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Minjoong Rim, Dongguk University Signals and Systems1
디지털통신
Signals and Systems
임 민 중
동국대학교정보통신공학과
Minjoong Rim, Dongguk University Signals and Systems2
Signals and Spectra
Minjoong Rim, Dongguk University Signals and Systems3
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)
Minjoong Rim, Dongguk University Signals and Systems4
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,...
Minjoong Rim, Dongguk University Signals and Systems5
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
Minjoong Rim, Dongguk University Signals and Systems6
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
Minjoong Rim, Dongguk University Signals and Systems7
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
Minjoong Rim, Dongguk University Signals and Systems8
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
Minjoong Rim, Dongguk University Signals and Systems9
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)
Minjoong Rim, Dongguk University Signals and Systems10
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))
Minjoong Rim, Dongguk University Signals and Systems11
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))
Minjoong Rim, Dongguk University Signals and Systems12
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
Time Domain Frequency Domain
Minjoong Rim, Dongguk University Signals and Systems13
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
Time Domain Frequency Domain
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
Minjoong Rim, Dongguk University Signals and Systems14
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
Minjoong Rim, Dongguk University Signals and Systems15
Fourier Transform - 2
• 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
Minjoong Rim, Dongguk University Signals and Systems16
Fourier Transform - 3
• 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
Minjoong Rim, Dongguk University Signals and Systems17
Fourier Transform Properties - 1
• Duality
t f
)()( fxtX
ft
dtetxfX ftj 2)()(
dfefXtx ftj 2)()(
Note that the two equations are similar
Time Domain Frequency Domain
Minjoong Rim, Dongguk University Signals and Systems18
Fourier Transform Properties - 2
• Scale Change
)/(||
1)( afX
aatx
Wide Narrow
Narrow Wide Fast change in time-domain wideband signals
Time Domain Frequency Domain
t f
t f
Minjoong Rim, Dongguk University Signals and Systems19
Fourier Transform Properties - 3
• 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
Minjoong Rim, Dongguk University Signals and Systems20
Fourier Transform Properties - 4
• 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
Minjoong Rim, Dongguk University Signals and Systems21
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
Minjoong Rim, Dongguk University Signals and Systems22
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
Minjoong Rim, Dongguk University Signals and Systems23
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
Minjoong Rim, Dongguk University Signals and Systems24
• 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