symmetric, it is reasonably intuitive that the probability of error given is transmitted is also,

.

Case where is transmitted

The conditional probability distribution function (PDF) of given was transmitted is :

.

Figure: Probability distribution function when the alphabet +1 is sent

The probability of error given is transmitted is (the area in green and red region):

Given that the constellation for +1 and 1 is symmetric, it is reasonably intuitive that the probability of error given is transmitted is also,

.


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Total probability of symbol error

Given that all the alphabets are equally likely, the total probability of symbol error is,

.

Assuming, and , are equally probable i.e. , the symbol error probability is,

.

Simulation ModelThe Matlab/Octave script for generating a 4PAM transmission, pass it through additive white Gaussian noise and demodulation at the receiver will be useful for understanding the concept further. The symbol error rate plots obtained from simulations compare well with the theoretical derivations.

Click here to download: Matlab/Octave script for simulating 4PAM symbol error rate

Figure: Symbol Error Rate for 4PAM modulation


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http://www.dsplog.com/2007/10/07/symbol-error-rate-for-pam/#Simulation Model


Symbol Error Rate for QPSK (4QAM)Given that we have discussed symbol error rate probability for a 4PAM modulation, let us know focus on finding the symbol error probability for a QPSK (4QAM) modulation scheme.

Background

Consider that the alphabets used for a QPSK (4QAM) is (Refer example 5

35 in [DIGCOMMBARRYLEEMESSERSCHMITT]).

Figure: Constellation plot for QPSK (4QAM) constellation

The scaling factor of is for normalizing the average energy of the transmitted symbols to 1,

assuming that all the constellation points are equally likely.

Noise modelAssuming that the additive noise follows the Gaussian probability distribution function,

with and .

Computing the probability of errorConsider the symbol

The conditional probability distribution function (PDF) of given was transmitted is:

.


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http://www.dsplog.com/2007/10/07/symbol-error-rate-for-pam/


Figure: Probability density function for QPSK (4QAM) modulation

As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the hashed region i.e.

.

Probability of real component of greater than 0, given was transmitted is (i.e. area outside the red region)

, where

the complementary error function, .

Similarly, probability of imaginary component of greater than 0, given was transmitted is (i.e. area outside the blue region).

.

The probability of being decoded correctly is,

.


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Total symbol error probability

The symbol will be in error, it atleast one of the symbol is decoded incorrectly. The probability of symbol error is,

.

For higher values of , the second term in the equation becomes negligible and the probability of error

can be approximated as,

.

Simulation ModelSimple Matlab/Octave script for generating QPSK transmission, adding white Gaussian noise and

decoding the received symbol for various values.

Click here to download: Matlab/Octave script for computing the symbol error rate for QPSK modulation

Figure: Symbol Error Rate for QPSK (4QAM) modulation


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http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/#Simulation Model


Observations

1. Can see good agreement between the simulated and theoretical plots for 4QAM modulation

2. When compared with 4PAM modulation, the 4QAM modulation requires only around 2dB lower

for achieving a symbol error rate of .

Symbol Error Rate for 16QAM modulationGiven that we have went over the symbol error probability for 4PAM and symbol error rate for 4QAM , let us extend the understanding to find the symbol error probability for 16QAM (16 Quadrature

Amplitude Modulation). Consider a typical 16QAM modulation scheme where the alphabets (Refer example 537 in [DIGCOMMBARRYLEEMESSERSCHMITT] ).

are used.

The average energy of the 16QAM constellation is (here). The 16QAM constellation is as shown in the figure below


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http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/




Figure: 16QAM constellation

Noise modelAssuming that the additive noise follows the Gaussian probability distribution function,

with and .

Computing the probability of errorConsider the symbol in the inside, for example


.


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As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the black hashed region i.e.

.

Using the equations from (symbol error probability of 4PAM as reference)

.

The probability of being decoded incorrectly is,

.

Consider the symbol in the corner, for example


.

As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the red hashed region i.e.

.

Using the equations from (symbol error probability of 4QAM as reference)

.


.

Consider the symbol which is not in the corner OR not in the inside, for example



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.

As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the blue hashed region i.e.

.

Using the above two cases are reference,

.


.

Total probability of symbol error

Assuming that all the symbols are equally likely (4 in the middle, 4 in the corner and the rest 8), the total probability of symbol error is,

.

Simulation modelSimple Matlab/Octave code for generating 16QAM constellation, transmission through AWGN channel and computing the simulated symbol error rate.

Click here to download : Matlab/Octave script for simulating 16QAM symbol error rate


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http://www.dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/#Simulation Model


Figure: Symbol Error Rate curve for 16QAM modulation

Observations

Can observe that for low values, the theoretical results seem to be 'pessimistic' 'optimistic'

compared to the simulated results. This is because for the approximated theoretical equation, the term was ignored. However, this approximation is valid only when the term is

small, which need not be necessarily true for low values.

Symbol Error Rate for 16PSK modulationConsider a general MPSK modulation, where the alphabets,

are used.

(Refer example 538 in [DIGCOMMBARRYLEEMESSERSCHMITT] )


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Figure: 16PSK constellation plot

Deriving the symbol error rateLet us the consider the symbol on the real axis, i.e

.

The received symbol .

Where the additive noise follows the Gaussian probability distribution function,

with and .

The conditional probability distribution function (PDF) of received symbol given was transmitted is:

.

As can be seen from the figure above, due to the addition of noise, the transmitted symbol gets spreaded. However, if the received symbol is present with in the boundary defined by the magenta lines, then the symbol will be demodulated correctly.

To derive the symbol error rate, the objective is to find the probability that the phase of the received symbol lies within this boundary defined by the magenta lines i.e. from to .


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For simplifying the derivation, let us make the following assumptions:

(a) The signal to noise ratio, is reasonably high.

For a reasonably high value of , then the real part of the received symbol is not afected by noise i.e.,

and

the imaginary part of the received symbol is equal to noise, i.e.

.

(b) The value of M is reasonably high (typically M >4 suffice)

For a reasonably high value of M, the constellation points are closely spaced. Given so, the distance of the

constellation point to the magenta line can be approximated as .

Figure: Distance between constellation points

Given the above two assumptions, it can be observed that the symbol will be decoded incorrectly, if

the imaginary component of received symbol isgreater than . The probability of

being greater than is,


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.

Changing the variable to ,

.

Note: The complementary error function, .

Similarly, the symbol will be decoded incorrectly, if the imaginary component of received symbol

is less than . The probability of being less than is,

.

The total probability of error given was transmitted is,

.

Total symbol error rate

The symbol will be in error, if atleast one of the symbol gets decoded incorrectly. Hence the total symbol error rate from MPSK modulation is,

.

Simulation modelSimple Matlab/Octave script for simulating transmission and reception of an MPSK modulation is attached. It can be observed that the simulated symbol error rate compares well with the theoretical symbol error rate.

Click here to download: Matlab/Octave script for simulating symbol error rate curve for 16 PSK modulation


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http://www.dsplog.com/2008/03/18/symbol-error-rate-for-16psk/#Simulation Model

http://www.dsplog.com/2008/03/18/symbol-error-rate-for-16psk/#Simulation Model


Figure: Symbol Error rate curve for 16PSK modulation

16QAM Bit Error Rate (BER) with Gray mappingLet us derive the theoretical 16QAM bit error rate (BER) with Gray coded constellation mapping in additive white Gaussian noise conditions. Further, the Matlab/Octave simulation script can be used to confirm that the simulation is in good agreement with theory.

Gray coded bit mapping in 16QAM modulationAs we discussed in the previous post on Binary to Gray code for 16QAM, the 4 bits in each constellation point can be considered as two bits each on independent 4PAM modulation on Iaxis and Qaxis respectively.

b0b1 I b2b3 Q

00 3 00 3

01 1 01 1

11 +1 11 +1

10 +3 10 +3

Table: Gray coded constellation mapping for 16QAM


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http://www.dsplog.com/2008/06/01/binary-to-gray-code-for-16qam/


Figure: 16QAM constellation plot with Gray coded mapping

Symbol Error and Bit Error probabilityAs can be seen from the above constellation diagram, with Gray coded bit mapping, adjacent constellation symbols differ by only one bit. So, if the noise causes the constellation to cross the decision threshold,

only 1 out of bits will be in error. So the relation between bit error and symbol error is,

.

Note:

For very low value of , it may so happen that the noise causes the constellation to fall near a

diagonally located constellation point. In that case, the each symbol error will cause two bit errors. Hence

the need for approximate operator in the above equation. However, for reasonably high value of , the

chances of such events are negligible.

Bit energy and symbol energyAs we learned from the post discussing Bit error rate for 16PSK, since each symbol consists of bits, the symbol to noise ratio k times the bit to noise ratio i.e,

where,


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http://www.dsplog.com/2008/05/18/bit-error-rate-for-16psk-modulation-using-gray-mapping/


.

16QAM BERFrom the post detailing the derivation of 16QAM Symbol error rate, we know that the symbol error is,

.

Combining the above two equations, the bit error rate for Gray coded 16QAM in Additive White Gaussian Noise is

.

Simulation modelThe Matlab/Octave script performs the following:

(a) Generation of random binary sequence

(b) Assigning group of 4 bits to each 16QAM constellation symbol per the Gray mapping

(c) Addition of white Gaussian Noise

(d) Demodulation of 16QAM symbols and

(e) Demapping per decimal to Gray conversion

(f) Counting the number of bit errors

(g) Running this for each value of Eb/No in steps of 1dB.

Click here to download : Script for computing 16QAM BER with Gray mapping


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http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/#Simulation Model



Figure: Bit Error Rate plot for 16QAM modulation with Gray mapping

BPSK BER with OFDM modulationIn this post, we will discuss a simple OFDM transmitter and receiver, find the relation between Eb/No (Bit to Noise ratio) and Es/No (Signal to Noise ratio) and compute the bit error rate with BPSK.

OFDM modulationLet us use the OFDM system loosely based on IEEE 802.11a specifications.

Parameter Value

FFT size. nFFT 64

Number of used subcarriers. nDSC 52

FFT Sampling frequency 20MHz

Subcarrier spacing 312.5kHz

Used subcarrier index {26 to 1, +1 to +26}

Cylcic prefix duration, Tcp 0.8us

Data symbol duration, Td 3.2us

Total Symbol duration, Ts 4us

You may refer to post Understanding an OFDM Transmission for getting a better understanding of the


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http://www.dsplog.com/2008/02/03/understanding-an-ofdm-transmission/


above mentioned parameters.

Cyclic prefixIn an OFDM transmission, we know that the transmission of cyclic prefix does not carry 'extra'

information in Additive White Gaussian Noise channel. The signal energy is spread over time whereas the bit energy is spread over the time i.e.

Simplifying,

Frequency spreadIn OFDM transmission, all the available subcarriers from the DFT is not used for data transmission. Typically some subcarriers at the edge are left unused to ensure spectrum roll off. For the example scenario, out of the available bandwidth from 10MHz to +10MHz, only subcarriers from 8.1250MHz (26/64*20MHz) to +8.1250MHz (+26/64*20MHz) are used.

This means that the signal energy is spread over a bandwidth of 16.250MHz, whereas noise is spread over bandwidth of 20MHz (10MHz to +10MHz), i.e.

Simplifying,

.

Relation between Eb/No and Es/No in OFDMCombining the above two aspects, the relation between symbol energy and the bit energy is as follows:

.

Expressing in decibels,

.

Simulation modelThe 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


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(c) Assigning to multiple OFDM symbols where data subcarriers from 26 to 1 and +1 to +26 are used, adding cyclic prefix, concatenation of multiple symbols to form a long transmit sequence

(d) Adding White Gaussian Noise

(e) Grouping the received vector into multiple symbols, removing cyclic prefix, taking the desired subcarriers

(f) Demodulation and conversion to bits

(g) Counting the number of bit errors

Click here to download: Script for BER computation of BPSK using OFDM

Figure: Bit Error Rate plot for BPSK using OFDM modulation

Can observe that the simulated bit error rate is in good agreement with the theoretical bit error rate for BPSK modulation i.e.

.


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http://www.dsplog.com/2008/06/10/ofdm-bpsk-bit-error/#Simulation Model


3. 3. ReferencesReferences[DIGITAL COMMUNICATION: PROAKIS] Digital Communications by John Proakis

[DIGCOMMBARRYLEEMESSERSCHMITT]

Digital Communication: by John R. Barry, Edward A. Lee, David G. Messerschmitt

[COMMUNICATION SYSTEMS: PROAKIS, SALEHI]

Fundamentals of Communication Systems, by John G. Proakis, Masoud Salehi


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e Book DspLog Error Rates Awgn