Wireless Communications From 5G and WiFi6 to Low Power IoT

Wireless Communications From 5G and WiFi 6 to Low Power IoT

Lecture 4: OFDMHaitham Hassanieh

Yesterday’s Lecture Was

A. Too Difficult: I understood nothing.

B. Difficult: I understood something but not everything.

C. Moderate: I understood most stuff but I have questions

D. Easy: I understood everything really well.

Multipath in the Wireless Channel is problematic since it creates:

A. Inter-Symbol-Interference

B. Pathloss

C. Frequency Selective Fading

D. Additive White Gaussian Noise

ISI is considered negligible if the delayed symbols arriving longer paths interfere with less than < 1% of the symbol length.

If the longest path in the channel delays the symbol by 10ns, what is the maximum bandwidth for which we can ignore ISI?

A. 1 MHz

B. 10 MHz

C. 100 MHz

D. You can never ignore ISI

Previous Lecture:ü Pulse Shaping

ü Matched Filter

ü Multipath Channel

ü Channel Estimation & Correction

ü Narrowband vs. Wideband Channels

ü Channel Equalization

q Multi-Carrier Modulation

q Orthogonal Frequency Division Multiplexing (OFDM)

q OFDM Time Synchronization

q OFDM Frequency Synchronization

q OFDM Channel Estimation & Correction

q OFDM Phase Tracking

This Lecture:

Wireless Communication System

Symbols-to-Bits Mapper

Bits

Synchronization Demodulation(Decoding)

MatchedFilter

TX RX

10110011001 10110111001

Bits-to-SymbolsMapper

LPF BPF

PA

PLL

Mixer

Pulse Shaping

DACModulation (Encoding)

LPFBPFLNA

PLL

Mixer

ADC ChannelEqualization

ℎ!" 𝑡

Training + Data Bits



Bits

Demodulation(Decoding)

MatchedFilter

TX RX

10110011001 10110111001


LPF BPF

PA

PLL

Mixer

Pulse Shaping


LPFBPFLNA

PLL

Mixer


ℎ!" 𝑡


Δ𝑓!Frequency

SynchronizationTime Sync.



Bits

Demodulation(Decoding)

MatchedFilter


LPF BPF

PA

PLL

Mixer

Pulse Shaping


LPFBPFLNA

PLL

Mixer


ℎ!" 𝑡


Δ𝑓!Frequency

SynchronizationTime Sync.

TX

RX

Single Carrier ModulationSymbols modulated on a single carrier frequency

𝑠 𝑛 cos 2𝜋𝑓!𝑡

Single Carrier ModulationSymbols modulated on a single carrier frequency

• Low Spectral Efficiency: sinc & raised cosine leakage

• ISI: Inter-Symbol-Interference limits performance

𝑓

−𝐵 𝐵

Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:

y(t) = hx(t) + n(t).

• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:

y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index

Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)

• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.

y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N

−80 −60 −40 −20 0 20 40 60 80Frequency in MHz

Figure 6: Frequency Selective Fading for 100 MHz channel

• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).


y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



Multi-Carrier Modulation

• Divide spectrum into many narrow bands

• Transmit symbols on different carriers in narrow bands

Symbols modulated on multiple Sub-carrier frequencies


y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



• Channel is Flat à No need to worry about ISI

𝑥 𝑡 =$#

𝑠# 𝑛 e$%&'(!)




y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz






𝑥 𝑡 =$#

𝑠# 𝑛 e$%&'(!)

𝑦 𝑡 =$#

ℎ#𝑠# 𝑛 e$%&'(!)




y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz





𝑥 𝑡 =$#

𝑠# 𝑛 e$%&'(!)


𝑦 𝑡 =$#

ℎ#𝑠# 𝑛 e$%&'(!)

Not That Simple!

Multi-Carrier ModulationSymbols modulated on multiple Sub-carrier frequencies


• Significant Leakage between adjacent subcarriers

• Need Guard Bands à Very inefficient!

𝑓 𝑓

Guard Bands

Solution: Make the Sub-Carriers Orthogonal

Multi-Carrier ModulationSymbols modulated on multiple Sub-carrier frequencies

Make the Sub-Carriers Orthogonal

OFDM: Orthogonal Frequency Division Multiplexing

• Subcarriers are orthogonal: At the sub-carrier frequency, the sampled value has zero leakage from other subcarriers.

• Subcarrier separation can be very small, for N subcarriers and bandwidth B:

Δ𝑓 =𝐵𝑁


• Subcarriers are orthogonal: At the sub-carrier frequency, the sampled value has zero leakage from other subcarriers.

• Subcarrier separation can be very small, for N subcarriers and bandwidth B:

Δ𝑓 =𝐵𝑁

How to Achieve This?


N-Point DFT:

Use DFT: Discrete Fourier Transform

𝑋 𝑓# =1𝑁$)*+

,$-

𝑥 𝑡 𝑒$%&'(!),

𝑥(𝑡) = $(!*+

,$-

𝑋 𝑓# 𝑒%&'(!),N-Point IDFT:

Send symbols in Frequency Domain𝑋 𝑓" = 𝑠 𝑛 → Compute and transmit 𝑥 𝑡 using IDFT


Send symbols in Frequency Domain𝑋 𝑓" = 𝑠 𝑛 → Compute and transmit 𝑥 𝑡 using IDFT

• 𝑁subcarrier à IDFT of length 𝑁

• Symbols 𝑠 𝑛 can come from any modulation: BPSK, QPSK, QAM…

• 𝑥 𝑡 is complex à need 𝐼 & 𝑄à No point using PAM or ASK …

• OFDM Symbol: 𝑁 samples of 𝑥 𝑡 generated from the same modulated symbols using IDFT.

• OFDM Symbol Time: 𝑇 = 𝑁/𝐵 where 𝐵 is the bandwidth.

• OFDM Frequency Bin Width: Δ𝑓 = 1/𝑇 = 𝐵/𝑁

Receiver

Transmitter

ModulationBits

Demodulation Bits


Para

llel t

o Se

rial

Seria

l to

Para

llel

IFFT

Para

llel t

o Se

rial

Seria

l to

Para

llel

FFT

LPFBPF

PLL

Mixer

LPFBPF Mixer

90"LNA

ADC

ADC

𝐼

𝑗𝑄+

LPF BPFMixerDAC

LPF BPF

PLL

Mixer

90"PA

DAC

ℜ𝔢{ }

ℑ𝔪{ }

+

OFDM Symbol in Frequency Domain


y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



0−𝑁2

𝑁2− 1

• FFT can be represented 0 to 𝑁 − 1 or 𝑁/2 to 𝑁/2 − 1.

• OFDM Symbol created in digital baseband à 0 bin corresponds to DC

𝑋 0 =1𝑁$)*+

,$-

𝑥 𝑡 𝑒$%&'+), =

1𝑁$)*+

,$-

𝑥 𝑡 = 𝐷𝐶



y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



0−𝑁2

𝑁2− 1



• DC of the circuits corrupts bits sent on the 0 bin à Do not use 0 bin



y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



−𝑁2

𝑁2− 1



• DC of the circuits corrupts bits sent on the 0 bin à Do not use 0 bin

0

DC Bin



y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



0−𝑁2

𝑁2− 1

• Subcarriers orthogonal to each other but not to near by channels.

• Need Guard Bins at sides of the channel à Transmit nothing there

DC Bin



y(t) = hx(t) + n(t).


y(t) =i=k∑

i=0

h(i)s(t− iτ)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

|H|2

Tap Index




−80 −60 −40 −20 0 20 40 60 80Frequency in MHz



0−𝑁2

𝑁2− 1

• Subcarriers orthogonal to each other but not to near by channels.

• Need Guard Bins at sides of the channel à Transmit nothing there

• Reduce Number of Guard band from 𝑁 to 2 à Very Spectrally Efficient

Guard BinsGuard Bins DC Bin

ModulationBits

Demodulation Bits


Para

llel t

o Se

rial

Seria

l to

Para

llel

IFFT

Para

llel t

o Se

rial

Seria

l to

Para

llel

FFT

LPFBPF

PLL

Mixer

LPFBPF Mixer

90"LNA

ADC

ADC

𝐼

𝑗𝑄+

LPF BPFMixerDAC

LPF BPF

PLL

Mixer

90"PA

DAC

ℜ𝔢{ }

ℑ𝔪{ }

+TX

RX

Transmit Symbols in Frequency Domain On Orthogonal Subcarriers

OFDM Symbol

0−𝑁2

𝑁2 − 1

Guard Bins

Guard Bins

DC

… ,+1,−1,+1,−1,+1,−1,−1,−1,+1,+1,−1,+1,+1,−1,−1,…

1 0 1 0 1 0 0 0 1 1 0 1 1 0 0Bits:

IFFT

Symbol in Time

OFDM Symbol

Symbol in Time Symbol in Time Symbol in Time Symbol in Time ⋯⋯FFT FFT FFT FFT

… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,… … ,+1,−1,−1,…

…101… …111… …001… …100…

Not That Simple

OFDM Symbol

S1 S2 S3 S4 ⋯⋯

FFT

… ,+1,−1,+1,…

…101…

FFT Window

OFDM Symbol

S1 S2 S3 S4 ⋯⋯

FFT FFT

… ,+1,−1,+1,… … ,+1,+1,+1,…

…101… …111…

FFT Window

OFDM Symbol

S1 S2 S3 S4 ⋯⋯

FFT FFT FFT

… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,…

…101… …111… …001…

FFT Window

OFDM Symbol

S1 S2 S3 S4 ⋯⋯

FFT FFT FFT FFT

… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,… … ,+1,−1,−1,…

…101… …111… …001… …100…

FFT Window

Assumes FFT window is perfectly aligned with symbol boundaries

OFDM Symbol

S1 S2 S3 S4 ⋯⋯

FFT

… ,+0.5 + 1i, −0.7 + 0.3i, …

FFT Window

FFT window is misaligned with symbol

Cannot decode!

Subcarriers are no longer orthogonal.

OFDM Cyclic Prefix

FFT Window

• DFT (FFT) assumes time samples are periodic of period 𝑁

𝑥[𝑡] → 𝑋[𝑓]

𝑥 𝑡 − 𝜏 mod 𝑁 → 𝑋 𝑓 𝑒0123456

• Circular Shift before taking FFT:

S1 S2 S3 S4 ⋯⋯

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

• DFT (FFT) assumes time samples are periodic of period 𝑁

𝑥[𝑡] → 𝑋[𝑓]

𝑥 𝑡 − 𝜏 mod 𝑁 → 𝑋 𝑓 𝑒0123456

• Circular Shift before taking FFT:

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

• Even if FFT window is misaligned, CP ensures that all samples come from the same symbol à Orthogonality is preserved!

• Cyclic Prefix can be created by:o Take first few samples and append them to end of symbol.o Take last few samples and prefix them to beginning of symbol.

• Simple Phase Shift à Can be corrected by lumping with channel 𝐻[𝑓]

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

Cyclic Prefix:

• Preserves orthogonality by allowing some misalignment in FFT Window

• Deals with Inter-Symbol-Interference

ISI ISI ISI

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

Cyclic Prefix:



ISI ISI ISI

NO ISI inFFT Window

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

Cyclic Prefix:



ISI ISI ISI

NO ISI inFFT Window

FFT Window

NO ISI inFFT Window

FFT Window

NO ISI inFFT Window

• Overhead: Send 𝐶𝑃 + 𝑁 samples for every 𝑁 samples

OFDM Cyclic PrefixCyclic Prefix:



Overhead =𝐶𝑃

𝐶𝑃 + 𝑁

e. g.WiFi 802.11n:𝑁 = 64, CP = 16 → Overhead = 20%

e. g. LTE:𝑁 = 1024, CP = 72 → Overhead = 6.5%

Progress so far?

A. I understand and I can follow

B. I understand most stuff but not everything

C. I can follow but I do not understand everything

D. I cannot follow and understood nothing about OFDM

OFDM Cyclic Prefix

S1 S2 S3FFT Window

S1 S2 S3

• Cyclic prefix is a not a bullet proof solution.

• Can still end up misaligned!

• Need a way to ensure we detect the beginning of the packet correctly.

• If we do, CP will ensure that even if we are not accurate, we can still decode.

OFDM Packet Detection

• Detect Beginning of packet to make sure we are within the CP

• Send Training Sequence: Preamble Symbols

• Preamble Symbols: Known Symbol Repeated at the beginning of packet

S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯

• No need for CP with preamble symbols

OFDM Packet Detection: Sliding Window

S1 S2CP1Preamble Preamble Preamble ⋯A B

• Two windows of 𝐿 (2𝑁) samples each.

• Compute:𝑃7𝑃8

=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1



• Two windows of 𝐿 (2𝑁) samples each.


=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1



• Two windows of 𝐿 samples each.


=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1





=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1





=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1





=∑9:;<=;<2= 𝑥[𝑘] 2

∑9:;;<= 𝑥[𝑘] 2

𝑃!𝑃"

1

Packet Start +L

Previous Lecture:ü Pulse Shaping

ü Matched Filter

ü Multipath Channel

ü Channel Estimation & Correction

ü Narrowband vs. Wideband Channels

ü Channel Equalization

ü Multi-Carrier Modulation

ü Orthogonal Frequency Division Multiplexing (OFDM)

ü OFDM Time Synchronization

q OFDM Frequency Synchronization

q OFDM Channel Estimation & Correction

q OFDM Phase Tracking

This Lecture:

ModulationBits

Demodulation Bits


Para

llel t

o Se

rial

Seria

l to

Para

llel

IFFT

Para

llel t

o Se

rial

Seria

l to

Para

llel

FFT

LPFBPF

PLL

Mixer

LPFBPF Mixer

90"LNA

ADC

ADC

𝐼

𝑗𝑄+

LPF BPFMixerDAC

LPF BPF

PLL

Mixer

90"PA

DAC

ℜ𝔢{ }

ℑ𝔪{ }

+TX

RX

Transmit Symbols in Frequency Domain On Orthogonal Subcarriers

So far, we assumed carriers generated by LOs are

perfectly synchronized!

Carrier Frequency Offset

TX RX

101101011 101101011

𝑥 𝑡 ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(𝑥 𝑡 ×𝑒#$%&'#( ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(×𝑒$%&'#(

ℎ 𝑡 ∗ 𝑥 𝑡

𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 + 𝑣 𝑡

Assumes TX & RX perfectly synched


TX RX

101101011 101101011

𝑥 𝑡 ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(𝑥 𝑡 ×𝑒#$%&'#( ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(×𝑒$%&'#$(

ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&)'#(

𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&)'#( + 𝑣 𝑡

TX & RX are not synched

Phase changes with time!

CFO: Δ𝑓0 = 𝑓0 − 𝑓01


+1−1 𝐼

𝑄

Consider BPSK Modulation.0 → −11 → +1

𝑥 𝑡 ℎ 𝑥 𝑡 − 𝜏 𝑒$%&'2(+) + 𝑣 𝑡

+1−1 𝐼

𝑄


+1−1 𝐼

𝑄

Consider BPSK Modulation.0 → −11 → +1

𝑥 𝑡 ℎ 𝑥 𝑡 − 𝜏 𝑒$%&'2(+) + 𝑣 𝑡

+1−1 𝐼

𝑄

Impossible to Decode!


𝐼

𝑄Consider 16 QAM Modulation

Need to estimate and correct CFO to decode!

OFDM CFO Estimation & Correction

• Use Preamble to estimate CFO

𝑦H 𝑡 = 𝑥 𝑡 𝑒0123I4!;

𝑦2 𝑡 = 𝑥 𝑡 𝑒0123I4! ;<6J"


𝑦H[𝑛] = 𝑥[𝑛]𝑒0123I4!KJ"

𝑦2[𝑛] = 𝑥[𝑛]𝑒0123I4! KJ"<6J"



• Compute: 𝐴 =$)*-

,

𝑦- 𝑛 𝑦&∗[𝑛] =$)*-

,

𝑥 𝑛 𝑥∗[𝑛]𝑒%&'2(+,4,

= 𝑒%&'2(+,4,$)*-

,

𝑥 𝑛 & Δ𝑓0 =∠𝐴

2𝜋𝑁𝑇5


𝑦H[𝑛] = 𝑥[𝑛]𝑒0123I4!KJ"

𝑦2[𝑛] = 𝑥[𝑛]𝑒0123I4! KJ"<6J"



• Compute: 𝐴 =$)*-

,

𝑦- 𝑛 𝑦&∗[𝑛] Δ𝑓0 =∠𝐴

2𝜋𝑁𝑇5

• Correct CFO: 𝑦 𝑛 ×𝑒123I4!KJ"


We use the following equation to estimate CFO: Δ𝑓0 =∠7

&',4*.

Suppose 𝑓0 = 5 GHz, the bandwidth = 10 MHz and the clock precision is 20ppm. For what values of N will the above equation

estimate the CFO incorrectly?

A. N < 10

B. N < 20

C. N > 50

D. N < 50

• Equation give wrong result when the phase of A wraps around 2𝜋! We need: ∠𝐴 ≤ 𝜋

Δ𝑓0 ≤1

2𝑁𝑇5

• Δ𝑓0 = 5 GHz × 20/1000000 = 100 kHz

• 𝑇5 = 1/10MHz = 0.1𝜇𝑠

OFDM Channel Estimation

• Use Preamble to estimate the channel

𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 ↔ 𝑌 𝑓 = 𝐻 𝑓 𝑋 𝑓

• Send 𝑋 𝑓 : −1,+1,−1,−1,−1,+1,…

• Receive: −𝐻(1), 𝐻(2), −𝐻(3), −𝐻(4), −𝐻(5), 𝐻(6), …

• Estimate: Z𝐻 𝑓 =𝑌 𝑓𝑋 𝑓

• Use two preambles to average noise: Z𝐻 𝑓 =𝑌H 𝑓 + 𝑌2 𝑓

2 𝑋 𝑓


Phase TrackingSo Far: Estimated and Corrected For Coarse Value of CFO

• Residual CFO:

𝑦 𝑡 = ℎ 𝑡 𝑥 𝑡 𝑒$%&'2(+) + 𝑣 𝑡

Δ𝑓0 = 𝑑𝑓0 + 𝛿𝑓0

Coarse CFO Residual CFO

We estimated and corrected for coarse CFO!

Even small residual can accumulate over time to create large phase: 𝑒$%&'8(+)

Need to track the phase

Phase Tracking• Residual CFO (Carrier Frequency Offset)

• Residual SFO (Sampling Frequency Offset)

Phase Tracking

𝑦[𝑛] = 𝑥[𝑛 + 𝑛𝛿𝑇L]𝑒0123M4!KJ"

= ]4:N

60H

𝑋 𝑓 𝑒1234(K<KMJ")

6 𝑒0123M4!KJ"

𝑌 𝑓 = 𝑋[𝑓]𝑒1234KMJ"

6 023M4!KJ"

• Residual CFO (Carrier Frequency Offset)


When we sample the signal there is a residual sampling offset: 𝑛𝛿𝑇5

Phase Tracking

𝑌H 𝑓 = 𝑋H[𝑓]𝑒1234KMJ"6 023M4!KJ"

𝑌2 𝑓 = 𝑋2[𝑓]𝑒1234(K<6<QR)MJ"6 023M4! K<6<QR J"

Δ𝜙 = 2𝜋𝑓𝑁 + 𝐶𝑃 𝛿𝑇5

𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5Phase accumulation:

• Residual CFO (Carrier Frequency Offset)


Frequency Bins−𝑁/2 𝑁/2

y-intercept: CFO

Slope: SFO


𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5• Phase accumulation:

Phase Tracking• Residual CFO (Carrier Frequency Offset)


OFDM Phase Tracking

Frequency Bins−𝑁/2 𝑁/2

y-intercept: CFO

Slope: SFO

• Sufficient to estimate slope & y-intercept to know the phase accumulated for all subcarriers.

• Use only few subcarriers as pilots & send known bits in them.


𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5• Phase accumulation:

OFDM Symbol

0−𝑁2

𝑁2− 1

Guard Bins

Guard Bins

DCPilots Pilots

−𝑁/2 𝑁/2

Δ𝜙Use Linear Regression to estimate phase accumulated

OFDM: Putting it Together

At TX:

• Create preamble symbol from training sequence (Uses BPSK)

• Repeat preamble symbol:

o 4 times for packet detectiono 2 times for CFO estimationo 2 times for channel estimationo Add CP for the last preamble

• Create data symbol from: o Data bits (Uses BPSK, QPSK, M-QAM)o Pilot bits (Uses BPSK)

• Add cyclic prefix to data symbols.

OFDM: Putting it Together

At RX:

• Detect beginning of packet.• Estimate & correct for CFO.• Jump ≈ 0.75 𝐶𝑃 samples into symbol to avoid ISI• Estimate the channel. • For each subsequent data symbol:

o Remove CPo Take FFT of Size No Correct for channelo Use linear regression to estimate residual CFO and SFOo Estimate accumulated phase Δ𝜙 𝑓 for each frequency bino Add Δ𝜙 𝑓 to channel estimate X𝐻 𝑓o Decode Bits

Progress so far?

A. I understand and I can follow

B. I understand most stuff but not everything

C. I can follow but I do not understand everything

D. I cannot follow and understood nothing about OFDM

Documents

Wireless Communications From 5G and WiFi6 to Low Power IoT