Lecture 5: Simulation of OFDM communication systems

Transcription

1 Lecture 5: Simulation of OFDM communication systems March 28 April 9 28 Yuping Zhao (Doctor of Science in technology) Professor, Peking University Beijing, China Yuping.zhao@pku.edu.cn

2 Single carrier communcation systems Frequency bandwidth = B Symbol period T=/B B (MHz) If τ_max>t intersymbol interference (ISI) T=/B (μs) Time Equalization 2

3 OFDM communication system B Frequency Δf (khz) Bandwidth is divided into N sub-band Symbol period is N time longer Time T (ms) Δf=B/N Symbol duration T=N/B; T=/ Δf 3

4 4 ( ) = < =., ), (,, 2 exp ), ( otherwise k t g T t t f j k t g k π = = = = T T T p k T dt k t g dt p t g k t g p k dt p t g k t g 2 * *., ), ( ), ( ), (,, ), ( ), ( for the kth sub-carrier Orthogonal condition 4

5 OFDM-BPSK signal:,,-,-, data Real part: cos(2πf k t) Imaginary part: sing(2πf k t) OFDM -5 symbol

6 OFDM spectrum 6

7 F( f ( t)) => F( j2 π f ) F f t e => F j π f Δf j2πδft ( ( ) ) ( 2 ( )) Δf 7

8 Block diagram of OFDM systems Binary data Encoding (BPSK ) X(k) x(n) x (n) s(t) Guard Lowpass IFFT insertion filtering Carrier modulation Binary data decoding (BPSK ) y (k) y(n) y (n) r(t) Lowpass Guard filtering FFT deleting A/D converting Channel Carrier demodulation Channel estimation OFDM symbol synchronisation 8

9 99 Using FFT and IFFT,,...,,,..., 2 exp ) ( = = = N n N k kn N j k g π = = = = =, 2 exp 2 exp, 2 exp 2 exp N n N n p k kn N j pn N j p k N kn N j pn N j π π π π

10 OFDM signal expressions Assume the information data be X(k), then the transmitted signal is s(n) N 2π s( n) = X( k)exp j nk k = N n N The received signals () t = s( t) w( t) r + w(t): AWGN signal.

11 The signal on the kth subcarrier 2 ( ) ( )exp N n Yk rn j kn N N π = =, 2 )exp ( ) ( = = N k kn N j n w N k W N n π 2 ( )exp ( ), N n sn j kn Wk k N N N π = = +

12 About the guard interval Multipath delay profile Time τ_max (5μs,7μs,...) 2

13 OFDM Symbol s(t) Guard interval G > τ_max h ( t ) OFDM Symbol s ( t ) Same signals 3

14 4

15 Problems in OFDM systems Peak to average power ratio (PAPR) Symbol synchronization Channel response estimation Impact of frequency error Impact of clock error 5

16 Peak to average power ratio (PAPR) Pdf function of the OFDM samples 3 Central Limit theorem, N= Figure The peak to average power ratio (PAPR ) is very large! 6

17 The impact of high PAPR to OFDM Normal amplifier response Output Input Linear range 7

18 Truncation of OFDM time domain signals caused by amplifier 3 Central Limit theorem, N=

19 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 9

20 Symbol synchronization 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 2

21 Symbol Symbol h ( t ) OFDM Symbol s ( t ) Same signals 2

22 Synchronization using guard interval Delay T conju gate Get the phase Frequen cy error Input time domain signals Integral Get the maximum synch 22

23 L 相关窗 23

24 L 相关窗 24

25 L 相关窗 25

26 L 相关窗 26

27 L 相关窗 27

28 The symbol synchronization of OFDM systems Figure 2 The correlation function for OFDM synchronization (ideal condition) 28

29 The correlation function for OFDM synchronization (SNR = 3dB, Multipath channel, carrier frequency error) 29

30 Frequency domain channel response Figure 3 Note: when transmitted signal on all sub carriers are, you get the frequency domain channel response 3

31 Pilot for OFDM frequency domain signals 3

32 Interpolation results Zero order interpolation 32

33 Interpolation results first order (linear) interpolation 33

34 Signal Constellation with different lowpass filters R=.5; % roll-factor delay=3; % number of side lobes 34

35 Signal Constellation with different lowpass filters R=.2; % roll-factor delay=3; % number of side lobes 35

36 Signal Constellation with different lowpass filters R=.2; % roll-factor delay=5; % number of side lobes 36

37 Exercise 2 Build OFDM system with 8 times upper sampling The channel are three types: Ch-: No AWGN and no multipath delay Ch-2: AWGN channel Ch-3: multipath delay without AWGN (channel parameters can be decided by yourself) 37

38 Exercise 2: Show the following figures and discuss the results Get time domain signal histograms for Ch-, Ch-2, Ch-3, after down sampling, three figures (figure -3, see instruction 2-) Perform time domain symbol synchronizations before or after down sampling for Ch-, Ch-2, Ch-3, three figures, (Figure 4-6, see instruction 2-2) Modulate signal on all sub-carriers, show the frequency domain response for Ch-, 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 38

39 Instruction 2-: for Figure -3 Binary data 6QAM modulat ion Binary data 6QAM demodu lation X(k) x(n) x (n) s(t) Guard insertion Up sampling N-points IFFT y (k) y(n) y (n) r(t) N-points FFT Guard deleting show Histogram Down sampling Lowpass filtering Channel Lowpass filtering A/D converting Suggestions and instruction are shows in the next page 39

40 Instruction 2-: How to make OFDM system OFDM systems can be made simply by adding the yellow color blocks to your first exercise -6QAM communication system (Assume FFT point N=52; length of guard interval = ) Build OFDM transmitter Generate binary data sequence, for example length=n 4 Perform 6QAM modulation Make 6QAM symbol as frames of length N, total 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) Up sampling and low pass filtering Build OFDM Receiver Make received symbol (after down sampling) as frames of length N+G, total frames Remove guard interval for each frame Perform IFFT to each frame Make a vector consist of all frames, length of vector=n In AWGN channel, the BER performance should be same as the original 6QAM system The histogram is shown in blue block for three channels 4

41 Instruction 2-2: synchronization for OFDM ---After down sampling Binary data 6QAM modulatio n Binary data 6QAM demodulation X(k) x(n) x (n) s(t) Guard insertion Up sampling N-points IFFT y (k) y(n) y (n) r(t) N-points FFT Guard deleting Down sampling OFDM symbol synchronisation Lowpass filtering Channel Lowpass filtering A/D converting The instruction for the synchronization is shown in next page 4

42 Instruction 2-2: synchronization for OFDM ---After down sampling FFT length = N; Guard length = G Delay N samples conju gate x(n) Input time domain signals (After down sampling) Sum up G signals y(n) Get the maximum The peak value -- synchroniz ation point In your report, you need to show y(n) value, where n is time domain index. y(n) can be expressed as (n=,2,3 ) y G i= ( ) ( ) * n = x n + i x ( n + i + N ) 42

43 Instruction 2-2: synchronization for OFDM ---After down sampling FFT length = N; Guard length = G Binary data 6QAM modulation X(k) x(n) x (n) s(t) Guard insertion Up sampling N-points IFFT Lowpass filtering Binary data 6QAM demodulation y (k) y(n) y (n) r(t) N-points FFT Guard deleting Down sampling Channel Lowpass filtering A/D converting OFDM symbol synchronisation 43

44 Instruction 2-2: synchronization for OFDM ---before down sampling FFT length = N; Guard length = G; Up sampling rate = K Delay N K samples conju gate x(n) Input time domain signals (Before down sampling) Sum up G K signals y(n) Get the maximum The peak value -- synchroniz ation point y G K ( ) ( ) * n = x n + i x ( n + i + N K ) i= 44

45 Instruction 2-3: Show Channel response of OFDM systems,,, 6QAM modulation X(k) x(n) x (n) s(t) Guard insertion Up sampling N-points IFFT Lowpass filtering,,, 6QAM demodulation y (k) y(n) y (n) r(t) N-points FFT Show Channel response (real and Imaginary parts), for one frame ONLY Guard deleting Down sampling Channel Lowpass filtering A/D converting For figure 7,8,9 45

46 Instructions 2-4: channel response estimation of OFDM,,,,,,,,,,,, X(k) x(n) x (n) s(t) Guard insertion Up sampling N-points IFFT Lowpass filtering,,, y (k) y(n) y (n) r(t) N-points FFT Interpolation to get channel response (real and imaginary), show one frame ONLY Guard deleting Down sampling Channel Lowpass filtering A/D converting Detail explanation is given in next page 46

47 Instructions 2-4: channel response estimation of OFDM systems Assume you have pilot subcarriers with carrier number,5,9,3, The pilot subcarriers are modulated by signal value 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 47

48 Interpolation results Accurate channel response Solid line: accurate channel response of your figure 9 Dotted line: Linear interpolation results 48

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