EE456 Digital Communications Professor Ha Nguyen September 2015 - - PowerPoint PPT Presentation

EE456 – Digital Communications

Professor Ha Nguyen September 2015

Introduction to Orthogonal Frequency-Division Multiplexing (OFDM)

Spectrum of M-FSK

f

f

f

f

f

f

⋯

f ∆

For FSK with N = 2λ frequencies, only one of N frequencies is activated over

• ne symbol duration of Ts = λTb, where Tb is the bit duration. What frequency

that is activated over any symbol duration is determined by the mapping from λ information bits to the frequency value. FSK is not a spectral-efficient modulation scheme! Why not using all the carriers to carry information at the same time since they are orthogonal? This leads to OFDM (orthogonal frequency-division multiplexing) technique.

OFDM (Orthogonal Frequency-Division Multiplexing)

Bandwidth

N W N f T ≈ ⋅∆ = f

f

f

f

f ∆ =

f

) ( f H 2 /

T

In OFDM, the spectrum is divided into overlapping but orthogonal subcarriers. Each sub-carrier is independently modulated by QAM. The minimum subcarrier separation is 1/TN, where TN is the OFDM symbol length. OFDM can be simply looked upon as a combination of amplitude, phase and frequency modulation techniques.

Key Features of OFDM

The data rate on each of the subchannels is much less than the total data rate, and the corresponding subchannel bandwidth is much less than the total system bandwidth. The number of subchannels can be chosen so that each subchannel has a bandwidth small enough so that the frequency response over each subchannel’s frequency range is approximately constant. This ensures that inter-symbol interference (ISI) on each subchannel is small. The subchannels in OFDM need not be contiguous, so a large continuous block

• f spectrum is not needed for high rate transmission.

The modulation formats on different subchannels need not be the same. In fact

• ne may adaptively choose different modulation schemes according to the

instantaneous quality of the subchannels. The most attractive feature of OFDM is that its modulator and demodulator can be efficiently implemented with DSP. The main technical issues that impair performance of OFDM are frequency offset and timing jitter, which degrade the orthogonality of the subchannels. Having QAM signals transmitted simultaneously over all carriers makes the peak-to-average power ratio (PAPR) of the OFDM signal significantly higher than that of single-carrier QAM signal. This is a serious problem when nonlinear amplifiers are used.

OFDM Viewed and Implemented as a Multiple QAM Systems

e

bits Infor. bps

r ( ) ( ) ( ) s t h t t ∗ + w

t kT =

e

Detected

• infor. bits

Review of DFT/IDFT

Let x[n], 0 ≤ n ≤ N − 1, be a DT sequence. The N-point DFT of x[n] is DFT{x[n]} = X[i]

N−1

• n=0

x[n]e−j 2πni

, 0 ≤ i ≤ N − 1. Given X[i], 0 ≤ i ≤ N − 1, the sequence x[n] can be recovered using the IDFT: IDFT{X[i]} = x[n] 1 N

N−1

• i=0

X[i]ej 2πni

, 0 ≤ n ≤ N − 1. When N is a power of two, the DFT and IDFT can be efficiently performed using the fast Fourier transform (FFT) and inverse FFT (IFFT) algorithms. When the discrete-time sequence x[n] is passed through a discrete-time linear time-invariant system whose impulse response is h[n], the output y[n] is the discrete-time convolution of the input and the channel impulse response: y[n] = h[n] ∗ x[n] = x[n] ∗ h[n] =

∞

• k=−∞

h[k]x[n − k] The N-point circular convolution of x[n] and h[n], both with length N, is y[n] = h[n] ⊗ x[n] = x[n] ⊗ h[n] =

N−1

• k=0

h[k]x[(n − k) mod N]. Circular convolution in time leads to multiplication in frequency: Y [i] = DFT{y[n] = x[n] ⊗ h[n]} = X[i]H[i] , 0 ≤ i ≤ N − 1

Example of Circular Convolution and FFT/IFFT in Matlab

Implementation of OFDM with DFT/IDFT

t nT =

t nT = IDFT

• r

IFFT Add cyclic prefix, and parallel- to-serial converter p(t) bits/sec

r [0] X [1] X [ 1] X N − [0] x [1] x [ 1] x N −

⋮

( )

x t ɶ

( )

cos 2 f t π

( )

cos 2 f t π

⋮

Remove prefix, and serial-to- parallel converter [0] y [1] y [ 1] y N −

⋮

[0] Y [1] Y [ 1] Y N −

⋮

FFT

• r

DFT P/S Converter p(t) ( )

x t ɶ

( )

sin 2 f t π ( ) s t ( ) t r

( )

sin 2 f t π [ ], [ 1], , [ 1] x x x N µ µ − − + − ɶ ɶ ɶ ⋯ ( ) p t − ( ) p t − [ ], [ 1], , [ 1] y y y N µ µ − − + − ⋯ (a) Transmitter (b) Receiver bits M0-QAM modulator S/P Converter M1-QAM modulator MN-1-QAM modulator M0-QAM demodulator M1-QAM demodulator MN-1-QAM demodulator

How Does the Cyclic Prefix Work in OFDM?

Equivalent discrete-time channel

{ }

[ ] n h n

, [ ], [ 1], , [ 1], x x x N µ µ − − + − ɶ ɶ ɶ ⋯ ⋯ ⋯ , [ ], [ 1], , [ 1], y y y N µ µ − − + − ⋯ ⋯ ⋯ [ ], [ 1], , [ 1] x N x N x N µ µ − − + − ⋯ [0], [1], [2], , [ 1] x x x x N µ − − ⋯⋯⋯ [ ], [ 1], , [ 1] x N x N x N µ µ − − + − ⋯ Append last symbols to the front µ Cyclic prefix (CP) of length µ Original signal sequence of length N

≈

One has ˜ x[n] = x[n mod N] for −µ ≤ n ≤ N − 1, which also means that ˜ x[n − k] = x[(n − k) mod N] for −µ ≤ n − k ≤ N − 1. Given ˜ x[n] is the input of the channel (ignoring noise), the channel output between 0 ≤ n ≤ N − 1 can be computed as: y[n] = ˜ x[n] ∗ h[n] =

µ

• k=0

h[k]˜ x[n − k] =

µ

• k=0

h[k]x[(n − k) mod N] = x[n] ⊗ h[n] where the third equality follows from the fact that, for 0 ≤ k ≤ µ, ˜ x[n − k] = x[(n − k) mod N] for 0 ≤ n ≤ N − 1. Taking into account AWGN, the DFT of the channel output yields Y [i] = DFT{y[n] = (x[n] + w[n]) ⊗ h[n]} = X[i]H[i] + W [i], 0 ≤ i ≤ N − 1,

[ ], [ 1], , [ 1] x N x N x N µ µ − − + − ⋯ [0], [1], [2], , [ 1] x x x x N µ − − ⋯⋯⋯ [ ], [ 1], , [ 1] x N x N x N µ µ − − + − ⋯ Append last symbols to the front µ Cyclic prefix (CP) of length µ Original signal sequence of length N

≈

[0], , [ 1] y y N − µ N [0], , [ 1] y y N − µ N [0], , [ 1] y y N − µ N CP CP CP Data block Data block Data block [0], , [ 1] x x N − [0], , [ 1] x x N − [0], , [ 1] x x N −

An OFDM symbol is basically a super-symbol obtained by multiplexing many M-QAM symbols in a complicated manner. The length of a super-symbol (TN) becomes longer and hence more resistent to multipath effect. One can also use zero padding to create a guard interval between consecutive OFDM symbols, hence avoiding ISI.

OFDM System with the Equivalent Discrete-Time Multipath Channel

[0] X [1] X [ 1] X N − [0] x [1] x [ 1] x N − [0] y [1] y [ 1] y N − [0] Y [1] Y [ 1] Y N − [ ], [ 1], , [ 1] x x x N µ µ − − + − [ ], [ 1], , [ 1] y y y N µ µ − − + −

{ }

[ ] n h n

It was shown that the input of the IFFT block in the transmitter and the output

• f the FFT block in the receiver is related by Y [i] = X[i]H[i] + W [i],

0 ≤ i ≤ N − 1, where W [i] is Gaussian noise component. The use of CP and IFFT/FFT decomposes the wideband channel H(f) into a set

• f narrowband orthogonal subchannels, each with a different QAM symbol.

The demodulator needs to know the channel gains H[i] to recover the original QAM symbols by dividing out these gains: X[i] = Y [i]/H[i]. This process is called frequency equalization. The benefits of adding a cyclic prefix come at a cost. Since µ symbols are added to the input data blocks, there is an overhead of µ/N and a resulting data-rate reduction of N/(µ + N). The transmit power associated with sending the cyclic prefix is also wasted since this prefix consists of redundant data.

16-QAM vs. “Constellation” of Time Samples (plotted with 4 random OFDM symbols, i.e., 4 × N = 64 samples)

−1 1 −2 −1 1 2

16-QAM of X[i], PAPR ≈ 2.55 dB

−0.5 0.5 −0.5 0.5 1

x[n], N = 16, PAPR ≈ 7.96 dB

“Constellation” of Time Samples (plotted with 4 random OFDM symbols) for N = 64 and N = 128

−0.5 0.5 −0.5 0.5 1

x[n], N = 64, PAPR ≈ 11.91 dB

−0.5 0.5 −0.5 0.5 1

x[n], N = 128, PAPR ≈ 14.33 dB

The peak powers appear to be the same, but the average power of the right constellation is much smaller than that of the left one, giving rise to a higher PAPR.

Communication Services using OFDM

Wireless Wireline IEEE 802.11a, g, n (WiFi) Wireless LANs ADSL and VDSL broadband access via POTS copper wiring IEEE 802.15.3a Ultra Wideband (UWB) Wireless PAN MoCA (Multi-media over Coax Alliance) home networking IEEE 802.16d, e (WiMAX), WiBro, PLC (Power Line Communication) and HiperMAN Wireless MANs IEEE 802.20 Mobile Broadband Wireless Access (MBWA) DVB (Digital Video Broadcast) terrestrial TV systems: DVB -T, DVB -H, T-DMB, and ISDB-T DAB (Digital Audio Broadcast) systems: EUREKA 147, Digital Radio Mondiale, HD Radio, T-DMB, and ISDB-TSB Flash-OFDM cellular systems 3GPP UMTS & 3GPP@ LTE (Long-Term Evolution), and 4G

Multipath Problem in High-Speed Wireless Transmission

• ✂ ☎

• ✍

• ✂

Example: Consider the symbol rate of 106 symbols/sec ⇒ The receiver expects a specific symbol within a window of 1 µs. If multi-path delays the signal by more than 1 µs (easily happen in real propagation environment), then the receiver will also receive the symbol in the next symbol period, causing inter-symbol-interference (ISI), hence severe performance degradation.

Application Example of OFDM: Wireless LAN (Wi-Fi) Standards

The IEEE 802.11a Wireless LAN standard, which occupies 20 MHz of bandwidth in the 5 GHz unlicensed band, is the first version of 802.11 family that is based

• n OFDM. Modulation choices are BPSK, QPSK, 16-QAM and 64-QAM. The

maximum bit rate is 54 Mbps. The 802.11g standard is virtually identical, but operates in the smaller and more crowded 2.4 GHz unlicensed ISM band. IEEE 802.11ac, released in December 2012, also operates in the 5 GHz band. It uses a wider RF bandwidth (80 or 160 MHz), multiple-input multiple-output (MIMO) technology (up to 8 MIMO streams), high-density modulation (up to 256 QAM) and could deliver a maximum data rate close to 7Gbps (on eight 256-QAM channels, each delivering 866.7Mbps). The latest development of 802.11 standard is 802.11ad, which operates in the tri-band 2.5/5.0/60 GHz.

The 802.11a Standard

N = 64 subcarriers: 48 are for data, the outer 12 are zeroed to reduce adjacent channel interference, and 4 carriers used as pilot symbols for channel estimation and synchronization. The CP has µ = 16 samples. The length of each OFDM symbol is 80 samples. The transmitter gets periodic feedback from the receiver about the packet error rate, and uses this information to pick an appropriate error correction code and modulation scheme. The error correction code is a convolutional code with one of three possible code rates: rc = 1/2, 2/3, or 3/4. The modulation types are BPSK, QPSK, 16-QAM, or 64-QAM. The bandwidth B (and sampling rate 1/Ts) is 20 MHz. There are 64 subcarriers evenly spaced, therefore the subcarrier bandwidth is: BN = 20 MHz 64 = 312.5 KHz. Since µ = 16 and 1/Ts = 20MHz, the maximum delay spread for which ISI is removed is Tm < µTs = 16 20 MHz = 0.8 µsec, The symbol duration per subchannel is T (+CP)

= TN + µTs = (N + µ)Ts = 80Ts = 80 20 × 106 = 4 µsec The data rate per subchannel is log2 M/T (+CP)

. Thus, the minimum data rate for this system, corresponding to BPSK (1 bit/symbol), an r = 1/2 code, and taking into account that only 48 subcarriers actually carry information data, is given by (rb)min = 48 sub. × 1/2 bit coded bit × 1 code bit

• sub. symbol × 1 sub. symbol

4 × 10−6 = 6Mbps The maximum data rate corresponds to 64-QAM and r = 3/4 code: (rb)max = 48 sub. × 3/4 bit coded bit × 6 code bit

• sub. symbol × 1 sub. symbol

4 × 10−6 = 54Mbps

Multipath Channels Simulated in the OFDM Lab (Fs = 32 MHz)

5 10 15 0.5 1 1.5 Frequency (MHz) Magnitude response Echo delay = 0, Echo gain = 0 5 10 15 0.5 1 1.5 Frequency (MHz) Magnitude response Echo delay = 8, Echo gain = 0.5 5 10 15 0.5 1 1.5 Frequency (MHz) Magnitude response Echo delay = 16, Echo gain = -0.33333 5 10 15 0.5 1 1.5 Frequency (MHz) Magnitude response Echo delay = 32, Echo gain = -0.25

For a single-echo channel where the echo delay is d samples and the echo gain is α, the squared magnitude frequency response is |H(ejω)|2 = (1 + α2) + 2α cos(dω).