EEE8077 SIMULATION OF WIRELESS COMMUNICATIONS

EEE8077 Simulation of Wireless Communications Dr. Charalampos C. Tsimenidis Newcastle University School of Electrical and Electronic Engineering

October 2013

Dr. Charalampos C. Tsimenidis

EEE8077 Simulation of Wireless Communications

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Motivation

Performance evaluation of wireless communication systems: Theoretical or analytical approach (desired solution), Semi-analytical approach (alternative solution), Simulation-based approach (practical solution).



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Example: Theoretical Performance Bit-error rate (BER) performance of Binary Shift Keying (BPSK) modulation in Rayleigh fading in the presence Additive White Gaussian Noise (AWGN) channel. System Model: xn = hn dn + wn

dn

hn

wn

where hn are the complex-valued Rayleigh fading coefficients with variance 2σ2h , dn = ±1 are the BPSK symbols, wn are the complex-valued AWGN samples. Dr. Charalampos C. Tsimenidis


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Theoretical Performance (Cont.) Theoretical BER (closed-form solution): ! r Γ 1 1− Pe (Γ) = 2 1+Γ where Γ is the average signal to noise ratio (SNR) given as Γ=

Eb 2 2σ N0 h

Eb is the bit energy N0 and σ2w are the noise energy and variance N0 = 2σ2w Dr. Charalampos C. Tsimenidis


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Theoretical Performance (Cont.) 0

10

BPSK in AWGN BPSK in Rayleigh

−1

10

−2

P e (Γ) =

Pe

10

1 2

q 1 2 Γ 1 − 1+Γ

−3

10

−4

10

−5

10

P e (γ) = Q

!√

2γ

"

−6

10

−5

0

10

20 30 Average SNR Γ (dB)


40

50


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Theoretical Performance (Cont.) 1 2 3 4

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

5 6 7 8 9

%--BPSK in Rayleigh-Gamma_dB=-4:0.1:40; Gamma_lin=10.ˆ(Gamma_dB/10); ber_Rayleigh=0.5*(1-sqrt(Gamma_lin./(Gamma_lin +1)));

10 11 12 13

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



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Semi-analytical Performance Semi-analytical BER with N sufficiently large: 1 N Pe (γ) = ∑ Q N n=1

q 2γ |hn |2

Average effect of hn on γ. BER in AWGN is given as Pe (γ) = Q

p 2γ

γ is the signal to noise ratio given as γ= Dr. Charalampos C. Tsimenidis

Eb N0


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Semi-analytical Performance (Cont.) 0

10

−1

10

−2

P e (γ) =

10

1 N

BPSK in AWGN BPSK in Rayleigh Semi−analytical N 1p 2 X 2γ |hn |2 Q

Pe

n=1

−3

10

−4

10

−5

10

P e (γ) = Q

!√

2γ

"

−6

10

−5

0

10



40

50


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Semi-analytical Performance (Cont.)

1 2 3 4 5 6 7 8 9 10

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



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Simulated Performance

Implement system using: Transmitter by generating bn and mapping it to dn Channel by generating hn and wn and computing for different SNR values xn = hn dn + wn Receiver by implementing decision rule and BER computation. Display simulation results.



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Simulated Performance (Cont.) xn = hn dn + wn

dn

hn BPSK Transmitter Data

bn

Tx

wn hn , wn

dn

Comm. Channel

Receiver

xn

Estimated Data

Rx

ˆbn

Compute BER

Pe = f (SN R)



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Simulated Performance (Cont.) 0

10

BPSK in AWGN BPSK in Rayleigh Simulation q 1 2 Γ P e (Γ) = 21 1 − 1+Γ

−1

10

−2

Pe

10

−3

10

−4

10

−5

10

P e (γ) = Q

!√

2γ

"

−6

10

−5

0

10



40

50


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Digital Modulation Digital information is transmitted by varying the information of either amplitude a(t), frequency fi (t) or phase 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) and phase φ(t). Dr. Charalampos C. Tsimenidis


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Bandpass and Complex-Baseband Signals The information bearing signal is referred to as the complex-baseband or lowpass equivalent signal, sl (t), given as sl (t) = a(t)e jφ(t) = xI (t) + jxQ(t) Bandpass signal is only required to reduce the effective antenna size required. It is given as s(t) = Re{sl (t)e j2π fct }

= Re{[xI (t) + jxQ (t)]e j2π fct } = xI (t) cos(2π fct) − xQ (t) sin(2π fct)

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


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Bandpass and Complex-Baseband Signals (Cont.) The entire system is simulated in most cases at the symbol rate to minimize simulation times. The symbol-rate, sampled complex-baseband signal is given as sl (nT ) = a(nT )e jφ(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 be also dropped for clarity. System/signal bandwidth and sampling frequency are only required for normalizing the times of multipath arrivals and for Doppler effect simulations in time-varying systems Dr. Charalampos C. Tsimenidis


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Typical Program Structure

1 2

% Short description of simulation clear, clc, close all

3 4 5 6

% Simulation parameter definitions M=16; % Constellation size ...

7 8 9

% Reset BER variable BER=zeros(1,?);



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Typical Program Structure (Cont.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

for m=... % SNR loop total_errors=0; for n=... % Averaging loop % Transmitter ... % Communications channel ... % Receiver ... % Bit error computation new_errors=... total_errors=total_errors+new_errors; end BER=... end Dr. Charalampos C. Tsimenidis


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Typical Program Structure (Cont.)

1 2 3

% Plot results semilogy(?,?) ...

4 5 6

% Store results in a file save ...



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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 = 2k bits = 24 = 16 points M-QAM alphabet: {±(2k − 1)d ± j(2k − 1)d}, where √ M k ∈ {1, . . . 2 } For 16-QAM, with M = 16 and k = 4 we obtain 4 values for the I and Q channel, i.e. I, Q ∈ {−3d, −d, d, 3d} d is an arbitrary value, typically d = 1 or d = This corresponds to 2 bits per channel. Dr. Charalampos C. Tsimenidis

1 3


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Constellation of Binary-Coded 16-QAM

4 2 3 4 5 6 7 8

C=[-3+3j, -3+1j, -3-1j, -3-3j, -1+3j, -1+1j, -1-1j, -1-3j, 1+3j, 1+1j, 1-1j, 1-3j, 3+1j, 3+1j, 3-1j, 3-3j];

3 2 Quadrature

1

1 0 −1 −2 −3 −4 −4


0000

0100

1000

1100

0001

0101

1001

1101

0010

0110

1010

1110

0011

0111

1011

1111

−3

−2

−1

0 1 In−phase

2

3


4

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Constellation of Gray-Coded 16-QAM

4 2 3 4 5 6 7 8

C=[-3+3j, -3+1j, -3-3j, -3-1j, -1+3j, -1+1j, -1-3j, -1-1j, 3+3j, 3+1j, 3-3j, 3-1j, 1+3j, 1+1j, 1-3j, 1-1j];

3 2 Quadrature

1

1 0 −1 −2 −3 −4 −4


0000

0100

1100

1000

0001

0101

1101

1001

0011

0111

1111

1011

0010

0110

1110

1010

−3

−2

−1

0 1 In−phase

2

3


4

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Constellation of Gray-Coded 16-QAM (Cont.) For a constellation point z = x + jy, the bit allocation, b0 b1 b2 b3 , is as follows First bit, b0 b0 = 0 if Re{z} ≤ 0 b0 = 1 if Re{z} > 0

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

Third bit, b2

b2 = 0 if Im{z} ≥ 0 b2 = 1 if Im{z} < 0

Forth bit, b3 b3 = 0 if Im{z} ≤ −2 or Im{z} ≥ 2 b3 = 1 if Im{z} > −2 or Im{z} < 2 Dr. Charalampos C. Tsimenidis


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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 the symbol index, e.g. k = 0, 1, 2, ..., M − 1. To generate bk use: randi(?,?,?) Help: >> doc randi



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Random Number Generation

MATLAB by default generates always the same random numbers at the start of each session. To obtain different random values in different MATLAB sessions include at the top of your code, right after clear, the following: 1 2 3

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



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Code Developement Lab 2 Develop a 16-QAM modulator using look-up table approach to generate a frame of 2048 16-QAM symbols. Add complex-valued random AWGN noise with small standard deviation (e.g. 0.25) and plot the resulting constellation. 4 3

Quadrature

2 1 0 −1 −2 −3 −4 −4

−3

−2


−1

0 1 In−phase

2

3

4


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Time-Frequency Relationship The time-bandwidth product is defined as: B T = 1 The Fourier spectrum of a rectangular pulse t FT pT (t) = A rect( ) ←→ PT ( f ) = A T sinc ( f T ) T sin(πx) x

where the sinc( .) function defined as: sinc(x) = 1

T T 4

0.75

PT (f)

T 4

0.5 0.25

T

0 −8 −7 −6 −5 −4 −3 −2 −1

0

1

2

3

4

5

6

7

8

f (Hz)



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Principle of OFDM B

Channel Frequen y Response

Single Carrier

Multi Carrier

∆f =


B N

f


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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.



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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 FFT leakage.



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OFDM Transmitter Model

Insert Pilots 16−QAM Symbols

[0, 1, . . . , 15]

Modulator 16−QAM

S/P

IFFT

Insert Cyclic Prefix

P/S

OFDM Symbol

x(n)

d(n)



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OFDM Signal Model Definitions: 2πkn 1 N−1 d[n] = √ ∑ D[k]e j N , n = 0, 1, . . ., N − 1 N k=0

N−1 2πkn ˆ = √1 ∑ d[n]e− j N , k = 0, 1, . . ., N − 1 D[k] N n=0

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 OFDM symbol. k is the subcarrier index in frequency domain. Dr. Charalampos C. Tsimenidis


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MATLAB Implementation

2πkn 1 N−1 D[k]e j N , n = 0, 1, . . ., N − 1 d[n] = ∑ N k=0 1

d=ifft(D);

N−1

ˆ = D[k]

∑ d[n]e− j

2πkn N

n=0 1

, k = 0, 1, . . ., N − 1

D_est=fft(d);



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MATLAB Implementation (Cont.)

IDFT vs IFFT and DFT vs FFT If N is a power of 2, MATLAB employs FFT, otherwise a slower DFT is used. ifft() and fft() accept a second a argument to specify the transform size and use zero-padding. No need to zero-pad manually. Useful oversampling of the channel impulse response, h(k) in order to obtain the correct frequency response H(n).



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Oversampling Effect (Channel Estimation, PAPR) H=fft(h,Nfft);

1.5

0.2 0.15

1

|d[n]|

|d[n]|

1

0.5 0

0.1 0.05

0

5 10 15 20 25 32 a) Without oversampling.


0

0

50 100 150 200 256 b) With oversampling by 8.


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Addition of Cyclic Prefix Copy last K samples CP

1 2

OFDM Symbol

CP

d=ifft(D); x=[d(?:?) d]; K is the cyclic prefix length added to reduce multipath

effects.



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Cyclic Prefix (Cont.)

DFT Property: replication in one domain corresponds to interpolation in the other. Thus, if a signal is replicated by M in time domain, its DFT is zero interpolated by M (and scaled). The use of CP converts a frequency selective channel into a set of parallel flat-fading independent channels. Some papers suggest alternatively zero-padding (ZP) instead of CP.



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Energy Normalization

Asymmetric energy transform implementation with respect to time-frequency domain transition. Problem with required noise power computation for given SNR. Signal energy normalization required. Implementation: √ either by multiplying x with N before adding Gaussian noise, or by scaling the noise √ standard deviation appropriately by 1/ N.



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Code Developement Lab 3

Develop the code to implement The 16-QAM OFDM transmitter, insertion of K-sample cyclic-prefix using concatenation, Energy normalization of the OFDM waveform.



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Channel Simulation x(n)

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

w(n) Complex additive Gaussian noise: 1

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

randn() generates zero-mean unity variance noise samples. Appropriate scaling using the standard deviation sigma(?) is required. Dr. Charalampos C. Tsimenidis


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Gaussian Probability Density Function (PDF) Models the statistics of thermal noise in receivers known as additive white Gaussian noise (AWGN). Appears in closed-form BER solutions for the performance over AWGN channels. The PDF of a Gaussian distribution is defined as − (x−µw ) 1 e 2σ2w fw (x) = √ 2πσw

2

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 or normal pdf fw (x), we write w → N (µw , σ2w ) Dr. Charalampos C. Tsimenidis


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Gaussian pdf

1.6 µw µw µw µw µw

1.4 1.2

R∞

−∞ fw (x)dx

=1

= 0, σw2 = 1 = 0, σw2 = 0.5 = 0, σw2 = 0.25 = 1, σw2 = 1 = −1, σw2 = 1

fw (x)

1 0.8 0.6 0.4 0.2 0 −4

−3

−2

−1

0

1

2

3

4

x Dr. Charalampos C. Tsimenidis


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Gaussian Cumulative Density Function (CDF) In general, the CDF is defined as Fw (x) = P(w ≤ x) = CDF properties

Z x

−∞

fw (u)du

Fw (x) ≥ 0, ∀x

Fw (−∞) = 0, Fw (∞) = 1 The CDF of a Gaussian distribution is given as 1 1 x − µw Fw (x) = + erf √ 2 2 2σw 1 1 x − µw = − erfc √ 2 2 2σ w x − µx = 1−Q σ Simulation of Wireless Communications Dr. Charalampos C. Tsimenidis EEE8077w

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Gaussian CDF and PDF Gaussian cdf and pdf for µ w = 0, σw2 = 1 1.5 fw (x) Fw (x)

fw (x), Fw (x)

Fw (∞) = 1 1

0.5

Fw (−∞) = 0 0 −4

−3

−2

−1

0

1

2

3

4

x Dr. Charalampos C. Tsimenidis


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The erf(x), erfc(x) and Q(x) Functions The error function x 2 2 erf(x) = √ e−t dt π 0 The complementary error function Z ∞ 2 2 erfc(x) = √ e−t dt π x Relationship: erfc(x) = 1 − erf(x)

Z

The Q( .) function:

∞ −t 2 1 Q(x) = √ e 2 dt 2π x Relationship between Q(x) and erfc(x): x 1 Q(x) = erfc √ 2 2

Z



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SNR Computation The SNR is computed as follows: SNR =

Eb N0

The average bit energy Eb computed from the constellation: Es Eb = log2 (M) The noise energy is computed as: N0 = 2σ2w We need to solve for σw = Dr. Charalampos C. Tsimenidis

√

...


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Es Computation Es is computed as follows: 1 M−1 |cn |2 ∑ M n=0 1 = 4 |3d + j3d|2 + 8 |3d + jd|2 + 4 |d + jd|2 16 160 2 1 72d 2 + 80d 2 + 8d 2 = d = 16 16 = 10 d 2

Es =

Reminder complex numbers: z ∈ C z = a + jb ⇒ |z|2 = Dr. Charalampos C. Tsimenidis

p

a2 + b2

2

= a2 + b2


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16-QAM BER in AWGN For M = 2 (2-QAM/BPSK) and M = 4 (4-QAM/QPSK): r 2Eb Pb = Q N0 For M >> 4, k = log2 (M) Pb =

PM k

where PM ≈ 1 − (1 − P√M )2 with

s ! 1 3k E b Q P√M = 2 1 − √ M − 1 N M 0 Dr. Charalampos C. Tsimenidis


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16-QAM BER in AWGN (Cont.) 0

10

BPSK in AWGN 16−QAM in AWGN

−1

10

−2

Pe

10

−3

10

−4

10

−5

10

P e (γ) = Q

!√

2γ

"

−6

10

0

2

4

6

8 γ=


Eb N0

10 (dB)

12

14

16


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Code Developement Lab 4

Develop MATLAB code that Computes the standard deviation for 16-QAM in AWGN, for a given SNR value given in dB. Generates random AWGN noise wn scaled by the correct σw and adds it to the OFDM signal. Generates the BER vs Eb /N0 plots for BPSK and 16-QAM in AWGN.



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Receiver Structure for AWGN

Received OFDM Symbol

Remove Cyclic Prefix

S/P

FFT

16−QAM Demodul.

P/S

16−QAM Symbols

CP is discarded at the receiver to avoid interblock interference. DFT/FFT is performed. Channel estimation is not required for AWGN channel. 16-QAM demodulation using custom code.



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Demodulator Input Signal

Eb N0

= 16 dB

γ= 5

4

4

3

3

2

2 Quadrature

Quadrature

γ= 5

1 0 −1

0 −1 −2

−3

−3

−4

−4 −4

−3

−2

−1

0 1 In−phase

2

3

4


5

= 6 dB

1

−2

−5 −5

Eb N0

−5 −5

−4

−3

−2

−1

0 1 In−phase

2

3


4

5

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16-QAM Demodulator Decision metric: bˆ k = arg min(|Ci − Dˆ k |), i = 0, 1 . . . , M − 1, M = 16 i

where |Ci − Dˆ k | is the Euclidean distance defined as q 2 2 ˆ Re(Ci ) − Re(Dˆ k ) + Im(Ci ) − Im(Dˆ k ) |Ci − Dk | = Im{C}

C0

C4

C12

C8 ˆk D

C1

C5

C13

C9

C3

C7

C15

C11

C2

C6

C14

C10


Re{C}


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Displaying Information about Simulation

1 2

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

3 4

SNR=10.5,BER=0.100000



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Measuring Simulation Time 1 2 3

tic ... toc

4 5 6 7 8

% or t_start=cputime; ... cputime-t_start

9 10 11

% or (not recommended) t_start = clock;

12 13

etime(clock, t_start)



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Parallel Configuration (Menu) Parallel −→ Manage Cluster Profiles



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Parallel Computing with parfor

1 2 3

% Start default worker pool using matlabpool open %...

4 5 6

% Outer loop parfor m=...

7 8

end

9 10 11

% Close worker pool matlabpool close



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Code Developement Lab 5 Develop MATLAB code that Removes the CP and computes the FFT of the received noisy signal. Estimates using the minimum Euclidean distance the transmitted 16-QAM symbols. Computes and displays the BER vs. SNR performance by averaging P-times per SNR point. Displays information about simulation during code execution. Measures the total simulation time elapsed. Simulates the performance using the parallel configuration. Dr. Charalampos C. Tsimenidis


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Multipath Channel Scenario

Reflector Cluster Single Reflector

Impulse Response

τ41

h(τ ; t) a1 e−j2πfc τ1 a2 e−j2πfc τ2 a3 e−j2πfc τ3

τ42 τ32

a4 e−j2πfc τ4

τ31 τ1 τ21

Base Station (BS)

τ22

τ1 τ2 τ3 τi = τi1 + τi2

τ4

τ

i = 2, 3, 4

Mobile Station (MS)



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Channel Impulse Response (CIR) The complex-baseband impulse response of the channel is given L

h(τ;t) =

∑ al (t)e− jφl (t)δ(t − τl (t))

l=1

The phase shifts are φl (t) are proportional to the corresponding time delays τl (t), that is φl (t) = 2π fc τl (t) where fc is the carrier frequency.



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Simplified CIRs Constant arrival times: τl (t) = τl L

h(τ,t) =

∑ al (t)e− jφl (t) δ(t − τl )

l=1

Static multipath: τl (t) = τl , al (t) = al , φl (t) = φl L

h(τ) =

∑ al e− jφl δ(t − τl )

l=1



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

x(n)

y(n) h(n)

z(n) = y(n) + w(n) = h(n) ∗ x(n) + w(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 is M = N +L−1 Dr. Charalampos C. Tsimenidis


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Multipath Channel Implementation (Cont.) Preferred MATLAB Implementation: 1 2 3 4 5 6

% Initial filter state zf=[]; ... % Multipath channel [y,zf]=filter(h,1,x,zf); z=y+w; % Add WGN

This implementation keeps track of the channel state between OFDM symbols. Interblock interference can be simulated. Documentation: >>doc filter In general, y=conv(h,x)+w; can be utilized, but not as useful. Dr. Charalampos C. Tsimenidis


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ITU Channel for Mobile WiMAX Multipath magnitudes hi need to be obtained from P = [0 -0.9 -4.9 -8.0 -7.8 -23.9];

Appropriate zero-padding is required between the paths. ITU Pedestrian B Ch. 103 0

P (dB)

−5

−10

−15

−20

−25

0

5

18

27

52

84

Delay spread (Samples) Dr. Charalampos C. Tsimenidis


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Channel Normalization The channel magnitudes are obtained as q hi =

Pi

10 10

Channel profile needs to be normalized to unity energy L

∑ |hi|2 = 1

i=1

{hi }Li=1

If are the unnormalized channel coefficients then in order to normalize we use: s L

U=

∑ |hi|2

i=1

hi =

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



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Theoretical Performance in Multipath for BPSK Identical to semi-analytical BER since channel is static. For BPSK that is q 1 N 2 Pe (γ) = ∑ Q 2γ |Hn | N n=1 where Hn is the frequency response of the ITU channel. N is here the FFT length. The frequency response is obtained applying the FFT on the vector {hi }Li=1 by oversampling by N. γ is the signal to noise ratio given as γ= Dr. Charalampos C. Tsimenidis

Eb N0


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Theoretical Performance in Multipath for 16QAM For 16QAM, with M = 16, k = log2 (M) = 4, that is s ! N 1 1 3k E b P√M = 2 1 − √ |Hn|2 Q ∑ N M − 1 N M 0 n=1 with PM ≃ 1 − (1 − P√M )2 and Pb ≃


PM k


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Code Developement Lab 6-7 Develop MATLAB code that with regard to the ITU multipath channel: Implements and normalizes its impulse response. Filters the OFDM signal through the channel and adds complex-valued AWGN noise for a given SNR value in dB. Computes and displays its frequency response. 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 a receiver without equalizer. Dr. Charalampos C. Tsimenidis


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Generic Receiver Structure

Received OFDM Symbol

Remove Cyclic Prefix

S/P

FFT

One−tap Equalizer

16−QAM Demodul.

P/S

16−QAM Symbols

CP is discarded at the receiver to avoid interblock interference. DFT/FFT is performed. One-tap per subcarrier zero-forcing design to remove channel effect. Channel estimation: impulse response known or unknown. 16-QAM demodulation using custom code (same code as in AWGN case). Dr. Charalampos C. Tsimenidis


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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), of the channel frequency response. Thus, the decision variable becomes W (k) ˆ D(k) = D(k) + , k = 0, 1, 2, ...N − 1 H(k) | {z } W˜ (k)

˜ (k) is no longer Gaussian. The noise term W Noise enhancement occurs if channel frequency response exhibits spectral nulls. Dr. Charalampos C. Tsimenidis


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Equalizer for Known Channel We assume h(n) is perfectly known. H(k) is derived from the impulse response h(n) via DFT/FFT. Oversampling of H(k) is required to match the length of an OFDM symbol. 1 2 3

N=2048; ... H=fft(h,N);

Detection variable after CP removal, FFT and zero forcing: Y (k) ˆ D(k) = H(k) ˜ (k), k = 0, 1, 2, ...N − 1 = D(k) + W Dr. Charalampos C. Tsimenidis


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Code Developement Lab 8-9

Develop MATLAB code that Computes the zero-forcing equalizer coefficients (computation is required only once as the channel is time-invariant). Implements the modified 16-QAM receiver using one-tap equalizer. Computes and displays the simulated BER vs Eb /N0 performance.



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Equalizer for Unknown Channel Insert 1 pilot for every 8 subcarriers at the transmitter: 1 Data Pilots Estimates

|Hest (k)|

0.8 0.6 0.4 0.2 0 −1

0

1

2

3

4

5

6 7 8 9 10 11 12 13 14 15 16 17 18 subcarrier index, n

Estimate unknown channel at pilot indices H(p). Dr. Charalampos C. Tsimenidis


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Linear Interpolation Interpolate to obtain in-between channel values. Line equations connecting two points: y − y1 x − x1 = y2 − y1 x2 − x1 y Hp (m + 1) y2 Hp (m)

y1

x1 n m Dr. Charalampos C. Tsimenidis

x2 n+L m+1

x n (OFDM index) m (Pilot index)


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Linear Interpolation (Cont.)

Interpolation equation: l Hest (k) = H p (m) + [H p(m + 1) − H p(m)] L Indices: m = 1, 2, . . ., N p − 1, l = 1, 2, . . ., L − 1, k = (m − 1)L + l + 1



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Linear Interpolation (Cont.) For the data subcarriers after the N p th pilot we need to extrapolate the equation from the last two pilots, e.g. H p (N p − 1) and H p(N p ) and l = l + L. Hest (n) extrapolate

Hp (Np ) Hp (Np − 1)

Np − 1 L (Np − 2)


Np L (Np − 1)

N

m (Pilot index) n (OFDM index)


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Estimated Frequency Response

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

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

2

1.5 1

0.5 0

0

500

1000 1500 subcarrier index


2000


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Equalizer for Unknown Channel

Zero forcing equalizer: Y (k) ˆ D(k) = , k = 0, 1, 2, ...N − 1, k 6= m Hest (k) BER should not be computed at the pilot locations, e.g. k = m Total number of bits should be adjusted to account for the pilot overhead.



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Code Developement Lab 10-12 Develop MATLAB code that Estimates the frequency response of the multipath channel using pilots and computes the zero-forcing equalizer coefficients. Implements the modified 16-QAM receiver using one-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 of SNR in dB between the estimated and actual frequency response of the channel.



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EEE8077 SIMULATION OF WIRELESS COMMUNICATIONS

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