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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
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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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