**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

### frequency 2.45GHz (Central frequency)

### 20MHz

### Capacity = BW * log(1+SNR)

**Basic Concept of OFDM **

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 1

Intuitive Guide to Principles of Communications www.complextoreal.com

**Orthogonal Frequency Division Multiplexing (OFDM) **

**Modulation - a mapping of the information on changes in the carrier phase, frequency or ***amplitude or combination. *

**Multiplexing - method of sharing a bandwidth with other independent data channels. **

OFDM is a combination of modulation and multiplexing. Multiplexing generally refers to

independent signals, those produced by different sources. So it is a question of how to share the spectrum with these users. In OFDM the question of multiplexing is applied to independent signals but these independent signals are a sub-set of the one main signal. In OFDM the signal itself is first split into independent channels, modulated by data and then re-multiplexed to create the OFDM carrier.

OFDM is a special case of Frequency Division Multiplex (FDM). As an analogy, a FDM channel is like water flow out of a faucet, in contrast the OFDM signal is like a shower. In a faucet all water comes in one big stream and cannot be sub-divided. OFDM shower is made up of a lot of little streams.

(a) (b)

**Fig. 1 – (a) A Regular-FDM single carrier – A whole bunch of water coming all in one stream. (b) **
**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 that if I put my thumb over the faucet hole, I can stop the water flow but I cannot do the same for the shower.

So although both do the same thing, they respond differently to interference.

**Fig. 2 – All cargo on one truck vs. splitting the shipment into more than one. **

Another way to see this intuitively is to use the analogy of making a shipment via a truck.

We have two options, one hire a big truck or a bunch of smaller ones. Both methods carry the exact same amount of data. But in case of an accident, only 1/4 of data on the OFDM trucking will suffer.

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 1

Intuitive Guide to Principles of Communications www.complextoreal.com

**Orthogonal Frequency Division Multiplexing (OFDM) **

**Modulation - a mapping of the information on changes in the carrier phase, frequency or ***amplitude or combination. *

**Multiplexing - method of sharing a bandwidth with other independent data channels. **

OFDM is a combination of modulation and multiplexing. Multiplexing generally refers to

independent signals, those produced by different sources. So it is a question of how to share the spectrum with these users. In OFDM the question of multiplexing is applied to independent signals but these independent signals are a sub-set of the one main signal. In OFDM the signal itself is first split into independent channels, modulated by data and then re-multiplexed to create the OFDM carrier.

OFDM is a special case of Frequency Division Multiplex (FDM). As an analogy, a FDM channel is like water flow out of a faucet, in contrast the OFDM signal is like a shower. In a faucet all water comes in one big stream and cannot be sub-divided. OFDM shower is made up of a lot of little streams.

(a) (b)

**Fig. 1 – (a) A Regular-FDM single carrier – A whole bunch of water coming all in one stream. (b) **
**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 that if I put my thumb over the faucet hole, I can stop the water flow but I cannot do the same for the shower.

So although both do the same thing, they respond differently to interference.

**Fig. 2 – All cargo on one truck vs. splitting the shipment into more than one. **

Another way to see this intuitively is to use the analogy of making a shipment via a truck.

We have two options, one hire a big truck or a bunch of smaller ones. Both methods carry the exact same amount of data. But in case of an accident, only 1/4 of data on the OFDM trucking will suffer.

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 1

### Intuitive Guide to Principles of Communications www.complextoreal.com

**Orthogonal Frequency Division Multiplexing (OFDM) **

**Modulation - a mapping of the information on changes in the carrier phase, frequency or ** *amplitude or combination. *

**Modulation - a mapping of the information on changes in the carrier phase, frequency or**

**Multiplexing - method of sharing a bandwidth with other independent data channels. **

**Multiplexing - method of sharing a bandwidth with other independent data channels.**

### OFDM is a combination of modulation and multiplexing. Multiplexing generally refers to

### independent signals, those produced by different sources. So it is a question of how to share the spectrum with these users. In OFDM the question of multiplexing is applied to independent

### signals but these independent signals are a sub-set of the one main signal. In OFDM the signal itself is first split into independent channels, modulated by data and then re-multiplexed to create the OFDM carrier.

### OFDM is a special case of Frequency Division Multiplex (FDM). As an analogy, a FDM channel is like water flow out of a faucet, in contrast the OFDM signal is like a shower. In a faucet all

### water comes in one big stream and cannot be sub-divided. OFDM shower is made up of a lot of little streams.

### (a) (b)

**Fig. 1 – (a) A Regular-FDM single carrier – A whole bunch of water coming all in one stream. (b) ** **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 that if I put my thumb over the faucet hole, I can stop the water flow but I cannot do the same for the shower.

### So although both do the same thing, they respond differently to interference.

**Fig. 2 – All cargo on one truck vs. splitting the shipment into more than one. **

### Another way to see this intuitively is to use the analogy of making a shipment via a truck.

### We have two options, one hire a big truck or a bunch of smaller ones. Both methods carry the exact same amount of data. But in case of an accident, only 1/4 of data on the OFDM trucking will suffer.

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 1

### Intuitive Guide to Principles of Communications

www.complextoreal.com**Orthogonal Frequency Division Multiplexing (OFDM) **

**Modulation - a mapping of the information on changes in the carrier phase, frequency or ***amplitude or combination. *

**Multiplexing - method of sharing a bandwidth with other independent data channels. **

### OFDM is a combination of modulation and multiplexing. Multiplexing generally refers to

### independent signals, those produced by different sources. So it is a question of how to share the spectrum with these users. In OFDM the question of multiplexing is applied to independent signals but these independent signals are a sub-set of the one main signal. In OFDM the signal itself is first split into independent channels, modulated by data and then re-multiplexed to create the OFDM carrier.

### OFDM is a special case of Frequency Division Multiplex (FDM). As an analogy, a FDM channel is like water flow out of a faucet, in contrast the OFDM signal is like a shower. In a faucet all water comes in one big stream and cannot be sub-divided. OFDM shower is made up of a lot of little streams.

### (a) (b)

**Fig. 1 – (a) A Regular-FDM single carrier – A whole bunch of water coming all in one stream. (b) **
**Orthogonal-FDM – Same amount of water coming from a lot of small streams. **

### So although both do the same thing, they respond differently to interference.

**Fig. 2 – All cargo on one truck vs. splitting the shipment into more than one. **

### Another way to see this intuitively is to use the analogy of making a shipment via a truck.

Copyright 2004 Charan Langton www.complextoreal.com

### Send a sample using the entire band

### Send samples concurrently using

### multiple orthogonal sub-channels

### Wide-band channel Multiple narrow-band channels

**Why OFDM is better? **

### • 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

### f f

### t t

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

**Importance of Orthogonality **

### • Why not just use FDM (frequency division multiplexing)

### Not orthogonal

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

### f Individual sub-‐channel Leakage interference from

### adjacent sub-‐channels

### f guard band

### Guard bands protect

### leakage interference

**Difference between FDM and OFDM **

### f guard band

### Frequency division multiplexing

### 2 OFDM 6

### (a) Adjacent sub-channels interfere

### (b) Guard bands protect leakage from adjacent frequencies

### Figure 9: Frequency Division Multiplexing

### Figure 10: Sub-carriers in OFDM

### It is easy to see that these sub-carriers are orthogonal, i.e. they do not interfere with each other.

### =

### N/2 1

### t= N/2

### e ^{j2}

^{N}

^{k}

^{t} e ^{j2}

^{N}

^{p}

^{t} = 0 (p ⇥= k) (6) Hence we can improve spectral e⇥ciency without causing interference between the sub-carriers.

### 2.3 OFDM Block Diagram

### At the transmitter, we have an input - a stream of D bits. Suppose we have nfft sub-carriers.

### Then we must transmit D/nfft = nsym symbols, where each symbol has nfft bits. Here we are assuming each signal value in our modulation represents one bit, e.g BPSK. Note that if we use 4-QAM : each symbol will have 2 nfft bits, if we use 16-QAM : each symbol will have 4 nfft bits, etc. The bits in each is then fed into a serial-to-parallel converter and modulated (BPSK/4-QAM/etc.). Note that it is possible for different sub-carriers to use differ- ent modulation schemes. An inverse fast Fourier transform (IFFT) is performed on the nfft complex numbers. The stream is fed to a parallel to serial converter. Hence the output signal is a sequence of nsym symbols where each symbol has nfft samples.

### Orthogonal sub-carriers in OFDM Don’t need guard bands

### f

**Orthogonal Frequency Division Modulation **

2 OFDM 6

(a) Adjacent sub-channels interfere

(b) Guard bands protect leakage from adjacent frequencies

Figure 9: Frequency Division Multiplexing

Figure 10: Sub-carriers in OFDM

It is easy to see that these sub-carriers are orthogonal, i.e. they do not interfere with each other.

=

N/2 1

t= N/2

e ^{j2} ^{N}^{k}^{t}e^{j2} ^{N}^{p} ^{t} = 0 (p ⇥= k) (6)

Hence we can improve spectral e⇥ciency without causing interference between the sub-carriers.

2.3 OFDM Block Diagram

At the transmitter, we have an input - a stream of D bits. Suppose we have nfft sub-carriers.

Then we must transmit D/nfft = nsym symbols, where each symbol has nfft bits. Here we are assuming each signal value in our modulation represents one bit, e.g BPSK. Note that if we use 4-QAM : each symbol will have 2 nfft bits, if we use 16-QAM : each symbol will have 4 nfft bits, etc. The bits in each is then fed into a serial-to-parallel converter and modulated (BPSK/4-QAM/etc.). Note that it is possible for different sub-carriers to use differ- ent modulation schemes. An inverse fast Fourier transform (IFFT) is performed on the nfft complex numbers. The stream is fed to a parallel to serial converter. Hence the output signal is a sequence of nsym symbols where each symbol has nfft samples.

Data coded in frequency domain

### f

^{IFFT }

* x[1]

* x[2]

* x[3]

…

### t

TransformaNon to Nme domain:

each frequency is a sine wave In Nme, all added up

Channel frequency response

### 3

## Orthogonal Frequency Division Modulation

### Data coded in frequency domain

N carriers

B

### Transformation to time domain:

### each frequency is a sine wave in time, all added up.

f

Transmit

Symbol: 8 periods of f_{0}
Symbol: 4 periods of f_{0}
Symbol: 2 periods of f_{0}

### +

### Receive

### time

### B

### Decode each frequency bin separately

### Channel frequency response

f

### f

### Time-domain signal Frequency-domain signal

## OFDM uses multiple carriers to modulate the data

### N carriers

### B

**Modulation technique**

**Modulation technique**

**A user utilizes all carriers to transmit its data as coded quantity at each ** **frequency carrier, which can be quadrature-amplitude modulated (QAM).**

**Intercarrier Separation = ** **1/(symbol duration)**

**– No intercarrier guard bands**

**– Controlled overlapping of bands**

**– Maximum spectral efficiency (Nyquist rate)** **– Very sensitive to freq. synchronization**

**– Easy implementation using IFFTs** **Features**

**Features**

### Data

### Carrier

### T=1/f

_{0}

### Time f

_{0}

### B

### Frequency

### One OFDM symbol **Time-frequency grid**

transmit

receive

Time domain signal Frequency domain signal

FFT

Decode each subcarrier separately

**OFDM Transmitter and Receiver **

### 2 OFDM 7

### The received signal is fed to a parallel to serial converter. An fast Fourier transform (FFT) is performed on the nfft complex numbers to produce the symbol. The bits are retrieved through demodulation and the stream is serialized and output. A basic outline of this chain is shown in Figure 11.

### Figure 11: Basic view of the OFDM Transmitter and Receiver

### 2.4 Inter Symbol Interference (ISI)

### Inter-symbol interference is caused when delayed and attenuated versions of the signal (e.g.

### produced due to multi-path) overlap with the signal. Hence one symbol’s delayed version may overlap with an adjacent symbol causing inter-symbol interference. One simple technique to avoid this is to introduce a guard-band between adjacent symbols as shown in Figure 12.

### Figure 12: Guard band between symbols in OFDM

### However, a better approach is to introduce a cyclic prefix where some trailing portion of the symbol is copied in front of it. This gives an illusion to the FFT algorithm that the signal is periodic (the delayed and attenuated signals will be exactly the way it should be if the signal was periodic). Hence equation (5) is valid. This is shown in Figure 13.

### The full OFDM transmitter and receiver including the cyclic prefix is shown in Figure 14.

### 2 OFDM 7

### The received signal is fed to a parallel to serial converter. An fast Fourier transform (FFT) is performed on the nfft complex numbers to produce the symbol. The bits are retrieved through demodulation and the stream is serialized and output. A basic outline of this chain is shown in Figure 11.

### Figure 11: Basic view of the OFDM Transmitter and Receiver

### 2.4 Inter Symbol Interference (ISI)

### Inter-symbol interference is caused when delayed and attenuated versions of the signal (e.g.

### produced due to multi-path) overlap with the signal. Hence one symbol’s delayed version may overlap with an adjacent symbol causing inter-symbol interference. One simple technique to avoid this is to introduce a guard-band between adjacent symbols as shown in Figure 12.

### Figure 12: Guard band between symbols in OFDM

### However, a better approach is to introduce a cyclic prefix where some trailing portion of the symbol is copied in front of it. This gives an illusion to the FFT algorithm that the signal is periodic (the delayed and attenuated signals will be exactly the way it should be if the signal was periodic). Hence equation (5) is valid. This is shown in Figure 13.

### The full OFDM transmitter and receiver including the cyclic prefix is shown in Figure 14.

**Orthogonality of Sub-carriers **

*x(t) =* *X[k]e*

^{j 2}^{π}

^{kt N}*k=−N 2*
*N 2−1*

### ∑

### Encode: frequency-domain samples à time-domain sample ^{IFFT }

### Time-domain Frequency-domain

### Decode: time-domain samples à frequency-domain sample ^{FFT }

*X[k] =* 1

*N* *x(t)e*

^{− j 2}

^{π kt N}*t=N 2*
*N 2−1*

### ∑

### Orthogonality of any two bins : ^{e}

^{e}

*− j 2π kt N*

*e*

*− j 2π pt N*

*t=N 2*

*N 2−1*

### ∑ *= 0, ∀p ≠ k*

**Example **

### • Say we use BPSK and 4 sub-carriers to transmit a stream of samples

### • Serial to parallel conversion of samples

### • Parallel to serial conversion, and transmit time- domain samples

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 5

### In OFDM we have N carriers, N can be anywhere from 16 to 1024 in present technology and depends on the environment in which the system will be used.

### Let’s examine the following bit sequence we wish to transmit and show the development of the ODFM signal using 4 sub-carriers. The signal has a symbol rate of 1 and sampling frequency is 1 sample per symbol, so each transition is a bit.

**Fig. 7 – A bit stream that will be modulated using a 4 carrier OFDM. **

### First few bits are 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1,…

### Let’s now write these bits in rows of fours, since this demonstration will use only four sub- carriers. We have effectively done a serial to parallel conversion.

### Table I – Serial to parallel conversion of data bits.

### c1 c2 c3 c4

### 1 1 -1 -1

### 1 1 1 -1

### 1 -1 -1 -1

### -1 1 -1 -1

### -1 1 1 -1

### -1 -1 1 1

### Each column represents the bits that will be carried by one sub-carrier. Let’s start with the first carrier, c1. What should be its frequency? From the Nyquist sampling Theorem, we know that smallest frequency that can convey information has to be twice the information rate. In this case, the information rate per carrier will be 1/4 or 1 symbol per second total for all 4 carriers. So the smallest frequency that can carry a bit rate of 1/4 is 1/2 Hz. But we picked 1 Hz for convenience.

### Had I picked 1/2 Hz as my starting frequency, then my harmonics would have been 1, 3/2 and 2 Hz. I could have chosen 7/8 Hz to start with and in which the harmonics would be 7/4, 7/2, 21/2 Hz.

### We pick BPSK as our modulation scheme for this example. (For QPSK, just imagine the same thing going on in the Q channel, and then double the bit rate while keeping the symbol rate the same.) Note that I can pick any other modulation method, QPSK, 8PSK 32-QAM or whatever.

### No limit here on what modulation to use. I can even use TCM which provides coding in addition to modulation.

### Carrier 1 - We need to transmit 1, 1, 1 -1, -1, -1 which I show below superimposed on the BPSK carrier of frequency 1 Hz. First three bits are 1 and last three -1. If I had shown the Q channel of this carrier (which would be a cosine) then this would be a QPSK modulation.

Copyright 2004 Charan Langton www.complextoreal.com

c1 c2 c3 c4 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

Frequency-‐domain signal Time-‐domain signal

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

IFFT

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, …

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 6

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

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.

**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.

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

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 6

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

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.

**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.

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

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 6

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

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.

**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.

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

Copyright 2004 Charan Langton www.complextoreal.com

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

bin1

bin2

bin3

bin4

t1 t2 t3 t4 t5 t6

**Multi-Path Effect **

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 12

The functional block diagram of how the signal is modulated/demodulated is given below.

**Fig 17 – The OFDM link functions **
**Defining fading **

If the path from the transmitter to the receiver either has reflections or obstructions, we can get fading effects. In this case, the signal reaches the receiver from many different routes, each a copy of the original. Each of these rays has a slightly different delay and slightly different gain.

The time delays result in phase shifts which added to main signal component (assuming there is one.) causes the signal to be degraded.

0 Tree

0

*Line of sight path gain*
*Path delay*

1 1

*Secondary path gain*
*Secondary path delay*

*k*
*k*

*Secondary path gain*
*Secondary path delay*

Faded path

Reflected multipath

1 0

0

0

( ) ( )

ex p

*K*

*c* *k* *k*

*k*

*k*

*k* *k*

*h t* *t*

*Compl* *ath gain*

*Normalized path delay relative to LOS*
*difference in path time*

**Fig. 18 – Fading is big problem for signals. The signal is lost and demodulation must have a way of **
**dealing with it. Fading is particular problem when the link path is changing, such as for a moving car **
**or inside a building or in a populated urban area with tall building. **

If we draw the interferences as impulses, they look like this

Copyright 2004 Charan Langton www.complextoreal.com

*y(t) = h(0)x(t) + h(1)x(t −1) + h(2)x(t − 2) +*

### = *h(*

Δ

### ∑ *Δ)x(t − Δ) = h(t) ⊗ x(t)* time-domain

⟺

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

### frequency-domain

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 13

0

1 *k*

1

*k*
0

**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 main signal, all components are reflected. **

In fading, the reflected signals that are delayed add to the main signal and cause either gains in the signal strength or deep fades. And by deep fades, we mean that the signal is nearly wiped out.

The signal level is so small that the receiver can not decide what was there.

The maximum time delay that occurs is called the delay spread 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 different types of degradations to the signal. The delay spread of a signal changes as the environment is changing as all cell phone users know.

Fig. 19 shows the spectrum of the signal, the dark line shows the response we wish the channel to have. It is like a door through which the signal has to pass. The door is large enough that it allows the signal to go through without bending or distortion. A fading response of the channels is

something like shown in Fig.20 b, we note that at some frequencies in the band, the channel does not allow any information to go through, so called deep fades frequencies. This form of channel frequency response is called frequency selective fading because it does not occur uniformly across the band. It occurs at selected frequencies. And who selects these frequencies.

Environment. If the environment is changing such as for a moving car, then this response is also changing and the receiver must have some way of dealing with it.

Rayleigh fading is a term used when there is no direct component and all signals reaching the receiver are reflected. This type of environment is called Rayleigh fading.

In general when the delay spread is less than one symbol, we get what is called flat fading. When delay spread is much larger than one symbol that it is called frequency-selective fading.

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial

### 8

### Fig. 13 – The generated OFDM signal. Note how much it varies compared to the underlying constant amplitude sub-carriers.

### In short-hand, we can write the process above as

1

### ( )

^{N}

_{n}### ( )sin(2 )

*n*

*c t* *m t* *nt* (4)

### Eq. 4 is basically an equation of an Inverse FFT.

**Using Inverse FFT to create the OFDM symbol **

### The equation 4 is essentially an inverse FFT. The IFFT block in the OFDM chain confuses people. So let’s examine briefly what an FFT/IFFT does.

Copyright 2004 Charan Langton www.complextoreal.com

### Current symbol + delayed-version symbol

### à Signals are deconstructive in only certain frequencies

**Frequency Selective Fading **

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 14

**Fig. 20 – (a) The signal we want to send and the channel frequency response are well matched. (b) A **
**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. **

An OFDM signal offers an advantage in a channel that has a frequency selective fading response.

As we can see, when we lay an OFDM signal spectrum against the frequency-selective response of the channel, only two sub-carriers are affected, all the others are perfectly OK. Instead of the whole symbol being knocked out, we lose just a small subset of the (1/N) bits. With proper coding, this can be recovered.

The BER performance of an OFDM signal in a fading channel is much better than the

performance of QPSK/FDM which is a single carrier wideband signal. Although the underlying BER of a OFDM signal is exactly the same as the underlying modulation, that is if 8PSK is used to modulate the sub-carriers, then the BER of the OFDM signal is same as the BER of 8PSK signal in Gaussian channel. But in channels that are fading, the OFDM offers far better BER than a wide band signal of exactly the same modulation. The advantage here is coming from the

diversity of the multi-carrier such that the fading applies only to a small subset.

In FDM carriers, often the signal is shaped with a Root Raised Cosine shape to reduce its bandwidth, in OFDM since the spacing of the carriers is optimal, there is a natural bandwidth advantage and use of RRC does not buy us as much.

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 14

**Fig. 20 – (a) The signal we want to send and the channel frequency response are well matched. (b) A **
**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. **

An OFDM signal offers an advantage in a channel that has a frequency selective fading response.

As we can see, when we lay an OFDM signal spectrum against the frequency-selective response of the channel, only two sub-carriers are affected, all the others are perfectly OK. Instead of the whole symbol being knocked out, we lose just a small subset of the (1/N) bits. With proper coding, this can be recovered.

The BER performance of an OFDM signal in a fading channel is much better than the

performance of QPSK/FDM which is a single carrier wideband signal. Although the underlying BER of a OFDM signal is exactly the same as the underlying modulation, that is if 8PSK is used to modulate the sub-carriers, then the BER of the OFDM signal is same as the BER of 8PSK signal in Gaussian channel. But in channels that are fading, the OFDM offers far better BER than a wide band signal of exactly the same modulation. The advantage here is coming from the

diversity of the multi-carrier such that the fading applies only to a small subset.

In FDM carriers, often the signal is shaped with a Root Raised Cosine shape to reduce its bandwidth, in OFDM since the spacing of the carriers is optimal, there is a natural bandwidth advantage and use of RRC does not buy us as much.

Copyright 2004 Charan Langton www.complextoreal.com

### Frequency selecNve fading: Only some sub-‐carriers get aﬀected

**Inter Symbol Interference (ISI) **

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

### • One simple solution to avoid this is to introduce a guard-band

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 15

**Delay spread and the use of cyclic prefix to mitigate it **

You are driving in rain, and the car in front splashes a bunch of water on you. What do you do?

You move further back, you put a little distance between you and the front car, far enough so that the splash won’t reach you. If we equate the reach of splash to delay spread of a splashed signal then we have a better picture of the phenomena and how to avoid it.

Delayed splash from front symbol

Symbol 1 Symbol 2

Fig. 21 – Delay spread is like the undesired splash you might get from the car ahead of you. In fading, the front symbol similarly throws a splash backwards which we wish to avoid.

Increase distance from car in front to avoid splash. The reach of splash is same as the delay

spread of a signal. Fig. 22a shows the symbol and its splash. In composite, these splashes become 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 the front of the symbol, we will move our symbol further away from the region of delay spread as shown below. A little bit of blank space has been added between

symbols to catch the delay spread.

**Fig 22 – Move the symbol back so the arriving delayed signal peters out in the gray region. No **
**interference to the next symbol! **

But we can not have blank spaces in signals. This is won’t work for the hardware which likes to crank out signals continuously. So it’s clear we need to have something there. Why don’t we just let the symbol run longer as a first choice?

Copyright 2004 Charan Langton www.complextoreal.com

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 15

**Delay spread and the use of cyclic prefix to mitigate it **

You are driving in rain, and the car in front splashes a bunch of water on you. What do you do?

You move further back, you put a little distance between you and the front car, far enough so that the splash won’t reach you. If we equate the reach of splash to delay spread of a splashed signal then we have a better picture of the phenomena and how to avoid it.

Delayed splash from front symbol

Symbol 1 Symbol 2

Fig. 21 – Delay spread is like the undesired splash you might get from the car ahead of you. In fading, the front symbol similarly throws a splash backwards which we wish to avoid.

Increase distance from car in front to avoid splash. The reach of splash is same as the delay

spread of a signal. Fig. 22a shows the symbol and its splash. In composite, these splashes become 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 the front of the symbol, we will move our symbol further away from the region of delay spread as shown below. A little bit of blank space has been added between

symbols to catch the delay spread.

**Fig 22 – Move the symbol back so the arriving delayed signal peters out in the gray region. No **
**interference to the next symbol! **

But we can not have blank spaces in signals. This is won’t work for the hardware which likes to crank out signals continuously. So it’s clear we need to have something there. Why don’t we just let the symbol run longer as a first choice?

Copyright 2004 Charan Langton www.complextoreal.com

Guard band

**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

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 17

Original symbol Portion added in

the front

Copy this part at front

Symbol 1 Symbol 2 Copy this part at front

Original symbol Extension

**Fig. 25 – Cyclic prefix is this superfluous bit of signal we add to the front of our precious cargo, the **
**symbol. **

This procedure is called adding a cyclic prefix. Since OFDM, has a lot of carriers, we would do this to each and every carrier. But that’s only in theory. In reality since the OFDM signal is a linear combination, we can add cyclic prefix just once to the composite OFDM signal. The prefix is any where from 10% to 25% of the symbol time.

Here is an OFDM signal with period equal to 32 samples. We want to add a 25% cyclic shift to this signal.

1. First we cut pieces that are 32 samples long.

2. Then we take the last .25 (32) = 8 samples, copy and append them to the front as shown.

**Fig. 26 – The whole process can be done only once to the OFDM signal, rather than doing it to each **
**and every sub-carrier. **

We add the prefix after doing the IFFT just once to the composite signal. After the signal has arrived at the receiver, first remove this prefix, to get back the perfectly periodic signal so it can be FFT’d to get back the symbols on each carrier.

However, the addition of cyclic prefix which mitigates the effects of link fading and inter symbol interference, increases the bandwidth.

Copyright 2004 Charan Langton www.complextoreal.com

In 802.11, CP:data = 1:4

### • Because of the usage of FFT, the signal is periodic

### • Delay in the time domain corresponds to rotation in the frequency domain

### • Can still obtain the correct signal in the frequency domain by compensating this rotation

**Cyclic Prefix (CP) **

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 16

### Fig. 23 – If we just extend the symbol, then the front of the symbol which is important to us since it allows figuring out what the phase of this symbol is, is now corrupted by the “splash”.

### We extend the symbol into the empty space, so the actual symbol is more than one cycle.

### But now the start of the symbol is still in the danger zone, and this start is the most important thing about our symbol since the slicer needs it in order to make a decision about the bit. We do not want the start of the symbol to fall in this region, so lets just slide the symbol backwards, so that the start of the original symbol lands at the outside of this zone. And then fill this area with something.

**Fig. 24 – If we move the symbol back and just put in convenient filler in this area, then not only we ** **have a continuous signal but one that can get corrupted and we don’t care since we will just cut it out ** **anyway before demodulating. **

### Slide the symbol to start at the edge of the delay spread time and then fill the guard space with a copy of what turns out to be tail end of the symbol.

### 1. We want the start of the symbol to be out of the delay spread zone so it is not corrupted and 2. We start the signal at the new boundary such that the actual symbol edge falls out side this zone.

### We will be extending the symbol so it is 1.25 times as long, to do this, copy the back of the

### symbol and glue it in the front. In reality, the symbol source is continuous, so all we are doing is adjusting the starting phase and making the symbol period longer. But nearly all books talk about it as a copy of the tail end. And the reason is that in digital signal processing, we do it this way.

Copyright 2004 Charan Langton www.complextoreal.com

### FFT(
)
=
exp(-‐2jπ _{Δ} f)*FFT(
)

delayed version original signal

**Cyclic Prefix (CP) **

original signal

### y(t) à FFT( ) àY[k] = H[k]X[k]

### w/o mulNpath

### w mulNpath

original signal + delayed-‐version signal

### y(t) à FFT( ) àY[k] = α(1+ exp(-‐2jπ

_{Δ}

### k))* X[k]

### = H’[k]X[k]

Lump the phase shid in H

**Side Benefit of CP **

### • Allow the signal to be decoded even if the packet is detected after some delay

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 16

### Fig. 23 – If we just extend the symbol, then the front of the symbol which is important to us since it allows figuring out what the phase of this symbol is, is now corrupted by the “splash”.

### We extend the symbol into the empty space, so the actual symbol is more than one cycle.

### But now the start of the symbol is still in the danger zone, and this start is the most important thing about our symbol since the slicer needs it in order to make a decision about the bit. We do not want the start of the symbol to fall in this region, so lets just slide the symbol backwards, so that the start of the original symbol lands at the outside of this zone. And then fill this area with something.

**Fig. 24 – If we move the symbol back and just put in convenient filler in this area, then not only we ** **have a continuous signal but one that can get corrupted and we don’t care since we will just cut it out ** **anyway before demodulating. **

### Slide the symbol to start at the edge of the delay spread time and then fill the guard space with a copy of what turns out to be tail end of the symbol.

### 1. We want the start of the symbol to be out of the delay spread zone so it is not corrupted and 2. We start the signal at the new boundary such that the actual symbol edge falls out side this zone.

### We will be extending the symbol so it is 1.25 times as long, to do this, copy the back of the

### symbol and glue it in the front. In reality, the symbol source is continuous, so all we are doing is adjusting the starting phase and making the symbol period longer. But nearly all books talk about it as a copy of the tail end. And the reason is that in digital signal processing, we do it this way.

Copyright 2004 Charan Langton www.complextoreal.com

### decodable

### undecodable

**OFDM Diagram **

Modulation

S/P IFFT Insert P/S

CP D/A

channel

noise

### +

A/D

De-mod

P/S FFT remove S/P

CP

### Transmitter

### Receiver

**Unoccupied Subcarriers **

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 14
**Fig. 20 – (a) The signal we want to send and the channel frequency response are well matched. (b) A **
**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. **

An OFDM signal offers an advantage in a channel that has a frequency selective fading response.

As we can see, when we lay an OFDM signal spectrum against the frequency-selective response of the channel, only two sub-carriers are affected, all the others are perfectly OK. Instead of the whole symbol being knocked out, we lose just a small subset of the (1/N) bits. With proper coding, this can be recovered.

The BER performance of an OFDM signal in a fading channel is much better than the

performance of QPSK/FDM which is a single carrier wideband signal. Although the underlying BER of a OFDM signal is exactly the same as the underlying modulation, that is if 8PSK is used to modulate the sub-carriers, then the BER of the OFDM signal is same as the BER of 8PSK signal in Gaussian channel. But in channels that are fading, the OFDM offers far better BER than a wide band signal of exactly the same modulation. The advantage here is coming from the

diversity of the multi-carrier such that the fading applies only to a small subset.

In FDM carriers, often the signal is shaped with a Root Raised Cosine shape to reduce its bandwidth, in OFDM since the spacing of the carriers is optimal, there is a natural bandwidth advantage and use of RRC does not buy us as much.

Copyright 2004 Charan Langton www.complextoreal.com

### • 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)?

**Packet Detection **

### • Double sliding window packet detection

### • Optimal threshold depends on the receiving power

### Packet

### A

_{n }

### B

_{n }

### threshold M

_{n}

### =A

_{n}

### /B

_{n }

### Packet Packet

**Packet Detection **

### • Use cross-correlation to detect the preamble

### A

_{n }

### B

_{n }

### preamble

_{ }

### preamble

_{ }

### threshold

**Synchronization **

### • DAC (at Tx) and ADC (at Rx) never have exactly the sampling period

### A slow shift of the symbol timing point, which rotates subcarriers

### Intercarrier interference (ICI), which causes loss of the orthogonality of the subcarriers

### DAC (Tx)

### ADC (Rx)

**Carrier Frequency Offset (CFO) **

### • The oscillators of Tx and Rx are not typically tuned to identical frequencies

### Up-convert baseband signal s

_{n}

### to passband signal y

_{n}

### =s

_{n}

### *e

^{j2πf}

^{tx}

^{nT}

^{s }

### Down-convert passband signal y

_{n}

### back to r

_{n}

### =s

_{n}

### *e

^{j2πf}

^{tx}

^{nT}

^{s}

### *e

^{-j2πf}

^{rx}

^{nT}

^{s}

### =s

_{n}

### *e

^{j2πf}

^{Δ}

^{nT}^{s }

### Error accumulates DAC (Tx)

### ADC (Rx)

### f

_{tx }

### f

_{rx }

**Correct CFO in Time Domain **

Symbol 1 Symbol 2

### s

_{n }

### S

_{n+N }

### r

_{n}

### =s

_{n}

### *e

^{j2πf}

^{Δ}

^{nT}^{s}

### r

_{n+N}

### =s

_{n+N}

### *e

^{j2πf}

^{Δ}

^{(n+N)T}^{s}

*r*

_{n}*r*

_{n+N}^{*}

*= s*

_{n}*e*

^{j 2}^{π}

^{f}^{Δ}

^{nT}

^{s}*s*

_{n+N}^{*}

*e*

^{− j 2}^{π}

^{f}^{Δ}

^{(n+N )T}

^{s}*= e*

^{− j 2}^{π}

^{f}^{Δ}

^{NT}

^{s}*s*

_{n}*s*

_{n+N}^{*}

*= e*

^{− j 2}^{π}

^{f}^{Δ}

^{NT}

^{s}*s*

_{n}^{2}

*z =* *r*

_{n}*r*

_{n+N}^{*}

*n=1*
*L*

### ∑

### = *e*

^{− j 2}^{π}

^{f}^{Δ}

^{NT}

^{s}*s*

_{n}*s*

_{n+N}^{*}

*n=1*
*L*

### ∑

*= e*

^{− j 2}^{π}

^{f}^{Δ}

^{NT}

^{s}*s*

_{n}^{2}

*n=1*
*L*

### ∑

*f*

_{Δ}

### = 1

### 2 π ^{NT}

^{NT}

_{s}^{∠z}

^{∠z}

**Sampling Frequency Offset (SFO) **

### • The transmitter and receiver may sample the signal at slightly different offset

### Rotate the signal

### • Y

_{i}

### =H

_{i}

### X

_{i}

### * e

^{j2πt}

^{Δ}

^{iN}

^{s}

^{/N}

^{fft }

### • All subcarriers experience the same sampling delay, but have different frequencies

### DAC (Tx)

### ADC (Rx)

### t

_{Δ }

**Sample Rotation due to SFO **

I Q

x x x x x

x x x x x

x x x x x x

Ideal BPSK signals (No rotaNon)

θ

x x x x x x x x

x x x

x x x x x

Symbol 1 Symbol 2

Symbol 3

### Signals keep rotaNng

**Correct SFO in Frequency Domain **

### • SFO: slop; residual CFO: intersection of y-axis

2πδfT_{s
}(Residual
CFO)_{
}
1

2πt_{Δ}N_{s}/N_{n
}(SFO)_{
}

### Change in phase between Tx and Rx ader CFO correcNon

**Data-aided Phase Tracking **

### • Using pilot bits (known samples) to compute H

_{i}

### *e

^{j2πtΔiN}

^{s}

^{/N}

^{fft}

^{=Y}

_{i/}

^{X}

_{i }

### • Find the phase change experienced by the pilot bits using regression

### • Update H

_{I = }

### H

_{i}

### *e

^{j2πtΔiN}

^{s}

^{/N}

^{fft}

### for every symbol

2πδfT_{s
}(Residual
CFO)_{
}
1

2πt_{Δ}N_{s}/N_{n
}(SFO)_{
}

Change in phase between Tx and Rx ader CFO correcNon x

x

x

x

regression

**After Phase Tracking **

I Q

x x x x x

x x x x x

x x x x x

x Symbol 1

### Ader correcNon

θ

Symbol 2

**Nondata-aided Phase Tracking **

I Q

x x x x x

x x x x x

x x x x x x

Symbol 1

θ

**OFDM Diagram **

Modulation

S/P IFFT Insert P/S

CP D/A

channel

noise

### +

A/D

De-mod

P/S FFT remove S/P

CP

### Transmitter

### Receiver

Correct CFO

Phase track