디지털통신 signals and systems - dongguk · 2015-06-22 · minjoong rim, dongguk university 6...

Minjoong Rim, Dongguk University Signals and Systems1

디지털통신

Signals and Systems

임 민 중

동국대학교정보통신공학과


Signals and Spectra


Classification of Signals

• Deterministic signal

there is no uncertainty with respect to its value at any time

• Random signal

there is some degree of uncertainty before the signal actually occurs

• Periodic signal

x(t) = x(t + T0) for - < t <

• Nonperiodic(Aperiodic) signal

• Analog (continuous-time) signal

continuous function of time, that is, uniquely defined for all t

• Discrete signal

exists only at discrete times; characterized by a sequence of numbers defined for each time

1 0 1 1 1 0

y = cos(2fct)


Special Functions

• Special Functions

sinc function

rectangular function

triangular function

impulse function (delta function)

- area = 1

- amplitude =

- pulse width = 0

t1t

tt

)sin()(sinc

otherwise0

2

11

)(t

t

otherwise0

11)(

ttt

t1/2

t1

1

1

1

2 3 4 5

t

1

)(t

sinc(0) = 1sinc(n) = 0 n = 1,2,...

-1,-2,...


Circuits and Equipments - 1

• ADC

Analog-to-Digital Converter

a device that converts a continuous physical

quantity to a digital number that represents the quantity's amplitude

• DAC

Digital-to-Analog Converter

a device that converts digital data

(usually binary) into an analog signal (current, voltage, or electric charge)

• Amp

Amplifier

an electronic device that increases the power of a signal

• Gain-controlled Amplifier

Voltage-controlled amplifier

an electronics amplifier that varies

its gain depending on a control voltage

discrete &quantized

analog

ADC

digital analog

DAC

Amp

Amp

Gain


Circuits and Equipments - 2

• Oscillator

an electronic circuit that produces a repetitive, oscillating electronic

signal, often a sine wave or a square wave

Examples: signals broadcast by radio and television transmitters,

clock signals that regulate computers and quartz clocks

• Voltage-controlled oscillator

an electronic oscillator designed to be controlled in oscillation

frequency by a voltage input

Modulation

Sinusoidal wave

Power

AmpMessage Signal

Carrier

antennabaseband signal

passband signal

Oscillator

Modulated Signal

ppm: part per million(Example) 20ppm at 2GHz

= 20 10-6 2 109 Hz = 40000 Hz


Linear Time-Invariant Systems

• Linear systems

if x1(t) y1(t) and x2(t) y2(t)

ax1(t) + bx2(t) ay1(t) + by2(t)

• Time-invariant systems

if x(t) y(t)

x(t -) y(t -)

• Causal

No output prior to the time, t = 0,

when the input is applied

Linear

networkInput

x(t)

X(f)

Output

y(t)

Y(f)h(t)

H(f)

input output

linear

time-invariant

input output non-causal

filter input [ filter ] filter output

transmitted signal [ communication channel ] received signal


Unit Impulse Function

• Unit impulse function

an infinitely large amplitude pulse, with zero pulse width, and unity

weight (area under the pulse), concentrated at the point where its

argument is zero

• Characteristics

(t) = 0 for t 0

(t) is unbounded at t = 0

• Impulse response

the response when the input is

equal to a unit impulse (t)

(t) h(t)

( )t dt

1

x t t t dt x t( ) ( ) ( )

0 0

Linear

Time-Invariant

Systemimpulse impulse response

t t

t t

continuous

discrete

input output


Convolution - 1

• Linear Time-Invariant System

• Convolution

In linear time-invariant system, system output can be calculated with

convolution of the input and the system impulse response

y t x t h t x h t d( ) ( ) ( ) ( ) ( )

Linear

Time-Invariant

Systemimpulse impulse response

Linear

Time-Invariant

System

input

(transmitted signal)

output

(received signal)

(t)h(t)

x(t) y(t) = x(t) h(t)


Convolution - 2

• Example

0 1

x(t) 1

0 1

h(t) 1

0 1

x()h(t-)

tt-1

0 1tt-1

0 1 tt-1

0 1 tt-1

Case 1

Case 2

Case 3

Case 4

0t

0 1t

1 2t

2 t0 1

y(t) 1

2

y t x t h t x h t d( ) ( ) ( ) ( ) ( )

input system impulse response

output

t t

t

)()()( ttt

0 < t-1 < 1

1 < t-1

0)( ty

tdtyt

0 11)(

ttdtyt

2)1(111)(1

1

0)( ty

h(t-)

h(t-)

h(t-)

0-1

h(-)

tt-1

h(t-)=h(-(-t))


Convolution - 3

• Example

0 1

x(t)

1

0 2

h(t)1

0 1

x()h(t-)

tt-2

0 1tt-2

0 1 tt-2

0 1 tt-2

Case 1

Case 2

Case 3

Case 4

0t

0 1t

1 2t

2 3t

0 1 tt-2

Case 5

3 t0 1

y(t)1

2 3

t t

input system impulse response

output

t

t-2 < 0 and t < 1

0 < t-2 < 1

1 < t-2

0)( ty

tdtyt

0 11)(

111)(1

0 dty

ttdtyt

3)2(111)(1

2

0)( ty

h(t-)

h(t-)

h(t-)

h(t-)

0-2

h(-)

tt-2

h(t-)=h(-(-t))


Frequency Domain - 1

• Frequency-domain Representation

02

0 0cos2 sin 2j f t

e f t j f t

0cos2 f t

0sin 2 f t

real

imaginary

)()( 0fffX tfjetx 02

)(

Time Domain Frequency Domain

ff0

)()( ffX 1)( tx

ff



Frequency Domain - 2

• Example: Frequency-domain representation of cosine signal

tf02cos f

f

f

)()( 02

102

1 ffff

tf022cos

tf032cos

)2()2( 02

102

1 ffff

)3()3( 02

102

1 ffff


0 0 02 2 21 10 2 2

cos2j f t j f t j f t

real e f t e e

real signal in the time domain

minus frequency

f0

2f0

3f0

minus frequency plus frequency

0.5 0.5

plus frequency


Fourier Transform - 1

• Fourier Transform for nonperiodic signal

specifies the frequency-domain description or spectral content of the

signal

time domain to frequency domain

• Inverse Fourier Transform

frequency domain to time domain

dtetxfX ftj 2)()(

dfefXtx ftj 2)()(

Fourier Transform

Inverse Fourier Transformt f

frequency-domaintime-domain

energy signal



• Fourier Transform of periodic signal

• Periodic signal

can be represented as a sum of complex exponentials

n

n tnfjctx )2exp()( 0

2

2

0

0

0

0)2exp()(

1T

Tn dtTnfjtxT

c

f

time-domain frequency-domain

=

+

+

periodic

discrete

sum of complex exponentials with period T0 / n (frequency = nf0)

T0

f0 2f0 3f0

f0=1/T0

t

n

n nffcfX )()( 0

)()2exp( 00 nfftnfj

Power signal: Note that only a single period is considered for integration



• Four types of Fourier Transform

t

t

t

t

Time Domain

Continuous

Aperiodic

Continuous

Periodic

Discrete

Aperiodic

Discrete

Periodic

f

f

f

f

Frequency Domain

Continuous

Aperiodic

Continuous

Periodic

Discrete

Aperiodic

Discrete

Periodic

Fourier Transform

Fourier Series

Discrete-Time

Fourier Transform

Discrete Fourier Transform

(Fast Fourier Transform)FFT is used in computers or digital devices


Fourier Transform Properties - 1

• Duality

t f

)()( fxtX

ft

dtetxfX ftj 2)()(

dfefXtx ftj 2)()(

Note that the two equations are similar




• Scale Change

)/(||

1)( afX

aatx

Wide Narrow

Narrow Wide Fast change in time-domain wideband signals


t f

t f



• Time Shift

time delay linear phase in the frequency domain

ft

02

0 )()(ftj

efXttx

( )X f

f

( )X f

ft

( )X f

f

( )X f

2X frequency → 2X phase delay

3X frequency → 3X phase delay



• Convolution

Convolution

Multiplication

- Convolution in the time-domain transforms to multiplication in the frequency

domain

Frequency transfer function (frequency response)

y t x h t d( ) ( ) ( )

Y f X f H f( ) ( ) ( )

H fY f

X f( )

( )

( )

)()()()( fYfXtytx

)()()()( fYfXtytx


Filter - 1

• Distortionless Transmission

The output signal from an ideal transmission line may have some time

delay compared to the input, and it may have a different amplitude than

the input, but otherwise it must have no distortion

Taking the Fourier transform

System transfer function

- Constant magnitude response

- Phase shift must be linear with frequency

y t Kx t t( ) ( ) 0

Y f KX f e j ft( ) ( ) 2 0

H f Ke j ft( ) 2 0

| ( )|H f

( )H f time delay → linear phase

f

f

t t

distortionlesstransmission


Filter - 2

• Convolution

in the time-domain transforms to multiplication in the frequency

domain

y t x h t d( ) ( ) ( )

Y f X f H f( ) ( ) ( )

Y f X f H f( ) ( ) ( )

)( fX

)( fH

f

f

f

passband stopband

input

system

output


Filter - 3

• Ideal low-pass filter

H f H f e j f( ) | ( )| ( )

| ( )|H f

1

0

for |f| < fu

for |f| fu

e ej f j ft ( ) 2 0

Low pass filter passes low frequency components and stops high frequency componentsFilter output is a smooth version of the input signal

| ( )|H f

( )H f

bandwidthf

f

t t

slope → delay

passband

stopband


• Ideal bandpass filter

• Ideal high-pass filter

Filter - 4

| ( )|H f

| ( )|H f

bandwidthf

f

Low-pass Filter

High-pass FilterTime Domain

Example: Low-pass, and high-pass filtering

passband

stopband

stopband

passband

디지털통신 signals and systems - dongguk · 2015-06-22 · minjoong rim, dongguk university 6...

Documents