Orthogonal Frequency Division Modulation (OFDM)

Orthogonal Frequency Division Modulation (OFDM) •  •  •  • 

OFDM diagram Inter Symbol Interference Packet detection and synchronization Related works

Motivation •  Signal over wireless channel  

y[n] = Hx[n]

•  Work only for narrow-band channels, but not for wide-band channels  

e.g., 20 MHz for 802.11 20MHz

Capacity = BW * log(1+SNR)

frequency 2.45GHz (Central frequency)

OFDM is a special case isofaFrequency Multiplex (FDM). As an OFDM special caseDivision of Frequency Division Multiplex (F a faucet, in contrast the in OFDM signal is like sig as likeofwater flow out of a faucet, contrast the OFDM (a) is like water flowisout (b) water comes in(b)one bigcomes stream be sub-divided. showO water in and one cannot big stream and cannot beOFDM sub-divided. Fig. 1little – (a) streams. A Regular-FDM single carrier – A whole bunch of water coming all in one strea little streams. Wide-band channel Multiple narrow-band channels

Basic Concept of OFDM

(a)

Fig. 1 – (a)Orthogonal-FDM A Regular-FDM single carrier – A whole bunchcoming of waterfrom coming allof insmall one stream. (b) – Same amount of water a lot streams. Orthogonal-FDM – Same amount of water coming from a lot of small streams.

Think about what the advantage might be of one over the other? One obvious one is tha

Think about what the advantage might be of one over the other? One obvious one is that if I put my thumb over the faucet hole, I can stop the water flow but I cannot do the same for th my thumb over the faucet hole, I can stop the water flow but I cannot do the same for the shower. So both although the they samerespond thing, differently they respond differently to interference. So although do theboth samedo thing, to interference.

(a)

(a)

(b)

(b)

Fig. 1 – (a) A Regular-FDM single carrier – A whole bunch o Fig. 1 – (a) A Regular-FDM single –carrier – A whole bunch of water comin Orthogonal-FDM Same amount of water coming from a lot

Orthogonal-FDM – Same amount of water coming from a lot of small strea Think about what the advantage might be of one over the o Fig. 2 – All cargo on one truck vs. splitting the shipment into more than one. Fig. 2Think – All cargo onwhat one my truck vs. splitting the shipment than one. about thethumb advantage might behole, ofinto one over thethe other? over the faucet Imore can stop waterOne flowob bu over isthe hole, can stop thething, water flow but I cannot do So although bothIofdo the same respond differentl Another way tomy see thumb this intuitively to faucet use the analogy making a shipment viathey a truck.

Another way tohire see athis is to of use the analogy of making acarry shipment via a truc We have two options, one big intuitively truck or asame bunch smaller ones.respond Both methods the to interfere So although both do the thing, they differently We have of two options, big truckonly or a1/4bunch ofon smaller ones.trucking Both methods car exact same amount data. But in one case hire of anaaccident, of data the OFDM exact same amount of data. But in case of an accident, only 1/4 of data on the OFDM tr will suffer. will suffer. Send a sample using the entire band Copyright 2004 Charan Langton

Send samples concurrently using multiple orthogonal sub-channels www.complextoreal.com

Why OFDM is better?

t

t

f

0 Wide-‐band

1

1 0

0

0

1 0 1 1 0 0 0 1 …........ Narrow-‐band

f

•  Multiple sub-channels (sub-carriers) carry samples sent at a lower rate  

Almost same bandwidth with wide-band channel

•  Only some of the sub-channels are affected by interferers or multi-path effect

Importance of Orthogonality •  Why not just use FDM (frequency division multiplexing)  

Not orthogonal

Individual sub-‐channel

Leakage interference from adjacent sub-‐channels

f guard band

Guard bands protect leakage interference

•  Need guard bands between adjacent frequency bands à extra overhead and lower throughput

f

Difference between FDM and OFDM guard band (b) Guard bands protect leakage from adjacent frequencies f Figure 9: Frequency Division Multiplexing Frequency division multiplexing

f

Orthogonal sub-carriers in OFDM Don’t need guard bands Figure 10: Sub-carriers in OFDM

Orthogonal Frequency Division Modulation

rd bands protect leakage from adjacent frequencies

ure 9: Frequency Division Multiplexing

rs

* x[1]

Symbol: 2 periods of f0

f

* xTransmit [2] transmit

IFFT

f

+

* x[3]


…

t

f

Data c10: oded in frequency domain TransformaNon to Nme domain: Figure Sub-carriers in OFDM each frequency is a sine wave In Nme, all added up


efrequency sub-carriers domain are orthogonal, i.e. they do not interfere each Transformation to timewith domain:

each frequency is a sine wave � p k in time, all added up. = e−j2π N t ej2π N t = 0 (p �= k) (6) FFT

al eﬃciency without causing interference between the sub-carriers.

ram

Channel frequency response Decode each subcarrier separately

N/2−1

receive t=−N/2

Channel frequency response

Decode each frequency bin separately

n input - a stream of Dtime bits. Suppose we have nfft sub-carriers. Time d omain s ignal fsignal fft = nsym symbols, where each symbol hasFrequency nfft bits.domain Here we

dnfft the stream is numbers serializedto and output. basic outline of this complex produce theAsymbol. The bits are and Receiver d the OFDM stream is Transmitter serialized and output. A basic outline of this

Figure 11: Basic view of the OFDM Transmitter and Receive Figure 11: Basic view of the OFDM Transmitter and Receive

Orthogonality of Sub-carriers IFFT

Encode: frequency-domain samples à time-domain sample N 2!1

x(t) =

"

X[k]e j 2 ! kt

N

k=!N 2

Time-domain

Frequency-domain N 2!1

1 ! j 2 ! kt X[k] = x(t)e " N t=N 2

N

FFT

Decode: time-domain samples à frequency-domain sample N 2!1

Orthogonality of any two bins :

" t=N 2

e! j 2 ! kt N e! j 2 ! pt

N

= 0, #p $ k

Example

•  Say we use BPSK and 4 sub-carriers to transmit g. 7 – A bit stream that will be modulated using a 4 carrier OFDM. a stream of samples

st few bits are 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1,…

•  Serial to parallel conversion of samples

t’s now write these bits in rows of fours, since this demonstration will use only four subFrequency-‐domain signal Time-‐domain signal rriers. We have effectively done a serial to parallel conversion.

c1 c2 c3 c4 IFFT 0 2 -‐ 2i 0 2 + 2i symbol1 1 1 -‐1 -‐1 ble I – Serial to parallel conversion of data bits. 2 0 -‐ 2i 2 0 + 2i symbol2 1 1 1 -‐1 c1 symbol3 c2 1 -‐c3 -‐2 2 2 2 1 -‐1 -‐1 c4 1 symbol4 1 -‐1 -1 -‐2 0 -‐ 2i -‐2 0 + 2i 1 -‐1 -‐1 -1 0 -‐2 -‐ 2i 0 -‐2 + 2i 1 symbol5 1 -‐1 11 1 -‐1 -1 0 -‐2 + 2i 0 -‐2 -‐ 2i 1 1 1-1 1 symbol6 -1 -‐1 -‐-1 -1 1 -1 -1

•  Parallel to serial conversion, and transmit time-1 1 1 -1 samples -1 domain -1 1 1 0, 2 -‐ 2i, 0, 2 + 2i, 2, 0 -‐ 2i, 2, 0 + 2i, -‐2, 2, 2, 2, -‐2, 0 -‐ 2i, -‐2, 0 + 2i, 0, -‐2 -‐ 2i, 0, -‐2 + 2i, 0, -‐2 + 2i, 0, -‐2 -‐ 2i, …

Fig. 8 – Sub-carrier 1 and the bits it is modulating (the first column of Table I)

t1

t2

t3

t4

Orthogonal Frequency Division Multiplex (OFDM) Tutorial

t5

t6

6

Carrier 2 - The next carrier is of frequency 2 Hz. It is the next orthogonal/harmonic to frequency of the first carrier of 1 Hz. Now take the bits in the second column, marked c2, 1, 1, -1, 1, 1, -1 and modulate this carrier with these bits as shown in Fig.

bin1


Carrier 2 - The next carrier is of frequency 2 Hz. It is the next orthogonal/harmonic to frequency of the first carrier of 1 Hz. Now take the bits in the second column, marked c2, 1, 1, -1, 1, 1, -1 and modulate this carrier with these bits as shown in Fig.


symbol1 1 1 -‐1 -‐1 symbol2 1 1 1 -‐1 symbol3 1 -‐1 -‐1 -‐1 symbol4 -‐1 1 -‐1 -‐1 symbol5 -‐1 1 1 -‐1 symbol6 -‐1 -‐1 1 1

Carrier 2 - The next carrier is of frequency 2 Hz. It is the next orthogonal/harmonic to frequency Fig. – Sub-carrier bitstake thatthe it isbits modulating (the 2nd column of Table of the9 first carrier of21and Hz.the Now in the second column, marked c2, I)1, 1, -1, 1, 1, -1 and modulate this carrier with these bits as shown in Fig. Carrier 3 – Carrier 3 frequency is equal to 3 Hz and fourth carrier has a frequency of 4 Hz. The third carrier is modulated with -1, 1, 1, -1, -1, 1 and the fourth with -1, -1, -1, -1, -1, -1, 1 from Table I.

bin2

Fig. 9 – Sub-carrier 2 and the bits that it is modulating (the 2nd column of Table I)

Carrier 3 – Carrier 3 frequency is equal to 3 Hz and fourth carrier has a frequency of 4 Hz. The third carrier is modulated with -1, 1, 1, -1, -1, 1 and the fourth with -1, -1, -1, -1, -1, -1, 1 from Table I.

bin3

Fig. 9 – Sub-carrier 2 and the bits that it is modulating (the 2nd column of Table I)

Carrier 3 – Carrier 3 frequency is equal to 3 Hz and fourth carrier has a frequency of 4 Hz. The third carrier is modulated with -1, 1, 1, -1, -1, 1 and the fourth with -1, -1, -1, -1, -1, -1, 1 from Table I.

bin4

Fig. 10 – Sub-carrier 3 and 4 and the bits that they modulating (the 3rd and 4th columns of Table I)

If the path from the transmitter to the receiver either has reflections or obstructions fading effects. In this case, the signal reaches the receiver from many different rou copy of the original. Each of these rays has a slightly different delay and slightly d The time delays result in phase shifts which added to main signal component (assu one.) causes the signal to be degraded.

Multi-Path Effect Faded path


D0 '0

D1 '1

D113 Secondary path gain W 1 Secondary path delay

Tree D 0 Line of sight path gain W 0 Path delay

Dk 'k

Reflected multipath

Dk Wk

Secondary path gain Secondary path delay

Fig. 19 – Reflected signals arrive at a delayed time period and interfere with the main line of sight signal, if there is one. In pure Raleigh fading, we have no Kmain signal, all components are reflected. 1

hc (t ) ¦ D k G (t W k ) k 0main signal and cause either gains in In fading, the reflected signals that are delayed add to the the signal strength or deep fades. And by deep fades, we mean thegain signal is nearly wiped out. D k Compl exthat p ath The signal level is so small that the receiver can not decide what was there. ⟺ W 0 Normalized path delay relative to LOS

y(t) = h(0)x(t) + h(1)x(t !1) + h(2)x(t ! 2) +! = # h(")x(t ! ") = h(t) $ x(t)

Y ( f ) = H ( f )X( f )

' k delay W k spread W 0 difference in path time The maximum time delay that occurs is called the of the signal in that environment. " This delay spread can be short so that it is less than symbol time or larger. Both cases, cause frequency-domain time-domain different types of degradations to the signal. delay spread of a signal thesignal is lost and demodulation must Fig. 18The – Fading is big problem forchanges signals.asThe environment is changing as all cell phone userswith know. dealing it. Fading is particular problem when the link path is changing, such as

Current symbol + delayed-version symbol à Signals are deconstructive in only certain frequencies

Orthogonal Frequency Division Multiplex (OFDM) Tutorial Orthogonal Frequency Division Multiplex (OFDM) Tutorial

Frequency Selective Fading

14 14

Fig. 20 – (a) The signal we want to send andO the channel frequency response are well matched. A Frequency selecNve fading: nly some sub-‐carriers get a(b) ﬀected fading channel has frequencies that do not allow anything to pass. Data is lost sporadically. (c) With OFDM, where we have many little sub-carriers, only a small sub-set of the data is lost due to fading.

Symbol

1

Symbol

2

Increase distance from car in front to avoid splash. The reach of splash is same as Fig. of 21a–signal. Delay Fig. spread like the splash you might get from the car sa spread 22aisshows theundesired symbol and its splash. In composite, these fading, the front similarly a splash backwards which we wish to noise and affect thesymbol beginning of the throws next symbol as shown in (b).

Inter Symbol Interference (ISI)

•  The delayed version of a symbol overlaps with the adjacent symbol

Increase distance from car in front to avoid splash. The reach of splash is same spread of a signal. Fig. 22a shows the symbol and its splash. In composite, these noise and affect the beginning of the next symbol as shown in (b).

Fig. 21 – The PSK symbol and its delayed version. (a) The delayed, attenuated signal and (b) composite interference.

To mitigate this noise at theto front avoid of the symbol, we willis move •  One simple solution this toour symbol further region as shown below. A little bit of blank space has been added Fig. of 21 delay – The spread PSK symbol and its delayed version. (a) a Thetoguard-band delayed, attenuated signal and (b) composite interference. symbols catch the delay spread. introduce

To mitigate this noise at the front of the symbol, we will move our symbol furth region of delay spread as shown below. A little bit of blank space has been adde symbols to catch the delay spread.

and symbol back so the arriving delayed signal peters out in the gray r Fig 22Guard – Movebthe interference to the next symbol!

Cyclic Prefix (CP) •  However, we don’t know the delay spread exactly  

The hardware doesn’t allow blank space because it needs to send out signals continuously

•  Solution: Cyclic Prefix  

Make the symbol period longer by copying the tail and glue it in the front


Symbol 2

Symbol 1

Copy this part at front

Copy this part at front

In 802.11, CP:data = 1:4

Portion added in the front

Original symbol

Extension

Original symbol

17

not want the start of the symbol to fall in this region, so lets just sli that the start of the original symbol lands at the outside of this zone something.

Cyclic Prefix (CP)

Fig. of 24 –the If weusage move the just put in convenient filler •  Because ofsymbol FFT, back the and signal is periodic

have a continuous signal but one that can get corrupted and we don’t anyway before demodulating.

FFT( ) = e xp(-‐2jπ f)*FFT( ) Δ Slide the symbol to start at the edge of the delay spread time and th copy ofvwhat symbol. delayed ersion turns out to be tail end of theoriginal signal

wanttime the start of the symbol to be out of the •  Delay1.inWethe domain corresponds to delay spread zon 2. We start the signal at the new boundary such that the actual sym rotation in the frequency domain zone.

•  Can still obtain the correct signal in the frequency We will extending the symbol so it is 1.25 times as long, to do t domain bybe compensating this rotation

symbol and glue it in the front. In reality, the symbol source is cont

Cyclic Prefix (CP) w/o mulNpath y(t) à FFT( ) àY[k] = H[k]X[k] original signal

w mulNpath y(t) à FFT( ) àY[k] = α(1+exp(-‐2jπΔk))*X[k]

= H’[k]X[k]

original signal + delayed-‐version signal

Lump the phase shid in H

We extend the symbol into the empty space, so the actual symbol i

Side Benefit of CP

• 

But now the start of the symbol is still in the danger zone, and this our symbol the slicereven needs if it inthe order to make a d Allowthing theabout signal to be since decoded not want the start of the symbol to fall in this region, so lets just sli packet is detected after some delay that the start of the original symbol lands at the outside of this zone something.

Fig. 24 – If we move the symbol back and just put in convenient filler undecodable decodable

have a continuous signal but one that can get corrupted and we don’t anyway before demodulating.

Slide the symbol to start at the edge of the delay spread time and th copy of what turns out to be tail end of the symbol.

OFDM Diagram Modulation

Transmitter S/P

IFFT

Insert CP

P/S

D/A

channel

De-mod

+ P/S

FFT

remove CP

Receiver

S/P

A/D

noise


Unoccupied Subcarriers

•  Edge sub-carriers are more vulnerable to errors under discrete FFT  

Frequency might be shifted due to noise or multi-path

•  Leave them unused  

In 802.11, only 48 of 64 bins are occupied bins

•  Is it really worth to use OFDM when it costs so many overheads (CP, unoccupied bins)?