EEE8077 Simulation of Wireless Communications · Γis the average signal to noise ratio ... BPSK in...

EEE8077 Simulation of WirelessCommunications

Dr. Charalampos C. Tsimenidis

Newcastle UniversitySchool of Electrical and Electronic Engineering

October 2013

Dr. Charalampos C. Tsimenidis EEE8077 Simulation of Wireless Communications 1 / 78

Motivation

Performance evaluation of wireless communicationsystems:

Theoretical or analytical approach (desired solution),

Semi-analytical approach (alternative solution),

Simulation-based approach (practical solution).


Example: Theoretical Performance

Bit-error rate (BER) performance of Binary Shift Keying(BPSK) modulation in Rayleigh fading in the presenceAdditive White Gaussian Noise (AWGN) channel.

System Model:

dn

hn wn

xn = hndn + wn

wherehn are the complex-valued Rayleigh fadingcoefficients with variance 2σ2

h,dn =±1 are the BPSK symbols,wn are the complex-valued AWGN samples.


Theoretical Performance (Cont.)

Theoretical BER (closed-form solution):

Pe(Γ) =12

(

1−√

Γ1+Γ

)

where

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

Γ =Eb

N02σ2

h

Eb is the bit energy

N0 and σ2w 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 AWGNBPSK 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=0.5 * (1- sqrt (Gamma_lin./(Gamma_lin

+1)));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) =1N

N

∑n=1

Q

(√

2γ |hn|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 =Eb

N0


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 AWGNBPSK in RayleighSemi−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

Implement system using:

Transmitter by generating bn and mapping it to dn

Channel by generating hn and wn and computing fordifferent SNR values

xn = hndn+wn

Receiver by implementing decision rule and BERcomputation.

Display simulation results.


Simulated Performance (Cont.)

dn

hn wn

xn = hndn + wn

Tx RxComm.

Transmitter Receiver

BER

Channel

Compute

BPSK

Estimated DataData

Pe = f(SNR)

b̂nbn

hn, wn

dn xn


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 AWGNBPSK in RayleighSimulation


Digital Modulation

Digital information is transmitted by varying theinformation of either amplitude a(t), frequency fi(t) orphase φ(t) of a sinusoid:

s(t) = a(t)sin[2π fi(t)t +φ(t)]

Modulation types:Amplitude Shift Keying (ASK):a(t) conveys information, while fi(t) and φ(t) constant.Frequency Shift Keying (FSK):fi(t) 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) andphase φ(t).


Bandpass and Complex-Baseband Signals

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

sl (t) = a(t)ejφ(t) = xI (t)+ jxQ(t)

Bandpass signal is only required to reduce theeffective antenna size required. It is given as

s(t) = Re{sl(t)ej2π fct}

= Re{[xI (t)+ jxQ(t)]ej2π fct}= xI (t)cos(2π fct)−xQ(t)sin(2π fct)

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


Bandpass and Complex-Baseband Signals(Cont.)

The entire system is simulated in most cases at thesymbol rate to minimize simulation times.The symbol-rate, sampled complex-baseband signalis given as

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

In simulations, we only need xI (nT) and xQ(nT) ora(nT) and φ(nT), with T being the symbol duration,which can be also dropped for clarity.System/signal bandwidth and sampling frequency areonly required for normalizing the times of multipatharrivals and for Doppler effect simulations intime-varying systems


Typical Program Structure

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=...15 end


Typical Program Structure (Cont.)

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

5 % Store results in a file6 save ...


16-Point Quadrature Amplitude Modulation(16-QAM)

The theoretical bandwidth efficiency is 4 bit/s/Hz

k = 4 bits are grouped to create a symbol

Constellation consists of M = 2kbits = 24 = 16 points

M-QAM alphabet: {±(2k−1)d± j(2k−1)d}, wherek∈ {1, . . .

√M2 }

For 16-QAM, with M = 16 and k = 4 we obtain 4values for the 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.


Constellation of Binary-Coded 16-QAM

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

Qua

drat

ure


Constellation of Gray-Coded 16-QAM

1 C=[-3+3j, -3+1j,2 -3-3j, -3-1j,3 -1+3j, -1+1j,4 -1-3j, -1-1j,5 3+3j, 3+1j,6 3-3j, 3-1j,7 1+3j, 1+1j,8 1-3j, 1-1j];

−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

Qua

drat

ure


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

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

First bit, b0

b0 = 0 if Re{z} ≤ 0b0 = 1 if Re{z}> 0

Second bit, b1

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

Third bit, b2

b2 = 0 if Im{z} ≥ 0b2 = 1 if Im{z}< 0

Forth bit, b3

b3 = 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 look-up table:

D(k) = C(bk +1)

where C is the constellation vector and sk is thesymbol index, e.g. k = 0,1,2, ...,M−1.

To generate bk use: randi(?,?,?)

Help: >> doc randi


Random Number Generation

MATLAB by default generates always the samerandom numbers at the start of each session.

To obtain different random values in differentMATLAB sessions 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-QAMsymbols.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

Qua

drat

ure


Time-Frequency Relationship

The time-bandwidth product is defined as: BT = 1

The Fourier spectrum of a rectangular pulse

pT(t) = Arect(tT

)FT←→ PT( f ) = ATsinc( f T)

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

T

4

T−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 80

0.25

0.5

0.75

1

f (Hz)

PT(f)

TT4


Principle of OFDM

B ChannelFrequen yResponseSingle CarrierMulti 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 PowerRatio (PAPR).

Very sensitive to carrier frequency offsets (CFO).

Very sensitive to intercarrier interference (ICI) due toFFT leakage.


OFDM Transmitter Model

SymbolOFDM

S/P16−QAM

Insert Pilots

Modulator IFFT P/SSymbols16−QAM

Insert

PrefixCyclic

[0, 1, . . . , 15] x(n)

d(n)


OFDM Signal Model

Definitions:

d[n] =1√N

N−1

∑k=0

D[k]ej 2πknN , n = 0,1, . . . ,N−1

D̂[k] =1√N

N−1

∑n=0

d[n]e− j 2πknN , k = 0,1, . . . ,N−1

D[n] is the 16-QAM modulated data sequence +Pilots 1+ j.

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

d[n] =1N

N−1

∑k=0

D[k]ej 2πknN , n = 0,1, . . . ,N−1

1 d=ifft (D);

D̂[k] =N−1

∑n=0

d[n]e− j 2πknN , k = 0,1, . . . ,N−1

1 D_est= fft (d);


MATLAB Implementation (Cont.)

IDFT vs IFFT and DFT vs FFT

If N is a power of 2, MATLAB employs FFT, otherwisea slower DFT is used.

ifft () and fft () accept a second a argument tospecify the transform size and use zero-padding.

No need to zero-pad manually.

Useful oversampling of the channel impulseresponse, h(k) in order to obtain the correctfrequency response H(n).


Oversampling Effect (Channel Estimation,PAPR)

1 H=fft (h,Nfft);

0 5 10 15 20 25 320

0.5

1

1.5

a) Without oversampling.

|d[n

]|

0 50 100 150 200 2560

0.05

0.1

0.15

0.2

b) With oversampling by 8.

|d[n

]|


Addition of Cyclic Prefix

CP CPOFDM Symbol

Copy last K samples1 d=ifft (D);2 x=[d(?:?) d];

K is the cyclic prefix length added to reduce multipatheffects.


Cyclic Prefix (Cont.)

DFT Property: replication in one domain correspondsto interpolation in the other.

Thus, if a signal is replicated by M in time domain, itsDFT is zero interpolated by M(and scaled).

The use of CP converts a frequency selectivechannel into a set of parallel flat-fading independentchannels.

Some papers suggest alternatively zero-padding (ZP)instead of CP.


Energy Normalization

Asymmetric energy transform implementation withrespect to time-frequency domain transition.

Problem with required noise power computation forgiven SNR.Signal energy normalization required.Implementation:

either by multiplying x with√

N before addingGaussian noise,or by scaling the noise standard deviationappropriately by 1/

√N.



Develop the code to implement

The 16-QAM OFDM transmitter,

insertion of K-sample cyclic-prefix usingconcatenation,

Energy normalization of the OFDM waveform.


Channel Simulation

y(n) = x(n) + w(n)

w(n)

x(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 receiversknown as additive white Gaussian noise (AWGN).Appears in closed-form BER solutions for theperformance over 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 Gaussianor normal pdf fw(x), we write

w→N (µw,σ2w)


Gaussian pdf

−4 −3 −2 −1 0 1 2 3 40

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) =

Z x

−∞fw(u)du

CDF propertiesFw(x)≥ 0, ∀x

Fw(−∞) = 0, Fw(∞) = 1

The CDF of a Gaussian distribution is given as

Fw(x) =12

+12

erf

(x−µw√

2σw

)

=12− 1

2erfc

(x−µw√

2σw

)

= 1−Q

(x−µx

σw

)


Gaussian CDF and PDF

−4 −3 −2 −1 0 1 2 3 40

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

Z x

0e−t2

dt

The complementary error function

erfc(x) =2√π

Z ∞

xe−t2

dt

Relationship: erfc(x) = 1−erf(x)

The Q( .) function:

Q(x) =1√2π

Z ∞

xe−t2

2 dt

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

Q(x) =12

erfc

(x√2

)


SNR Computation

The SNR is computed as follows:

SNR=Eb

N0

The average bit energy Eb computed from theconstellation:

Eb =Es

log2(M)

The noise energy is computed as:

N0 = 2σ2w

We need to solve for σw =√

. . .


Es Computation

Es is computed as follows:

Es =1M

M−1

∑n=0|cn|2

=116

(4|3d+ j3d|2+8|3d+ jd|2+4|d+ jd|2

)

=116

(72d2+80d2+8d2)=

16016

d2

= 10d2

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

(√2Eb

N0

)

For M >> 4, k = log2(M)

Pb =PM

k

wherePM ≈ 1− (1−P√M)2

with

P√M = 2

(

1− 1√M

)

Q

(√

3kM−1

Eb

N0

)


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

Pe(γ) = Q(√

2γ)

γ = Eb

N0(dB)

Pe

BPSK in AWGN16−QAM in AWGN



Develop MATLAB code that

Computes the standard deviation for 16-QAM inAWGN, for a given SNR value given in dB.

Generates random AWGN noise wn scaled by thecorrect σw and adds it to the OFDM signal.

Generates the BER vs Eb/N0 plots for BPSK and16-QAM in AWGN.


Receiver Structure for AWGN

16−QAMSymbolsRemove

S/P FFTOFDMReceived

SymbolCyclicPrefix

P/SDemodul.

16−QAM

CP is discarded at the receiver to avoid interblockinterference.

DFT/FFT is performed.

Channel estimation is not required for AWGNchannel.

16-QAM demodulation using custom code.


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

Qua

drat

ure

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

Qua

drat

ure

γ = Eb

N0= 6 dB


16-QAM Demodulator

Decision metric:

b̂k = argi

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

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

|Ci− D̂k|=√[Re(Ci)−Re(D̂k)

]2+[Im(Ci)− Im(D̂k)

]2

Re{C}Im{C}

C7

C2

C3

C6

C1

C0 C4

C5

C8

C9

C12

C10

C11

C13

C15

C14

D̂k


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 Configuration

(Menu) Parallel −→ Manage Cluster Profiles


Parallel Computing with parfor

1 % Start default worker pool using2 matlabpool open3 %...4

5 % Outer loop6 parfor m=...7

8 end9

10 % Close worker pool11 matlabpool close




Removes the CP and computes the FFT of thereceived noisy signal.

Estimates using the minimum Euclidean distance thetransmitted 16-QAM symbols.

Computes and displays the BER vs. SNRperformance by averaging P-times per SNR point.

Displays information about simulation during codeexecution.

Measures the total simulation time elapsed.

Simulates the performance using the parallelconfiguration.


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 thechannel is given

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) = 2π fcτl(t)

where fc is the carrier frequency.


Simplified CIRs

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

h(τ) =L

∑l=1

ale− jφl δ(t− τl )


Multipath Channel Implementation

Multipath channel treated as linear time-invariantsystem:

Multipath Channel

w(n)

x(n)h(n)

y(n) z(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 ofy(n) ∈C1×M is

M = N+L−1


Multipath Channel Implementation (Cont.)

Preferred MATLAB Implementation:

1 % Initial filter state2 zf=[];3 ...4 % Multipath channel5 [y,zf]= filter (h,1,x,zf);6 z=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 notas useful.


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 thepaths.

0 5 18 27 52 84−25

−20

−15

−10

−5

0

Delay spread (Samples)

P(dB)

ITU Pedestrian B Ch. 103


Channel Normalization

The channel magnitudes are obtained as

hi =

√

10Pi10

Channel profile needs to be normalized to unityenergy

L

∑i=1|hi|2 = 1

If {hi}Li=1 are the unnormalized channel coefficientsthen in order to normalize we use:

U =

√L

∑i=1|hi|2

hi =hi

U, i = 1, 2, 3, . . . , L


Theoretical Performance in Multipath forBPSK

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

Pe(γ) =1N

N

∑n=1

Q

(√

2γ |Hn|2)

whereHn is the frequency response of the ITU channel.N is here the FFT length.The frequency response is obtained applying the FFTon the vector {hi}Li=1 by oversampling by N.γ is the signal to noise ratio given as

γ =Eb

N0


Theoretical Performance in Multipath for16QAM

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

P√M = 2

(

1− 1√M

)1N

N

∑n=1

Q

(√

3kM−1

Eb

N0|Hn|2

)

withPM ≃ 1− (1−P√M)2

andPb≃

PM

k


Code Developement Lab 6-7

Develop MATLAB code that with regard to the ITUmultipath channel:

Implements and normalizes its impulse response.

Filters the OFDM signal through the channel andadds complex-valued AWGN noise for a given SNRvalue in dB.

Computes and displays its frequency response.

Computes and displays the theoretical BER vs.Eb/N0 performance for BPSK and 16-QAM throughthis channel.

Computes and displays the simulated BER vs. Eb/N0

performance for 16-QAM through this channel usinga receiver without equalizer.


Generic Receiver Structure

16−QAMSymbolsRemove

S/P P/SFFT16−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 orunknown.16-QAM demodulation using custom code (samecode as in AWGN case).


Mathematical Model of the Received Signal

The received signal at the FFT output can be given as

Y(k) = D(k)H(k)+W(k), k = 0,1,2, ...N−1

To remove the channel effects, we need to divide Y(k)by the complex-valued channel coefficient, H(k), ofthe channel frequency response.Thus, the decision variable becomes

D̂(k) = D(k)+W(k)H(k)︸︷︷︸

W̃(k)

, k = 0,1,2, ...N−1

The noise term W̃(k) is no longer Gaussian.Noise enhancement occurs if channel frequencyresponse exhibits spectral nulls.


Equalizer for Known Channel

We assume h(n) is perfectly known.H(k) is derived from the impulse response h(n) viaDFT/FFT.Oversampling of H(k) is required to match the lengthof an OFDM symbol.

1 N=2048;2 ...3 H=fft (h,N);

Detection variable after CP removal, FFT and zeroforcing:

D̂(k) =Y(k)H(k)

= D(k)+W̃(k), k = 0,1,2, ...N−1




Computes the zero-forcing equalizer coefficients(computation is required only once as the channel istime-invariant).

Implements the modified 16-QAM receiver usingone-tap equalizer.

Computes and displays the simulated BER vs Eb/N0

performance.


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 17 180

0.2

0.4

0.6

0.8

1

|Hest(k)|

subcarrier index, n

DataPilotsEstimates

Estimate unknown channel at pilot indices H(p).


Linear Interpolation

Interpolate to obtain in-between channel values.Line equations connecting two points:

y−y1

y2−y1=

x−x1

x2−x1

y

x2x1

y1

y2

x

n n + L

m m + 1n (OFDM index)

Hp(m + 1)

Hp(m)

m (Pilot index)Dr. Charalampos C. Tsimenidis EEE8077 Simulation of Wireless Communications 73 / 78

Linear Interpolation (Cont.)

Interpolation equation:

Hest(k) = Hp(m)+lL[Hp(m+1)−Hp(m)]

Indices:m= 1,2, . . . ,Np−1,

l = 1,2, . . . ,L−1,

k = (m−1)L+ l +1


Linear Interpolation (Cont.)

For the data subcarriers after the Npth pilot we needto extrapolate the equation from the last two pilots,e.g. Hp(Np−1) and Hp(Np) and l = l +L.

extrapolate

Np − 1 Np m (Pilot index)n (OFDM index)

Hest(n)

N

Hp(Np − 1)

Hp(Np)

L (Np − 2) L (Np − 1)


Estimated Frequency Response

0 500 1000 1500 20000

0.5

1

1.5

2

2.5

subcarrier index

|H(k

)|,|H

est(k

)|

|H(k) ||Hest(k) |


Equalizer for Unknown Channel

Zero forcing equalizer:

D̂(k) =Y(k)

Hest(k), k = 0,1,2, ...N−1, k 6= m

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

Total number of bits should be adjusted to account forthe pilot overhead.




Estimates the frequency response of the multipathchannel using pilots and computes the zero-forcingequalizer coefficients.

Implements the modified 16-QAM receiver usingone-tap equalizer and channel estimation.

Computes and displays the BER vs Eb/N0

performance with channel estimation.

Computes the mean-squared error as the function ofSNR in dB between the estimated and actualfrequency response of the channel.


EEE8077 Simulation of Wireless Communications · Γis the average signal to noise ratio ... BPSK in...

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