Principles of Communication

Signals and Spectra (1)

2-1, 2-7, 2-9

Lecture 3, 2008-09-12

Lecture 3 2

Introduction

Practical WaveformsTime Average OperatorsBandlimited SignalsBandwidth

Lecture 3 3

Properties of Signals and NoiseIn communication systems, the received waveform is usually categorized into the desired part containing the information and the extraneous or undesired part. The desired part is called the signal, and the undesired part is called noise.

Lecture 3 4

Analysis of Signal and Noise

The signal is referred to a deterministic waveform, modeled and analyzed by mathematical tool “Signals and Systems”.The noise is referred to a random waveform, modeled and analyzed by mathematical tool “Random Processes”

Lecture 3 5

Review of “Signals and Systems”Fourier Transform and Spectra (LC 2-2)Power Spectral Density and Autocorrelation Function (LC 2-3)Orthogonal Series Representation (LC 2-4)Fourier Series (LC 2-5)Linear Systems (LC 2-6)Sampling Theorem (LC 2-7)Discrete Fourier Transform (LC 2-8)

Lecture 3 6

Practical WaveformsPractical waveforms that are physically realizable (i.e., measurable in a laboratory) satisfy several conditions:

The waveform has significant (nonzero) values over a composite time interval that is finite.The spectrum of the waveform has significant values over a composite frequency interval that is finite.The waveform is a continuous function of time.The waveform has a finite peak value.The waveform has only real values. That is, at any time, it cannot have a complex value a + jb, where b is nonzero.

Lecture 3 7

Mathematical ModelsMathematical models that violate some or all of the conditions listed above are often used, and for one main reason – to simplify the mathematical analysis.The physical waveform is said to be an energy signal because it has finite energy, whereas the mathematical model is said to be a power signal because it has the property of finite power (and infinite energy).

Lecture 3 8

Time Average OperatorThe time average operator is given by

[ ] [ ]dtT

T

TT ∫

−∞→

⋅>=⋅<2

2

1lim

A waveform w(t) is periodic with period T0 if

Linear Operator, the average of the sum of two quantities is equal to the sum of their averages

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )a w t a w t a w t a w t+ = +

[ ] [ ]dtT

a

a

T

T∫+

+−

⋅>=⋅<2

0

200

1

THEOREM: If the waveform involved is periodic with period T0, the time average operator can be reduced to

0( ) ( ) for all w t w t T t= +where T0 is the smallest positive number that satisfies the relationship.

Lecture 3 9

DC ValueThe dc (“direct current”) value of a waveform w(t) is given by its time average, <w(t)>.

For any physical waveform, we are actually interested in evaluating the dc value only over a finite interval of interest, say, from t1 to t2, so that the dc value is

dttwT

twWT

TTdc ∫

−∞→

==2

2

)(1lim)(

dttwtt

Wt

tdc ∫−=

2

1

)(1

12

Lecture 3 10

PowerLet v(t) denote the voltage across a set of circuit terminals, and let i(t) denote the current into the terminal. The instantaneous power associated with the circuit is given by

)()()( titvtp =

><=><= )()()( titvtpP

><= )(2 twWrms

rmsrmsrmsrms IVRIR

VRtiR

tvP ===>=<><

= 22

22

)()(

The average power is

The root-mean-square (rms) value of w(t) is

If the load is resistive, the average power is

Lecture 3 11

PowerThe concept of normalized power is often used by communication engineers. In this concept, R is assumed to be 1 Ω, although it may be another value in the actual circuit. The average normalized power is

><= )(2 twP

∫−

∞→=

2

2

)(lim 2T

T

dttwET

w(t) is a power waveform if and only if the normalized average power P is finite and nonzero. (0 < P < ∞)The total normalized energy is

w(t) is an energy waveform if and only if the total normalized energy E is finite and nonzero. (0 < E < ∞)

Lecture 3 12

Power

Physically realizable waveforms are of the energy type.Basically, mathematical models are of the power type.Mathematical functions can be found that have both infinite energy and infinite power, e.g., w(t) = e -t.

Lecture 3 13

Decibel

The decibel gain of a circuit is

The decibel power level with respect to 1 milliwatt is

noiserms

signalrms

noiserms

signalrms

noise

signaldB

VV

VV

tnts

PP

NS

log20log10

)()(log10log10)(

2

2

2

2

==

><><

==

dBWwattswattsP

wattswattsPdBm +=+== − 30

)(1)(log1030

)(10)(log10 3

10log out

in

PdBP

⎛ ⎞= ⎜ ⎟

⎝ ⎠

The decibel signal-to-noise ratio is

Lecture 3 14

Bandlimited SignalA waveform w(t) is said to be (absolutely) bandlimited to B hertz if

BffW ≥= for,0)(

Tttw >= for ,0)(

A waveform w(t) is said to be (absolutely) time limited if

An absolutely bandlimited waveform cannot be absolutely time limited, and vice versa. This raises an engineering paradox. This paradox is resolved by realizing that we are modeling a physical process with a mathematical model and perhaps the assumption in the model are not satisfied – although we believe them to be satisfied.If a waveform is absolutely bandlimited, it is analytic function.

On Bandwidth, D. Slepian, Proceedings of IEEE, vol. 64, no. 3, 1976

Lecture 3 15

BandwidthWhat is bandwidth? There are numerous definitions of the term.In engineering definitions, the bandwidth is taken to be the width of a positive frequency band. In other words, the bandwidth would be f2 – f1, where f2 ≥ f1 ≥ 0 and is determined by the particular definition that is used.ABSOLUTE BANDWIDTH is f2 – f1 , where the spectrum is zero outside the interval f1 < f < f2 along the positive frequency axis.3-dB BANDWIDTH (or HALF-POWER BANDWIDTH) is f2 – f1 , where for frequencies inside the band f1 < f < f2, the magnitude spectra, say, |H(f)|, falls no lower than 1/ times the maximum value of |H(f)| , and the maximum value occurs at a frequency inside the band.EQUIVALENT NOISE BANDWIDTH is the width of a fictitious rectangular spectrum such that the power in that rectangular band is equal to the power associated with the actual spectrum over positive frequencies. Let f0 be the frequency at which the magnitude spectrum has a maximum, then

2

∫∫∞∞

=⇒=0

22

00

220 )(

)(1)()( dffHfH

BdffHfHB eqeq

Lecture 3 16

BandwidthNULL-TO-NULL BANDWIDTH (or ZERO-CROSSING BANDWIDTH) is f2 –f1 , where f2 is the first null in the envelope of the magnitude spectrum above f0 and , for bandpass systems, f1 is the first null in the envelope below f0, where f0 is the frequency where the magnitude spectrum is a maximum. For baseband systems f1 is usually zero.BOUNDED SPECTRUM BANDWIDTH is f2 – f1 such that outside the band f1 < f < f2, the PSD must be down at least a certain amount, say, 50dB, below the maximum value of the PSD.POWER BANDWIDTH is f2 – f1, where f1 < f < f2 defines the frequency band in which 99% of the total power resides. FCC BANWIDTH is an authorized bandwidth parameter assigned by the FCC to specify the spectrum allowed in communication systems. For operating frequencies below 15 GHz, in any 4kHz band, the center frequency of which is removed from the assigned frequency by P% (50 < P < 250) of the authorized bandwidth (B, in MHz), the attenuation below the mean output power level is given by the following equation:

(dB)log10)50(8.035 BPA +−+=

Lecture 3 17

ExampleBANDWIDTH FOR A BPSK SIGNAL ttmts cωcos)()( =

Lecture 3 18

ExampleThe worst-case (widest-bandwidth) spectrum occurs when the digital modulating waveform has transitions that occur most often. In this case, m(t) would be a square wave.

∑∑

∫

∑

∞

≠−∞=

∞

≠−∞=

−

∞

−∞=

−=−=

⎪⎩

⎪⎨⎧

=⋅⋅

==

−=

0

2

0

02

00

2/

2/0

02

)2

()2/()()2/()(

)2/()(2

0)(1

)()(

0

0

nn

nn

m

bb

T

Tn

nnm

RnfnSanffnSafP

nSaTnfSaTf

dttmTc

nffcfP

δπδπ

ππ

δ

Lecture 3 19

Example

12

1 12 21212

( ) ( ) ( ) ( ) cos ( ) cos ( )( ) ( ) [cos cos(2 )]

( ) ( ) cos ( ) ( ) cos(2 )( ) ( ) cos

( )cos

s c c

c c c

c c c

c

m c

R s t s t m t t m t tm t m t t

m t m t m t m t tm t m t

R

τ τ ω τ ω ττ ω τ ω ω ττ ω τ τ ω ω ττ ω τ

τ ω τ

=< + >=< ⋅ + + >

=< + ⋅ + + >

= < + > + < + + >

= < + >

=

The autocorrelation of s(t).

1 12 4

214

0

( ) [ ( )] [ ( ) cos ] [ ( ) ( )]

( / 2)[ ( ) ( )]2 2

s s m c m c m c

c cnn

P f F R F R P f f P f fR RSa n f f n f f n

τ τ ω τ

π δ δ∞

=−∞≠

= = = − + +

= − − + + −∑

The PSD of s(t) is obtained by taking Fourier transform of both sides

Lecture 3 20

Example

Lecture 3 21

ExampleTo evaluate the bandwidth for the BPSK signal, the shape of the PSD for the positive frequencies is needed, it is

∑∞

≠−∞=

−−−=

0

241 )

2()]([)(

nn

ccbsRnffffTSafP δπ

Lecture 3 22

Absolute Bandwidth

∞=−∞=−= 012 ffBabs

Lecture 3 23

3-dB Bandwidth

2 12 22

2 3 2 1 2

[ ( )] ( ) 1.41.4 0.446 0.446 2( ) 0.891

b c b c

c dB cb b

Sa T f f T f f

f f R B f f f f RT T

π π

π

− = ⇒ − ≈ ⇒

− = ≈ = ⇒ = − = − =

21

Lecture 3 24

Equivalent Noise Bandwidth

RRdffHfH

BffTSa

RRxdxSaRdfffTSadfffTSa

eqffcb

f cbcb

c

c

00.11

)()(

11)]([

222)]([2)]([

0

22

0

2

0

22

0

2

===⇒=−

=⋅==−≈−

∫

∫∫∫∞

=

∞∞∞

π

πππ

ππ

Lecture 3 25

Null-to-Null Bandwidth

RRBRfTffRfTff

ffTffTSa

nullcb

ccb

c

cbcb

00.221,1)(0)]([

12

2

==⇒−=−=+=+=⇒

±=−⇒=− πππ

Lecture 3 26

Bounded spectrum bandwidth (50dB)

RRBRT

ffffT

ffTffT

ffTSa

dBb

ccb

cbcb

cb

32.2016584.10026584.100101010)]([

50)](log[10)]([

1log10)]([log10

50

5510

502

22

2

≈×=⇒≈≥−⇒=≥−

−≤−−=−

≤−

ππ

ππ

π

dB50−

Lecture 3 27

Power bandwidth (20dB)

RfBRxT

fxdxxSa

dxxSa

fTx

powerb

xb

56.202278.10129.3299.0)(

)(

Let

0000

0

2

0

20

≈=⇒≈=⇒≈⇒=

=

∫∫∞ π

π

area total theof %99

Lecture 3 28

FCC bandwidthIn this example, the FCC bandwidth parameter B = 30 MHz. Assume that the PSD is constant across the 4-kHz bandwidth (4 kHz / 30 MHz = 0.013%), the decibel attenuation of the BPSK signal

)()(4000log10

P)(Plog10)( 4

ceqtotal

kHz

fPBfPffA −=−=

Plugging in Beq = R Hz, P(fc) = 1 Watt/Hz, we have

)]([4000log10)( 2cb ffTSa

RfA −−= π

The envelope of A(f)

2262

22

log1046.83)]1030%([

4000log10)]%([

4000log10

)]([4000log10

)]([14000log10)(

PR

PR

BPR

ffR

ffTRfA

ccbenv

−≈××

−=−=

−−=

−−=

ππ

ππ

Lecture 3 29

FCC BandwidthFCC envelope A for B = 30 MHz

2

122.33 8010

35 0.8( 50) 10log35 0.8( 50) 10log30 49.77 0.8( 50)

80 49.7780 50 87.790.8

87.79 ( ) 83.46 10log87.79

122.33 10log 80 10 17100 0.0171

env

A P BP P

A dB P

RP A f

R R bps Mbps−

= + − += + − + ≈ + −

−= ⇒ = + ≈

= ⇒ = −

≈ − ≥ ⇒ ≤ ≈ ≈

Lecture 3 30

FCC Bandwidth

Lecture 3 31

Homework2-4, 2-10, 2-32, 2-45, 2-92