Digital Communications Tutorial

Digital Communications Tutorial

Cognitive Radio Communications @ Virginia Tech

NSF Research Experiences for Undergraduates (REU) Site

Ratchaneekorn (Kay) [email protected]

Outline Introduction Fourier Transform Sampling Pulse Amplitude Modulation (PAM) InterSymbol Interference (ISI) Digital Bandpass Modulation

Analog vs. Digital

Transmitted bits can be detected and regenerated, so noise does not propagate additively.

More signal processing techniques are available to improve system performance: source coding, channel (error-correction) coding, equalization, encryption, filtering,…

Digital ICs are inexpensive to manufacture Digital communications permits integration of voice, video, and data on a

single system (ISDN) Implementation of various algorithms can be done by software instead of

hardware Security is easier to implement.

Simple Digital Communication System Diagram

Digital communications system block diagram

Fourier Transform

F() is the continuous-time Fourier transform of f(t).

The Fourier transformation F(ω) is the frequency domain representation of the original function f(t). It describes which frequencies are present in the original function.

Example 1:

Ex A: Find the Fourier Transform of x(t) = (t)

Ex B: Find the Fourier Transform of x(t) = 0.5cos(500t)

t=0

x(t)

Ex C: Find the Fourier Transform of f(t) = rect(t/)

2sinc

2

2sin

2sin2

1

rect

2/2/

2/

2/

jj

tjtj

eej

dtedtet

F

sinc function

sinc(x) = sin(x) x

even function zero crossings at Amplitude decreases proportionally to 1/x

0

1

x

sinc(x)

... ,3 ,2 , x

Ex D: Pulsed Cosine:cos(t)rec(t/T) <=> (T/2) sinc(sinc(

2 2

Linear Time-Invariant (LTI) system

Convolution: y(t) = x(t)*h(t)

Its Fourier Transform:

Y(ω) = X(ω)H(ω)

where H(ω) is a frequency response or a transfer function of a system h(t).

h(t)

Ideal filters

A filter is used to eliminate unwanted parts of the frequency spectrum of a signal.

A filter is LTI system with an impulse response h(t).

The output y(t) of a filter can be founded in time domain using a convolution.

However, it is easier to do it in a frequency domain: Y(ω) = X(ω)H(ω)

Low Pass Filterwith a cutoff frequency c High Pass Filter

Example 2: Given x(t) = cos(500t)cos(1000t), find an

impulse response h(t) of a low-pass filter that passes the

low frequency component of the signal.

x(t) y(t) = low freq.

component of x(t)

Low-pass filterh(t)

H()

Y() = H()X3() = /2[(-500) + (+500)]

=> y(t) = 0.5cos(500t)

H( ) = rect( => h(t) = 1000sinc(1000t)

Sampling Continuous-Time signals

Sampling – generating of an ordered number of sequence by taking values of f(t) as specified instants of time i.e. f(t1), f(t2), f(t3), … where tm are instants at which sampling

occurs.

Sampling operation is implemented in hardware by an analog-to-digital converter (ADC) – electronic device used to sample physical voltage signals.

In most cases, continuous-time signals are sampled at equal increments of time. The sample increment, called sample period, is usually denoted as Ts.

Impulse sampling

Define the continuous time impulse train as:

p(t) is an infinite train of continuous time impulse functions, spaced Ts seconds apart.

Let x(t) be a continuous time signal we wish to sample. We will model sampling as multiplying a signal x(t) by p(t).

Sampling Theorem

let P(ω) be a Fourier Transform of p(t), X(ω) be a Fourier Transform of x(t), Xs(ω) be a Fourier Transform of xs (t),

Sincexs(t) = x(t)p(t)

by a multiplication property (Fourier Transform),

where Ck are the Fourier Series coefficients of the periodic signal.

21

We see that an impulse train in time, p(t), has a Fourier Transform that is an impulse train in frequency, P().

The spacing between impulses in time is Ts, and the spacing

between impulses in frequency is ω0 = 2/Ts.

Note: If we increase the spacing in time between impulses, this will decrease the spacing between impulses in frequency, and vice versa.

Spectrum of a sampled signal

replicated scaled versions of X(), spaced every 0 apart in frequency

Time-domain

Frequency-domainω0 = 2/Ts

If c < c , ALIASING (overlap area) occurs

If c ≥ c ,

Note: if ω0 - ωc ≥ ωc or ω0 ≥ 2ωc, there is no aliasing

Sampling Theorem

Let x(t) be a band-limited signal with X(ω) = 0 for |ω| > ωc.

Then x(t) is uniquely determined by its samples x(nTs), n =

0, ±1, ± 2, … if

ω0 ≥ 2ωc

where ω0 = 2/Ts.

This is how to choose a sampling frequency (fs = 1/Ts) or period (Ts) such that an original continuous-time signal x(t) can be recovered from a sampled version xs(t).

=> a sampling rate (ω0) MUST be at least twice the highest frequency (ωc) of a signal to avoid aliasing problem.

To recover x(t) from its sampled version xs(t), we use a low pass filter (reconstruction filter) to recover the center island of Xs():

Ex: Given a signal x(t) with Fourier Transform with cutoff frequency ωc as shown:

Given three different pulse trains with periods

Draw the sampled spectrum in each case. Which case(s) experiences aliasing?

Aliasing Phenomenon Sampling theorem: the signal is strictly band-limited (c).

However, in practice, no information-bearing signal is strictly band-limited.

Aliasing is the phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on the identify of a lower frequency in the spectrum of its sampled version.

To prevent the effects of aliasing in practice Prior to sampling : a low-pass anti-alias filter is

used to attenuate those high-frequency components of a message signal that are not essential to the information being conveyed by the signal.

The filtered signal is sampled at a rate slightly higher than the Nyquist rate.

Example: Why 44.1 kHz for Audio CDs? Sound is audible in 20 Hz to 20 kHz range:

fmax = 20 kHz and the Nyquist rate 2fmax = 40 kHz

What is the extra 10% of the bandwidth used?Rolloff from passband to stopband in the magnitude response of the anti-aliasing filter.

Okay, 44 kHz makes sense. Why 44.1 kHz?At the time the choice was made, only recorders capable of storing such high rates were VCRs.

NTSC: 60-Hz video (30 frames/s) - 490 lines per frame or 245 lines per field, 3 audio samples per line the sampling rate is 60 X 245 X 3 = 44.1 KHz

Pulse-Amplitude Modulation (PAM)

The amplitude of regularly spaced pulses are varied in proportion to the corresponding sample values of a continuous message signal.

Two operations involved in the generation of the PAM signal Instantaneous sampling of the message signal m(t) every

Ts seconds, Lengthening the duration of each sample, so that it

occupies some finite value T.

Sample-and-Hold Filter : Analysis

The PAM signal is

The h(t) is a standard rectangular pulse of unit amplitude and duration

The instantaneously sampled version of m(t) is

n

ss nTthnTmts )()()(

otherwise ,0

,0,2

1

0 ,1

2)( Ttt

Tt

T

Tt

rectth

n

ss nTtnTmtm )()()(

To modify mδ(t) so as to assume the same form as the PAM signal:

The PAM signal s(t) is mathematically equivalent to the convolution of mδ(t) , the instantaneously sampled version of m(t), and the pulse h(t).

Its Fourier Transform:

nss nTthnTm

dthmthtmts

)()(

)()()()()(

)()()( fHkffMffSk

ss

One benefit of PAM

It enables the simultaneous transmission of multiple signals using time-division multiplexing (TDM).

User 1

User 2

39

Quantization Process Amplitude quantization: The process of transforming the

sample amplitude m(nTs) of a baseband signal m(t) at time t=nTs into a discrete amplitude v(nTs) taken from a finite set of possible levels.

LkmmmI kkk ,...,2,1},{: 1

It will be represented by binary number(s)

Outline Introduction Fourier Transform Sampling Pulse Amplitude Modulation (PAM) MATLAB! InterSymbol Interference (ISI) Digital Bandpass Modulation

Baseband Transmission of Digital Data

The transmission of digital data over a physical communication channel is limited by two unavoidable factors

1.Intersymbol interference2.Channel noise

43

The level-encoded signal and the discrete PAM signal are

The transmitted signal is

The channel output is

The output from the receive-filter is

0 symbol is input theif1

1 symbol is input theif1

k

kk b

ba

k

bk kTtgats )()(

)()()( thtstx

)(*)(*)()()()( tqthtstqtxty

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The InterSymbol Interference (ISI) Problem

We may express the receive-filter output as the modified PAM signal

where

After sampling:

k

bk kTtpaty )()(

)()()()( tqthtgtp

k

bkb iTkipaiTy ,...2,1,0],)[()(

)( bi iTyy )( bi iTpp

kkiki ipay ,...2,1,0,

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ISI (cont.)

Definewhere E is the transmitted signal energy / bit (symbol).

What we desire is

However, from

Epp )0(0

ik

kkikii ipaaEy ,...2,1,0,

Residual phenomenon, intersymbol interference (ISI)

iaEy ii allfor ,

kkiki ipay ,...2,1,0,

Pulse-shaping

Given the channel transfer function, determine the transmit-pulse spectrum and receive-filter transfer function so as to satisfy two basic requirements:

1.Intersymbol interference (ISI) is reduced to zero.

2.Transmission bandwidth is conserved.

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The Nyquist Channel The optimum solution for zero ISI at the minimum transmission

bandwidth possible in a noise-free environment For zero ISI, it is necessary for the overall pulse shape p(t) and the

inverse Fourier transform of the pulse spectrum P(f) to satisfy the condition

0 allfor ,0

0for ,)(

i

iEiTpp bi

i

itBB

iptp )2(csin

2)( 0

0bT

B2

10

tB

tBEtBEtP

0

00opt 2

)2sin()2(csin)(

48

The overall pulse spectrum is defined by the optimum brick-wall function:

The brick-wall spectrum defines B0 as the minimum transmission bandwidth for zero intersymbol interference.

The optimum pulse shape is the impulse response of an ideal low-pass channel with an amplitude response Popt(f) in the passband and a bandwidth B0

otherwise ,0

for,2)( 00

0opt

BfBB

EfP

Symbol 1 Symbol 2 Symbol 3

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Two difficulties that make its use for a PAM system impractical:

1. The system requires that the spectrum P(f) be flat from –B0 to B0, and zero elsewhere

2. The time function p(t) decreases as 1/|t| for large |t|, resulting in a slow rate of decay

52

Raised-Cosine Pulse Spectrum

To ensure physical realizability of the overall pulse spectrum P(f), the modified P(f) decreases toward zero gradually rather than abruptly1. Flat portion, which occupies the frequency band 0≤|f| ≤f1 for

some parameter f1 to be defined

2. Roll-off portion, which occupies the frequency band f1 ≤|f| ≤2B0-f1

2 ,0

2,)(2

)(cos1

4

0 ,2

)(

10

10110

1

0

10

ffB

fBfffB

ff

B

E

ffB

E

fp

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The roll-off factor:

Time-domain of the overall channel

The amount of intersymbol interference resulting from a timing error ∆t decreases as the roll-off factor is increased form zero to unity.

0

11B

f

22

02

00 161

)2cos()2(csin )(

tB

tBtBEtp

54

Frequency domain P(f)

Time domain p(t)

55

Transmission-Bandwidth Requirement The transmission bandwidth required by using the raised-cosine

pulse spectrum is

Excess channel The transmission bandwidth requirement of the raised-cosine

spectrum exceeds that of the optimum Nyquist channel by the amount

1. When the roll-off factor is zero, the excess BW is reduced to zero

2. When the roll-off factor is unity, the excess BW is increased to B0.

102 fBBT

)1(0 BBT

0Bfv

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Summary (ISI)

The intersymbol interference problem, which arises due to imperfections in the frequency response of the channel

ISI refers to the effect on that pulse due to cross-talk or spillover from all other signal pulses in the data stream applied to the channel input

A corrective measure widely used in practice is to shape the overall pulse spectrum of the baseband system, starting from the source of the message signal all the way to the receiver.

ISI is a signal-dependent phenomenon, it therefore disappears when the information-bearing signal is switched off.

Noise is always there, regardless of whether there is data transmission or not.

Another corrective measure for dealing with the ISI: channel equalization.

Outline Introduction Fourier Transform Sampling Pulse Amplitude Modulation (PAM) InterSymbol Interference (ISI) Digital Bandpass Modulation Techniques

58

Digital band-pass modulation techniques

Baseband Communication: Signals are transmitted without any shift in the range of frequency of the signal.

Band-pass Communication: Uses modulation to shift the frequency spectrum of a (carrier) sinusoidal signal. Usually, one of the basic parameters (amplitude, frequency, or phase) of the carrier signal is varied in proportion to the baseband signal (information-bearing data stream).

Why modulate signals? Convert signals to a form that is suitable for transmission Sharing the frequency band with other stations

Three basic modulation schemes: Amplitude-shift keying (ASK) Phase-shift keying (PSK) Frequency-shift keying (FSK)

Given a binary source The modulation process involves switching or keying the

amplitude, phase, or frequency of a sinusoidal carrier wave between a pair of possible values in accordance with symbol (bit) 0 and 1.

Examples of a band-pass process1. Binary amplitude shift-keying (BASK)

The carrier amplitude is keyed between the two possible values used to represent symbols 0 and 1

2. Binary phase-shift keying (BPSK) The carrier phase is keyed between the two possible

values used to represent symbols 0 and 1.3. Binary frequency-shift keying (BFSK)

The carrier frequency is keyed between the two possible values used to represent symbols 0 and 1.

)2cos()( ccc tfAtc

61

In digital comm., the usual practice is to assume that the carrier c(t) has unit energy measured over one symbol (bit) duration (Tb).

where

Decreasing the bit duration Tb has the effect of increasing the transmission bandwidth requirement of a binary modulated wave. (Fourier Transform property).

bc TA

2)2cos(

2)( cc

b

tfT

tc

62

Band-Pass Assumption

The spectrum of a digital modulated wave s(t) is centered on the carrier frequency fc

where b(t) is an incoming binary stream with bandwidth W.

Assumption: fc>> BW,There will be no spectral overlap in the generation of s(t)

The transmitted signal energy per bit can be approximated as:

)()()( tctbts

)2cos()(2

)( tftbT

ts cb

bb

b

T

b

T

cb

T

b

dttbT

dttftbT

dttsE

0

2

0

22

0

2

)(1

)2(cos)(2

)(

Binary Amplitude-Shift Keying (BASK)

The ON-OFF signaling variety

The average transmitted signal energy is (the two binary symbols must be equi-probable)

0 symbolbinary for ,0

1 symbolbinary for ,)( bEtb

0 symbolfor ,0

1 symbolfor ),2cos(2

)(tf

T

Ets c

b

b

2avbEE

65

fc = 8 Hz, Tb = 1s

66

fc = 8 Hz, Tb = 0.5 s

67

From figures:

The spectrum of the BASK signal contains a line component at f=fc

When the carrier is fixed and the bit duration is halved, the width of the main lobe of the sinc function defining the envelope of the BASK spectrum is doubled, which, in turn, means that the transmission bandwidth of the BASK signal is doubled.

Tb halved <= => W is doubled

The transmission bandwidth of BASK, measured in terms of the width of the main lobe of its spectrum, is equal to 2/Tb, where Tb is the bit duration.

68

Phase-Shift Keying

Binary Phase-Shift Keying (BPSK) The pair of signals used to represent symbols 1 and 0,

An antipodal signals A pair of sinusoidal wave, which differ only in a relative

phase-shift of radians.

Note: The transmitted energy per bit, Eb, is constant. Equivalently, the average transmitted power is constant.


)2cos(2


)(

tfT

Etf

T

E

tfT

E

ts

cb

bc

b

b

cb

b

i

Signal Space diagram of BPSK

71

fc = 8 Hz, Tb = 1s

72

fc = 8 Hz, Tb = 0.5 s

73

From figures:

BASK and BPSK signals occupy the same transmission bandwidth (2/Tb), which defines the width of the main lobe of the sinc-shaped power spectra.

The BASK spectrum includes a carrier component, whereas this component is absent from the BPSK spectrum.

74

Quadriphase-Shift Keying (QPSK) An important goal of digital communication is the

efficient utilization of channel bandwidth.

In QPSK, the phase of the sinusoidal carrier takes on one of the four equally spaced values, such as /4, 3/4, 5/4, and 7/4

Each one of the four equally spaced phase values corresponds to a unique symbol which is a pair of bits (00, 01, 10, 11).

Symbol duration

elsewhere ,0

0 ,4

)12(2cos2

)( TtitfT

Ets c

i

bTT 2

75

1. In reality, the QPSK signal consists of the sum of two BPSK signals.

2. One BPSK signal, represented by the first term defined the product of modulating a binary wave by the sinusoidal carrier

3. The second binary wave

),2cos(4

)12(cos/2 tfiTE c

3 2for 2/

4 1for 2/

4)12(cos

,iE

,iEiE

),2sin(4

)12(sin/2 tfiTE c

4 3for 2/

2 1for 2/

4)12(sin

,iE

,iEiE

)2sin(4

)12(sin2

)2cos(4

)12(cos2

)( tfiT

Etfi

T

Ets cci

Signal Space diagram of QPSK

QPSK Transmitter

80

QPSK Receiver

81

fc = 8 Hz, Tb = 1s

BW = 1/Tb

82

fc = 8 Hz, Tb = 0.5 s

BW = 1/Tb

83

Frequency-Shift Keying

Binary Frequency-Shift Keying (BFSK) Each symbols are distinguished from each other by

transmitting one of two sinusoidal waves that differ in frequency by a fixed amount:

Sunde’s BFSK When the frequencies f1 and f2 are chosen in such a way that

they differ from each other by an amount equal to the reciprocal of the bit duration Tb



)(

2

1

tf

T

E

tfT

E

ts

b

b

b

b

i

85

fc = 8 Hz, Tb = 1s

f = fc ± 1/(2Tb)

BW = 3/Tb

86

2/Tb

2/Tb

3/Tb forf1,2 = fc ± 1/(2Tb)

Bandwidth

QPSK 1/Tb

87

M-ary Digital Modulation Schemes We send any one of M possible signals during each

signaling interval of duration T.

The requirement is to conserve bandwidth at the expense of both increased power and increased system complexity.

When the bandwidth of the channel is less than the required value, we resort to an M-ary modulation scheme for maximum bandwidth conservation

88

M-ary Phase-Shift Keying If we take blocks of m bits to produce a symbol and

use an M-ary PSK scheme with M=2m and symbol duration T=mTb

The bandwidth required is proportional to 1/(mTb).

The use of M-ary PSK provides a reduction in transmission bandwidth by a factor m=log2M over BPSK.

90

M-ary Quadrature Amplitude Modulation (QAM)

The mathematical description of the new modulated signal

The level parameter for in-phase component and quadrature component are independent of each other for all i.

M-ary QAM is a hybrid form of M-ary modulation.

M-ary amplitude-shift keying (M-ary ASK)

0

1,...,1,0),2sin(

2)2cos(

2)( 00

Tt

Mitfb

T

Etfa

T

Ets cicii

1,...,1,0)2cos(2

)( 0 MitfaT

Ets cii

Signal-Space Diagram

Figure 7.21 is the signal-space representation of M-ary QAM for M=16

Unlike M-ary PSK, the different signal points of M-ary QAM are characterized by different energy levels

Each signal point in the constellation corresponds to a specific quadbit

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Bit Error Rate Average bit error rate (BER) Let n denote the number of bit errors observed in a

sequence of bits of length N; then the relative frequency definition of BER is

BER goal: For data transmission over wireless channels, a bit

error rate of 10-5 to 10-6

For video transmission, a BER of 10-7 to 10-12 depending upon the quality desired and the encoding method.

N

nNlimBER

Signal to Noise Ratio (SNR)

The ratio of the modulated energy per information bit to the one-sided noise spectral density; namely,

The reference SNR is independent of transmission rate. Since it is a ratio of energies, it has essentially been normalized by the bit rate.

0

digital

ref

density spectral Noise

bitper energy ModulatedSNR

N

Eb

95

x

dssxQ )2/exp(2

1 2

where

dyyQ )2/)(exp(

2

1 22

Pe = 0.5P(0 decided| 1 is trans.) + 0.5P(1 decided|0 is trans.)

/2 for 2

5.05.0

QQQPe

BASK

TEATTNN

EQQP b

be 2 and 2/ where

2 00

Real-world use (Tidbits)

The wireless LAN standard, IEEE 802.11b-1999, uses a variety of different PSKs depending on the data-rate required.

- Basic-rate of 1 Mbit/s, DBPSK - Extended-rate of 2 Mbit/s, DQPSK- 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is used with complementary code keying.

The higher-speed wireless LAN standard, IEEE 802.11g-2003[1][3] has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s.

- The 6 and 9 Mbit/s modes, OFDM modulation where each sub-carrier is BPSK modulated.

- The 12 and 18 Mbit/s modes use OFDM with QPSK. - The fastest four modes use OFDM with QAM

http://en.wikipedia.org/wiki/Phase-shift_keying#cite_note-ref80211-0

http://en.wikipedia.org/wiki/Phase-shift_keying#cite_note-80211g-2

BPSK is appropriate for low-cost passive transmitters, and is used in RFID standards.

Bluetooth uses /4-DQPSK for the rate 2 Mbit/s and 8-DPSK at its higher rate (3 Mbit/s)

IEEE 802.15.4 (the wireless standard used by ZigBee) also relies on PSK. It has two frequency bands: - 868–915 MHz using BPSK and- 2.4 GHz using OQPSK

References:

Simon Haykin and Michael Moher, “Introduction to Analog and Digital Communications,” 2nd ed., John Wiley & Sons, Inc., 2007.

Charles L. Phillips, John M. Parr, Eve A. Riskin, “Signals, Systems, and Transforms,” 4th ed., Pearson/Prentice Hall, 2008.

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