Communications II

Lecture 6: Introduction to Digital Communications

Professor Kin K. LeungEEE and Computing Departments

Imperial College London© Copyright reserved

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Outline

• PCM (A/D)

• Quantization noise

• Companding

• Line coding

• Baseband digital transmission

• Reference: Lathi, Chap. 6-7.

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Noise in Digital Communication Systems

Different goals between analog and digital communication systems:

• Analog communication systems: to reproduce the transmitted waveform accurately.⇒ Use signal to noise ratio to assess the quality of the system

• Digital communication systems: the transmitted symbol to be identified correctly by the receiver ⇒ Use the probability of error of the receiver to assess the quality of the system

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Digital transmission

For digital transmission: Analog signals are converted to digital

How densely should we sample an analog signal so that we can reproduce its form accurately?

Sampling TheoremA signal the spectrum of which is band-limited to W Hz, can be reconstructed exactly from its samples, if they are taken uniformly at a rate of R ≥ 2W Hz.

Nyquist frequency: fs = 2W Hz

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Maximum Information Rate

What is the maximum rate of distortionless information transfer over a channel of bandwidth B Hz, in the absence of noise?

It can pass a signal of bandwidth B Hz without distortion.

This signal can be exactly represented by its Nyquist samples, which occur at a (sampling) rate of 2B Hz

A channel of bandwidth B Hz can transmit 2B pieces of information. At most we can transmit without distortion 2 bits of information per second per hertz

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What is Pulse-Code Modulation (PCM)?

A baseband scheme that can be used to represent any analog signal in digital form.

The analog signal is converted into a discrete-valued discrete signal.

Advantages:① digital signals are more immune to channel noise by using channel coding ② repeaters along the transmission path can detect a digital signal and

retransmit a new noise-free signal ③ PCM signals derived from all types of analog sources can be represented

using a uniform format

Caution: PCM isn’t modulation in the usual sense; it’s a type of Analog-to-Digital Conversion.

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PCM modulator

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• Sample the message signal above the Nyquist frequency

• Quantize the amplitude of each sample

• Encode the discrete amplitudes into a binary codeword

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The PCM process

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Quantization Noise

Quantization: a destructive process

It introduces the quantization noise

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∆: gap between quantizing levels

q: Quantization error = a random variable in the range

Assume that it is uniformly distributed over this range:

22

q

otherwise,022

,1)(

qqp

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Mean square error:

12

24)(

241

31

1

)(}{

2

33

2/

2/

3

2/

2/

2

22

q

dqq

dqqpqeEPN

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Assume: the encoded symbol has n bits the maximum number of quantizing levels is L = 2n

maximum peak-to-peak dynamic range of the quantizer=2n∆

P: power of the message signal mpeak = max |m(t)|: maximum absolute value of the message signal

Assume: the message signal fully loads the quantizer:

12221 nn

peakm (136)

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SNR at the quantizer output:

From (136):

2212

12/

PPPPSNR

N

So

npeak

npeak

npeak mmm

2

22

1 24

22

2

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In decibels,

n

peakn

peako m

Pm

PSNR 22

2

2 23

2412

)(3log1002.6

3log102log20

3log10)2(log10)(

210

21010

2102

10

dBm

Pn

mPn

mPdBSNR

peak

peak

peak

no

(139)

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Hence, each extra bit in the encoder adds 6 dB to the output SNR of the quantizer.

Implication: It’s easy to increase the output SNR by increasing n, if there is enough bandwidth (which is not always true).

The audio signal is sampled at 8 kHz, quantized into 256 levels (8 bits). Thus, a telephone PCM signal requires 64 kbps.

DPCM (differential PCM): quantize the difference between consecutive samples; can go down to 32 kbps. ADPCM (Adaptive DPCM) with nonuniformquantization can go further down to 16 kbps.

Delta modulation: 1-bit DPCM with oversampling; has even lower symbol rate.

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Example: Sinusoidal message signal: m(t) = Am cos(2 fmt).

Average signal power:

Maximum signal value: mpeak = Am.Substitute into (139):

In decibels

Audio CDs: n = 16 ⇒ SNR > 90 dB

2

2mAP

n

m

mo A

ASNR 22

2

223

dBndBSNRo 76.102.6)(

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Bandwidth Requirements

A band limited signal to W Hz It is sampled with frequency 2W It requires 2W symbols per second

Use an n-bit quantizer Each symbol is represented using n bits We must transmit 2nW bits per second.

At most we can transmit (without distortion) 2 bits per second per hertz Minimum transmission bandwidth required:

BT = nW

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Output SNR due to quantization (139):

n = BT/W, then

SNR increases exponentially with the transmission bandwidth BT

A small increase in bandwidth yields a large increase in SNR

n

peako m

PSNR 22 23

WB

peako

T

mPSNR /2

2 23

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Companding

Problem with uniform quantization: the output SNR is adversely affected by peak to average power ratio (cf. (140))

Companding is the corresponding to pre-emphasis and de-emphasis scheme used for FM.

Predistort a message signal in order to achieve better performance in the presence of noise, and then remove the distortion at the receiver

How is it done?

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Typically small signal amplitudes occur more often than large signal amplitudes.

The signal does not use the entire range of quantization levels available with equal probabilities

Small amplitudes are not represented as well as large amplitudes, as they are more susceptible to quantization noise

Solution: Nonuniform quantization that uses quantization levels of variable spacing, denser at small signal amplitudes, broader at large amplitudes

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A practical solution to nonuniform quantization:• Compress the signal first • Quantize it • Transmit it • Expand it

Companding=Compressing+Expanding

The exact SNR gain obtained with companding depends on the exact form of the compression used.

With proper companding, the output SNR can be made insensitive to peak to average power ratio.

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Two standards: the A-law used in most countries of the world and the µ-law used in North America and Japan.

Compression characteristics

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Line coding

The bits of PCM, DPCM etc need to be represented by some electrical signals.

Line coding encodes the bit stream for transmission through a line, or a cable.

Line coding was used former to the invention of the channel coding and modulation techniques.

Nowadays, it is used for communications between the CPU and peripherals, and for short-distance baseband communications, such as the Ethernet.

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on-off signaling

Nonreturn-to-zero signaling

Return-to-zero signaling

Bipolar return-to-zero signaling

Manchester encoding

Differential encoding

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Baseband Data Transmission

Effect of additive noise on digital transmission: at the receiver, symbol 1 may be mistaken for 0, and vice versa.⇒ bit errors

What is the probability of such an error?

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Model of a binary baseband data communication system

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Unipolar system: 0 → 0 and 1 → A

Bipolar system: 0 → −A and 1 → A

Assume:• The channel noise is additive white Gaussian, with a double-sided PSD of N0/2.• The LPF is an ideal filter with unit gain on [−W, W ].• The signal passes through the LPF without distortion (approximately).

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After the LPF, the predetection signal isy(t) = s(t) + n(t) (147)

wheres(t): the binary-valued function (either 0 or A volts)n(t): additive white Gaussian noise with zero mean and variance

Reminder:A sample value N of n(t) is a Gaussian random variable drawn from a probability density function (the normal distribution):

WNdfNW

W 002 2/

),0(2

exp2

1)( 22

2

NnnpN

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Probability density functions for binary data transmission in noise: (a) symbol 0 transmitted, and (b) symbol 1 transmitted

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Y : a sample value of y(t)

If a symbol 0 were transmitted: y(t) = n(t) ⇒ Y will have a PDF of N (0, σ2)

If a symbol 1 were transmitted: y(t) = A + n(t) ⇒ Y will have a PDF of N (A, σ2)

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Maximum likelihood detector:

Use as decision threshold T :

if Y < T, choose symbol 0 if Y > T, choose symbol 1

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Two cases of decision error:

(i) a symbol 0 was transmitted, but a symbol 1 was chosen

(ii) a symbol 1 was transmitted, but a symbol 0 was chosen

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Probability of (i) occurring =(Probability of an error, given symbol 0 was transmitted) ×(Probability of a 0 to be transmitted in the first place):

p(i) = Pe0 × p0

where:p0: the a priori probability of transmitting a symbol 0Pe0: the conditional probability of error, given that symbol 0 wastransmitted:

Te dnnP 2

2

0 2exp

21

(151)

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Probability of (ii) occurring =(Probability of an error, given symbol 1 was transmitted) ×(Probability of a 1 to be transmitted in the first place):

p(ii) = Pe1 × p1

where:p1: the a priori probability of transmitting a symbol 1Pe1: the conditional probability of error, given that symbol 0 was transmitted:

T

e dnAnP 2

2

1 2)(exp

21

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Total error probability:

Choose T so that Pe(T ) is minimum:

T

T

ee

iiie

dnnp

dnAnp

PpPpppTP

2

2

1

2

2

1

0111

)()(

2exp

21)1(

2)(exp

21

)1()(

0)(

dTTdPe

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Leibnitz rule of differentiating an integral with respect to a parameter: if

then

)(

)();()(

b

adxxfI

)(

)(

);());(()());(()()(

b

adxxfaf

ddabf

ddb

ddI

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02

exp2

1)1(2

)(exp2

1)(2

2

12

2

1

TpATp

dTTdPe

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2

2

222

2

22

1

1

2)2(exp

22exp

2)(exp

1

ATA

TAATT

ATTp

p

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Equi-probable symbols (p1 = p0 = 1 − p1)⇒ T = A/2.

21ln

21

ln2

)2(1

ln2

2)2(

1ln

1

12

1

12

1

12

21

1

Ap

pA

T

ATp

pA

ATAp

p

ATAp

p

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For equi-probable symbols, it can be shown that Pe0 = Pe1

Probability of total error:

where the last line follows from p0 = p1 = 1/2, and Pe0 = Pe1.

10

1100

)()(

ee

ee

iiie

PPPpPp

ppP

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Calculation of Pe

Define a new variable of integration in (151):

When n = A/2, z = A/(2σ). When n = ∞, z = ∞Then

dzdndndznz

1

dze

dzeP

A

z

A

ze

)2/(

2/

)2/(

2/0

2

2

21

21

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We may express Pe0 in terms of the Q-function:

Then:

tdttxQ

2exp

21)(

2

2AQPe

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Example 1: A/σ = 7.4 ⇒ Pe = 10−4

⇒ For a transmission rate is 105 bits/sec, there will be an error every 0.1 secondsExample 2: A/σ = 11.2 ⇒ Pe = 10−8

⇒ For a transmission rate is 105 bits/sec, there will be an error every 15 mins

A/σ = 7.4: Corresponds to 17.4 dBA/σ = 11.2: Corresponds to 21 dB⇒ Enormous increase in reliability by a relatively small increase in SNR (if that is affordable).

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Probability of bit error for binary detection in Gaussian noise

Observe the threshold effect