Receiver Designs for the Radio Channel

Receiver Designs for theRadio Channel

COS 463: Wireless NetworksLecture 15

Kyle Jamieson[Parts adapted from C. Sodini, W. Ozan, J. Tan]

1. Delay Spread and Frequency-Selective Fading

2. Time-Domain Equalization

3. Orthogonal Frequency Division Multiplexing

2

Today

Last Time: Multipath causes Delay Spread

3

ReceiverTransmitter a1,τ1

a2,τ2

Power Delay Profile P(t)

tτ1

a1

τ2

a2

Channel delay spread Td

• Interference between reflected and line-of-sight radio waves results in frequency dependent fading

Last Time: Multipath causes Frequency Selectivity

-400 -200 0 200 400Frequency (kHz)

15

20

25

30

35

Rec

eive

d Po

wer

(dBm

)

Problem: Inter-symbol interference (ISI)

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• Transmitted signal• Received signal with ISI

• Transmitted signal• Received signal with ISI

• ISI at one symbol depends on the value of other symbols

Problem: Inter-symbol interference (ISI)

6




7

Today

Wideband System Design

8

p(t)

Transmit pulse filter

Data dk

h(t)

p*(−t) heq(t)

yt y[n]

Received data !"#Matched filter Equalizer

Transmitter

Receiver

Channel

$ % = ' ∗ ) % ∗ '∗(−%)• Composite channel f between data and received signal y[n] is made up of

the transmit pulse shape, radio channel, and matched filter

Zero-Forcing Equalizer

9

p*(−t) heq(t)

yt y[n]

!"#Matched filter Equalizer

Receiver:

$%& ' = 1*(')

+

Receiver front-end noise: nk

Received

ReceivedInformation

Noise




10

Today

• Choose symbol time Ts≫ channel delay spread Td

11

Simple Solution: Increase symbol time

Transmitted signal Received signal

1

(double) TsNo ISI

(at receiver)

Symbol time determinesfrequency bandwidth

12

t

Symbol timeT

2T

t f

W/2

f

W

Slowing down by a factor of two halves the frequency bandwidth of the sender’s signal

• Over what frequency range is the channel approximately the same? This is the coherence bandwidth!" ≈ $

%&'

A narrowband signal “fits into” the coherence bandwidth

-400 -200 0 200 400Frequency (kHz)

15

20

25

30

35

Rec

eive

d Po

wer

(dBm

)

Coherence bandwidth Wc

Wideband versus OFDM

Time

Frequency

Frequency selective fadingdistorts wide-band signals

Multipath causes ISI

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Wideband Signal

Narrowbandsignals

Longer symbol time Ts Time

Frequency OFDM Signal

FDMA: Frequency Division Multiple Access • Channel frequency bandwidth divided into smaller sub-bands

• Each data gets own frequency sub-band (sinusoidal carrier)

• But need to add quiet guard sub-bands in between each sub-band

– To mitigate inter-sub band interference

15

freq

uenc

y ba

nds time

FDM signal

(guard)

(guard)

(guard)

16

Long sinusoids, different frequenciesBaseband signal (In-Phase part)

• Modulate amplitude/phase of each above subcarrier to send information

• Different frequencies on different subcarriers don’t mutually interfere

Symbol time Ts

• Suppose N subcarriers and N complex valued data symbols

– X[k] (0 ≤ k < N) are the complex-valued data we want to send

• Goal: for integers k from 0 to N−1:

– cosine at frequency 2πk/N on the I channel

– sine at frequency 2πk/N on the Q channel

• So (as before) let each subcarrier be the time-domain (indexed by n) signal:

! " # cos '()*+ + - sin '()*

+

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Integer-Multiple Frequencies

OFDM Subcarriers are “Orthogonal”• Integer-multiple frequencies so peaks of each subcarrier coincide in frequency

with zeros of other subcarriers

– No guard bands required!

3

• How to generate this type of signal:

! " = $%&'

()*+ , - cos 22,"

3 + 5 sin 22,"3

• By Euler’s formula, ! " = ∑%&'()*+ , 9:;<=%/(

– This is the (familiar) Inverse DFT

• Transmit time domain signal x[n] = DFT−1{ X[k] }

• So has an efficient hardware implementation

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Generating an OFDM Signal

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Transmitting one OFDM symbol in time

OFDM symbol time Ts

• Apply the DFT to each OFDM symbol individually, to recover data:

! " = 1/&'(

)*+,[.]0*12345/)

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Receiving an OFDM signal

Time

Frequency

One OFDM symbolx[0]…x[N−1]

• Channel impulse response h(t) is a function of time– Impulse has an infinite bandwidth

• Effect of a very narrowband channel is one complex number h– Subcarrier eventually arrives on all paths, e.g. h = h1+h2:

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What about the wireless channel?

Channel impulse response h(t)

tτ1

a1

τ2

a2

Delay spread Td

ReceiverTransmitter h1

h2

• Channel used at each subcarrier’s discrete frequency location

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What about the wireless channel?

-400 -200 0 200 400Frequency (kHz)

15

20

25

30

35R

ecei

ved

Pow

er (d

Bm)

e.g. with three subcarriers: k=1 k=2 k=3

Estimating the Channel• Transmit known a OFDM preamble symbol p[n]

– In frequency domain on frequency k denote preamble P[k]

• After DFT on the preamble, the receiver hears frequency domain value Y[k]

• Receiver computes channel estimate on kth subcarrier: H[k] = Y[k] / P[k]

• Aside: To compute channel impulse response: DFT−1{ H[k] } = h[n]

3

Problem: Inter-OFDM Symbol Interference

Problem: Receiver synchronization

1 2 3

Receiver: 1 2 3

Receiver’s view of Symbol 1 contains actual Symbols 1 and 2

Interference solution:Inter-symbol guard interval

Guard interval between adjacent symbols mitigates adjacent symbol interference

Synchronization solution: Cyclic prefix

Receiver: Symbol OK!

Ns, Ts

Nfft, Tfft

5

OFDM signal: Frequency-Domain view

Uniform power in the frequency domain over the OFDM signal bandwidth

• Many low-frequency sinusoids in the time domain• Occasionally in time, many will all constructively interfere

– Result: High ratio of peak power / average power30

OFDM signal: Time Domain View

0 2000 4000 6000 8000 10000 12000 14000 16000−40

−30

−20

−10

0

10

20

30

Time Sample Index n

Re(

x[n]

)

Re(x[n]) vs n

Large Peak

(a) Real Part of an OFDM Signal

0 2000 4000 6000 8000 10000 12000 14000 16000−40

−30

−20

−10

0

10

20

30

40

Time Sample Index n

Im(x

[n])

Im(x[n]) vs n

(b) Imaginary Part of an OFDM Signal

Figure 3-17: Rare Occurrence of Large Peaks in OFDM Signal

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• Transmit power amplifier sits just before the transmit antenna

• Peak power in non-linear region causes signal distortion

– So lower input signal level:

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Peak to Average Power & Transmit AmplifiersTypical Transmit Power

Amplifier Response (Vout / Vin)

• High peak to average power ratio (PAPR) àLow average power level à

• Signal mostly uses fewer levels in discrete representation, so high quantization error (another form of distortion)

Put peak power here

An OFDM Modem

P/SQAM

decoder

invert channe

l=

frequencydomain

equalizer

S/P

quadrature amplitude

modulation (QAM) encoder

N-IFFTadd

cyclic prefix

P/SD/A +

transmit filter

N-FFT S/Premove cyclic prefix

TRANSMITTER

RECEIVER

N subchannels N complex samples

N complex samplesN subchannels

Receive filter

+A/D

multipath channel

Bits

00110

• OFDM uses two identical, repeated symbols s1, s2 in the preamble for packet detection:

• Receiver radio is always listening, receiving samples

– Call this received sample stream r[n]

33

Packet detection

s2

Preamble Packet body

s1

• Suppose each preamble symbol is of length L

• Receiver computes c n = ∑%&'()* + , + . +∗[, + . + 1]

• Angle of each term in the sum is random

• Sum of complex numbers with random angle ≈ 0– c[0] ≈ 0

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Searching for the preamble in noise

s2s1background noise Packet bodyr[n]:

L

Computing c[0]:

• Suppose preamble at position n0

• Receiver computes c n = ∑%&'()* + , + . +∗[, + . + 1]

• ∡(zz*) = 0, so angle of each term in the sum is ≈ 0

• Sum of complex numbers with ≈ 0 angle is large– c[n0] is large

35

Search window encounters preamble


L

Computing c[n0]:

n0

• c n = ∑%&'()* + , + . +∗[, + . + 1]

• Normalize power fluctuations in r[n], by measuring power:– p n = ∑%&'()* + , + . 4

• Schmidl-Cox Packet Detection signal:m[n] = c[n] / p[n]

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Schmidl-Cox Packet Detection

Packet detection metric m[n]

Time samples n

• Limited precision of frequency oscillators

• Up-convert baseband signal sn to passband signal !" = $"%&'()*+",-

• Down-convert passband signal yn back to baseband:

." = $"%&'()*+",-%&'()/+",- = $"%&'(∆)",- (∆1 = 123 − 153)

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A Closer Look at Carrier Frequency Offset

6.082 Spring 2007 Fourier Series and Fourier Transform, Slide 15

y(t) = 2cos(2π(fo+ε)t)

z(t) r(t)

LowpassFilter

w(t)H(j2πf)

Impact of Frequency Offset

• Baseband signal is corrupted!– Filtering cannot fix this

0

1

ffo+ε

Y(j2πf)1

-(fo+ε)0f

Z(j2πf)

0f

fo

W(j2πf)

-fo

A A

fo-fo

A A

2fo+ε-2fo-ε

2A

• Because of carrier frequency offset, !" = !$%&"'∆)*+,– c ./ = ∑12/34$ 5 ./ + 7 5∗[./ + 7 + :]

• Consider the kth term in sum: 5 ./ + 7 5∗ ./ + 7 %&"'∆)*+,– This is equal to %&"'∆)*+, 5[./ + 7] "

• So all terms have the same angle 2=∆>?@A

• So, carrier frequency offset estimator B∆C = ∡E[FG]HIJKL

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Estimating Carrier Frequency Offset


L

Computing c[n0]:

n0

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Sample Clock Offset

• The transmitter and receiver may sample the signal at slightly different rates, leading to a sample time offset ζ

• All subcarriers experience the same sampling delay, but travel over different frequencies Figure 3-8: OFDM symbol window drift due to sampling frequency offset

degradation DCFO in dB to the current SNR caused by CFO δf can be

expressed as the following[11]

DCFO =10

3ln10(πδfTFFT )2SNR (3.5)

From equation 3.5, it can be seen that as the SNR requirement is higher,

the sensitivity to CFO increases. In the context of the WiGLAN, the

highest SNR requirement occurs when 256-QAM is used (28.6 dB for BER

= 10−3). With DCFO set to a negligible loss of 0.5 dB, the maximum CFO

tolerated by the WiGLAN system is 6 kHz.

• ICI from SFO: The SNR degradation DSFO;k from SFO on the kth sub-

carrier with SNRk is given by [12]

DSFO;k = 10log10(1 +1

3SNRk · (

πk(f ′s − fs)

fs)2) (3.6)

From Equation 3.6, the degradation increases when k increases or SNRk

increases. To obtain the maximum degradation DSFO;k, the highest SNR

requirement of 28.6 dB for the use of 256-QAM and k of -64 are considered.

Hence for the WiGLAN with fs of 128 MHz and the sampling clocks of 100

parts per million (PPM), the maximum degradation Dk from Equation 3.6

46

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Correcting Sample Clock Offsetin the Frequency Domain

• Sample clock offset : slope• Residual CFO: intersection with y-axis

f

⦟ H "#$ %&%''(

• OFDM with up to 48 subcarriers

– Subcarrier spacing is 312.5 KHz

– Subcarriers modulated: BPSK, QPSK, 16-QAM, or 64-QAM

• Uses a convolutional code at a rate of ½, 2/3, ¾ , or 5/6 to provide forward error correction

• Results in data rates of 6, 9, 12, 18, 24, 36, 48, and 54 MBps

• Cyclic prefix is 25% of a symbol time (16 vs 64)

Example: IEEE 802.11a, 802.11g

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Per-subcarrier Bit Rate Choice5 10 15 20 25 30 35

10−4

10−3

10−2

10−1

SNR

BER

Adaptive Modulation Scheme

BPSK4 QAM16 QAM64 QAM256 QAM

NULL BPSK 4 QAM 16 QAM 64 QAM 256 QAM

Figure 1-3: Frequency Adaptive Modulation Used: Optimization of Data Rate

−50 0 500

10

20

30SNR vs Freq

Freq (MHz)

SNR

(dB

)

256QAM64QAM16QAM4QAMBPSKNULL

Figure 1-4: Measurement: Frequency Adaptive Modulation In Action

coherence bandwidth of the channel.

Frequency Diversity Order =W

Wc(1.3)

Chapter 5 will explore empirically the goodness of adaptive modulation under various

indoor channel conditions. When performing Frequency Adaptive Modulation, the

criterion of optimization can be one of the following:

1. Data Rate.

2. Average Transmitting Energy.

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Tuesday Topic:MIMO I: Spatial Diversity

Friday Precept:Lab 4: Single-carrier transceiver

on the HackRF hardware

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Receiver Designs for the Radio Channel

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