Chapter 5 OFDM 1 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
2 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 1 1 2 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]
OFDM Application 802.11 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 3
4
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) 5
Chapter 5 OFDM 5.1 Wireless Channel 6 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
Single Carrier Transmission Baseband: N 1 s t s p t kt k0 Passband: x t Re k 1 t p t s 0, T j2 fct s t e s 1, t 0, Ts 0, otherwise. 1.2 (a) (b) 1 1 0.8 0.6 0.8 0.6 0.4 0.2 0 7 0.4 0.2 0-0.2-1 0 1 2 3 4 5 6 7 8 9 Time -0.2-0.4-0.6-0.8-1 -1 0 1 2 3 4 5 6 7 8 9 Time
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. 8
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 0.3T 0.1 t 0.5T h2 t 0.5 t 0.2 t 0.7T 0.3 t 1.5T 0.1 t 2.3T h t i0 (b) (a) (b) v s s s i i 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0-0.2-0.2-0.2-0.4-0.4-0.4-0.6-0.6-0.6 9-0.8-1 6 7 8 9-1 0 1 2 3 4 5 6 7 8 9 Time -0.8-1 -1 0 1 2 3 4 5 6 7 8 9 Time -0.8-1 0 2 4 6 8 10 12 Time
Frequency Domain The transmitted signal (envelope) 1 P(f) 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f H 1 (f) 1.5 Channel with weak multipath 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f H 2 (f) 1.5 Channel with strong multipath 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f 10
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. 11 [Fazel and Kaiser, 2008, Table 1-1]
3GPP LTE Channel Modelss 12 [Fazel and Kaiser, 2008, Table 1-3]
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. 13 [3GPP TS 45.005 3GPP; Technical Specification Group GSM/EDGE Radio Access Network; Radio Transmission and Reception (Release 7)]
Wireless Propagation [Bahai, 2002, Fig. 2.1] 14
Three steps towards modern OFDM 1. Solve Multipath Multicarrier modulation (FDM) 2. Gain Spectral Efficiency Orthogonality of the carriers 3. Achieve Efficient Implementation FFT and IFFT 15
Chapter 5 OFDM 5.2 Multi-Carrier Transmission 16 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
Single-Carrier Transmission [Karim and Sarraf, 2002, Fig 3-1] 17
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. 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. 18 [Fazel and Kaiser, 2008, Fig 1-4]
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. 19
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. 20
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. 21
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. 22
23 Orthogonality Two vectors/functions are orthogonal if their inner product is zero. The symbol a b a b a b a b Vector: 1 1 n *, k k 0 k 1 a n b n 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 2 17 2t 3 and 5 t t on 1,1 9 on 0, T
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 1 2 24 The C k are non-overlapping in freq. domain.
OFDM Let S 1, S 2,, S N be the information symbol. The discrete baseband OFDM modulated symbol can be expressed as N 1 1 2 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 25 Note that: N 1 1 2kt 2kt Re s( t) ReSkcos ImSksin N k 0 Ts Ts
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 1 2 1 2 26 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
Frequency Spectrum N 1 1 2 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 sin ct 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 27 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
Subcarrier Spacing S N 1 1 2 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. 28 Spectrum Overlap in OFDM The spectrum of each QAM signal is of the form with nulls at the center of the other subcarriers.
Normalized Power Density Spectrum Flatter when have more sub-carriers 29 [Fazel and Kaiser, 2008, Fig 1-5]
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] 30 N 1 1 2 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
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! 31
Chapter 5 OFDM 5.3 DFT and FFT 32 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
Discrete Fourier Transform (DFT) Transmitter produces N 1 1 2 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 1 1 2 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! 33
Discrete Fourier Transform (DFT) The -1 are there because we start from row 1 and column 1. 34 Key Property: 1 N N is a unitary matrix
35 DFT
36 DFT
DFT: Example 37 [http://www.fourier-series.com/fourierseries2/dft_tutorial.html]
Efficient Implementation: (I)FFT [Bahai, 2002, Fig. 2.9] 38 An N-point FFT requires only on the order of NlogN multiplications, rather than N 2 as in a straightforward computation.
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 1024. References: E. Oran Brigham, The Fast Fourier Transform, Prentice-Hall, 1974. 39
DFT Samples N 1 1 2 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 0.6 0.4 0.2 0-0.2-0.4 s n Ts s n N N 1 1 2 kn Sk exp j N k 0 N 0 nn N IDFT S n -0.6-0.8 0 1 2 3 4 5 6 7 8 40 T s
Oversampling 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 0 1 2 3 4 5 6 7 8 41
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 42 N 1 L 1 2 s n Sk exp j N k 0 Ts k T 1 s N 1 2 kn n Sk exp j LN N k 0 LN N 1 1 1 2 kn LN Sk exp j LN k 0 LN N 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
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 0.6 0.6 0.4 0.4 0.2 0.2 0 0-0.2-0.2-0.4-0.4 43-0.6-0.8 0 1 2 3 4 5 6 7 8-0.6-0.8 0 1 2 3 4 5 6 7 8
OFDM implementation by IFFT/FFT T s L L s n sn L N IFFT S n LN S k R k NS k 44 r n = N IFFT S n This form of OFDM is often referred to as Discrete Multi-Tone (DMT).
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 N S W k k k Additive white Gaussian noise 45 Sub-channel are independent. (No ICI)
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. 2.2.3] where hn 0 for n 0 and n i. i. d. ~ CN0, N w n 0 We will assume that N Remarks: 46
OFDM Architecture 47 [Bahai, 2002, Fig 1.11]
Chapter 5 OFDM 5.4 Cyclic Prefix (CP) 48 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
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 49
Cyclic Prefix: Motivation (1) Recall: Multipath Fading and Delay Spread 50
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. 51 [Jiang]
52 Cyclic Prefix
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 53 m
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 54 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
Circular Convolution: Example Find 1 2 34 5 6 1 2 3 4 5 6 1 2 3 0 0 4 5 6 0 0 55
Discussion 56 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.
Example Suppose x (1) = [1-2 3 1 2] and h = [3 2 1] [1-2 3 1 2] * [3 2 1 0 0] = [8-2 6 7 11] [1 2 1-2 3 1 2] * [3 2 1] = [3 8 8-2 6 7 11 5 2] Suppose x (2) = [2 1-3 -2 1] [2 1-3 -2 1] * [3 2 1 0 0] = [6 8-5 -11-4] [-2 1 2 1-3 -2 1] * [3 2 1] = [-6-1 6 8-5 -11-4 0 1] [ 1 2 1-2 3 1 2-2 1 2 1-3 -2 1] * [3 2 1] = [ 3 8 8-2 6 7 11 5 2] + [-6-1 6 8-5 -11-4 0 1] = [ 3 8 8-2 6 7 11-1 1 6 8-5 -11-4 0 1] 57
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 58
59 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 sn wn FFT y n p N p N r r N r n R k By circular convolution property of DFT, Rk H ksk W No ICI! k
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. 60 [Tarokh, 2009, Fig 2.9]
Chapter 5 OFDM 5.5 Remarks about OFDM 61 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
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. 62
Example: 802.11a 1 4 0.8 s 63
Summary: 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. 64
65 Complex-valued Matrix (1)
66 Complex-valued Matrix (2)
67 Complex-valued Matrix (3)
Chapter 5 OFDM 5.6 OFDM-Based Multiple Access 68 Office Hours: BKD 3601-7 Tuesday 14:00-16:00 Thursday 9:30-11:30
OFDM-based Multiple Access Three multiple access techniques 1. OFDMA, 2. OFDM-TDMA, and 3. OFDM-CDMA 69
OFDM-TDMA A particular user is given all the subcarriers of the system for any specific OFDM symbol duration. Thus, the users are separated via time slots. 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. Different OFDM symbols can be allocated to different users based on certain allocation conditions. 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 70
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. 71
OFDM-TDMA vs. OFDMA OFDM-TDMA OFDMA 72
OFDMA Block Diagram Combined subcarrier, bit, and power allocation Channel state information for all K users x N subcarriers/user User 1 data User 2 data User K data Subcarrier and bit allocation Adaptive mod. 1 Adaptive mod. 2 Adaptive mod. N Adaptive demod. 1 IFFT Add guard interval Freq. selective fading channel for User k User k data Extract bits for User k Adaptive demod. 2 FFT Remove guard interval 73 Adaptive demod. N
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. 74