EEE8128 Communications and Signal Processing: Part 1 ...

EEE8128 Communications and Signal Processing:Part 1, Simulation Techniques

Dr. Charalampos C. Tsimenidis

Newcastle UniversitySchool of Engineering

October 2019

Dr. Charalampos C. Tsimenidis EEE8128 Communications and Signal Processing 1 / 94

Introduction

Aim:The aim of this module is to simulate a wirelesscommunication link.Objectives:The objectives of this module are to

Map the components of a theoretical system model intosoftware (MATLAB).Evaluate the performance under various channel modelsusing theoretical, semi-analytical and simulationapproaches.Efficient and fast software development.


Coursework Assessment: Software Development (15%)

Software Development and Demonstration (15%):Matlab m-files with comments written for this assignmentshould be submitted electronically on the last teachingday (Friday) of Semester 1 2020 as one zip file stgrp-x,where x is your group number, by email to:

[email protected]

Break down of marks:The marks are assigned with respect to the developedsoftware code: i.e.

Correctness: software produces the correct results and theresults are the same as those included in the report (30%)Completeness: all objectives of the course work are met(30%)Documentation/comments (30%)Efficiency (10%)


Coursework Assessment: Report (35%)

One formal assignment report per group is required(minimum of ∼4500 words, 1500 words per student) whichmust be submitted to General Office on the last teachingday (Friday) of Semester 1.As a general guideline, the report should at least include:

Abstract (5%).Short background theory section (10%).Detailed description of your algorithm implementation(30%).Performance results (30%).Discussions, conclusions and reference sections (10%).Complete set of references (10%).Correctly formatted program listing appendix (5%).

Students may submit individual reports or indicate parts ofthe submitted report as their contribution to obtainindividual marks. This will only happen at request and afterprior agreement.


Time Plan and Module Support

Lab and Lectures: 4hrs per Week (see timetable)Self-directed study whenever cluster available.Textbooks available in Library.Module support on BlackboardPersonal NCL webpage:http://www.staff.ncl.ac.uk/

charalampos.tsimenidis/EEE8128/EEE8128.html

Surgery hour: TBC


System Model

Tx RxData Comm. Estimated Data

Transmitter

16−QAM OFDM

Receiver

BER

Channel

Compute

Pe = f(SNR)

0 2 4 6 8 10 12 14

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb

N0(dB)

BER


Transmitter Structure and Tasks

Develop MATLAB code to perform:Random information bit generation for an OFDM symbolsize of 2048.Mapping of the information bit sequence to Gray-coded16-QAM constellation.OFDM symbol generation without and withcomb-interleaved pilots.Cyclic prefix insertion: No CP, 16, 32, and 64 samples.Loop for Monte Carlo simulation of the communication linkthat runs for at least 106 16-QAM symbols persignal-to-noise ratio (SNR) point.


Channel Specifications and Tasks

Two different channels to be implemented:AWGN: SNR = 0-16 dB in steps of 1 dBFrequency selective multipath channel with impulseresponse h(n) and AWGN:

ITU Pedestrian B, Channel 103.SNR: 0-40 dB in steps of 2 dB.

Perfect symbol timing it is assumed.Develop MATLAB code to perform:

Computation of the theoretical standard deviation for theAWGN noise samples.Frequency response of the multipath channel.Channel filtering of the transmitted signal.


Receiver Structure and Tasks

Develop MATLAB code to perform:FFT and 16-QAM demodulation using the minimumEuclidean distance approach.Zero Forcing equalization with perfect channel knowledge.ZF equalization with channel estimation using a combdistributed pilots, e.g. 1 pilot for every 8 subcarriers.Constellation plots at the output of the equalizer for specificSNR point, e.g. 5, 15 and 25 dB.Performance of mean square error (MSE) of channelestimation, eMSE = E{|H(n)− H(n)|2}, vs. SNR fordifferent cyclic prefix lengths and the two investigatedmultipath channels.BER vs. SNR in dB for the 2 channels and 2 receivers.For the AWGN channel no cyclic prefix is required.For the multipath channel + AWGN the ZF receiver mustbe evaluated with or without channel knowledge and forthe different cyclic prefix lengths.


References

1 B.P. Lathi and Z. Ding, Modern Digital and AnalogCommunications Systems, 4th Ed., OUP USA, 2013.

2 J.G. Proakis, M. Salehi and G. Bauch , ModernCommunication Systems Using MATLAB, 3rd Ed., NelsonEngineering, 2011.

3 J.G. Proakis and M. Salehi, Digital Communications, 5thEd., McGraw-Hill, 2008.


Motivation

Performance evaluation of wireless communication systems:Theoretical or analytical approach (desired solution),Semi-analytical approach (alternative solution),Simulation-based approach (practical solution),Full hardware implementation followed by trials (impracticalsolution).


Example: Theoretical Performance

Bit-error rate (BER) performance of Binary Phase Shift Keying(BPSK) modulation in Rayleigh fading in the presence AdditiveWhite Gaussian Noise (AWGN) channel.System Model:

h[n] w[n]

Mod.

d[n]b[n] r[n] = h[n]d[n] + w[n]

whereh[n] are the complex-valued Rayleigh fading coefficientswith variance 2σ2

h,d[n] = ±1 are the BPSK symbols,w[n] are the complex-valued AWGN samples.


Theoretical Performance (Cont.)

Theoretical BER (closed-form solution):

Pe(Γ) =

∫ ∞0

p(x)Q(x)dx =1

2

(1−

√Γ

1 + Γ

)

whereΓ is the average signal to noise ratio (SNR) given as

Γ =EbN0

2σ2h

Eb is the bit energyN0 and σ2

w are the noise energy and variance

N0 = 2σ2w



−5 0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Pe(Γ) =12

(

1−√

Γ

1+Γ

)

Pe(γ) = Q(√

2γ)

Average SNR Γ (dB)

Pe

BPSK in AWGN

BPSK in Rayleigh



1 %--BPSK in AWGN--2 gamma_dB=-4:0.1:10;3 gamma_lin=10.ˆ(gamma_dB/10);4 ber_AWGN=qfunc(sqrt(2*gamma_lin));5

6 %--BPSK in Rayleigh--7 Gamma_dB=-4:0.1:40;8 Gamma_lin=10.ˆ(Gamma_dB/10);9 ber_Rayleigh=(1-sqrt(Gamma_lin./(Gamma_lin+1)))/2;

10

11 % Plot results12 semilogy(gamma_dB,ber_AWGN,’r-’,...13 Gamma_dB,ber_Rayleigh,’b--’);


Semi-analytical Performance

Semi-analytical BER with N sufficiently large:

Pe(γ) =1

N

N∑n=1

Q(√

2γ |h[n]|2)

Average effect of hn on γ.BER in AWGN is given as

Pe(γ) = Q(√

2γ)

γ is the signal to noise ratio given as

γ =EbN0


Semi-analytical Performance (Cont.)

−5 0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Pe(γ) =1N

N∑

n=1

Q(

√

2γ |hn|2)

Pe(γ) = Q(√

2γ)

Average SNR Γ (dB)

Pe

BPSK in AWGN

BPSK in Rayleigh

Semi−analytical


Semi-analytical Performance (Cont.)

1 N=1e6;2 gamma_dB_2=-5:5:45;3 gamma_lin=10.ˆ(gamma_dB_2/10);4 L=length(gamma_dB_2);5 ber_Semi_Rayleigh=zeros(1,L);6 for i=1:L7 h=1/sqrt(2)*(randn(1,N)+1j*randn(1,N));8 Pe=qfunc(sqrt(2*gamma_lin(i)*abs(h).ˆ2));9 ber_Semi_Rayleigh(i)=sum(Pe)/N;

10 end


Simulated Performance

Simulated system includes implementations of:Transmitter: generates b[n] and maps it to d[n],Channel: generates h[n] and w[n] and computes fordifferent SNR values

r[n] = h[n]d[n] + w[n]

Receiver: implements the decision rule and BERcomputation.

d[n] = Re{h∗[n]r[n]}d[n] ≤ 0 =⇒ b[n] = 1

d[n] > 0 =⇒ b[n] = 0

Visualization: displays probe signals and simulationresults.


Simulated Performance (Cont.)

h[n] w[n]

Mod.

d[n]b[n] r[n] = h[n]d[n] + w[n]

Tx RxComm.

Receiver

BER

Channel

Compute

Data

BPSKTransmitter

Estimated Data

b[n]d[n]

h[n], w[n]

r[n]b[n]

Pb = f(SNR)

d[n] = Re{h∗[n]r[n]}


Simulated Performance (Cont.)

−5 0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Pe(Γ) =12

(

1−√

Γ

1+Γ

)

Pe(γ) = Q(√

2γ)

Average SNR Γ (dB)

Pe

BPSK in AWGN

BPSK in Rayleigh

Simulation


Digital Modulation

Digital information is transmitted by varying the information ofeither amplitude a(t), frequency fi or phase φ(t) of a sinusoid:

s(t) = a(t) sin(

2πfi t+ φ(t))

Modulation types:Amplitude Shift Keying (ASK):a(t) conveys information, while fi and φ(t) constant.Frequency Shift Keying (FSK):fi conveys information, while a(t) and φ(t) constant.Phase-shift keying (PSK):φ(t) conveys information, while a(t) and fi(t) constant.Quadrature Amplitude Modulation (QAM):Two parameters are varied, i.e. amplitude a(t) and phaseφ(t).


Bandpass and Complex-Baseband

The information bearing signal is referred to as thecomplex-baseband or lowpass equivalent signal, sl(t),given as

sl(t) = a(t)ejφ(t) = a(t) cos(φ(t)) + ja(t) sin(φ(t))

= xI(t) + jxQ(t)

Bandpass signal is only required to reduce the effectiveantenna size required. It is given as

s(t) = Re{sl(t)ej2πfct} = Re{[xI(t) + jxQ(t)] ej2πfct}= xI(t) cos(2πfct)− xQ(t) sin(2πfct)

sl(t) is much lower frequency (typically ≤ 20 MHz) thans(t), which can be up to 60 GHz.Thus, it is computationally inefficient to include the carrierin the simulations.


Bandpass and Complex-Baseband (Cont.)

The entire system is simulated in most cases at the symbolrate to minimize simulation times.The symbol-rate, sampled complex-baseband signal isgiven as

sl(nT ) = a(nT )ejφ(nT ) = xI(nT ) + jxQ(nT )

In simulations, we only need xI(nT ) and xQ(nT ) or a(nT )and φ(nT ), with T being the symbol duration, which can bealso dropped for clarity.System/signal bandwidth and sampling frequency are onlyrequired for normalizing the times of multipath arrivals andfor Doppler effect simulations in time-varying systems


Typical Program Structure

This template can be downloaded from the website.

1 % Short description of simulation2 clear, clc, close all3

4 % Simulation parameter definitions5 M=16; % Constellation size6 ...7

8 % Reset BER variable9 BER=zeros(1,?);


Typical Program Structure (Cont.)

1 for m=... % SNR loop2 total_errors=0;3 for n=... % Averaging loop4 % Transmitter5 ...6 % Communications channel7 ...8 % Receiver9 ...

10 % Bit error computation11 new_errors=...12 total_errors=total_errors+new_errors;13 end14 BER(m)=...15 end


Typical Program Structure (Cont.)

1 % Plot results2 semilogy(?,?)3 ...4

5 % Store results in a file6 save ...


16 Quadrature Amplitude Modulation (QAM)

The theoretical bandwidth efficiency is 4 bit/s/Hzζ = 4 bits are grouped to create a symbolConstellation consists of M = 2ζ bits = 24 = 16 pointsM-QAM alphabet: {±(2ζ − 1)d± j(2ζ − 1)d}, whereζ ∈ {1, . . .

√M2 }

For 16-QAM, with M = 16 and ζ = 4 we obtain 4 values forthe I and Q channel, i.e.

I, Q ∈ {−3d, −d, d, 3d}

d is an arbitrary value, typically d = 1 or d = 13

This corresponds to 2 bits per channel.


Binary-Coded 16-QAM Constellation

0 C=[-3+3j,1 -3+1j,2 -3-1j,3 -3-3j,4 -1+3j,5 -1+1j,6 -1-1j,7 -1-3j,8 1+3j,9 1+1j,

10 1-1j,11 1-3j,12 3+3j,13 3+1j,14 3-1j,15 3-3j];

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

40000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

In−phase

Quadrature


Gray-Coded 16-QAM Constellation

0 C=[1

2

3

4

5

6

7

8

9

10

11

12

13

14

15 ];

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

40000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

In−phase

Quadrature


Constellation of Gray-Coded 16-QAM (Cont.)

For a constellation point z = x+ jy, the bit allocation, b0b1b2b3,is as follows

First bit, b0b0 = 0 if Re{z} ≤ 0b0 = 1 if Re{z} > 0

Second bit, b1b1 = 0 if Re{z} ≤ −2 or Re{z} ≥ 2b1 = 1 if Re{z} > −2 or Re{z} < 2

Third bit, b2b2 = 0 if Im{z} ≥ 0b2 = 1 if Im{z} < 0

Forth bit, b3b3 = 0 if Im{z} ≤ −2 or Im{z} ≥ 2b3 = 1 if Im{z} > −2 or Im{z} < 2


MATLAB Implementation for 16 QAM Tx

By writing our own code using the look-up table approach:

D[k] = C(b[k] + 1

)where

C is the constellation vector,k = 0, 1, 2, ..., N − 1 is the subcarrier index, andb[k] = {0, 1, . . .M − 1}.

To generate b[k] use: randi(?,?,?)To obtain help: >> doc randi


Random Number Generation

MATLAB by default generates always the same randomnumbers at the start of each session.To obtain different random values in different MATLABsessions include at the top of your code, right after clear,the following:

1 RN=sum(100*clock);2 RS=RandStream(’mt19937ar’,’seed’,RN);3 RandStream.setGlobalStream(RS);


Code Developement Lab 2

Develop a 16-QAM modulator using look-up tableapproach to generate a frame of 2048 16-QAM symbols.Add complex-valued random AWGN noise with smallstandard deviation (e.g. 0.25) and plot the resultingconstellation.

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

In−phase

Quadrature


Time-Frequency Relationship

The time-bandwidth product is defined as: B T = 1

The Fourier spectrum of a rectangular pulse

gT (t) = A rect(t

T)

FT←→ GT (f) = AT sinc (f T )

where the sinc( .) function defined as: sinc (x) = sin(πx)x

T

4

Tf (Hz)

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

GT(f)

0

0.25

0.5

0.75

1

TT4


Principle of OFDM

B

Channel

Frequen y

Response

Single Carrier

Multi Carrier

f∆f =B

N


Advantages of OFDM

Efficient spectrum usage.Resilience to frequency-selective multipath channels.Simplified receiver design: FFT + one-tap equalizer.Less sensitive to symbol timing and impulsive noise.Bit and power loading at subcarrier level possible.


Disadvantages of OFDM

OFDM signal exhibits high Peak-to-Average Power Ratio(PAPR).Very sensitive to carrier frequency offsets (CFO).Very sensitive to intercarrier interference (ICI) due to FFTleakage.


OFDM Transmitter Model

Symbol

OFDM

S/P

Insert Pilots

IFFT P/SModulator

16−QAM

Insert

CyclicPrefix

Symbols

16−QAM

x[n]D[n] d[n]b[n]


OFDM Signal Model

Definitions:

d[n] =1√N

N−1∑k=0

D[k]ej2πknN , n = 0, 1, . . . , N − 1

D[k] =1√N

N−1∑n=0

d[n]e−j2πknN , k = 0, 1, . . . , N − 1

D[k] is the 16-QAM modulated data sequence + Pilots3 + 3j.N is the length of the IDFT / DFT transform.n is the time sample index of the transmitted OFDMsymbol.k is the subcarrier index in frequency domain.


MATLAB Implementation of IFFT and FFT

IFFT equation (transmitter):

d[n] =1

N

N−1∑k=0

D[k]ej2πknN , n = 0, 1, . . . , N − 1

MATLAB implementation:

1 d=ifft(D);

FFT equation (receiver):

D[k] =

N−1∑n=0

d[n]e−j2πknN , k = 0, 1, . . . , N − 1

MATLAB implementation:

1 D_est=fft(d);


MATLAB Implementation (Cont.)

IDFT vs IFFT and DFT vs FFTIf N is a power of 2, MATLAB employs FFT, otherwise aslower DFT is used.ifft() and fft() accept a second a argument to specifythe transform size and use zero-padding.Useful oversampling of the channel impulse response, h(k)in order to obtain the correct frequency response H(n).No need to zero-pad manually.You can write your own code for IDFT/IDFT: two for loopsfor k and n.IFFT/FFT implementation more difficult.


Addition of Cyclic Prefix

CP CPOFDM Symbol

Copy last K samples

1 d=ifft(D);2 x=[d(?:?) d];

K is the cyclic prefix length added to reduce multipatheffects.The use of CP converts a frequency selective channel intoa set of parallel flat-fading independent channels.Alternatively zero-padding (ZP) can be used instead of CP.


Signal Energy Normalization

Matlab implementation utilizes an asymmetric energytransform implementation with respect to time-frequencydomain transition.Problem with required noise power computation for a givenSNR.Signal energy normalization required.Solution:

either by multiplying x with√N before adding Gaussian

noise,or by scaling the noise standard deviation appropriately by1/√N .



Develop the code to implementThe 16-QAM OFDM transmitter,Insertion of K-sample cyclic-prefix using concatenation,Energy normalization of the OFDM waveform.Original signal recovery by removing CP and applyinginverse transform.


Channel Simulation

w(n)

x[n] r[n] = x[n] + w[n]

Complex additive Gaussian noise:

1 w=sigma(?)*(randn(?,?)+1j*randn(?,?))

randn() generates zero-mean unity variance noisesamples.Appropriate scaling using the standard deviationsigma(?) is required.


Gaussian Probability Density Function (PDF)

Models the statistics of thermal noise in receivers knownas additive white Gaussian noise (AWGN).Appears in closed-form BER solutions for the performanceover AWGN channels.The PDF of a Gaussian distribution is defined as

fw(x) =1√

2πσwe− (x−µw)2

2σ2w

where µw is the mean value and σ2w is the variance.

σw is called the standard deviation.It is also referred to as normal, N .To indicate that a random variable w has a Gaussian ornormal pdf fw(x), we write

w → N (µw, σ2w)


Gaussian pdf

−4 −3 −2 −1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

fw(x)

∫∞

−∞fw(x)dx = 1

µw = 0, σ2w = 1µw = 0, σ2w = 0.5µw = 0, σ2w = 0.25µw = 1, σ2w = 1µw = −1, σ2w = 1


Gaussian Cumulative Density Function (CDF)

In general, the CDF is defined as

Fw(x) = P (w ≤ x) =

∫ x

−∞fw(u)du

CDF propertiesFw(x) ≥ 0, ∀x

Fw(−∞) = 0, Fw(∞) = 1

The CDF of a Gaussian distribution is given as

Fw(x) =1

2+

1

2erf

(x− µw√

2σw

)=

1

2− 1

2erfc

(x− µw√

2σw

)= 1−Q

(x− µxσw

)Dr. Charalampos C. Tsimenidis EEE8128 Communications and Signal Processing 49 / 94

Gaussian CDF and PDF

−4 −3 −2 −1 0 1 2 3 4

0

0.5

1

1.5

x

fw(x),Fw(x)

Gaussian cdf and pdf for µw = 0,σ2w = 1

Fw(−∞) = 0

Fw(∞) = 1

fw(x)Fw(x)


The erf(x), erfc(x) and Q(x) Functions

The error function

erf(x) =2√π

∫ x

0e−t

2dt

The complementary error function

erfc(x) =2√π

∫ ∞x

e−t2dt

Relationship: erfc(x) = 1− erf(x)

The Q( .) function:

Q(x) =1√2π

∫ ∞x

e−t22 dt

Relationship between Q(x) and erfc(x):

Q(x) =1

2erfc

(x√2


Example: BPSK in AWGN

w[n]

d[n]Mod.

x[n] = d[n] + w[n]b[n]

0 → +11 → −1

Assuming equi-probable d[n] symbols, their PDF is given as

pd(χ) =1

2δ(χ−

√Eb) +

1

2δ(χ+

√Eb)

The PDF of w[n] is given as

pw(χ) =1√

2πσwe− x2

2σ2w

The PDF of x[n] is then given as

px(χ) =1

2

1√2πσw

e− (χ+

√Eb)

2

2σ2w +1

2

1√2πσw

e− (χ−

√Eb)

2

2σ2w


Example: BPSK in AWGN (cont.)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2

The probability of a correct decision is given as

Pc = P{x[n] ≤ 0|b[n] = 1} = F (0) = 1−Q(

0− (−√Eb)σw

)Thus, the probability for a wrong decision is

Pe = Pb = 1− Pc = 1− 1 +Q

(√Ebσw

)= Q

(√Ebσw


Example: BPSK in AWGN (cont.)

With N0 = 2σ2w, we can write√Ebσw

=

√Eb√N0/2

=

√2EbN0

=√

2γb

Thus, with Pe = Pb, the probability of a bit error for BPSK isgiven as

Pb = Q(√

2γb

)or in terms of the complementary error function

Pb =1

2erfc (

√γb)


SNR Computation

The SNR, γ, per bit is computed as follows:

γb =EbN0

The average energy per bit, Eb, is computed from thesymbol energy, Es, as:

Eb =Es

log2(M)

The noise energy is computed as:

N0 = 2σ2w

We need to solve for σw =√. . .


Es Computation

Es is computed from the 16-QAM constellation as follows:

Es =1

M

M−1∑m=0

|C[m]|2 =?

Reminder complex numbers: z ∈ C

z = a+ jb⇒ |z|2 =(√

a2 + b2)2

= a2 + b2


16-QAM BER in AWGN

For M = 2 (2-QAM/BPSK) and M = 4 (4-QAM/QPSK):

Pb = Q

(√2EbN0

)

For M >> 4, ζ = log2(M)

Pb =PMζ

wherePM ≈ 1− (1− P√M )2

with

P√M = 2

(1− 1√

M

)Q

(√3ζ

M − 1

EbN0

)


16-QAM BER in AWGN (Cont.)

0 2 4 6 8 10 12 14 1610

-6

10-5

10-4

10-3

10-2

10-1

100

BPSK in AWGN

16-QAM in AWGN



Develop MATLAB code thatComputes the standard deviation, σw of thecomplex-valued AWGN samples for 16-QAM in AWGN, fora given SNR value in dB.Generates random AWGN noise wn scaled by the correctσw and adds it to the OFDM signal.Generates the BER vs Eb/N0 plots for BPSK and 16-QAMin AWGN.


Receiver Structure for AWGN

16−QAM

SymbolsRemoveS/P FFTOFDM

Received

Symbol

CyclicPrefix

P/SDemodul.

16−QAM

CP is discarded at the receiver to avoid interblockinterference.DFT/FFT is performed.Channel estimation is not required for the AWGN channel.16-QAM demodulation using the minimum Euclideandistance approach, implemented via custom MATLABcode.


Demodulator Input Signal

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

In−phase

Quadrature

γ =Eb

N0

= 16 dB

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

In−phase

Quadrature

γ =Eb

N0

= 6 dB


16-QAM Demodulator

Decision metric:

bk = argi

min(|Ci − Dk|), i = 0, 1 . . . ,M − 1, M = 16

where |Ci − Dk| is the Euclidean distance defined as

|Ci − Dk| =√[

Re(Ci)− Re(Dk)]2

+[Im(Ci)− Im(Dk)

]2

Re{C}

Im{C}

C7

C2

C3

C6

C1

C0 C4

C5

C8

C9

C12

C10

C11

C13

C15

C14

Dk


Displaying Information about Simulation

1 str=sprintf(’SNR=%0.1f,BER=%0.6f\n’,10.5,0.1);2 disp(str);3

4 SNR=10.5,BER=0.100000


Measuring Simulation Time

1 tic2 ...3 toc4

5 % or6 t_start=cputime;7 ...8 cputime-t_start9

10 % or (not recommended)11 t_start = clock;12

13 etime(clock, t_start)


Parallel Computing with parfor

Use parfor for the outer loop only.Iterations will be distributed on the CPU cores (workers)available.The first time parfor is called may take some time.

1 % Outer SNR loop2 parfor m=...3

4 ...5

6 end



Develop MATLAB code thatRemoves the CP and computes the FFT of the receivednoisy signal.Estimates using the minimum Euclidean distance thetransmitted 16-QAM symbols.Computes and displays the BER vs. SNR performance byaveraging Navg-times per SNR point.Displays information about simulation during codeexecution.Measures the total simulation time elapsed.Simulates the performance using the parallel configuration.


Multipath Channel Scenario

Mobile Station (MS)

Reflector Cluster

Base Station (BS)

Impulse ResponseSingle Reflector

τ21τ22

τ32

τ1

τ31

τ4τ3τ2τ1

τi = τi1 + τi2 i = 2, 3, 4

τ41 τ42

τ

a1e−j2πfcτ1

a2e−j2πfcτ2

a3e−j2πfcτ3

a4e−j2πfcτ4

h(τ ; t)


Channel Impulse Response (CIR)

The complex-baseband impulse response of the channel isgiven as

h(τ ; t) =

L∑l=1

al(t)e−jφl(t)δ(t− τl(t))

The phase shifts are φl(t) are proportional to thecorresponding time delays τl(t), that is

φl(t) = 2πfcτl(t) + θ(t)

where fc is the carrier frequency and θ(t) is a randomphase component.


Simplified CIR Models

Constant arrival times:

τl(t) = τl

h(τ, t) =

L∑l=1

al(t)e−jφl(t)δ(t− τl)

Static multipath:

τl(t) = τl, al(t) = al, φl(t) = φl, θ(t) = 0

h(τ) =

L∑l=1

ale−jφlδ(t− τl)


Multipath Channel Implementation

Multipath channel treated as linear time-invariant system:Multipath Channel

x[n] y[n]

w[n]

h[n]r[n] = y[n] + w[n] = h[n] ∗ x[n] + w[n]

Discrete convolution (filtering):

y[n] = h[n] ∗ x[n] =

∞∑m=−∞

h′[m]x[n−m]

If x[n] ∈ C1×N and h[n] ∈ C1×L then length of y[n] ∈ C1×M

isM = N + L− 1


FIR-Based Multipath Channel Model

z−1x[n− 1]x[n]

y[n]

hL−1hL−2h1h0

x[n− L− 2]z−1

x[n− L− 1]


MATLAB Implementation

Preferred MATLAB Implementation:

1 % Initial filter state2 zf=[];3 ...4 % Multipath channel5 [?,?]=filter(?,?,?,?);6 r=y+w; % Add WGN

This implementation keeps track of the channel statebetween OFDM symbols.Interblock interference can be simulated.Documentation: >>doc filter

In general, y=conv(h,x)+w; can be utilized, but not asuseful.


ITU Channel for Mobile WiMAX

Multipath magnitudes hi need to be obtained fromP = [0 -0.9 -4.9 -8.0 -7.8 -23.9];

Appropriate zero-padding is required between the paths.

l, Normalized Delay spread0 2 9 14 26 42

P[l](dB)

-25

-20

-15

-10

-5

0ITU Pedestrian B Ch. 103


Channel Energy Normalization

The channel magnitudes are obtained as

h[l] =

√10

P [l]10

Channel profile needs to be normalized to unity energy

L∑l=0

|h[l]|2 = 1

If {h[l]}Ll=0 are the unnormalized channel coefficients thenin order to normalize we use:

U =

√√√√ L∑l=0

|h[l]|2

h[l] =h[l]

U, l = 0, 1, 2, . . . , L− 1


Performance of BPSK in Multipath

Identical to semi-analytical BER performance since channel isstatic. For BPSK that is

Pb(γb) =1

N

N−1∑k=0

Q(√

2γb |H[k]|2)

whereγb = Eb

N0is the signal to noise ratio per bit.

N is the number of OFDM subcarriers.k = 0, 1, . . . , N − 1 is the subcarrier index.The frequency response of the channel, H[k], is obtainedfrom {h[l]}Ll=1 as

H[k] =

L∑l=0

h[l]e−j2πlkN


BER Performance of 16-QAM in Multipath

For 16-QAM, with M = 16, ζ = log2(M) = 4, that is

Pb 'PMζ

withPM ' 1− (1− P√M )2

and

P√M = 2

(1− 1√

M

)1

N

N−1∑k=0

Q

(√3ζ

M − 1γb |H[k]|2

)


Code Developement Lab 6-7

Develop MATLAB code with regard to the ITU multipathchannel that:

Implements and normalizes its impulse response.Filters the OFDM signal through the channel and addscomplex-valued AWGN noise for a given SNR value in dB.Computes and displays its frequency response. Note: usethe oversampling property of FFT to obtain the frequencyresponse for N subcarriers.Computes and displays the theoretical BER vs. Eb/N0

performance for BPSK and 16-QAM through this channel.Computes and displays the simulated BER vs. Eb/N0

performance for 16-QAM through this channel using areceiver without equalizer.


Expected Performance Results

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

Eb

N0(dB)

Pb

BPSK AWGN

BPSK MP

16−QAM AWGN (Theory)

16−QAM MF AWGN (Sim.)

16−QAM MP (Semi.)

16−QAM MF MP (Sim.)


Generic Receiver Structure

16−QAM

SymbolsRemoveS/P P/SFFT

16−QAMOne−tap

Equalizer Demodul.OFDM

ReceivedCyclicPrefixSymbol

CP is discarded at the receiver to avoid interblockinterference.DFT/FFT is performed.One-tap per subcarrier zero-forcing design to removechannel effect.Channel estimation: impulse response known or unknown.16-QAM demodulation using custom code (same code asin AWGN case).


Effect of Cyclic Prefix

Reminder: the OFDM signal can be written as

d = FHD

where D ∈ CN×1 are the 16-QAM modulated symbols andF is the DFT matrix with

Fm,n =1√Ne−j2π

mnN , m, n = 0, 1, . . . , N − 1

The received signal after linear convolution and CPremoval is given as

r = Hd + w

where H ∈ CN×N is the circulant channel matrix.After FFT at the receiver, we can write

Y = Fy = F(Hd + w) = FHFH︸︷︷︸Λ

D + Fw︸︷︷︸W

= ΛD + W


Mathematical Model of the Received Signal

Λ is a diagonal matrix, where the main diagonal is theCFR.Circulant matrices exhibit the following property

FHFH = diag(H[0], H[1], . . . , H[N − 1]) = Λ

The FFT output can be given as

Y [k] = H[k]D[k] +W [k], k = 0, 1, 2, ...N − 1

To remove the channel effects, we need to divide Y [k] bythe complex-valued channel coefficient, H[k], of thechannel frequency response (CFR).Hence, the equalizer one-tap weight are given as

CZF [k] =1

H[k]=

H∗[k]

|H[k]|2 , k = 0, 1, 2, ...N − 1


Example: Proakis Channel B

OFDM system with NFFT = 4, NCP = 2, thus,N = NFFT +NCP = 6

Channel impulse response:

h(z−1) = 0.407 + 0.815z−1 + 0.407z−2

Circulant channel matrix (time-domain):

H =

0.407 0 0 0 0.407 0.8150.815 0.407 0 0 0 0.4070.407 0.815 0.407 0 0 0

0 0.407 0.815 0.407 0 00 0 0.407 0.815 0.407 00 0 0 0.407 0.815 0.407


Circulant Matrix Demo

1 % OFDM system specs2 Nfft=4; Ncp=2; N=Nfft+Ncp;3

4 % Proakis B channel CIR5 h=[0.407 0.815 0.407]; L=length(h);6

7 % Circulant matrix8 H=gallery(’circul’,[h zeros(1,N-L)]).’;9

10 % DFT matrix11 F=1/sqrt(N)*dftmtx(N);12

13 % Channel frequency response (CFR)14 Lambda=diag(F*H*F’);15

16 % CFR via FFT17 Hf=fft(h,N);


Equalizer for Known Channel

After CP removal, FFT and zero forcing equalizer, thedecision variable becomes

D[k] = CZF [k]Y [k] =Y [k]

H[k]=H[k]D[k] +W [k]

H[k]

= D[k] +W [k]

H[k], k = 0, 1, 2, ...N − 1

Noise enhancement occurs if channel frequency responseexhibits spectral nulls and the noise term W [k]/H[k] is nolonger Gaussian.To prevent this, a Minimum Mean-Squared Error (MMSE)based equalizer can be be used instead designed as

CMMSE [k] =H∗[k]

|H[k]|2 +2σ2W

σ2D

, k = 0, 1, 2, ...N − 1

however, BER performance will only improve for codedsystems.


Computation of CFR

At this point, we assume that the CIR is perfectly known.The CFR can be computed directly from the CIR as

H[k] =

N∑n=0

h[n]e−j2πnkN , k = 0, 1, . . . , N − 1

where h[n] = 0, ∀n ≥ L− 1

In MATLAB, we can use H=fft(h,N) to obtain the CFR.The second argument applies zero-padding to h, which isequivalent to oversampling of the CFR in frequencydomain to N samples.



Develop MATLAB code thatComputes the ZF and MMSE equalizer coefficients(computation is required only once as the channel istime-invariant).Implements the modified 16-QAM receiver using theone-tap weights.Computes and displays the simulated BER vs Eb/N0

performances for both equalizers.


Equalizer for Unknown Channel

Insert 1 pilot for every 8 subcarriers at the transmitter:

−1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0

0.2

0.4

0.6

0.8

1

|Hest(k)|

subcarrier index, k

Data

Pilots

Estimates

Estimate unknown channel H[p] at pilot indices p.


Linear Interpolation

Interpolate to obtain in-between channel values.The line equation connecting two points is given as

y − y1

y2 − y1=

x− x1

x2 − x1

y

x2x1

y1

y2

x

p p+ 1 p (Pilot index)

k k (OFDM index)

H(p+ 1)

H(p)

k + I


Linear Interpolation (Cont.)

Interpolation equation:

H[k] = H[p] +i

I

(H[p+ 1]− H[p]

)Indices:

p = 0, 1, . . . , P − 2,

i = 1, 2, . . . , I − 1,

k = p I + i

with P = 256 and I = 8


Linear Interpolation (Cont.)

For the data subcarriers after the P -th pilot we need toextrapolate the equation from the last two pilots, e.g.H(P − 2) and H(P − 1) and p = p+ P .

extrapolate

Np (Pilot index)

k (OFDM index)

H[P − 1]

H[P − 2]

P − 1P − 2

(P − 1)I(P − 2)I

H[k]


ZF Equalizer for Unknown Channel

Computing the zero-forcing equalizer

CZF [k] =1

H[k], k = 0, 1, 2, ...N − 1, k 6= p

Filtering the received signal

D[k] = CZF [k]D[k] =D[k]

H[k], k = 0, 1, 2, ...N − 1, k 6= p

BER should not be computed at the pilot locations, e.g.k = p

Thus, the total number of bits should be adjusted toaccount for the pilot overhead.


Estimated and True Frequency Response

0 500 1000 1500 20470

0.5

1

1.5

2

2.5

Subcarrier Index k

H[k],H[k]

γ = 30 dB, K = 64

H[k]

H [k]


MSE vs. SNR For Different CPs

0 5 10 15 20 25 30−40

−35

−30

−25

−20

−15

−10

Eb

N0(dB)

MSE

(dB)

K=0

K=8

K=16

K=32

K=64

Mean Squared Error computation:

MSE = 10 log10

(1

N

N∑n=0

|Hn − Hn|2)

dB



Develop MATLAB code thatEstimates the frequency response of the multipath channelusing pilots and computes the zero-forcing equalizercoefficients.Implements the modified 16-QAM receiver using one-tapequalizer and channel estimation.Computes and displays the BER vs Eb/N0 performancewith channel estimation.Computes the mean-squared error (MSE) as the functionof SNR in dB between the estimated and actual frequencyresponse of the channel.

For higher marks:Quadratic, Cubic and spline based interpolation may beimplemented.More sophisticated pilot schemes, e.g. comb and shift.


EEE8128 Communications and Signal Processing: Part 1 ...

Documents

Transcript of EEE8128 Communications and Signal Processing: Part 1 ...