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Lecture 5: Simulation of OFDM communication systems

March 28 – April 19 2008

Yuping Zhao (Doctor of Science in technology)

Professor, Peking UniversityBeijing, China

Yuping.zhao@pku.edu.cn

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Time

Single carrier communcation systems

T=1/B (1μs)

Frequency

B(1MHz)

If τ_max>T

→intersymbol interference (ISI)

bandwidth = BSymbol period T=1/B

Equalization

2

3

Frequency

Time

T (1ms)

B

OFDM communication system

Δf(1kHz)

•Bandwidth is divided into N sub-band•Symbol period is N time longer

Δf=B/N

Symbol duration T=N/B; T=1/ Δf

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4

( )⎩⎨⎧

=<≤=

.,0),(,0,2exp),(

otherwisektgTttfjktg kπ

⎪⎩

⎪⎨⎧

===⋅

≠=⋅

∫∫∫

TT

T

pkTdtktgdtptgktg

pkdtptgktg

0

2

0

*0

*

.,),(),(),(

,,0),(),(

for the kth sub-carrier

Orthogonal condition

4

5

OFDM-BPSK signal: 1，1，-1，-1，1

data Real part: cos(2πfkt) Imaginary part: sing(2πfkt)

1

1

-1

-1

1

OFDM symbol 5

0 100 200 300 400 500 6000

1

2

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-5

0

5

0 100 200 300 400 500 6000

1

2

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-1

0

1

0 100 200 300 400 500 600-5

0

5

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OFDM spectrum

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77fΔ

2

( ( )) ( 2 )( ( ) ) ( 2 ( ))j ft

f t F j ff t e F j f fπ

π

πΔ

=>

⋅ => − Δ

FF

0 490 500 510 520 530 540 550 5

1+ 1+ 1+

1− 1−

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Block diagram of OFDM systems

Encoding (BPSK…) IFFT Guard

insertionLowpass filtering

Carrier modulation

Binary data

Channel

Carrier demodulation

Lowpass filteringA/D converting

Guard deletingFFTdecoding

(BPSK…)Binary data

OFDM symbol synchronisation

Channel estimation

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

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Using FFT and IFFT

1,...1,0,1,...1,0,2exp)( −=−=⎟⎠⎞

⎜⎝⎛= NnNkkn

Njkg π

⎪⎪⎩

⎪⎪⎨

⎧

≠=⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

==⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

∑

∑−

=

−

=

1

0

1

0

,02exp2exp

,2exp2exp

N

n

N

n

pkknN

jpnN

j

pkNknN

jpnN

j

ππ

ππ

1010

1

0

2( ) ( ) expN

ks n X k j nk

Nπ−

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

∑

OFDM signal expressions

10 −≤≤ Nn

Assume the information data be X(k), then the transmitted signal is s(n)

The received signals

( ) ( ) ( )twtstr +=w(t): AWGN signal.

1111

The signal on the kth subcarrier

1

0

1 2( ) ( ) expN

nY k r n j kn

N Nπ−

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

∑

10,2exp)(1)(1

0−≤≤⎟

⎠⎞

⎜⎝⎛−= ∑

−

=

NkknN

jnwN

kWN

n

π

1

0

1 2( )exp ( ), 0 1N

n

s n j kn W k k NN N

π−

=

⎛ ⎞= − + ≤ ≤ −⎜ ⎟⎝ ⎠

∑

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About the guard interval

Time

τ_max (5μs,17μs,...)

Time

τ_max (5μs,17μs,...)

Multipath delay profile

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OFDM Symbol s(t)

Guard interval G > τ_max

OFDM Symbol s(t)

Guard interval G > τ_max

O F D M S y m b o l s ( t )

S a m e s i g n a l s

h ( t ) O F D M S y m b o l s ( t )

S a m e s i g n a l s

h ( t )

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1515

Problems in OFDM systems

• Peak to average power ratio (PAPR)• Symbol synchronization• Channel response estimation• Impact of frequency error • Impact of clock error

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-10 -8 -6 -4 -2 0 2 4 6 8 100

50

100

150

200

250

300Central Limit theorem, N=60

Peak to average power ratio (PAPR)Pdf function of the OFDM samples

The peak to average power ratio (PAPR ) is very large !

Figure 1

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The impact of high PAPR to OFDM Normal amplifier response

Input

Output

Linear range

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Truncation of OFDM time domain signals caused by amplifier

-10 -8 -6 -4 -2 0 2 4 6 8 100

50

100

150

200

250

300Central Limit theorem, N=60

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About the OFDM time domain signals

• It appears as Gaussian distribution• The truncation of amplifier may cause

distortion of the received signals• The incorrect A/D conversion range may

cause the distortion of the received signals• The special time domain synchronization

procedure should be introduced

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Why the synchronization is needed?

•To get the starting point of the FFT window

•The guard period is used for the symbol synchronization

•The guard period is not used for the signal demodulation

Symbol synchronization

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Symbol Symbol

O F D M S y m b o l s ( t )

S a m e s i g n a l s

h ( t ) O F D M S y m b o l s ( t )

S a m e s i g n a l s

h ( t )

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×

Delay T conjugate

Integral synch

Input time domain signals

Get the maximum

Frequency error

Get the phase

Synchronization using guard interval

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28The correlation function for OFDM synchronization (ideal condition)

Figure 2

The symbol synchronization of OFDM systems

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The correlation function for OFDM synchronization

(SNR = 3dB, Multipath channel, carrier frequency error)

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Frequency domain channel response

Note: when transmitted signal on all sub carriers are 1, you get the frequency domain channel response

Figure 3

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Pilot for OFDM frequency domain signals

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Interpolation results

Zero order interpolation

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Interpolation results

first order (linear) interpolation

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Signal Constellation with different lowpass filters

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

R=0.5; % roll-factordelay=3; % number of side lobes

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Signal Constellation with different lowpass filters

R=0.2; % roll-factordelay=3; % number of side lobes

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

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Signal Constellation with different lowpass filters

R=0.2; % roll-factordelay=5; % number of side lobes

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

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Exercise 2

• Build OFDM system with 8 times upper sampling

• The channel are three types:– Ch-1: No AWGN and no multipath delay– Ch-2: AWGN channel– Ch-3: multipath delay without AWGN (channel

parameters can be decided by yourself)

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Exercise 2: Show the following figures and discuss the results

• Get time domain signal histograms for Ch-1, Ch-2, Ch-3, after down sampling, three figures (figure 1-3, see instruction 2-1)

• Perform time domain symbol synchronizations before or after down sampling for Ch-1, Ch-2, Ch-3, three figures, (Figure 4-6, see instruction 2-2)

• Modulate signal 1 on all sub-carriers, show the frequency domain response for Ch-1, Ch-2, Ch-3, in real and imaginary parts separately, three figures (figure 7-9, see instruction 2-3)

• Assume the pilot signals are given in page 32 and 33, using Ch-3 channel, perform interpolation for channel response estimation. Draw the figure together with Figure 9 see instruction 2-4)

• The instructions are given in the following pages

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Instruction 2-1: for Figure 1-3

16QAM modulation

N-points IFFT

Guard insertion

Lowpass filtering

Binarydata

Channel

Lowpass filteringA/D converting

Guard deleting

N-points FFT

16QAM demodulation

Binarydata

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

Up sampling

Down sampling

Suggestions and instruction are shows in the next page

show Histogram

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Instruction 2-1: How to make OFDM system

• OFDM systems can be made simply by adding the yellow color blocks to your first exercise -16QAM communication system (Assume FFT point N=512; length of guard interval = 100)

• Build OFDM transmitter– Generate binary data sequence, for example length=N×4×10– Perform 16QAM modulation– Make 16QAM symbol as frames of length N, total 10 frames– Perform IFFT to each frame– Add guard interval to each frame, then, frame length = N+G– Make a vector consist of all frames, length of vector=(N+G) ×10– Up sampling and low pass filtering

• Build OFDM Receiver– Make received symbol (after down sampling) as frames of length N+G, total 10 frames– Remove guard interval for each frame– Perform IFFT to each frame– Make a vector consist of all frames, length of vector=N ×10

• In AWGN channel, the BER performance should be same as the original 16QAM system

• The histogram is shown in blue block for three channels

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Instruction 2-2: synchronization for OFDM---After down sampling

16QAM modulation

N-points IFFT

Guard insertion

Lowpass filtering

Binary data

Channel

Lowpass filteringA/D converting

Guard deleting

N-points FFT

16QAM demodulation

Binary data

OFDM symbol synchronisation

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

Up sampling

Down sampling

The instruction for the synchronization is shown in next page

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×

conjugate

Sum up G signals

The peak value --synchronization pointInput time

domain signals(After down sampling)

Get the maximum

FFT length = N; Guard length = G

DelayN samples

y(n)

x(n)

In your report, you need to show y(n) value, where n is time domain index.y(n) can be expressed as (n=1,2,3…)

( ) ( ) ( )∑−

=

+++=1

0

*G

iNinxinxny

Instruction 2-2: synchronization for OFDM---After down sampling

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16QAM modulation

N-points IFFT

Guard insertion

Lowpass filtering

Binarydata

Channel

Lowpass filteringA/D converting

Guard deleting

N-points FFT

16QAM demodulation

Binary data

OFDM symbol synchronisation

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

Up sampling

Down sampling

FFT length = N; Guard length = G

Instruction 2-2: synchronization for OFDM---After down sampling

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×

conjugate

Sum up G×Ksignals

The peak value --synchronization pointInput time

domain signals(Before down sampling)

Get the maximum

DelayN×K samples

FFT length = N; Guard length = G; Up sampling rate = K

Instruction 2-2: synchronization for OFDM---before down sampling

( ) ( ) ( )∑−×

=

×+++=1

0

*KG

iKNinxinxny

x(n)

y(n)

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Instruction 2-3: Show Channel response of OFDM systems

16QAM modulation

N-points IFFT

Guard insertion

Lowpass filtering

1,1,1,1…

Channel

Lowpass filteringA/D converting

Guard deleting

N-points FFT

16QAM demodulation

1,1,1,1…

Show Channel response (real and Imaginary parts), for one frame ONLY

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

Up sampling

Down sampling

For figure 7,8,9

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Instructions 2-4: channel response estimation of OFDM

N-points IFFT

Guard insertion

Lowpass filtering

1,0,0,0,1,0,0,0,1,0,0,0,1…

Channel

Lowpass filteringA/D converting

Guard deleting

N-points FFT

1,1,1,1…

Interpolation to get channel response (real and imaginary), show one frame ONLY

X(k) x(n) x’(n) s(t)

y’(k) y(n) y’(n) r(t)

Up sampling

Down sampling

Detail explanation is given in next page

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Instructions 2-4: channel response estimation of OFDM systems

• Assume you have pilot subcarriers with carrier number 1,5,9,13…,

• The pilot subcarriers are modulated by signal value 1 and rest of subcarriers are empty

• You get received signals at those pilot subcarriers (in fact it is frequency domain channel response)

• You need to get the frequency domain channel response of the empty subcarriers, you can do the interpolation as shown in the next page

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Interpolation results

Accurate channel response

Solid line: accurate channel response of your figure 9Dotted line: Linear interpolation results

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