digital communications (by b. sklar)

College of Information & Communications

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Wireless everywhere!

Hanyang University

Digital Communications(by B. Sklar)

Sung Ho Cho

(Voice) 02‐2220‐0390(Cellular) 011‐412‐5178

E‐mail: [email protected]

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College of Information & Communications Hanyang University


Contents

Signals and SpectraFormatting and Baseband TransmissionBandpass Modulation and DemodulationChannel CodingModulation and Coding Trade‐offsSpread‐Spectrum Techniques

College of Information & Communications

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Hanyang University

Signals and Spectra

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Analog & Digital Communications

Analog Communications

Digital Communications

Modulation

AnalogInformation

Channel Demodulation

AnalogInformation

DigitalInformation

Channel

DigitalInformation

ADCModulation DAC

Waveguide

WaveguideWire

Free space

Demodulation

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Why Digital?

More reliableMore flexibleSimpler

Example: A Regenerative Repeater

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A Typical Digital Communication System

Format SourceEncode Encrypt Channel

Encode Multiplex Modulate FrequencySpread

MultipleAccess XMT

Format SourceDecode Decrypt Channel

DecodeDe‐

multiplexDe‐

modulateFrequencyDespread

MultipleAccess RCV

InformationSource

SourceBits

ChannelBits

From OtherSources

To OtherDestinations

ChannelBits

SourceBits

Channel

InformationSink

DigitalInput

DigitalOutput

SynchronizationBitStream

DigitalWaveform

Adaptation CarrierErrorCorrection

Protection PrivacyCompanion SocietyMake‐up WeightControl

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Brief Description of Digital Communication Systems

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Notations (1/3)

Information Source:AnalogDiscrete

Textual Message:A sequence of charactersEx)

Character:A member of an alphabet or set of symbolsASCII, EBCDIC, MorseEx)

Binary Digit (Bit):The fundamental information unit“1” and/or “0”

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Notations (2/3)

Bit stream:A sequence of binary digitsOften called a “baseband signal”Ex) 7‐bit ASCII

Symbol:A digital messageA group of k bits M = 2k (M = size of the alphabet)Baud = symbol rateEx)

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Notations (3/3)Digital Waveform:

A voltage of current waveform that represents a digital symbolEx)

Data Rate:Bit rate Rb: Rb = 1/Tb (bits/s) (Tb = Bit duration)Symbol rate (Baud) R : R = 1/T (symbols/s) (T = Symbol duration)Relationship between Rb and R:

R = Rb / k or Rb = kR = R log2M

Performance Criteria (Figure of Merit):Analog communications:

SNR, Percent distortion, Mean‐square errorDigital Communications:

BER, SEP, Bandwidth efficiency

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Classification of Signals (1/3)

Deterministic Signal vs. Random Signal

Periodic Signal vs. Nonperiodic Signal

Analog Signal vs. Discrete Signal

Energy Signal vs. Power SignalEnergy signal:

Power signal:

Finite energy, but zero average power

Finite power, but infinite energy

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Impulse FunctionsDirac Delta Function:

Kronecker Delta Function

Unit Step Functions

⎩⎨⎧

≠=

≠0001

)(tt

tδ

∫∞∞−

=1)( dttδ

t

1

)(tδ

⎩⎨⎧

≠=

=0001

)(nn

nδn

1

)(nδ

⎩⎨⎧

<≥

=−== ∫∫∞

∞− 0001

)()()( 0 tt

dtdtu t ττδττδ

⎩⎨⎧

<≥

=−== ∑∑∞

=−∞= 0001

)()()(0 n

nknknu

k

n

kδδ

u(n)

t

u(t)

1

n

1...

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Some Mathematics:

For any x(t) or x(n)

)()( ttudtd δ= )()( 00 ttttu

dtd

−=− δαα

)()1()( nnunu δ=−−

)()( tt −= δδ )()( nn −= δδ

)()()()( 000 tttxtttx −=− δδ

)()()()( 000 nnnxnnnx −=− δδ

ττδτ dtxtx )()()( −= ∫∞∞−

)()()( knkxnxk

−= ∑∞

−∞=δ

)()( tt −=− τδτδ )()( nkkn −=− δδ

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Spectral Density

Energy Spectral Density (ESD):

Total Energy:

Power Spectral Density (PSD):

For periodic signals

For nonperiodic signals

Average Power:

Fourier series coefficient

Truncated version

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Autocorrelation

For energy signals:

For power signals:

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Random Variables

Random Variable X: “A functional relationship between a random event and a real number”

Distribution Function:

Probability Density Function (pdf):

Computation of Probability Using pdf:

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Ensemble Averages

Mean Value (or Expected Value):

Moment:

Variance:

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Random ProcessesX(t): A representation of the random variable as a function of time

Ensemble of the Random Processes:

Mean of the Random Process:

Autocorrelation Function:

Ensemble Average

Time Average

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StationarityStrict‐Sense Stationary (SSS): All the statistics do not change with time.

Wide‐Sense Stationary (WSS):

Autocorrelation of a WSS Process:

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Time Averaging and Ergodicity

Ergodic Process: Time Average = Statistical Average

Ergodic in the Mean:

Ergodic in the Autocorrelation Function:

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PSD and Autocorrelation (1/3)

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Example 1:

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Example 2:

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Noise in Communication Systems

A zero‐mean Gaussian random process n(t):

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White Noise

PDF:

Autocorrelation:

Average Power:

digital communications (by b. sklar)

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