1Dept. of EE, NDHU

Chapter One

Signals and Spectra

2Dept. of EE, NDHU

Why Digital ?

• Advantages

– Digital signals are more easily regenerated

– Digital circuits are more reliable and can be produced at lower cost

– Different types of digital signals can be treated as identical signals in transmission and

switching

– Digital techniques are naturally to signal processing functions that protect against

interference and jamming, or provide encryption

• Costs

– Very signal-processing intensive

– Need to synchronize at various levels

– Non-graceful degradation

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Pulse Degradation and Regeneration

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Typical Digital Communication System

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Digital Communication Transformations

• Formatting

– Analog source: audio, speech, video signal

– Digital source: computer data, digital image

– Convert the source into a sequence of binary sequence

• Source encoding

– Efficiently convert the digital symbol into a sequence of binary digits

– Data compression: MEG encode, JPEG, Huffiman coding, MP3

• Channel encoder

– Introduce some redundancy in the binary information sequence that can be used at the

receiver to overcome the effects of noise and encounter the channel

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Digital Communication Transformations

• Pulse modulation

– Map the binary information sequence into signal waveform

• Bandpass signaling

– Coherent: PSK, FSK, GMSK

– Non-coherent: DPSK, FSK

7Dept. of EE, NDHU

Basic Digital Communication Nomenclature

(Textual messages)

(Characters)

(7-bit ASCII)

(Symbol)

(Bandpass digital waveform)

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Performance Criteria

• Analog communication systems

– The figure of merit is a fidelity criterion

– For example signal-to noise ratio, percent distortion, or

expected mean-square error between the transmitted and

received waveforms

• Digital communication systems

– Probability of incorrectly detecting a digit, or PE

9Dept. of EE, NDHU

Classification of Signals

• Deterministic and Random signals

– Deterministic signal means that there is no uncertainty with respect to its value at any t

ime, for example x(t)=5 cos 10t

– Random signal means that there is some degree of uncertainty before signal actually o

ccurs

– Random waveform is NOT possible to write an explicit expression, can be described b

y probabilities and statistical averages

• Periodic and Non-periodic signals

– A signal x(t) is periodic in time if there exits a constant T0 such that

– No value of T0 that satisfies equation (1.2) is called non-periodic signal

(1.2) for )()( 0 tTtxtx

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Classification of Signals

• Analog and Discrete signals

– x(t) and x(kT)

• Energy and Power signals

– Energy signal is defined by the signal has nonzero but finite energy for all time

– Power signal is defined by the signal has finite but nonzero power for all the time

– Periodic signal and random signal are generally classified as power signals

– Both deterministic and non-periodic signals are generally classified as energy signals

(1.7) )(2/

2/

2lim

T

TTx dttxE

(1.8) )(1 2/

2/

2lim

T

TTx dttx

TP

11Dept. of EE, NDHU

Spectral Density

• Energy spectral density

– Where is defined as energy spectral density (ESD) of the signal x(t)

• Power spectral density

– The power spectral density (PSD) is

– See Example 1.1

(1.13) )()( 22

dffdfX(f)dttxE xx

(1.17) )(1 2

2/

2/

2

0

0

0

CdttxT

Pn

n

T

Tx

(1.18) )()( 02 nffCfG

nnx

)( fGx

2)()( fXfx

12Dept. of EE, NDHU

Autocorrelation

• A measure of how closely the signal matches a copy of itself as the copy is

shifted in the time

)()(

allfor )0()(

)()(

fR

RR

RR

xx

xx

xx

,)()()( dttxtxRx

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Random Process

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Random Process

• Stationary

– Strict-sense stationary if none of statistics are affected by a shift in the time origin

– Wide-sense stationary if

• Ergodic

– Time averages equal ensemble averages

– For example,

– The statistical properties of the process can be determined by time averaging over a sin

gle sample function

)(),( 2121 ttRttR xx

constant a )]([ xmtxE

2/

2/

)(/1limT

TTx dttxTm

15Dept. of EE, NDHU

Some Useful Probability Distributions

• Binormial Distribution

– Let X be a discrete random variable X=1 or X=0, with probability p an 1-p

• Uniform Distribution

• Gaussian (normal) Distribution

• Chi-square (exponential) Distribution

• Rayleigh Distribution

• Ricean Distribution

• Lognormal Distribution

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Autocorrelation and Power Spectral Density

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Autocorrelation and Power Spectral Density

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Normalized Gaussian Probability Density Function

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White Noise

Figure 1.8 (a) Power spectral density of white noise.(b) Autocorrelation function of white noise.

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Linear Systems

• Frequency response

• Power spectral density

• Distortionless transmission

)}(Re{

)}(Im{)( where,

)()(

)()(

1

)(

fH

fHTanf

efHfX

fYfH fj

)()()( 2 fGfHfG xY

)()( 02 0 ttkekfH ftj

21Dept. of EE, NDHU

Ideal Filter

• Transfer function

• Impulse response

)(2sin2)()( 0

222 0 ttfcfdfeedfefHth uu

f

f

ftjftjftju

u

02)(

)(

and

for 0

for 1)( where

)()(

ftjfj

u

u

fj

ee

ff

fffH

efHfH

22Dept. of EE, NDHU

Impulse Response of the Ideal Low-pass Filter

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Realizable Filter

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Butterworth Filter

• Magnitude frequency response for the n-th order

1 )/(1

1)(

2

n

fffH

nu

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RC Filtering an Ideal Pulse

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Baseband versus Bandpass

mDSB fW 2

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Bandwidth Dilemma

Strictly bandlimited signal

Strictly time limited signal

• For all bandlimited spectra, the waveform are not realizable,

and for all realizable waveforms, the absolute bandwidth is infinite.

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Bandwidth Criteria

Fig. Bandwidth of digital data. (a) Half-power. (b) Noise equivalent. (c) Null to null. (d) 99% of power. (e) Bounded PSD (defines attentuation outside bandwidth) at 35 and 50 dB.