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# Orthogonal Frequency Division Modulation (OFDM)

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### 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.

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.

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 1

### 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.

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.

### (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.

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### t t

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

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Nk

Np

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### Orthogonal Frequency Division Modulation

2 OFDM 6

(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 Nktej2 Np 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

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

## Orthogonal Frequency Division Modulation

N carriers

B

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

f

Transmit

Symbol: 8 periods of f0 Symbol: 4 periods of f0 Symbol: 2 periods of f0

f

## OFDM uses multiple carriers to modulate the data

0

0

### One OFDM symbol Time-frequency grid

transmit

Time  domain  signal   Frequency  domain  signal

FFT

Decode  each  subcarrier   separately

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j 2πkt N

k=−N 2 N 2−1

− j 2π kt N

t=N 2 N 2−1

− j 2π kt N

### e

− j 2π pt N t=N 2

N 2−1

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

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 5

### 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.

### 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.

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,  …

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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 3rd and 4th columns of Table I)

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 3rd and 4th columns of Table I)

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 3rd and 4th columns of Table I)

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

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### Multi-Path Effect

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 12

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

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

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

Δ

⟺

### 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.

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Orthogonal Frequency Division Multiplex (OFDM) Tutorial

1

N n

n

### à Signals are deconstructive in only certain frequencies

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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.

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.

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### •  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?

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?

Guard  band

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

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 17

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.

In  802.11,   CP:data  =  1:4

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### Cyclic Prefix (CP)

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 16

### FFT(                      )    =      exp(-­‐2jπ Δ f)*FFT(                    )

delayed  version   original  signal

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### Cyclic Prefix (CP)

original  signal

### w  mulNpath

original  signal  +  delayed-­‐version  signal

Δ

### =  H’[k]X[k]

Lump  the  phase  shid  in  H

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

Orthogonal Frequency Division Multiplex (OFDM) Tutorial 16

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### 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

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### 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.

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n

n

n

n

n

(23)

n

n

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



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

n

n

n

j2πftxnTs



n

n

n

j2πftxnTs

-j2πfrxnTs

n

j2πfΔnTs



tx

rx

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### Correct CFO in Time Domain

Symbol  1   Symbol  2

n

n+N

n

n

j2πfΔnTs

n+N

n+N

j2πfΔ(n+N)Ts

n

n+N*

n

j 2

fΔnTs

n+N*

− j 2

fΔ(n+N )Ts

− j 2

fΔNTs

n

n+N*

− j 2

fΔNTs

n 2

n

n+N*

n=1 L

− j 2π fΔNTs

n

n+N*

n=1 L

− j 2π fΔNTs

n 2

n=1 L

Δ

s

(27)



i

i

i

j2πtΔiNs/Nfft

Δ

(28)

### 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

(29)

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

2πδfTs  (Residual  CFO)   1

2πtΔNs/Nn  (SFO)

(30)

i

### *e

j2πtΔiNs/Nfft=Yi/Xi

I =

i

j2πtΔiNs/Nfft

### for every symbol

2πδfTs  (Residual  CFO)   1

2πtΔNs/Nn  (SFO)

Change  in  phase  between  Tx  and  Rx  ader  CFO  correcNon     x

x

x

x

regression

(31)

### After Phase Tracking

I   Q

x  x   x  x  x

x   x   x   x   x

x   x  x  x   x

x   Symbol  1

θ

Symbol  2

(32)

### 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

(33)

### 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

Correct CFO

Phase track

In contrast, FARA exploits this frequency diversity via a frequency-aware rate adaptation scheme that picks different bitrates for different frequencies depending on their SNRs. 6.1

In this way, we can take these bits and by using the IFFT, we can create an output signal which is actually a time-domain OFDM signal.. The IFFT is a mathematical concept and does

Isakov [Isa15] showed that the stability of this inverse problem increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to

• This program constructs an instance of the device and prints out its properties, such as detected.. daughterboards, frequency range,