Introduction to OFDM Systems

Transcription

1 Introduction to OFDM Systems Dr. Prapun Suksompong June 23,

2 Outline 1. Overview of OFDM technique 2. Wireless Channel 3. Multi-carrier Transmission 4. Implementation: DFT and FFT 5. Concluding remarks 6. More advanced topics (if interest and time permitted) i. Cyclic Prefix (CP) and Circular Convolution ii. OFDM Drawbacks: PAPR and its solutions iii. OFDM-based Multiple Access Systems 2

3 Video Attendance Check Say your name into the camera Make sure that your voice is loud enough 3

4 4 OFDM: Overview Let S 1, S 2,, S N be the information symbol. The discrete baseband OFDM modulated symbol can be expressed as Some references may use different constant in the front Note that: N kt s( t) S k exp j, 0 t T N k 0 Ts 1 2 kt N 1 Sk 1 0, texp j T s k 0 N Ts N 1 1 2kt 2kt Re s( t) ReSkcos ImSksin N k 0 Ts Ts c k t s Some references may start with different time interval, e.g. [-T s /2, +T s /2]

5 OFDM Application Wi-Fi: a and g versions DVB-T (the terrestrial digital TV broadcast system used in most of the world outside North America) DMT (the standard form of ADSL - Asymmetric Digital Subscriber Line) WiMAX 5

6 6 Single-User OFDM

7 Motivation First, we study the wireless channel. There are a couple of difficult problems in communication system over wireless channel. Also want to achieve high data rate (throughput) 7

8 OFDM 1. Wireless Channel 8

9 Single Carrier Transmission Baseband: N1 s t S p t kt k0 Passband: k 1 t p t Rectangular waveform (a) 1.2 s 0, T s 1, t 0, Ts 0, otherwise. 1 Carrier frequency j2 f ct c x t Re s t e s t cos 2 f t Valid when s(t) is real-valued (b) Time Time

10 Frequency-Domain Analysis j2 ft0 j2 f0t Shifting Properties: g t t e G f e g t G f f Modulation: mtcos2 f t M f f M f f c c c

11 Spectrum P f 1 0, p t A t T A A1, T f [Hz] [S k ] = [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1] Can you sketch S f? A -A st T t t f [Hz] 11

12 Spectrum P f This is also the spectrum of c t kt for any k. 1 0, p t A t T A A1, T f [Hz] [S k ] = [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1] S f A -A st T t t f [Hz] N1 n1 12 k s t S p t kt S f P f S e k0 k0 k j2 fkt

13 Multipath Propagation In a wireless mobile communication system, a transmitted signal propagating through the wireless channel often encounters multiple reflective paths until it reaches the receiver We refer to this phenomenon as multipath propagation and it causes fluctuation of the amplitude and phase of the received signal. We call this fluctuation multipath fading. 13

14 Wireless Comm. and Multipath Fading The signal received consists of a number of reflected rays, each characterized by a different amount of attenuation and delay. r t x t h t n t x t n t v i0 i i t s s s h1 t 0.5 t 0.2 t 0.2T 0.3 t 0.3 T 0.1 t 0.5 T h2 t 0.5 t 0.2 t 0.7T 0.3 t 1.5 T 0.1 t 2.3 T h t i0 (b) (a) (b) v s s s i i Weak Strong Time Time Time

15 Frequency Domain The transmitted signal (envelope) Sinc function P(f) f H 1 (f) 1.5 Channel with weak multipath f H 2 (f) 1.5 Channel with strong multipath f 15

16 Tradeoff We want weak multipath fading. So, we want to make the symbol period (T) large with respect to the delay(s). But larger symbol period means lower rate! Hence, there is a tradeoff between the raw data rate and the quality of the received signal. Alternatively, we may allow the system to experience strong multipath fading. Use complicated equalizer at the receiver. Not our approach. 16

17 COST 207 Channel Model Based on channel measurements with a bandwidth of 8 10MHz in the 900MHz band used for 2G systems such as GSM. 17 [Fazel and Kaiser, 2008, Table 1-1]

18 3GPP LTE Channel Modelss 18 [Fazel and Kaiser, 2008, Table 1-3]

19 3GPP 6-tap typical urban (TU6) Delay profile and frequency response of 3GPP 6-tap typical urban (TU6) Rayleigh fading channel in 5 MHz band. 19 [3GPP TS GPP; Technical Specification Group GSM/EDGE Radio Access Network; Radio Transmission and Reception (Release 7)]

20 Equalization Chapter 11 of [Goldsmith, 2005] Delay spread causes ISI In a broad sense, equalization defines any signal processing technique used at the receiver to alleviate the ISI problem caused by delay spread. [Goldsmith, 2005] Higher data rate applications are more sensitive to delay spread, and generally require high-performance equalizers or other ISI mitigation techniques. Signal processing can also be used at the transmitter to make the signal less susceptible to delay spread. Ex. spread spectrum and multicarrier modulation 20

21 Equalizer design Balance ISI mitigation with noise enhancement Both the signal and the noise pass through the equalizer Nonlinear equalizers suffer less from noise enhancement than linear equalizers, but typically entail higher complexity. Most equalizers are implemented digitally after A/D conversion Such filters are small, cheap, easily tuneable, and very power efficient. The optimal equalization technique is maximum likelihood sequence estimation (MLSE). Unfortunately, the complexity of this technique grows exponentially with the length of the delay spread, and is therefore impractical on most channels of interest. Viterbi algorithm 21

22 Simple Analog Equalizer 22 xt H f Heq f nt 1 H f Remove all ISI Disadvantages: If some frequencies in the channel frequency response H( f ) are greatly attenuated, the equalizer H eq (f ) = 1/ H( f ) will greatly enhance the noise power at those frequencies. If the channel frequency response H( f ) has a spectral null (= 0 for some frequency), then the power of the new noise is infinite. Even though the ISI effects are (completely) removed, the equalized system will perform poorly due to its greatly reduced SNR. x t n t

23 Wireless Propagation [Bahai, 2002, Fig. 2.1] 23

24 Three steps towards modern OFDM 1. Solve Multipath Problem Multicarrier modulation (FDM) 2. Gain Spectral Efficiency Orthogonality of the carriers 3. Achieve Efficient Implementation FFT and IFFT 24

25 OFDM 2. Multi-Carrier Transmission 25

26 Single-Carrier Transmission [Karim and Sarraf, 2002, Fig 3-1] 26

27 Multi-Carrier Transmission Convert a serial high rate data stream on to multiple parallel low rate sub-streams. Each sub-stream is modulated on its own sub-carrier. Time domain perspective: Since the symbol rate on each sub-carrier is much less than the initial serial data symbol rate, the effects of delay spread, i.e. ISI, significantly decrease, reducing the complexity of the equalizer. 27 [Fazel and Kaiser, 2008, Fig 1-4]

28 Frequency Division Multiplexing Frequency Domain Perspective: Even though the fast fading is frequency-selective across the entire OFDM signal band, it is effectively flat in the band of each low-speed signal. [The flatness assumption is the same one that you used in Riemann approximation of integral.] 28 [Myung and Goodman, 2008]

29 Frequency Division Multiplexing To facilitate separation of the signals at the receiver, the carrier frequencies were spaced sufficiently far apart so that the signal spectra did not overlap. Empty spectral regions between the signals assured that they could be separated with readily realizable filters. The resulting spectral efficiency was therefore quite low. 29

30 Multi-Carrier (FDM) vs. Single Carrier Single Carrier Single higher rate serial scheme Multipath problem: Far more susceptible to inter-symbol interference (ISI) due to the short duration of its signal elements and the higher distortion produced by its wider frequency band Complicated equalization Multi-Carrier (FDM) Parallel scheme. Each of the parallel subchannels can carry a low signalling rate, proportional to its bandwidth. Long duration signal elements and narrow bandwidth in sub-channels. Complexity problem: If built straightforwardly as several (N) transmitters and receivers, will be more costly to implement. BW efficiency problem: The sum of parallel signalling rates is less than can be carried by a single serial channel of that combined bandwidth because of the unused guard space between the parallel subcarriers. 30

31 FDM (con t) Before the development of equalization, the parallel technique was the preferred means of achieving high rates over a dispersive channel, in spite of its high cost and relative bandwidth inefficiency. 31

32 OFDM OFDM = Orthogonal frequency division multiplexing One of multi-carrier modulation (MCM) techniques Parallel data transmission (of many sequential streams) A broadband is divided into many narrow sub-channels Frequency division multiplexing (FDM) High spectral efficiency The sub-channels are made orthogonal to each other over the OFDM symbol duration T s. Spacing is carefully selected. Allow the sub-channels to overlap in the frequency domain. Allow sub-carriers to be spaced as close as theoretically possible. 32

33 Vector: 33 Orthogonality Two vectors/functions are orthogonal if their inner product is zero. The symbol a b 1 1 n *, k k 0 k 1 a n b n a b a b a b Time-domain: *, 0 a b a t b t dt Frequency domain: * A, B A f B f df 0 is used to denote orthogonality. Example: sin t t 2 k 1 and cos 2 k 2 on 0, T T T e t j2 n T Example: Complex conjugate t 3 and 5 t t on 1,1 9 on 0, T

34 Orthogonality in Communication CDMA TDMA FDMA 1 1 s t S c t S f S C f k k k k k0 k0 1 1 s t S c t kt S f C f S e k s k k0 k0 where c(t) is time-limited to [0,T]. This is a special case of CDMA with c t ct kt 1 S f S C f k f k 0 k where C(f) is frequency-limited to [0,f]. This is a special case of CDMA with C f C f kf k k s where j2 fkt s The c k are non-overlapping in time domain. c k c k The C k are non-overlapping in freq. domain.

35 OFDM Let S 1, S 2,, S N be the information symbol. The discrete baseband OFDM modulated symbol can be expressed as N kt s( t) S k exp j, 0 t T N k 0 Ts 1 2 kt N 1 Sk 1 0, texp j T s k 0 N Ts c k t Another special case of CDMA! s 35 Note that: N 1 1 2kt 2kt Re s( t) ReSkcos ImSksin N k 0 Ts Ts

36 OFDM: Orthogonality Ts * 2k1t 2 2 k exp exp T 0 s Ts k t ck tc t dt j j dt 1 2 T s 0 2 k1 k2 t Ts, k k exp j dt Ts 0, k k When k k, 1 2 When k k, 1 2 T s * k 1 1 k 2 s 0 c t c t dt dt T T 2 k k t c t c t dt j k * k 1 2 s 1 2 exp j2 k1 k2 Ts 0 T 11 0 j2 k k s 1 2 T s

37 Frequency Spectrum N kt s( t) S k 1 0, texp j T s k 0 N Ts s 1 1 ct 0, Ts s s N N 2 kt k ck t c t exp j Ck f C f C f kf Ts Ts j2 f 2 1 t C f T e sinct f c k t 1 t T sin c T f T T s s s s, 2 2 T f 1 T This is the term that makes the technique FDM. s 37 N1 N1 s( t) S c t S( f ) S C f k k k k k0 k0 N 1 s 1 j2 f kf N k 0 S k 2 e T sin c T s T s f kf

38 Subcarrier Spacing S N kt s() t S k 1 0, texp j T s k 0 N Ts f 1 N N 1 k 0 j2 S e T sin c T f kf k Ts f kf 2 s s f 1 T s Each QAM signal carries one of the original input complex numbers. OFDM N separate QAM signals, FDM at N frequencies separated by the signalling rate. 38 Spectrum Overlap in OFDM The spectrum of each QAM signal is of the form with nulls at the center of the other subcarriers.

39 Normalized Power Density Spectrum Flatter when have more sub-carriers 39 [Fazel and Kaiser, 2008, Fig 1-5]

40 Time-Domain Signal Real and Imaginary components of an OFDM symbol is the superposition of several harmonics modulated by data symbols [Bahai, 2002, Fig 1.7] 40 N kt s( t) S k exp j, 0 t Ts N k 0 Ts N 1 1 2kt 2kt Re s( t) ReSkcos ImSksin N k 0 Ts Ts in-phase part quadrature part

41 Summary So, we have a scheme which achieve Large symbol duration (T s ) and hence less multipath problem Good spectral efficiency One more problem: There are so many carriers! 41

42 OFDM 3. Implementation: DFT and FFT 42

43 Discrete Fourier Transform (DFT) Transmitter produces N kt s( t) S k exp j, 0 t T N k 0 Ts Sample the signal in time domain every T s /N gives T N 1 s 1 2 sn sn Sk exp j N N k 0 Ts N k exp N k 0 k n T s kn S j N IDFT S n N N s We can implement OFDM in the discrete domain! 43

44 Discrete Fourier Transform (DFT) The -1 are there because we start from row 1 and column Key Property: 1 N N is a unitary matrix

45 45 DFT

46 46 DFT

47 DFT: Example 47 [

48 Efficient Implementation: (I)FFT [Bahai, 2002, Fig. 2.9] 48 An N-point FFT requires only on the order of NlogN multiplications, rather than N 2 as in a straightforward computation.

49 FFT The history of the FFT is complicated. As with many discoveries and inventions, it arrived before the (computer) world was ready for it. Usually done with N a power of two. Not only is it very efficient in terms of computing time, but is ideally suited to the binary arithmetic of digital computers. From the implementation point of view it is better to have, for example, a FFT size of 1024 even if only 600 outputs are used than try to have another length for FFT between 600 and References: E. Oran Brigham, The Fast Fourier Transform, Prentice-Hall,

50 DFT Samples N kt s( t) S k exp j, 0 t T N k 0 Ts Here are the points s[n] on the continuous-time version s(t): s s n Ts s n N N kn Sk exp j N k 0 N 0 nn N IDFT S n T s

51 Oversampling

52 Oversampling (2) Increase the number of sample points from N to LN on the interval [0,T s ]. L is called the over-sampling factor. T L s n s s sn s n n LN N 0 nn 0 n LN Ts 52 N 1 1 L 2 k T s N kn s n S k exp j n Sk exp j N k 0 T LN s N k 0 LN N kn LN Sk exp j N LN k 0 LN N 1 NL1 1 2kn 2kn L N Sk exp j 0exp j LN k 0 LN k N LN NL1 1 2 kn L N Sk exp j L N IDFT S n LN k 0 LN Zero padding: S k Sk, 0k N 0, N k LN

53 Oversampling: Summary T s N points LN points L sn sn N IDFTSn s n sn L N IDFT S n N LN 0 nn 0 n LN Ts Zero padding: S k Sk, 0k N 0, N k LN

54 OFDM implementation by IFFT/FFT T s L s n sn L N S n LN IFFT L S k R k N S k 54 r n = N IFFT S n This form of OFDM is often referred to as Discrete Multi-Tone (DMT).

55 OFDM with Memoryless Channel h t t should be ht t Sample every T s /N y t h t s t w t s t w t y n s n w n FFT FFT = IFFT s n N S n Y y n NS W k k k Additive white Gaussian noise 55 Sub-channel are independent. (No ICI)

56 Channel with Finite Memory Discrete time baseband model: * y n h s n w n h m s n m w n v m0 [Tse Viswanath, 2005, Sec ] where hn 0 for n 0 and n i. i. d. ~ CN0, N w n 0 We will assume that N Remarks: 56

57 OFDM Architecture 57 [Bahai, 2002, Fig 1.11]

58 OFDM 4. Remarks about OFDM 58

59 Summary: OFDM Advantages For a given channel delay spread, the implementation complexity is much lower than that of a conventional single carrier system with time domain equalizer. Spectral efficiency is high since it uses overlapping orthogonal subcarriers in the frequency domain. Modulation and demodulation are implemented using inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT), respectively, and fast Fourier transform (FFT) algorithms can be applied to make the overall system efficient. Capacity can be significantly increased by adapting the data rate per subcarrier according to the signal-to-noise ratio (SNR) of the individual subcarrier. 59

60 Example: a s 60

61 OFDM Drawbacks High peak-to-average power ratio (PAPR) The transmitted signal is a superposition of all the subcarriers with different carrier frequencies and high amplitude peaks occur because of the superposition. High sensitivity to frequency offset: When there are frequency offsets in the subcarriers, the orthogonality among the subcarriers breaks and it causes intercarrier interference (ICI). A need for an adaptive or coded scheme to overcome spectral nulls in the channel In the presence of a null in the channel, there is no way to recover the data of the subcarriers that are affected by the null unless we use rate adaptation or a coding scheme. 61

62 OFDM 5. Cyclic Prefix (CP) 62

63 Three steps towards modern OFDM 1. Mitigate Multipath (ISI) Multicarrier modulation (FDM) 2. Gain Spectral Efficiency Orthogonality of the carriers 3. Achieve Efficient Implementation FFT and IFFT Completely eliminate ISI and ICI Cyclic prefix 63

64 Cyclic Prefix: Motivation (1) Recall: Multipath Fading and Delay Spread 64

65 Cyclic Prefix: Motivation (2) When the number of sub-carriers increases, the OFDM symbol duration T s becomes large compared to the duration of the impulse response τ max of the channel, and the amount of ISI reduces. Can we eliminate the multipath (ISI) problem? To reduce the ISI, add guard interval larger than that of the estimated delay spread. If the guard interval is left empty, the orthogonality of the sub-carriers no longer holds, i.e., ICI (inter-channel interference) still exists. To prevent both the ISI as well as the ICI, OFDM symbol is cyclically extended into the guard interval. 65 [Jiang]

66 66 Cyclic Prefix

67 N n L L-1 N N-1 Convolution x m Flip Shift Multiply Add h m h m = h 0 m m m m h 1 m x hn xmhn m m h n m h N 1 m h N m m m m m h N + L 2 m 67 m

68 N n n L L-1 L L-1 N N-1 N N-1 Circular Convolution x m x m m m h m h m m m h m = h 0 m h m = h 0 m m m h 1 m h 1 m m m h n m h n m m m h N 1 m h N 1 m m m 68 Replicate x (now it looks periodic) Then, perform the usual convolution only on n = 0 to N-1 h N m m h N + L 2 m m

69 Circular Convolution: Example Find

70 Discussion 70 Circular convolution can be used to find the regular convolution by zero-padding. In modern OFDM, it is another way around. CTFT: convolution in time domain corresponds to multiplication in frequency domain. DFT: circular convolution in (discrete) time domain corresponds to multiplication in (discrete) frequency domain. We want to have multiplication in frequency domain. So, we want circular convolution and not the regular convolution. Real channel does regular convolution. With cyclic prefix, regular convolution can be used to create circular convolution.

71 Example Suppose x (1) = [ ] and h = [3 2 1] [ ] * [ ] = [ ] [ ] * [3 2 1] = [ ] Suppose x (2) = [ ] [ ] * [ ] = [ ] [ ] * [3 2 1] = [ ] [ ] * [3 2 1] = [ ] + [ ] = [ ] 71

72 Circular Convolution: Key Properties Consider an N-point signal x[n] Cyclic Prefix (CP) insertion: If x[n] is extended by copying the last samples of the symbols at the beginning of the symbol: xn Key Property 1: Key Property 2: x n, 0 n N 1 x n N, v n 1 h x n h* x n for 0 n N 1 FFT h x n H X k k 72

73 73 OFDM with CP for Channel w/ Memory We want to send N samples S 0, S 1,, S N-1 across noisy channel with memory. First apply IFFT: Then, add cyclic prefix This is inputted to the channel. The output is Remove cyclic prefix to get Then apply FFT: Sk IFFT s n,, 1, 0,, 1 s s N s N s s N,, 1, 0,, 1 rn hn snwn FFT y n p N p N r r N r n R k By circular convolution property of DFT, Rk HkSk W No ICI! k

74 OFDM System Design: CP A good ratio between the CP interval and symbol duration should be found, so that all multipaths are resolved and not significant amount of energy is lost due to CP. As a thumb rule, the CP interval must be two to four times larger than the root mean square (RMS) delay spread. 74 [Tarokh, 2009, Fig 2.9]

75 Reference A. Bahai, B. R. Saltzberg, and M. Ergen, Multi-Carrier Digital Communications: Theory and Applications of OFDM, 2nd ed., New York: Springer Verlag,

76 OFDM 6. OFDM-Based Multiple Access 76

77 OFDM-based Multiple Access Three multiple access techniques 1. OFDMA, 2. OFDM-TDMA, and 3. OFDM-CDMA 77

78 subcarriers OFDM-TDMA Users are separated via time slots. A particular user is given all the subcarriers of the system for any specific OFDM symbol duration. All symbols allocated to all users are combined to form a OFDM-TDMA frame. Allows MS to reduce its power consumption, as the MS shall process only OFDM symbols which are dedicated to it. Since the OFDM-TDMA concept allocates the whole bandwidth to a single user, a reaction to different subcarrier attenuations could consist of leaving out highly distorted subcarriers time 78

79 OFDMA Available subcarriers are distributed among all the users for transmission at any time instant. The subcarrier assignment is made at least for a time frame. Based on the subchannel condition, different baseband modulation schemes can be used for the individual subchannels The fact that each user experiences a different radio channel can be exploited by allocating only good subcarriers with high SNR to each user. The number of subchannels for a specific user can be varied, according to the required data rate. 79

80 subcarriers subcarriers OFDM-TDMA vs. OFDMA OFDM-TDMA OFDMA time time 80

81 OFDMA Block Diagram C o m b i n e d s u b c a r r i e r, b i t, a n d p o w e r a l l o c a t i o n C h a n n e l s t a t e i n f o r m a t i o n f o r a l l K u s e r s x N s u b c a r r i e r s / u s e r U s e r 1 d a t a U s e r 2 d a t a U s e r K d a t a S u b c a r r i e r a n d b i t a l l o c a t i o n A d a p t i v e m o d. 1 A d a p t i v e m o d. 2 A d a p t i v e m o d. N A d a p t i v e d e m o d. 1 I F F T A d d g u a r d i n t e r v a l F r e q. s e l e c t i v e f a d i n g c h a n n e l f o r U s e r k U s e r k d a t a E x t r a c t b i t s f o r U s e r k A d a p t i v e d e m o d. 2 F F T R e m o v e g u a r d i n t e r v a l 81 A d a p t i v e d e m o d. N

82 OFDM-CDMA User data are spread over several subcarriers and/or OFDM symbols using spreading codes, and combined with signal from other users. Several users transmit over the same subcarrier. In essence this implies frequency-domain spreading, rather than time-domain spreading, as it is conceived in a DS-CDMA system. 82