通訊原理
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Communication principleTRANSCRIPT
Chap. 2 11
Signal Models
Deterministic signals vs. Random signals Deterministic Signals
Completely being specified by functions of time. Predictable
CHAP. 2 Signals & Linear Systems 0
, no ascertainty. e.g. cos ,
11, 2
0, otherwise. Random signals
take on random values at any given time instance mus
x t A t t
tt
t be modeled probabilistically.
Chap. 2 22
0
0
Periodic vs. Aperiodic Peridic signals: A signal is periodic iff. there
exist a fundamental period , such that , , and
Aperiodic signals: Sig
x t
Tx t nT x t t n
0 0 00 0
nals excluding the periodic ones.
e.g.
2 2 cos cos , x t A t A t T
Chap. 2 33
0 0
Phasor signals and their Spectra
A special peridic function
,~ complex sinusoidal fn.
rotating phasor;
0 : magnitude : phasor with
j t j tj
j
x t Ae Ae e
x t
AAe
0*
0 0
, Note: : phase
More on the phasor signal
1. Given a fixed frequency or , information can be contained in and . 2.The related real sinusoidal fu
j tx t Ae
fA
*0
1
nction:
Re 2
x t Acos t x t x t x t
Chap. 2 44
3. Graphical Representation: Two ways of relating a phasor signal to a real sinusoidal signal
a projection of a rotating phasor onto the real axis.
b Addition of complex conjugate rotating phasors. 0j tx t Ae 12
x t 12
x t
Chap. 2 55
0 4. Frequency domain representation: cos 2A f t single sided spectra double sided spectra
Chap. 2 66
0 0
0 0
0
0
Singular Functions.singularity; 0
Unit Impulse function , 0 ; otherwise
1. Defined by
0 0 ;
1.
2. Sh
tt
x t t dt x t t dt x t dt x
t dt
0 0
ifting property:
x t t t dt x t 0 t
t0 t
0t t 0t
Chap. 2 77
2
1
2
1
0 1 0 2
0 0 0
0
0
3. Other properties:
1 1
2 even function
, 3
0 , otherwise
4
5 1
t
n
t
t
t n n
at ta
t t
x t t t tx t t dt
x t t t x t t t
x t t t dt x t
t
0 1 0 2,
where n
n
t t t
dt tdt
Chap. 2 88
2 2
2
1, 0 Unit step function,
0, 0
.
Signal Classifications:
Energy: lim Jouels
1 Power: lim Watts2
o
,
:
r
T
t
TT
T
TT
tu t
t
E x t dt x t
du tu t d t
dt
dt
P x t dtT
x t
Energy signal: iff. 0 , 0
Power signal: iff. 0 ,
E P
P E
Chap. 2 99
0
1
2
0
Examples: , 0, Energy signal
, power signal
, power signal
Periodic signals, Power signals
cos , power signal
t
j tj
p
x t Ae u t
x t Au t
x t Ae e
x t A t
Chap. 2 1010
0 0
0
0 0 0 00
0
periodic sig
Fourier series Synthesis & Analysis
nal
1
Eular's Formula: cos sin1 Given a , let , 2 ,
, where
j
jn tn n
n
jn t
T
e j
x t x t T f fT
X xx t eT
e dtt X
0
0
.
Frequency components Double-sided amplitude and phase line spectra. ,~ a complex number
: D.C. component: Amplitude of :
:: phase of .
n
n
j Xn n
n nn
nn n
X X e
XX XX
XX X harmonics
Chap. 2 1111
0
0
2
0
Power Spectral Density : , Define the time-average :
1lim , if is aperiodic2 1 , if is periodic
Parseval's theorem:
n
T
TT
T
jn tn
n
X n
v t dt v tTv tv t dt v t
T
x t X e
0
2 2 2
0
Ex. Please prove the Parseva
1
l's Thm.
nTn
P x t x t dt XT
Chap. 2 1212
Chap. 2 1313
*
Symmetric properties of Fourier coefficients
1. If , then its magnitude spectrum is even symmetric
and its phase spectrum is odd symmetric.
is real
, , nn n
x t
X Xx t X X even
odd
0
2. is a real, and ,
3. is a real, and , is imaginary,
1 4. If , halfwave symmetry, eg. square wa2
n n
n n
n
n
n
n
x t x t x t X X
x t x t x t X X X
x t x
X
T
X
t
ve
0, 0, 2, 4,...nX n 0
2T 0T
A
A
Chap. 2 1414
0
2T 0T
A
A 0
00
, 02. The square wave ,
, 2
Find the Fourier series expansion of .
TA tEx x t
TA t T
x t
Chap. 2 1515
Chap. 2 1616
0
12
1
1 harmonic:j nf t
n
t
n
s
X e 0
32
3
Sum up to the 3 harmonic
j nf tn
n
rd
X e 0
72
7
Sum up to the 7 harmonic
j nf tn
n
th
X e 0
112
11
Sum up to the 11 harmonic
j nf t
th
nn
X e 02j nf tn
n
x t X e
Chap. 2 1717
21 2
The Fourier transform Analysis & Synthesis: ~ an function
~ Fourier transform
aperiodi
cj ft
j ft
x t
x t X f x t e dt
X f x t X f e df
F
F ~ Inverse Fourier transform
Amplitude and phase spectra
, 0,
If is real,
j fX f X f e X f f X f
X f X fx t
f f
Chap. 2 1818
2
2 2
Symmetry Properties
1 and , and
is imaginary 2 and ,
Energy Spectral Density:
Energy
Rayleigh's energy
x t x t x t X f X f X f
X fx t x t x t
X f X f
E x t dt X f df G f
G f
df
X f
1 2 1 2 1 2
thm. Parseval's thm for Fourier transforms Convolution:
x t x t x x t d x t x d
Chap. 2 1919
0
1 1 2 2 1 1 2 2
20
Transformation theorems
Superposition:
Time-delay:
1 Scale-change:
Duality: ,
Frequency
j ft
a x t a x t a X f a X f
x t t X f e
fx at Xa a
x t X f X t x f
F
F
F
F F 020Translation:
The sinc functionsin
sin: c
j f t
z
x t e X f f
zz
F
Chap. 2 2020
-1 0 1-1012
Time (ms)-5 0 5
0
1
2
Frequency (k Hz)
-1 0 1-1012
Time (ms)-5 0 5
0
1
2
Frequency (k Hz)
-1 0 1-1012
Time (ms)-5 0 5
0
1
2
Frequency (k Hz)
t 1X f
1 2
0.5 1x t
1 2tx 1 2x t
1 fx at Xa a
F
Chap. 2 2121
0 0 0
1 2 1
lim 0
2
1 1 Modulation: cos 22 2
Differentiation: 2
1 Integration: 02 2
Convolution:
Multiplic
t
nn
xn
t
t
x t f t X f f X f f
d x tj f X f
dt
X fx d X f
j f
x t x t X f X f
F
F
F
F 1 2 1 2ation: x t x t X f X fF Fourier transform pairs see Appendix G.5 P.724
Chap. 2 2222
1 2 1 2
. Prove the following 1 a. scaling theorem:
b. Convolution theorem:
Exfx at X
a a
x t x t X f X fF
Chap. 2 2323
Chap. 2 2424
22
2
Example: 40sinc 20 , ?
1, ;2
0, o.w.
sin sinc2
Duality: ,
40sinc 2
s c
0
n
i
j f j fj ft
x t t X f
tt
t e e fe dt fj f f
x t X f
t
t
f
F
F
F 2
2
2
220
Energy Spectra Density: 420
Energy: 1600sinc 20 4 80 .20
x t
f
fG f X f
fE t dt G f df df J
Chap. 2 2525
Fourier transforms of periodic signal
Strictly speaking, the Fourier transform of a periodic signal does not exist since a periodic signal is a power signal with infinit
e energy.
By taking advantage of the Fourier series expansion, and incorporating with the use of delta function, the Fourier transform of a periodic signal can be expressed as
0
0
0
0
2
20
20
below ,~ periodic
, Fourier series expansion
,
jn f
jn f tn
n
jn f tn n
n n
te f n
x t x t T
x t X e
X f X e X f nf
fF
F
Chap. 2 2626
2
20
§ Power Spectral Density and Correlation Power Spectral Density PSD :
,
is a real, even-symmetric and non-negative function of .
Given , j nfn
S f
P S f df x t
S f f
x t x t T x t X e 0
20
0
2 2
,
so that is satisfied.
Example:
cos( ) ,2 2
4 4
,
o o
t
n
j t j t
o o
nn
A Ax t
S f X f nf
A t e e
A AS f f f f f
Chap. 2 2727
1 1 *
*
2
Time-average autocorrelation function for
,
For a real , ,
i
l
0
m
T
TT
G f X f X f
X f xx x x
X f x
x
x t x t dt x t x t dt
energy signals
F F t dt E
Chap. 2 2828
00
2
Time-average autocorrelation function for
1
1lim2
If is periodic,
0 , ~ PS
Wiener-Khinchine theor
D
e
T
T
TTR x t x t
x t R x t x t dtT
R x t P S f df S
x t x t dtT
f
power signals ★ 2
1 2
m:
= ,
=
j f
j f
R S f
S f R R e d
R S f S f e df
F
F
F
Chap. 2 2929
2
2
0 0
Properties of :
1. 0 , .
2. . even symmetric to
3. lim , if contains no periodic components.
4.
5. 0, .
R
R x t R
R x t x t R
R x t x t
x t x t T R R T
S f R f
F
Chap. 2 3030
0
2 2
0 0
21
0
2
0
Example: cos , ?
Sol.
1. By the Wiener-Khinchine thm.
,4 4
cos2
2. By direct calculation, 1 cos
x t A t R
A AS f f f f f
AR S f
R AT
F 0
0 0
0 0 0
2
0 0 00
2
0
cos
cos cos 2 22
cos2
T
T T
t t dt
A dt t dtT
A
Chap. 2 3131
1 1 2 2 1 1 2 2
Linear-Time-Invariant Systems
1.
Linear and Time-Invariant LTI systems: Linear Systems:
Satisfy the superposition principle:
y t H x t
y t H x t x t H x t H x t § 1 1 2 2
0 0
Time-Invariant Systems:
Delayed inputs produce the delayed outputs
,
y t y t
y t H x t H x t t y t t
x t y t H x t H
Chap. 2 3232
0
0
LTI systems:
Impulse response:
lim
lim
n
n
h t H t
x t x t t x t d
x n t n
y t H x t H x n t n
time-invariant, ( )
0linear, = lim
=
n
h t n
x n H t n
x h t d x t h t
t h tLTI x t y t x t h t LTI
Chap. 2 3333
Convolution theorem : For a LTI system, *
, ~ Transfer function.
BIBO Stability :
BIBO : Bounded Input Bounded Output
A LTI sys
y t x t h t Y f X f H f
H f h t
F
F Bounded Input
finite
tem is BIBO ( ) < ,
h t dt
y t x h t d
x h t d x d h t d
Chap. 2 3434
0
Causality
1. A system is causal if current output does not depend on future input. 2. For a causal system, 0, for 0
h t t
y t h x t d
h x t d
Symmetry properties of
: amplitude-response 1. ,
: phase-response
2. is real,
j H f
H f
H fH f H f e
H f
H f H fh t
H f H f
t h t
t
t
x t
0 t h tLTI
Chap. 2 3535
. 1. Find the transfer function of the following RC circuit.
12 2. Let the input ( ) , ?
Ex H f
t Tx t A y t
T
Chap. 2 3636
R
C x t y t
i t
Chap. 2 3737
Chap. 2 3838
0 T A x
0 t T 1 u t
0t 1 u t t T 1 u t
Chap. 2 3939
2
* *
2 2
Input-Output Relations for Spectra
1. ,
y
Y f H f X f
G f Y f Y f Y f
H f X f H f X f
H f X f
2
2
2. A similar relationship hold for power signal and spectra :
x
y x
H f G f
S f H f S f
Chap. 2 4040
0
0
0
0 0
2
2
20
20
Reponse to Periodic Inputs
1. For a complex exponential sinusoidal input =
. . is a scaled and
j f t
j f tss
j f t
j f t H f
ss
x t Ae
y t h Ae d
H f Ae
H f Ae
i e y t
0
0
0
0
2
2
2
amplitude scaling
20 0
phase-shifted copy of the input .
2. For a period
eigen-signal of LTI systems
ic input
.
,
n
j f tn
n
j nf t X H nj f tn n
n
j f t
x t
x t X e
y t X H nf e
Ae
X H nf e 0
phaseshift
f
n
H f
02= j f t
x t
Ae 0
ssy t
H f x t
Chap. 2 4141
/10
0
: Consider a filter having the transfer functon
2 ,42
what is its reponse to a unit-amplitude triangular signal with period 0.1 sec. and amplitude 1.
j ffH f e
TA
Example
2 2
the Fourier coefficients of the 4 ,
triangular signal . 0,
n
A n oddX n
n even 2121 2
f
H f
2121 H f
f
A0
2T0
2T x t
tA
Chap. 2 4242
2121 2
f
H f
2121 H f
f
Chap. 2 4343
0
020
20 0 0
1. Distionless system:
( )
1. The amplitude response has to be a constant for all frequecies.
j
j
f
ft
ty t H x t t Y f H X f e H f X f
H f H e
Distortionless Transmission
F 2. The phase response is a linear function of , for all frequencies. f
f
H f
0H
f
02H f t f 02slope t
Chap. 2 4444
2. Three types of distortions in communication channel:
1. Amplitude distortion: Linear system, but the amplitude
response is not a constant .
2. Phase delay distortion : Linear system, but the phase
shift is not a linear function of frequency.
3. Nonlinear distortion : nonlinear system
Chap. 2 4545
1
2
3
4
1. cos 10 cos 12
2. cos 10 cos 26
3. cos 26 cos 34
4. cos 32 cos 34
x t t t
x t t t
x t t t
x t t t
Δ Example H f iy t ix t
( )H f H f f ( )f Hz ( )f Hz
1515 00
2
2
21
201020 10( )
20 40f fH f , 15
30
, 152
, 15 2
f f
fH f
f
Chap. 2 4646
1
1
2
1 1 1. 2cos 10 2cos 126 5
1 1 1 2cos 10 2cos 12 2 , distortionless60 60 60
1 13 2. 2cos 10 cos 266 30
y t t t
t t x t
y t t t
3
1 1 2cos 10 cos 26 , amplitude distortion60 60
13 3. cos 26 cos 3430 2
1 1 cos 26 cos 3460 68
t t
y t t t
t t
4
, phase distortion
1 1 4. cos 32 cos 342 2
1 1 cos 32 cos 34 , phase distortion64 68
y t t t
t t
Chap. 2 4747
21 2
1 1 2 2
- Consider a nonlinear system with the input-output relation If the input signal is cos cos then the resultant output
y t a x t a
x
x
t A t A t
t Non linear Distortion 1 2
1 1 1 2 2
2 2 2 22 1 2 2 1 2
2 1 2 1 2 1 2
cos cos
1 1 cos 2 cos 22 2
cos cos
The system has produced frequencies in the output
y t a A t A t
a A A a A t A t
a A A t t
other than the original ones.
Chap. 2 4848
0
0
20
0 0
20
1. Ideal Low-pass Filter:
2
2 sinc 2 ,
non-causal
2. Ideal High-pass Filter
12
j ftLP
LP
j ftHP
fH f H eB
h t BH B t t
fH f H eB
• Filters 0 0 0 2 sinc 2HPh t H t t B B t t f
fBB LPH f LPH ff
fBB HPH f HPH f
Chap. 2 4949
021 0 1 0
1 0
0 0 0 0
3. Ideal Band-pass Filter
where
2 sinc cos 2
j ftBP
BP
H f H f f H f f e
fH f HB
h t H B B t t f t t
f
fBB
BPH f BPH f0f 0f
Chap. 2 5050
Chap. 2 5151
Chap. 2 5252
SAMPLING THEORY 1. Ideal instantaneous sampling waveform
s s sn n
s s s sn n
s t t nT S f f f nf
x t x t s t x t t nT x nT t nT
F x t sx t s
n
s t t nT LPH f 1 2LP
s
fH ff B
, if
R
s
x t x tW B f W
x t
t sx t
tsT
Chap. 2 5353
,
Uniform sampling theorem for Low-pass signals
A low-pass signal with finite bandwidth can be co
mple
tel
s
s
s sn
s sn
x t x t s t
X f X f S f
X f f f nf
x
f X f n
t W
f ★ y
1 described by , if .2
The original signal can be exactly reconstructed by passing through an ideal lowpass filter with bandwidth satisf
1 2 Nyquist sampling frequency
yi
s s
s
ss
x t TW
x tB
f WT
ng
sW B f W
Chap. 2 5454
BB s sX f x tWW (0)sf X
sf W sfsf Wsf W
sf W sf X f x tWW 0X
f LPH f1sf R RX f x t
WW 0X
f
f
f
( )x t( )sx t s
n
s t t nT LPH f 12LP
s
fH ff B
, if
R
s
x t x tW B f W
,
2s
s
W B f Wf W
Chap. 2 5555
Aliasing: If is not band-limited on 2 , distortion aliasing of overlapped spectra is inevitable.
sx t f W 1 Increase the sampling freq.
2 Pre-filter the signal with a low-pass filter anti-aliasing filter
solution :solution :
sf
0 fsf s s sn
X f f X f nf
Chap. 2 5656
1, 2 2
0, o.w.
1 , . Please verify 0, o.w.
1
tt
tt tEx
t t
Chap. 2 5757
t 2
21
1
2t
2t
2 2
2
t 2
t t
Chap. 2 5858
2 2
2
t 2
t t 2
2
2t 2
t t
Chap. 2 5959
t t
2
2
2t
2t t
2
sinc
sinc
t f
t f
F
F
Chap. 2 6060
Property:
0, only in the interval , ,
0, only in the interval ,
0, only in ,
x t t a b
y t t c d
x t y t t a c b d
Chap. 3 11
CHAPTER 3 BASIC MODULATION TECHNIQUES Analog modulation Continuous-wave CW modulation: Linear: AM Exponential angle : FM, PM
Pulse modulation sampled data 3.1 Linear Modulation
G
§
eneral form: cos
: one-to-one correspondence with message : fixed cos : carrier
c c
c
c
x t A t t
A t
t
Chap. 3 22
m t cx t rx t d t
cosc cA t 2cos ct Low pass filter
Dy t
Modulator
Demodulator
Double Side-band Modulation Suppressed Carrier : DSB or DSBSC
1.
cos
, 2 2 2
c c c
c c cc c c c
x t A m t t
A AX f M f f M f f f
F
M f
W Wf cM f f cM f f cX f
cf
0
fcf 0
Chap. 3 33
2
the phase and the fre2 Coherent synchronous Demodulator detector :
The receiver exactly knows of the
carrier signal.
2 cos cos
1 cos 2
quenc
2
y
cos 2
c c c
cc c c
d t A m t t t
A m t t A m t
同調 2 = cos 2
message pass the LPF; high-freq. term rejected
. : Upper sideband and lower sideband:
c c c
A B
t
A m t A
A
m t t
Def
B
0
cX f
cff
cf cf Wcf W: upper sideband
: lower sideband
rx t d t
2cos ct Low pass filter
Dy t
Demodulator
Chap. 3 44
M f
W Wf cM f f cM f f cX f
cf D f
2 cf2 cf 0
f
f
cf 0
0
m t cx t rx t d t
cosc cA t 2cos ct Low pass filter
Dy t
Modulator
Demodulator
Chap. 3 55
3 What if the receiver reference is ? A phase error occurs:
cos 2cos
cos + c
non-coherent
os 2
cos , 1 cos 1
is mult
c c c
c c c
D
d t A m t t t t
A m t t A m t t t
y t m t t t
m t
iplicatively distorted by a time-varying
factor cos .t rx t d t Low pass filter 2cos ct t Dy t
Chap. 3 66
2 2 2 2
2 2 2 2
4 Carrier recovery: regenerate the carrier from the receive signal at receiver cos
1 1 cos 2 2 2
r c c
c c c
x t A m t t
A m t A m t t
rx t cos ct cos 2 ct 2 Narrow-band
BPF at 2 cf2f H f
2 cf2 cf
Chap. 3 77
Amplitude Modulation AM :
Also known as DSB with carrier, DSB-Large Carrier (DSB-LC) 1
,~ original message,
,~
co
normalized messmin
s 1 cos ;
c c n c c c
n
x t A m t
m t
m tm t
m t
A t am t A t A AA age;
. min 1, if min 0
min,~ modulation index
nEx m t m t
m ta
A
cx t m t A m tcos cA t
,~ D.C. biasA 1, min ;
over-modulation.
a m t A
Chap. 3 88
2 AM waveforms and spectra
1 cosc n c cx t am t A t M f
f0
cfcf cX f
f0
2 2c c
AA Af f M f f cx t m t A m tcos cA t
,~ D.C. biasA
Chap. 3 99
3 AM demodulation a. Coherent detection: precise but requires carrier recovery circuit very complex . b. Incoherent detection: envelope detection. Simple receiver LPF , but requires sufficient carrier power and
1 . Envelope detection
ca f W
Chap. 3 1010
Modulated carrier and envelope detector outputs for various values of the modulation index
0.5a 1.0a 1.5a
Chap. 3 1111
22 2 2
4 Efficiencypower of the information-bearing signal Efficiency
total power of the transmitted signal For AM: Total power:
1 cos
=
ff
c n c c
E
P x t am t A t 22 2
2 2
22 2
2
1 2 1 cos 22
, 1 2Assume
Assume 02
,
, 1
2c
cn n
cn
nn
c
nc
Aam t
f W
m AP a m
a m t t
AP a m t a m t
t t
Chap. 3 1212
22 2
22 2
max
2 m x
2
a
2
2 22 100% =
12
If max min and 1, then 50%.
Ex. square-wave, 1, = 50%, for 1
If is sinus
10
oi
0%1
d
cn
ffc
n
ff
n ff
n
n
Aa m tE
Aa m t
m t m t a E
m t m t E a
a m t
a t
m
m
t
2 max
2 2 2 2
2 2 2
22
1al, , 33.3%, for 1.2
For DSB-SC: cos
1 1 cos 2 , ,2
, 100% 2
n ff
c c c
c c c c
cff
m t E a
x t A m t t
x t A m t t assume f W
A m t E
Chap. 3 1313
Example:
Modulation index 0.5 Carrier power = 50 Watt.
4cos 2 2sin 49
Find 1. The efficiency =?, given that min 4.364 occurs
at 0.435 by numeric
m m
ff t
m
a
m t f t f t
E m t
f t
al method .
2. The output spectrum for the AM modulation.
Chap. 3 1414
Chap. 3 1515
Chap. 3 1616
Chap. 3 1717
Single-Sideband Modulation SSB
1. For DSB modulation, two sidebands contain the same information. Therefore, it is a good idea to convey message by only a single LSB or USB
sideband for saving power and the transmission bandwidth. 2. Generation of SSB signals
Method 1: sideband filteringcosc cA t Sideband
filter c SSBx t x t DSBx t m t a
0
DSBX f
cff
cf USB
LSB2W2W
cM f f cM f f M f
f02W
Chap. 3 1818
0
0
f
fcf
cf sgn cf f sgn cf f 1
1 cf cf
12
f
1
1
1 M f
f
DSBX f cX f
LH f
W W
f
f
cM f f cM f f 1, 0 For a lower-sideband SSB signal, define sgn
1, 01 Sideband filter: sgn sgn2
1 1 2 2
L c c
DSB c c c c
ff
f
H f f f f f
X f A M f f A M f f
Chap. 3 1919
Lower-sideband SSB signal
1 sgn sgn4
sgn sgn
c
c
c L DSB
c c c c c
c c
M f f
M f
c
f
c
X f H f X f
A M f f f f M f f f f
M f f f f M f f f f
.
b.
1 = +41 sgn sgn4
?
c c c
c c c c c
c c
a
A M f f M f f
A M f f f f M f f f f
X f x t
F
0 fcf sgn cf f 1
1 cM f fcf
Chap. 3 2020
Method 2: Phase-shift modulator: Generate directly from its time-domain representation.
1 . 4ˆ ˆ . sgn , sgn
1ˆ , Hilber
cos 22
t
c
c
cc c c
x t
a A M f f M f f
b m t j f M f jm t f M f
m t m
A m
tt
t f t
F
F F 2 2
transform of
1 sgn sgn4
1 ˆ ˆ 41 ˆ 2 sin 24
1 ˆ sin 22
c c
c c c c c
j f t j f tc
c c c cA m t f t
m t
A M f f f f M f f f f
A jm t e jm t e
A jm t j f t
-1F
Chap. 3 2121
Similarly , for a upper-sideband SSB USB/SSB
ˆ LSB/SSB: c
ˆUSB/SSB
os 2 sin 2
: cos 2 sin 22
:
2
cc c c
cc c c
Ax t m
Ax t m t f t m t f
t f t m t
t
t f
Chap. 3 2222
0
2 2 2
2 2 20
Hilbert transform:
1, 0 , lim sgn
1, 01 1 4 2 2 4
4 sgn lim4
Duality theorem of Fourie
1ˆ ˆs
r
gn
t t tx t e u t e u t x t t
tj fX f
M f j f M f m t m
j f j f fj f jt
f f
tt
-1F
F Transform: sgn ,
1 sgn sgn , sgn
jtf
j f f j ft t
F
F F
tt
x t1
10 sgn t
Chap. 3 2323
1ˆ The Hilbert transform of :
1ˆ ˆ sgn
Ex. cos 21 1 2 2
1 1ˆ sgn2 2
1 2
ˆ 2
o
o o
o o
o o
j
x t x t x tt
X f x t x t j f X ft
x t f t
X f f f f f
X f j f f f f f
j f f j f f
jx t e
F F F 2 2 sin 2 o of t j f toe f t f
0f0f sgn f
Chap. 3 2424
ˆEx. sin 2 , ?ox t f t x t
Chap. 3 2525
Chap. 3 2626
2 2 2
2
ˆ
Properties of Hilbert Transformˆ 1. Energy or power of Energy or power of
Pf.
ˆ sgn
ˆ
ˆr 2. If is , teal n andhe
XX
x t x t
X f j f X f X f
E X f d E
x t x t
f , i.e.,
ˆ 0, for energy signals , or 1 ˆlim 0, for power signals
2 P
are orthogonal
lease prove it.
T
TT
x t x t dt
x t x t
x t
T
Chap. 3 2727
Chap. 3 2828
1 1
lowpass highpassnon-overlapping sp
3. Let and being a and a signal. If the two signals have , then
ˆ
pf.
ectra
lm t c t
m t c t m t c t
m t c t M f C f
M
F F 0 02 20
2
2
2
: sgn
, please verify i
sgn
tj f t j f t
j f f t
j f f t
j f f t
e j f e
f C f e dfdf
m t c t M f C f e dfdf
M f C f j f f e dfdf
Chap. 3 2929
region or
0, for lowpass Non-overlapping spectral: assume
0, for highpass
is nonzero only for and .
M f f W
C f f W
M f C f f W f W
A B
A B
2( ) ( ) sgn j f f tm t c t M f C f j f f e dfdf f WW
WW f
B
AW
W
WWf
f region of 'f f A
region of 'f f Bsgn( ) 1f f sgn( ) 1f f
Chap. 3 3030
2
2 2
By observing the function behavior in region and ,, .
sgn
sgn
in this case sgn sgnj f f t
j ft j f t
m t c t M f C f j f e dfdf
M f e df j f C
f f f
f e df
A B
A B 0 0
0 0
0
Example For a lowpass signal with 0 for , with , we conclude
ˆ
cos 2 sin 2
sin 2 cos 2
m t c t
m t f t m t f t
m t f t m t f
m t M f Wf
t
fW
Chap. 3 3131
*
ˆ Analytic signals: , is real
ˆ
ˆ ,1 Re2
sgn sgn
2 , 0
0 , 0ˆ
p
n
p
p p p
p
p
n
x t x t jx tx t
x t x t jx t
x t x t jx t
x t x t x t x t
X f X f j j f X f X f f X f
X f fX f
f
x t x t j
,
0 , 0 1 sgn
2 , 0n
x t
fX f X f f
X f f
Chap. 3 3232
0
0 0 0 0
21 0 1 0
1 0
0 0 0 0 0
Example Ideal bandpass filter with bandwidth ,
2 sinc cos
,
where
, 2 sinc sin
BP
j ftBP
BP
P
B
h t H B B t t t t
H f H f f H f f e
fH f HB
B f h t H B B t t t t
x t h
2 20 0
ˆ 2 sinc .
BP BP
P BP BP
t jh t
x t h t h t H B B t t
Chap. 3 3333
f0 0f0f0 0f
f0
B
B
X f pX f X f B
A
2A
2A
f
complex baseband
Complex Envelope Representation of Bandpass Signal For a bandpass signal x t X f ˆ
sgn
p
p
x t x t jx t
X f X f f X f
02
0
,~ complex baseband
j f tp
p
x t x t e
X f X f f
Chap. 3 3434
0
0
2
2
real
Comple
bandpass
x envelope of :
,
Given the signal , the
signal can be expressed as
Re Re
complex base
Re
b
cos
and
2
j f tP
R
I
I
j f tP R
x t
x t x t e x t j
x t
x t
x t x t x t e
x t jx t
x
f
t 0 0
0 0
sin 2
,~ the band-pass signal real
Bandpass Systems: Consider a bandpass system with impulse respose . T
cos 2 s
he impulse response can be repr
in
en
2
es
R Ix t f t
t
x t f
j f t
t
h t
02
ted in
terms of a complex envelop as
Re ,
where
j f t
R I
h t
h t h t e
h t h t jh t
h t y t x t H f
f0 0f0ff0
B
B
X f X f A
2Acomplex baseband
Chap. 3 3535
0
0
0
0
2
2
2
2 *
The response of the system to a bandpass signal is
Re
Re
Re
1 2
j f t
j f t
j f t
j f t
x t
Y f X f H f
x t x t e
h t h t e
H f h t h t e
h t e h t e
F F
F 02
*0 0
*0 0
1 2
also1 2
j f t
H f f H f f
X f X f f X f f
h t y t x t
Chap. 3 3636
0 0 0 0
0 0
non-overlapping spectra
non-overlapping sp
0
e
0ctra
1 4
Y f H f X f
X f f H f f X f f H f f
X f f H f f
0 0
0 0 0 01 4
X f f H f f
X f f H f f X f f H f f
f0f0f 0H f f 0X f f
Chap. 3 3737
0
0
0
0 0
*2
2
2
1 ,4
1
Define
1 R
4
e2
j f
j t
t
f t j f
Y f H f X f h t x t
y
Y f Y f f
t e
Y f f
y t y t e y t e Ff0 0f0ff0
X f X f A
2Acomplex baseband
f0 0f0ff0
H f H fcomplex baseband
Chap. 3 3838
0
0
cos 2
cos 2
?
t
tx t f t
h t e u t f t
y t
Example x t y t h t
0
0
0
2
2
2
12
Re Re
Re
j f t
j f t
j f t
y t x t h tx t x t ey t y t e
h t h t e
Chap. 3 3939
y t
y tt
t
t1 e 22
2
Chap. 3 4040
4. Demodulation of SSB signals Coherent detection with possibly phase error
+ : LSB-SSB1 1 ˆ cos sin ,: USB-SSB2 2c c c c cx t A m t t A m t t
LPF cx t Dy t d t cos , phase error
let 4 cK t t t
K
Chap. 3 4141
1 1 ˆcos sin 4cos2 2
2 cos cos
ˆ 2 sin cos
cos cos 2
c c c c c
c c c
c c c
c c c
d t A m t t A m t t t t
A m t t t t
A m t t t t
A m t t A m t t t
message crosstalk
ˆ ˆ sin 2 sin
After lowpass filtering and amplitude scalingˆ cos sin
If 0, .
c c c
D
D
A m t t t A m t t
y t m t t m t t
t y t m t
LPF cx t Dy t d t 4cos ct t
Chap. 3 4242
2 2
1
cos
1 1 ˆ cos sin2 2
What is the output of envelope detector ?
cos sin
Let tan
r c
c c c c
c c
b ta t
e t x t K t
A m t K t A m t t
e t a t t b t t
R t a t b t
bt
Carrier re-insertion : cos, ;
sin
a t R t tt b t R t t
a t
EnvelopeDetector
rx t Dy t e t cos cK t in pahsedirect
quadrature-axis R t t a t
b t
Chap. 3 4343
envelope
2 2
cos cos sin sin
cos
Assumming that the bandwidth of is much smaller than
1 1 2 2
c c
c
c
D c c
e t R t t t R t t t
R t t t
m t f
y t R t A m t K A m t
2
2 2
1 ˆ1 21 122
1 1 If is chosen large enough such that , 2 2
1 1 then , providing min2 2
c
c
c
c c
D c c
A m tA m t K
A m t K
K A m t K A m t
y t A m t K K A m t
Chap. 3 4444
1 1 1
1 1 1
2 2 1
Example
cos 0.4cos 2 0.9cos 3
ˆ sin 0.4sin 2 0.9sin 3 For SSB:
ˆ cos sin cos2
where
ˆˆ , and tan
2
cc c c c
c
m t t t t
m t t t t
Ax t m t t m t t R t t t
m tAR t m t m t tm t
Chap. 3 4545
1 Let 1Hz, 50Hz, 2. The time-domain waveform of ,
ˆ , , USB in d and LSB in e are shown below:c c
c
f f A m t
m t R t x t
Chap. 3 4646
Problems of SSB: 1 Need nearly perfect sideband filter with sharp cut-off frequency response.
2 Realistically, to avoid baseband interferen
Vestigial - Sideband Modulation
殘邊帶調變 ce due to the transition region at the cut off frequency of the sideband filter, the message signal is prohibited to have low frequency components. 3 Cannot transmit carrier to reduce the complexity . Allowing a vestige of the unwanted sideband to appear at the output of the modulator. VSB
Modification
H f
f
Chap. 3 4747
Idea:
DSB signaling
fcf cf0
cM f f cM f f H ff
cM f f H f cM f f H f cX f
0 f
demodulation cos ct rx t d tLPF
Dy t DY f
f DY f Assume an ideal channel:
r c c cX f X f M f f M f f H f cfcf f0
H f
Chap. 3 4848
D
]
c c c c
c c c c
c c
c c
D f d t
M f f M f H f f M f M f f H f f
M f H f f H f f M f f H f f
M f f H f f
Y f M f H f f H f f
1 1
2 22 2
1 12
2 2
12
2
1
2
F D ,
if constant at
the vestige band .c c
y t m t
H f f H f f
f cf2
cf2
cH f f cH f f 0 f
f0
Chap. 3 4949
D
Solution Let be an lowpass anti-symmetric filter, i.e.,
and for .
, Set ,
,
,
c c
c c
U f H f U f H f
c c
H f
H f f
U f f H f f for fH f
U f f H f f for f
H f
H f H f
f H f f
Y f
0
0
0
1 M f1
212
12
H ff cfcf cf cf 1
12
f0
H f
Chap. 3 5050
Example: cos cos
cos
cos
cos
cos
cos
DSB c
c
c
c
c
m t A t B t
x t m t t
A t
A t
B t
B t
1 2
1
1
2
2
1
2
1
2
1
2
1
2
Chap. 3 5151
The VSB signal Tx. :
cos cos
cos
The demoulated signal Rx. : , ?
cos
cos
cos co
s
c DSB
c c c
c
D
c c
c
x t X f H f
x t A t A t
B t
d t y t
d t x
t
t t
A tA A t
1 1
2
11 1
1 11
2 2
4
2 1
1
2 cos coscos
cos cos co
.
sc c
D
B t
A t A t B t
A t B t
y t m t
1 22
1 1 21
1 2 2 cos ct4
cx t d t LPF Dy t
Chap. 3 5252
cosdx t
m t t 1
e t ( )cosx t m t t 2
Local Oscillator
BPF cos[ ]t 1 22
2Center Freq. : Mixer The carrier freq. of the band-pass signal is now translated to .dx t f
2
cos cos , +: high-side t
uning
: low-sid
cos cos
e tuning
dx t
e t m t t t
m t t m t t
1 1 2
2 1 2
2
2
Frequency Translation and Mixing
Chap. 3 5353
Problem: signals at , for example
cos , will appear at the output of BPF.
cos cos
cos cos
. ., When pa
imagx t
k t t
k t t t
k t t k t t
image frequenc s
e
e
i
i 1 2 1
2
1
1
2
2 2
2 2
3
2
2 imag
ssing through the mixer, the desired signals
and the image signal will be coupled to each other at mixer output.
dx t
x t
Chap. 3 5454
IF 2
IF 2
C 1
Illustration of the image signals high-side tuning
LO 1 2 1 22
c IF 1 22 2 1 22 3
Desired signal
Localoscillator
Imagesignal
Imagesignal atmixer output
Signal atmixer output
Passbandof the IF filter
f
f
f
f
f
dX f E f e t imagX f
Chap. 3 5555
Super-heterodyne Receiver A super-heterodyne receiver has two "amplification-and- filtering" sections prior to demodulation.
OutputDemo-
dulator
Localoscillator
R.F. filter andAmplifier
I.F. filter andAmplifier
Mixer
near-by imagdx t x t x t
Chap. 3 5656
near-by imag
RF amplifier generally has limited amplification gain. Therefore, receive signal amplification is mainly done at the IF-amplifier.
At the antenna:
: td
d
x t x t x t
x t
imag
near-by
near-by
he desired signal at
: image signal at
: other unwanted signals in the channel near by .
The carrier frequency of does not confuse the IF m
c
c IF
c
x t
x t
x t
2 near-by
near-by
ixer.
After the RF amplifier: .
The RF filter is a wideband tunable filter. Although
can pass the RF-filter also, they will be removed by the succ
dx t x t
x t
eeding IF mixer
After the mixer: dx t
Chap. 3 5757
IF
2
c 1
c IF 2
Received signal
RF-filteroutput
Signal atIF filter output
f
dX f
Illustration of the superheterodyne receiver high-side tuning
i c IF 2
imagX f near-byX f
c 1
f
f IF IF
IF
IF
is . 1 AM I.F.: 455KHz; IF-BW ; RF: 540K~1.6MHz
2 FM I.F.: 10.7MHz; IF-BW ; RF: 88~108MHz High-side tuning
Low-side tuning
fixed
c
c
LO c IF
LO c IF
ff KHz f
f KHz f
2
10
200
Chap. 3 5858
LORelation between and in terms of :c if f f
IFf2
LOf
f
i LO IFf f f i LO IFf f f Desire signal
1 c
2 IF
Desire signal: , d
f fX f
f f
Image signal
Image signal: imagX f
c LO IFf f f c LO IFf f f a low-side tuning b high-side tuning
IFf2
f
LOf
Chap. 3 5959
Which one is better? High-side tuning vs. Low-side tuning For AM receiver, RF frequency range: 540 KHz 1600 KHz
I.F. 455 KHz ~ fixed
1 Low-side tuning: Local oscillator freque
L F
c
O c If f f
f ncy range: KHz KHz
KHz KHz min : max : .
2 High-side tuning: Local oscillator frequency range: KHz KHz
LO
LO L
LO
LO
O
c
O
IF
L
f
f
f
f
f f
f
f
540 455 1600 455
85 1145 1 13 47
540 455 1600 455 KHz KHz min : max : . Smaller local oscillator frequency range is preferred easy to implemen High-side tuning t , is better
LO LO LOf f f 995 2055 1 2 07
Chap. 3 6060
§ Angle Modulation
General form: cos
Instantaneous phase:
Instantaneous frequency:
Phase deviation:
Frequency deviation:
i
c c c
i c
ic
t
i
x t A t t
t t t
d t d tt
dt dtt
d
Phase modulation PM
; Phase deviation message.
: deviation constant, : message,
cos
p
c c c p
p
t k m t
x t A t
tdt
k
m
t
k t
m
Chap. 3 6161
0
0
0
0
0
Frequency modulation (FM)
; Frequency deviation message
2
= : Frequency deviation constant2
EX.
1
c
F r
os 2
o
t t
f
t
c c c
f dt t
fd
d t
d tk m t
dt
x t A t
t k m d f m d
kf
f m d
, , ;2
PM: cos2
FM: cos
a phase discontinuity occurs at 0 for the PM signal
p f d
c c c
c c c d
m t u t k k
x t A t u t
x t A u t t
t
Chap. 3 6262
PM: cos cos2 For cos , ;
FM: cos sin
c c c p m
m fc c c m
m
x t A t k tm t t k
x t A t t
Chap. 3 6363
2
carrier s
Narrowband Angle Modulation
cos
Re Re 12!
If ( ) 1, then
Re cos sin
c c
c
c
c
c c c
j tj t j tc c
j t j tc c c c c c c
m t A
x t A t t
tA e e A e j t
t
x t A e A t je A t A t t
( ) 1
( ) 1
in
narrowband angle modulation AM-like modulation
BW of , BW of 2W.
c
t
t c
t
t W x t
Chap. 3 6464
0 0
. A FM system with cos , 2
Let 0, cos sin
cos sin
m f d
td
f m mm
dc
t
c c mm
Ex m t A t k fAft t k A d tf
Afx t A t tf narFIG rowURE 3. band a24 Generatio ngle modulatn of ion: m t
dtdf2
pk
FM
PM
tsin ct
cos ct cA cx t
carrieroscillator
o90 phase shifter
Chap. 3 6565
2 2 2
cos sin sin
cos cos cos2
1, si
Re 12
n 1,
c m m
dc c c c m c
m
c dc c c m c m
m
j f t j f t j f td
d dm
m
m
c
m
Afx t A t A t tf
A
Af Aft
AfA t t tf
AfA e e ef
tf f NB-FM
NB-FM
AM
2
Compare to the AM signal with modulation index ,
1 cos 2 cos 2
cos 2 cos 2 cos 22
Re 12
c
c
c
c c m c
cc c c m c m
j f t jc
x t
x t a
x t A a f t f t
AA f t a f f t f f t
aA e e 2 2m mf t j f te
Chap. 3 6666
Spectrum of an Angle-Modulated Signal
wideband spectrum of single-tone message
Assume a single-tone message,
sin
: modulation index controls the maximum phase deviation
mt t si
in
n
s cos sin Re
Since is a with period , it can be
expressed by its Fourier series representation with coefficients
periodic function
mc
m
j tj tc
j
c
t
c
m
c mx t A t t A e e
e
2 / / sinsin
/
si
/
n
Let ,
m m m mm m
m m
j t n tj t jn tm mn
mj x nx
n n
C e e dt e dt
x Jt C e dx
2 2
1
2
Chap. 3 6767
sin
is the Bessel-function of the first-kind of orde
c
r and argument .
,
Re ,
os ,
m m
c m
n
j t jn tn n n
c n c
n
j t jn t
n
cn
m
c n
J n
C J e J
A J n
e
x t A e J e
t
An angle-modulated signal of a single-tone message has infinite number of line spectra, each spaced with , centered at the carrier frequency .
n
m
c
J
ff
Chap. 3 6868
Amplitude
cA phase rad
f
J 4
J 3
J 2
J 1 J 0
J 1 J 2 J 3 J 4
cm
ff4
cm
ff2 cf
cm
ff2
cm
ff4
cm
ff3
cm
ff1 cf cm
ff
cm
ff3 single-sided amplitude spectrum
single-sided phase spectrum
f
Chap. 3 6969
sin
Properties of :
1 is real-valued.
2
3
4 Recursive relation:
Given and , we can re
j x nxn
n
nn n
nn n
n n n
J e dx
J
J J
J J
nJ J J
J J
1 1
0 1
1
2
1
1
2 cursively find
all the for .nJ n 2
Chap. 3 7070
5 When is small eg., narrow-band FM, PM ,
!
, , for .
6 When is real and fixed, lim
7 , for all
Ex. Prove 1 and 7
n
n n
n
nn
nn
Jn
J J J n
J
J
0 1
2
2
1 0 12
0
1
Chap. 3 7171
Chap. 3 7272
, , , the narrow band angle-modulated signals:
cos
cos cos
n
c c n c mn
c c c c m
J J J n
x t A J n t
A J t A J t
0 1
0 1
1 2
Chap. 3 7373
Chap. 3 7474
Carrier nulls: The values for which
, . n
n
J
n J J
0
0
0 0 0 0 1
Chap. 3 7575
The above analysis is based on the single-tone modulation case in which sin , For PM, this means sin
sin sin
= or
Fo
m
m
p p m m
p p
t t
m t A t
t k m t k A t t
k A kA
r FM, this means cos
sin sin
m
td
d m mm
m t A tf At f m d t tf
2
Chap. 3 7676
= : given a fixed , is reverse propotional to
; The amplitude spectrum of a FM signal increase with a decreasing .
dd m
m
m m
m
f A f ff
f f
f
f
f
f
f
f
cf
. 1 0
. 0 5
. 2 0
. 5 0
. 10 0
mf
mf
mf
Chap. 3 7777
Power in an Angle-Modulated signal
single-tone case: cos
cos cos
c c n c mn
c c n k c
k n
m c mn k
c nn
x t A J n t
x t A J J n t k t
A J
2 2
1
2
1
2
2
1
2
1
2 frequencydeviation
.
general cases: cos ,
cos
, cos , .
Power of the angle modulated signal is constant
c c c
c
c
cc
c c
c c
x t A
A
t tAx t t t
d t t t xd
tt
A
2
2
22
2
1 2 2
02
2
2 2 . indep. of t
Chap. 3 7878
Bandwith of angle-modulated signals
. Single-tone t :
series expansion of : see appendix G3
, !
for moderate .
For
n
n
n n
m
J
Jn n n n
2 4
2 4
1
12 2 1 2 2 1 2 fixed large , , lim
! Strictly speaking, the bandwidth of an angle-modulated signal is infinitely large. However, the bandwidth can be approxiamtely specified
n
n nn nn J J
n 0
2
by the frequency band of the major power distribution.
Chap. 3 7979
20
single-to
Power ratio:1
component power 2 = 1total power2
for a angle modulated signal,
. single-underscored terms i
ne
n Table 3.2 :
c n k
r nn
n k
m
r
k
c
BW
A JkP J J
k
A
P
f
k
2 2
2
2 1
2
2
0 7 . double-underscored terms in Table 3.2 For 0.989, by observing Table 3.2,
integer part of +1 , i.e., +1 ,
+1 m
r
r
PP
k k
BW f 2
0 98
Chap. 3 8080
FM case
.) Arbitrary :
Define: Frequency deviation ratio
maxpeak frequency deviation max
bandwidth of
,
1
Carson's rule
,
td
m t
d tdt fD m t
m t W W
D
B W
W
D
B
W
2
1
2
1
2 1
2 narrow-band angle-modulated signal
2 , max
wide-band angle-modulated signal
d
W
D BW DW f m t
1 2 2
Chap. 3 8181
Ex. Consider a FM modulation system below with
cos
, cos , ,
BPF:
please determine the p
cc
c
d m
BP
Ax t t t
f
f m t t A f
fH f
100100 2 1000
1000
8 5 16 5 8
1000
56
ower at the filter output. m t cx t output
outx t FMModulator
BPH f
,c df f 1000 8
Chap. 3 8282
Chap. 3 8383
f
cX f
13.126.1
36.5
4.7
32.817.8
32.8
4.7
36.539.1
26.113.1
952 96
096
897
698
499
210
0010
0810
1610
2410
3210
4010
48
1000972
1
1028 f
39.1 BPH f
c nA J
Chap. 3 8484
sin sin
Ex. Given cos cos ,
sin sin , For FM mod.,
,
cos sin sin
Re
c
d d
c c c
j t j tj tc
m t A t B t
t t tAf Bff f
x t A t t t
A e e e 1 1 2 2
1 2
1 1 2 2
1 2
1 2
1 1 2 2 sin sin , and
Re
= cos
c
j t jn t j t jm tn m
n m
jn t jn tj tc c n m
n m
c n m cn m
e J e e J e
x t A e J e J e
A J J n m t
1 1 1 2 2 2
1 2
1 2
1 2
1 2 1 2
Chap. 3 8585
Narrowbandfrequency modulator
Localoscillator
Bandpass filter
Mixer
1
0
1
Narrowband FM signal : Carrier frequencyPeak frequency deviation :
Deviation ratio
maxd df f m t
f
D
0
2 1
2 1
Wideband FM signal :Carrier frequency Peak frequency deviation :Deviation ratio :
d d
nff nf
D nD
cx t Frequency Multiplier n Narrowband-to-wideband Conversion Indirect FM
(i.e., Figure 3.24)
FIGURE 3.31 Frequency modulation utilizing narrowband-to-wideband conversion
LOe t x t y t
y t m t
Chap. 3 8686
0
0
LO LO
LO
2 Contains two stages: 1 Narrowband FM, Frequency multiplier
cos narrowband FM output
After the freq. multiplier,
cos
local oscillator: 2cos
c
c
x t A t t
y t A n t n t
e t t
y t y t e t
0 LO 0 LO
c 0 LO c 0 LO
c
cos cos
The BPF may choose to select or
as its passband center frequency.
= cos wideband FM output
c c
c c
A n t n t A n t n t
n n
x t A t n t
x t n
z t
x t y t
Frequency multiplier is often a memoryless nonliner device followed by a
BPF.
Chap. 3 8787
0 0
1
Ex. A narrowband-to-wideband converter as in Fig. 3.31 Narrowband FM :
cos , 2 100,000
peak frequency deviation of : 50
bandwidth of : 500 Wideband
c
d
x t A t t Hz
t f Hz
t W Hz
2
output : carrier frequency 85 deviation ratio 5 Determine 1. ? ,
2. the two possible local oscillator frequencies,
3. the center frequency and
cf MHzD
n
bandwidth of the BPF .
cx t Freq.Mul.
n LOe t x t y t
y t m t
Chap. 3 8888
Chap. 3 8989
0
output the frequency-deviation of
Demodulation of Angle-Modulated s
the input signal.
ignals
Frequency discriminator :
Received signal
2 , for FM; cos ;
t
dr c
f m dx t A t t t
, for PMpk m t
Dy t
input frequencycf
slope DK Hzf
FrequencyDiscriminator
rx t D D
d ty t K
dt
Chap. 3 9090
Discriminator output:
, : discriminator constant.
For FM,
For PM,
2
2
1D
D D d
D p
D D
D
K
y t
d t
K f m tK k dy t m t
d
d
t
y t Kt
FrequencyDiscriminator :r
FMx t D D dy t K f m t
FrequencyDiscriminator :r
PMx t
dt 2D pK k
y t m t Dy t
Chap. 3 9191
Dy t e t ddt
EnvelopeDetector
rx t Envelope
Approximation to Ideal Discriminator
cos
sin
Envelope of
If for all , 0
r c c
c c c
c c
c c
x t A t t
d te t A t t
dt
d te t y t A
dt
d t d tt
dt dt
2 , for FM 2D c c d D c
d ty t A A f m t k A
dt
Chap. 3 9292
Due to interference perturbation and channel noise, may sometimes be time-varying , use bandpass limiter.
c
c
AA t
Limiter
Bandpass Filter
EnvelopeDetector
rx t
Band pass limiter
Dy t ddt
K
K limiter BPF
Chap. 3 9393
ddt
Discriminator implementation: Ideal differentiator : x t y t
H f
f0 2 , High-pass filter.H f j f 0 0
For small
Implementation: 1 Time-delay:
,
lim lim
, .
r r
r r
r r
e t x t x t
e t x t x t
dx t dx te t
dt dt
rx t e t
Time-delay
Chap. 3 9494
2 RC network:2 ;1 1 2
21 , 2 differentiator with gain .
2
R j fRCH fj fRCR
j fC
f H f j fRC RCRC
C
R
H f
f0
10.707
cf1
2 RC
Chap. 3 9595
The resultant Discriminator: 2D CK A RCDifferentiator Envelope detector
D D dy t K f m t :r
FMx t
1Disadvantage : 2
If freq. 10.7 for commercial FM,
, ,D D
fRC
MHzRC K y t
Chap. 3 9696
3 BPF-based Differentiator linear region differentiator Disadvantages : 1. small linear region 2. dc bias
Linearregion
H f
f 4 Balanced Discriminator Advantages : 1. wider linear region 2. no dc bias BPF at
Cf fBPF at
Cf f Envelopedetector
Envelopedetector
Dy t rx t
Chap. 3 9797
Chap. 3 9898
desired signal undesired interfere
Interference Interference in Linear Modulation
eg., DSB, AM, SSB
Interference : cos
cos cos cos cos
i c i
r c c m m c i c i
A t
x t A t A t t A t
§ message
nce
Coherent demodulation cos cos
Envelope detection Nonlinear detector phasor diagram :
cos cos cos cos
D m m i i
r c c m m c i c i
m t
y t A t A t
x t A t A t t A t
Dy tLPF 2cos ct rx t
Chap. 3 9999
Re2 2
Re2 2
c c m c m c i
c m m i
j t j t j t j t j t j t j tm mr c i
j t j t j t j tm mc i
j tR t e
A Ax t A e e e e e Ae e
A Ae A e e Ae
Chap. 3 100100
Envelo
Envelope detector output: 1 :
cos cos cos
+ cos cos sin sin
cos cos
c i
r c c m m c
i i c i c
c m m i i
A A
x t A t A t t
A t t t t
A A t A t
pe
cos
sin sin
If , with a DC-blocker to remove term cos cos , same as the
c
i i c
c i
A t
c
D m m i i
B t
t
A t t
A A Ay t A t A t
coherent demod.
12
02 2
cos sin cos ,
, A t B tA t
z t A t t B t t R t t t
R t A t B t R t A t
Chap. 3 101101
2 If
cos cos cos cos let
cos cos
cos cos
cos cos si
c i
r c c m m c i c i
c c i i i c i
m m c i i
c c i
c i
i
c i
A A
x t A t A t t A t
A t t A t
A t t t
A t t
Envelope
n sin
cos cos cos cos
sin sin
cos cos cos cos
c i i
i c i m m c i i
c i i
i c i m m i
A
c i
c
t
t t
A t A t t t
t t
A A t A t t t
A
sin cos sin sinB t
i m m i c it A t t t
Chap. 3 102102
cos cos2
If , then ,
cos + cos cos , The message is lost
Interference in Angle Modulation
Unmodulated carrier interference at
m i m i
c i
D c i m m i
t t
c i
A A R t A t
y t A t A t t
cos cos
cos cos cos sin sin
cos
r c c i c i
c c i i c i i c
c
x t A t A t
A t A t t A t t
R t t t
Chap. 3 103103
2 2
1
cos sin sin
tancos
cos if , ;
sin
1 cos cos sin
c tA
c i i i i
i i
c i i
c i i
c i ii
c
i ir c i c i
c c
R t A A t A t
A tt
A A t
R t A A tA A At t
A
A Ax t A t t tA A
1 3 5
1
Taylor's series:1 1tan3 5
1, tan
u u u u
u u u
Chap. 3 104104
Ideal discriminator output:
sin , for PM
cos , for FM2
Since th interference output at the FM discriminator is
proportio
e
nal to
i
D D ic
DiD
D ic
i
AK t K tA
y td t AK K t
df
t A
, for a small , the interference affects
less on the FM system than the PM system, and vice ver
sa.i if f
Dy t ddt
EnvelopeDetector
rx t dt Dy t
PM
FM
Chap. 3 105105
The interference analysis of angle modulation becomes difficult, if the preceding assumption is not hold. However, an approximation can be made through the use of the
phasor diagram. pleas
c iA A
interference impacting on the FM system is
proportioal to t
e refer to the textbook Generally, the
. One approach for interference
alleviation is to use the pre-emphas
he
is
frequenc
and the e
y
de-if
mphasis filters
in FM systems. pH f
Pre-emphasis Filter
Mod. cosi c iA t m t rx t dH fDe-emphasis Filter
Dy t
m tDiscriminator.
ddt
Chap. 3 106106
32
3
3
Deemphasis filter: 1st-order RC LPF1 1 , set
21
1 , ,
,
assume in the case of
While suppressing the interference wi
d
d
iD D
c
c i
H f f WRCf
f
f f H ff
Ay t KA
A A
th large , the de-emphasis filter distort the message signal as well, so we need to deploy a
1 pre-emphasis filter with at the transmitter.
i
pd
f
H fH f
dH f
fif3f
0.7071
R
C
Chap. 3 107107
Chap. 3 108108
§ Feedback DemodulatorsPhase-Lock Loops PLL for FM demodulation
Main buliding blocks: 1. Phase Dector PD
2. Loop Filter
3. Loop Amplifier
4. Voltage-Controlled Oscillator VCO
rx t Phasedetector
Loopfilter
Loopamplifier
VCO
de t 0e t e tDemodulated output
gain =
Chap. 3 109109
0
Phase Detector PD : Assume the input of PD:
cos ,
sin
, : constant
the corresponding output is a funct
1 si
ion of phase-dif
n2
ferenc
d
r c c
d
v c
c de t
x t A t t
e t A
A A K t t
t t
K e of the inputs
Ex. An implementation of PD
LPF rx t de t 0e t
1 rx tPD de t 0e t
Chap. 3 110110
phase di
VCO : Frequency-deviation of VCO output is proportional to the magnitude of its input.
rad sec ; : VCO constant
.
1 Since sin2
v v v
t
v v
d c d
d tK e t K
dt
t K e d
e t A A K t t fference only
,
it follows that the PLL can be modeled without taking the carrieN
r terms. onlinear PLL model. VCO e t 0
sinv c
e t
A t t
Chap. 3 111111
t tK dt t e t
Demodulated output
Loopfilter
Amplifiergain
: Nonilnear PLL modelFigure 3.46
sin 12 c dA A K
Phase Detector
de t Furthermore, when the VCO operating in lock, ,
sin
Nonlinear PLL model Linear PLL m, odelt t
t t
t t t t
Chap. 3 112112
: Linear PLL modelFigure 3.47
t tK dt t e t
Demodulated output
Loopfilter
Amplifiergain
Phase Detector
de t 12 c dA A K The loop filter can be
assumed a short-circuitfor simplification.
for FM
VCO input demodulated mes
When the PLL is
sage
in lock,
,
1 ;
t t
d t d tm t K e t
dt dt
e t m tK
Chap. 3 113113
,Phase Error Loop Gain
1VCO: sin2
sin
nonlinea
sin
r 1st-or
,
v v c v d v
t
t
ttK
d tK e t A A K K t t
dt
d t d t d tK t
dtd t d t
K tdt d
dt
t
dt
der differential equation
t tK dt t e t
Demodulated output
Loopfilter
Amplifiergain
: Nonilnear PLL modelFigure 3.46
sin 12 c dA A K
Phase Detector
de t
Chap. 3 114114
Assume that the input to a FM modulator:
s sin , 0
si
in ,
n , 0
tt
t
d tu t
dtd t
K t d K tdt
d K tdt
ordt
0d t
dt 0
d tdt increase
decrease ,
sint
dydt
y K
ss
ss
Under the condition , 0, and the trajectory of
staggers around ; 0,t
t t t
K
t
Chap. 3 115115
For 0, is monotonically ascending
going right on the -axis, so that the opearting point
will move from left to right along the "sin" curve as .
For 0,
d tt
dt
td t
dt
being monotonically descending,
the operating point will move in opposite direction. Point is a locally stable point.Note:
1 Steady-state error: At point , 0, no frequency erro
t
A
dAdt
1ss
r
but also at point , sin 0, phase error existstA K
Chap. 3 116116
is the lock
2 Lock range: The system converge to point if .
If , sin 0, the operating
curve will not intersect range
with -axis
t t
t
t
A
K K t
K
K
ddt tK Ex.: 1
ss
ss
Phase error: sin ,
.t
t
K
K
Chap. 3 117117
1st order linear model PD: Assuming that t is very small eg. ,
sin ,2
VCO: 1st-order linear model
Using the Laplace tr
t
c v dv
t
t t KA A Kt t t t e t t t
d tK t t
dt
,
ans
,
form,
,
,
.
t
t
t
t
t
t
K tt
s KH ss s K
h t K e u
Ks ss K
K h t t
t t h t t
t
signal
t
FM
. ., output of the VCO
t
i e
tK td t t
t
d tK t t
dt
Chap. 3 118118
Ex. The output to a FM modulator is ,
cos , 0cos
cos , 0
By using a first-order linear PLL, determine the demodulated output.
t c c
c c c fc c f
m t Au t
A t tx t A t K A u d
A K A t t
Sol.
Chap. 3 119119
1st order approximated PLL RC-LPF with a 3-dB bandwidth tK
Chap. 3 120120
ss
Summary of the 1st order approximated PLL:1. limited lock range:
.12. phase error: 0
3. The loop gain,1
drawbacks
2
is a
of
function
1st orde
of t
r PLLt
t
t c v d v
K
K
K A A K K
he amplitude of input signal. 4. 1st-order approximated PLL is equivalent to a RC LPF with 3-dB bandwidth . 5. As , 1. However, it is impractical.
c
t
t
A
KK H s
Chap. 3 121121
The two drawbacks of 1st order PLL can be solved by the 2nd order PLL: Perfect 2nd order PLL:
Linear model is assumed. i.e., is small
Loop filter transfer function:
.
s aF ss
t t For 1st order PLL: 1F s s1VCO: s Demodulated output
Loop Filter F s
: Linear PLL model freq. domain representationFigure 3.46 s s Loop gain tK
Chap. 3 122122
2
2 2
2 2 2
standard form of 2nd- order s
Linear model:
,
Phase error: 1 ,
12
n
t
t t
t t t
t t n
s aF s
sK F s
s s sss K F s K s a
H ss s K F s s K s K a
s s s s H s
s s sH ss s K s K a s s ystems
1 damping factor, 02 where
natural frequency
t
n t
Ka
K a
Chap. 3 123123
2
2
2 2 2
2 2
For step frequency input FM signal ,
,
Phase error: 2
2
n n
n n
d tu t s s s
dt s s
sss s s
s s
s
22 2
2
2
ss
1
Using inverse Laplace transform,
si
, 0, no steaty-state
n 1 , 1
phase e rror
1n
n n
tn
n
t e t u t f
t t
or 2 2
Laplace-transform pair: sin
bte at u t
as b a
Chap. 3 124124
Ex.: Some computer simulation results 0.707, 10 , 88.9, 44.4n tf Hz K a t d t
dt
ss 8 four cycle phase slipped ss 6 three cycle phase slipped
Chap. 3 125125
20f Hz 35f Hz 40f Hz 45f Hz
ss 6 3 cycles ss 8
4 cycles Features of the 2nd order PLL:
1. lock range ,
lim 0,
for arbitrary 2. Has cycle-slipping.
t
d tdt
Chap. 3 126126
Analog Pulse Modulation Pulse modulation is used to represent uniformly sampled signals Analog Pulse Modulation: A pulse train is used as the carrier wave in which amplitud
§ , or of each pulse can be used to represent the sampled message.
Pulse Amplitude Modulation PAM
Pulse Width Modulation PWM
Pulse Posit
ewidth p
ion Mod
os
ul
ition
ation PP
M
sT sT2 sT3
amplitude
width
positiont
Chap. 3 127127
for zero-value samplessT2
Chap. 3 128128
impulse-train s
holding networkampling
=
Pulse Amplitude Modulation PAM
Generation of PAM waveform :
*
c
s s c
tn th
c
m t
m t m t m nT t nT m t
m t m t h t
m nT
2 *
=
s sn
s sn
t nT t
m nT t nT
2
2
h t sn
t nT m t m t cm t = th t
2
Chap. 3 129129
, and sinc ,
sinc
j fs
n s
j fc s
n s
nm t f M f H f f eT
nM f f M f f eT
F
Chap. 3 130130
Pulse Width Modulation PWM
proportional to the sample values of a message signal.
Nonlinear modulation. Complicated spectrum.
Used extensively for DC motor con
Pulse
trol.
widt
The pulse
h
amplitude is a constant. Therefore, the pulse area of a PWM
waveform is proportional to the mes
Lowpass filters or integrators can be used as a demodulato
sage value.
Pulse Position M u
r.
od l
ation PPM
Pulse position proportional to the sample values of a message signal.
The spectrum is very similar to that of a PWM signal.
Chap. 3 131131
Delta Modulation and PCM Delta Modulation DM
The message signal is en- coded into a sequenc
p
e of binary symbols represented by the fun
olaritctions
y of impulse.
Operations:
§ 0
0
0
;
sgn
,
,
sd t m t m t
t d t
d t
d t
0
0
Chap. 3 132132
a stairstep approximation of .
For a specific time interval
c s s sn n
t
s s sn
s s
s s skn
x t t t nT nT t nT
m t nT nT d
m t
k T t kT
m t nT nT d
1 .
At the time instance ,
, if
, if
s
t
T
s s
s
s s ss s
s s s
m k T
t kT
m k T kTm kT
m k T kT
1
0 0
0 0
1
1
1
Chap. 3 133133
Chap. 3 134134
0
0
Slope overload:
Occurs when the message signal has a slope greater than that can be followed by the stairstep approximation . . : If step size , Maximum slo
s
s
d m tdt T
m t
m tEx
0
0
pe:
in the case , can exactly catch up the
variation of .
The slope overload implies a bandwidth limitation on .
m s
ss
S Td m t m tdt T
m t
m t
Chap. 3 135135
Digital Pulse Modulation: Signal samples are represented by discrete values (a countable set of values). Pulse-Code Modulation PCM
The message is
first sampled analog amplitude and then quantized into discrete values. The quantized values are encoded into a binary sequence.
011 ,111 ,110 ,011 , 010
10
Chap. 3 136136
A rough discussion of bandwidth Quantization levels: use bits to represents a quantized value Bits per sample: log Message bandwidth: Minimum sampling rate: Nyquist
nq nn q
WW
2
2
2 max
sampling rate. 2 pulse/sec. or bits/sec. Each pulse carries 1 bit message.
Maximum pulse width minimum bandwidth :
recalling sinc
PCM Bandwidt
nW
nWt f
1
2
F
max
h:
Major error quantization error.
quantization error
'
q
B k knW
n B
1 1 1 2 2
sinc ff
Chap. 3 137137
frequency, time, spatial,...
3.7 Multiplexing By using the variety of of communication channel,
multiple number of messages can be transmitted in a common
nature features
commu
§ ncation link. Frequency-Division Multiplexing FDM Several message signals are first translated, using modulation, to different spectral locations and then added to form a base- ban
d signal. Bandwidth ,
: bandwidth of the message
: gard-band of the message.
i i
thi i
thi
W G
W i m t
G i
1G1W 2G2W
frequency
Chap. 3 138138
: DSB
: SSB
: Angle modulationN
m t
m t
m t12
Chap. 3 139139
cos v.s. sin
Quadrature Multiplexing QM - are used for frequency translations.
- Two messages are transmitted in same frequency band.
- Strictly speaking,
Quadra
QM
ture carri
is
es
c ct t not a frequency-division technique. - Modulation:
co s n s ic c c cx t A m t t m t t 1 2
sinc cA t sin ct 2
m t1 m t2
cosc cA t cos ct 2 ..... cx t rx t
DDy t DQy tLowpass filter
QDSB modulator QDSB demodulator
Lowpass filter
Dx t Qx t
Chap. 3 140140
local osc.
- Demodulation: Assume ,
Local oscillator has a phase error
cos
cos cos sin cos
cos co
r c
D r c
c c c c c
c
x t x t
x t x t t
A m t t t m t t t
A m t m t
1 2
1 1
2
2 s
sin sin
cos sin
Phase error on local oscillator causes both attenuation and crosstalk at
c
c
DD c
t
m t t m t
y t A m t m t
2 2
1 2
2
2 detection output. , DD cy t A m t 10
sin ct 2
cos ct 2 rx t
DDy t DQy tLowpass filter
Lowpass filter
Dx t Qx t
Chap. 3 141141
Time-divission Multiplexing TDM - Each message signal occupies a time-slot in every data frame of second. - Suitable for sampled digital signals.
T 分時多工
BW W 1
Nii
B W 1
BW W 2
NBW W s2
s1
Ns
Chap. 3 142142
- The minimum bandwidth of a TDM baseband signal: The message BW. Assuming Nyquist rate sampling, the total number of samples
for the TDM baseband signal in seconds is
thi
s
i W
T n
Assume the TDM baseband signal is a lowpass signal with bandwidth .
,
N
ii
N
s ii
N
ii
WT
B
n BT WT B W
1
1 1
2
2 2
Chap. 3 143143
- A TDM example: The Digital Telephone System T1-line PCM TDM Bandwidth of human voice : 4 voice channel: 8000 samples sec, 8 bits sample data rate
W KHz
bps
8000 8 64000 sec. .
A T1 frame contains 24 8
data bits plus
24 voice channels
one frame synchronous bit in 0.125 ms interval.
data rate
yield one T1 ca
of
rrier.
T ms sampling period 10 125
8000
T1-line . Mbps.
bitsms
24 8 11 544
0 125
10011100
time
1channel11110010
2channel00010010
24channel
Chap. 3 144144
Chap. 5 1
Chap5: Review of Probabilities and Random Variables Set Theory: Set collection of elements
Ex. All possible outcomes of tossing a dice. 1,2,3,4,5,6 Subsets:
S
B
, Venn diagram: always be used to express the relationships between sets.
Ex. : 1 10 , .
: 1 10 , , and is even
A x B x A
A X X X Z
B X X X Z X
B A 1, 3, 5, 7, 9,
2, 4, 6, 8, 10
A
B
Chap. 5 2
1. Intersection (production)2. Union (Addition)
Set operations : 3. Complement4. Difference
1. Intersection: , or
and .
2. Union : or
or
A B A B
A B x x A x B
A B A B
A B x x A x
.
3. Complement :
; and
4. Difference: and
B
A
A S A S A x x S x A
A B x x A x A B
Chap. 5 3
, De Morgan s Law :
Ex. , 2,4,9
5,6,7,8
1,3
5,6,7,8 1,3
A B A B
A B A B
A B S A B
A S A
A B
A B
A B
2 4 9
S
5 6
7 8
BA1 3
Chap. 5 4
1 2
1
1
1
Mutually Exclusive or disjoint , and are mutually exclusive. Exhaustive:
, , .
Partition: ; 1, , ;
; ,
, ,
N
N ii
i
i j
N
ii
A B A B
A A A A S
A S i NA A i j
A S
A A
form a partition of .N S
1A 2A
3ANA
S
Chap. 5 5
Review of Probabilities and Random Variables Given a chance experient,
: Total number of equally likely and mutually exclusive outcomes : number of outcomes included in an event .
A
NN A
2
: headEx. Tossing a fair coin twice
: tail
,
two tossing with the same outcome1 4 , =1, ( ) ,4
1 , , , , .2B
A
A
N
NP AN
HT
A H T
B
N N A P A
B H H T T P B ,T H
,H H ,T T
,H TA
B
Chap. 5 6
Probability : Relative frequency of occurance
lim .
Sample space collection of all possible
outcomesof a chance experiment
Axioms of Proba
A
N
NP AN bilities:
. 0 ; , ~ non-negativity
. 1,~ Normalization
. ,
i P A A
ii P
iii A B P A B P A P B
Chap. 5 7
Ex. Tossing a fair dice 1,2,3,4,5,6
event 1 ,
even points ,
odd points1 61 2
1 1 2 , 6 2 31 , , 2
A
B
C
P A
P B P C
P A B A B
P A C A C P A C P A P C
1P B C P A B
C
Chap. 5 8
Some useful probability relationships:
1 , 1
Conditional Probability :
Probability of , given that event has been known.
, or
P A B P A P B P A B
P A A P P A P A P A P A
P A B
P A B A B
P ABP A B P AB P A
P B
B P B
Chap. 5 9
Ex. Tossing a fair dice: 1 , odd points , even points , 2
116 ; 1 3
20
, , are inde
012
16 1.
penden
16
t.
A B C D
P A BP A B
P B
P A B P
P A CP A C
P C
P A BP B A
P A
A A B
A
BC
D
Chap. 5 10
,T H ,T T
A
B
, ,H T ,H H
if , are independent, ,
.
1 tossing is Ex. Tossing a fair coin twice, define ,
2 tossing is
Are A, B inde
st
nd
P ABA B P A B P A
P B
P AB P A P B
A H
B T
p.?
Chap. 5 11
1
1
1
Bay's theorem:
; is a partition of , i.e., ,
event ,
.
i jN
Ni i
ii
jj
j jN
i ii
A A i jA
A
B
P A BP A B
P B
P B A P A
P B A P A
B
1A
2A
iA
NA
Chap. 5 12
Ex. A binary symmetric channel BSC is shown,
0 0.3
given 0 0 0.8 .
1 1 0.8
Find 1 , 1 1 , 1 ,
and error transmission ?
S
R S
R S
S S R R
P
P
P
P P P
P
1S
0S 0R
1R
(0 0 ) 0.8R SP (1 1 ) 0.8R SP
Chap. 5 13
Chap. 5 14
Random Variables r.v. :
A ramdom variable is a mapping or function which assigns a real number to each outcome of a chance experiment.
: , X
12
q X
Chap. 5 15
Ex. Flipping a fair coin twice , you would possiblely win $20, if you have , , you would loose $5 if you have , or
, ; otherwise, you would loose $10. Let being the . which represents the
H H H T
T H X r v money you win from the game, please find 5 ?, 10 ?, 20 ?.
.1 5 , 4
3 10 , , , 4 20 1
P X P X P X
Sol
P X P T T
P X P H T T H T T
P X P
,H T ,T H ,T T ,H H
20
510
Chap. 5 16
1
1. Discrete r.v., exists only on discrete points on the
real lin
Probability mass fu
e, ,
.
2. Continuous r.v. could be any values on th r.
nc
v.:
tion X i
qi
i
i
X
X x
X
P x P X x
Cumulative distribution function . . :
Probability density function . . :
e real line.
,
= .
X
X X
c d f
F x P X x
p d fdfx
X
x F xd
Chap. 5 17
1 For discrete r.v.:
Cumulative distribution function . . .
Properties of :
1. 0 1, and
0
. righ
q
X X i ii
X i X i X i
X
XX
X
X
f x P x x x
P x F x F x
c d f
F x
Fi F x
F
F
i
x
i
x P X
0 0
2 1 2 1
t continuous :
. non-decreasing: ,X X
X X
F x F x
iii x x F x F x
Chap. 5 18
Ex. Find the of the previous coin-tossing gamble.XF x ,H T ,T H ,T T ,H H
20
510
Chap. 5 19
20510 x14
12
14
1 1 110 5 204 2 4XF x u x u x u x
Chap. 5 20
1 2 2 1
0 0 0
0
0
=
Probability density function . . .
lim
lim
Proper
X X
X X
X XX X x
x
P x X x F x F x
P X x F x F x
p d f
F x x F xdf x F xdx x
P x X x xx
2
1
1 2 2 1
.
. ties of :
. 1
x
X X
X X
xXXx
X X
a F x f d
b P x X x F x F xf x
f d
c f x dx F
Chap. 5 21
0
2
0
Ex. Rotating a spinning pointer, denotes the angle readings1 , 0 2
~ 0,2 , ~ uniform distribution2 0 , o.w.
0, 0 01 , 0 2 =
21 , 2
2
f
F d
d
, 0
, 0 22 1 , 2
F 2 1
f 2 1
2
Chap. 5 22
Ex. For the previous flipping coin example 1 1 1 10 5 204 2 4
1 1 1 10 5 204 2 4
Joint ' and ' In the sense of event probability,
X
X X
F x u x u x u x
df x F x x x xdx
cdf s pdf s
2
event and both occur Similarily, in . . , and ,~ joint . . . and
, , , ~ joint . . .
XY
XYXY
P A B P A Br v s
F x y P X x Y y c d f
F x yf x y p d f
x y
Chap. 5 23
2 2
1 11 2 1 2 , ,
, , ( ) 1
, ,
, ,
, , ,
,~,
,
XY Y
XY X
y x
XYy x
XY
XY
XY X
XY Y
P x X x y Y y f x y dxdy
F P X y P
f x y dxdy P x dx X x y dy Y y
F x P X x Y P X x F x
F y P X Y y P Y y F y
f x y dx f y
f x y dy f x
Marginal densities.
Chap. 5 24
Independence: , r.v.s
, or
and are independent , or for discrete r.v.s
,
Conditional density function:
,, ,
XY X Y
XY X Y
XY X Y
XY YX YXY
X YY
X Y
f x y f x f y
X Y F x y F x F y
P x y P x P y
f x yf x y
f yf x y f x y f y
, are indep , . . XX YX Y f x y f x
Chap. 5 25
(2 )
Ex. Given two . . , with joint .
, 0, 0 ,
0 , o.w.
. ? , . ?, . ?, . Are , indep.?
. ?, . ?, and . ?
x y
XY
X Y
X Y X Y
r v s X Y pdf
Ae x yf x y
i A ii f x iii f y iv X Y
v F x vi F y vii f x y
Chap. 5 26
Chap. 5 27
0
2
1 , 0, 0.
0 , 0 0 , 0
. , indep., 2
I. 1 rv. to 1 rv. Transformation of random variables:
II. 2 rvs. to 2 rv
y yy YY Y
xXX Y
e yf d yvi F y f d
yy
vii X Y f x y f x e u x
1
1
s.
I. 1 rv. to 1 rv. transformation:
Given ~ , let , ?
1. From
Steps:
, Find ( ), 1,
,
2.
i
NX i
Y
X
i i
Y
Y g X x g y i
X f x Y g X
N
f y
f xf y
g x
Chap. 5 28
1
1 ;0 2
1 , Ex. Given ~ , 2
0 , o.w.
1, ?
Sol.
1. 1, ,
1 , 0 2, 1=
1 , 0 2 2. 2
0,
Y
Yy
y
f
Y f y
g y
g y y y
dg gd
f yf y
g . . o w
Chap. 5 29
1
2
12
2
1 2
21 2
1 1 2
Ex.
~ 0.5
, ?
Sol.
1. , ,
2 , and 2
2. 2
0
xX
Y
x y
yX i X X
Yi i
X f x e
Y X f y
x yy x
x y
dyg x y g x ydx
f x f x f x ef y u yg x g xg x y
y
Chap. 5 30
?
II. Two r.v.s. to two r.v.s. transformation:
, , , , , . . ; , ,
,
Steps: 2.
, ,1. From , find , 1, ,
, ,
Jacobian:
XY ZW XY
i
i
i
Z g X YX Y f x y r v s f z w f x y
W h X Y
z g x y x M z wi n
w h x y y N z w
g gx y
Jhx
,,
1
, , ,3. ,
ii
x M z wy N z w
ni iXY
ZWi i
hy
f M z w N z wf z w
J
Chap. 5 31
2 2
1
2 2 2 2
2 2 2 2cos cossin sin
Ex. , , , , , ?Tan
Sol.
cos 1. ,
sin
2.
XY R
x r x ry r y r
R X YX Y f x y f rY
X
x ry r
x yR Rx y x y x yJ
y xx y x y x y cos sin
1 sin cos
3. , cos , sinR XY
rr r
f r r f r r
Chap. 5 32
2 22
2
1 1 12 22 2 2 2 2
2 2 22 2
0,. , independent; ;
0,
. , ?, . ?, . ?, . , indep.?
Sol.
. , ;
,1 1 1 ,
,1 1
ZW Z W
Z X YX NEx X Y YY N W
Xa f z w b f z c f w d Z W
ya w y wxx
z wzx y wxw wz x y x wz wzx y wxw w
Chap. 5 33
Chap. 5 34
Chap. 5 35
2 2
, independent Gaussian r.v.s.,
distributed
distributed
, independent
X Y
Z X Y RayleighYW CauchyX
Z W
Chap. 5 36
Statistical Averages Mean or statistical average
~ ,
,~ mean value of
For diserete . . ,
X
X
X X i ii
X i X ii
X f x
E X X x f x dx X
r v s f x P x x x
E X x f x dx x P x
Y
~ . .
, , , ~ , ,
, ,
Y
Y
XY
X
X
g X r v
E Y E g X g x f x dx y f y dy
Z g X Y X Y
E Z g x y f x y dxd
f y
y
x
Chap. 5 37
2
2
1 ; Ex. ~ , cos 2
0 ; o.w
02 .
1 cos 02
Ex.1 1 1,2,4 , 1 , 2 4 ; log2 4
1 1 12 4 22 4 4 log
X X X
f X
E f d d
E X d
X P P P Y X
E X
E Y
2 21 1 1 31 log 2 log 4 .2 4 4 4
Chap. 5 38
1 1
,
pf. ,
= , ,
, , ~ . . , , , ,~ cons
X Y
XY
XY XY
f x f y
N N
Z X Y E Z E X E Y
E Z x y f x y dxdy
x f x y dydx y f x y dxdy
E X E Y
X X r v s a a
1 1
, are indep
tants
, . ., ,
.
N N
i i i ii i
XY X Yi e f x y f
E a X a E X
X Y
E XY E X
x y
E
f
Y
Chap. 5 39
2
2 2
2 2 2
1
Variance: ~ ,
,~ Variance of
{ 2 }
{ }, are randomstatistically independent
X x
X
x x
X X
N
X f x E X
Var X E X X
E X X
E XX X
1 2 1 2
1
2
1
1
2
1
variables,
. ., ( , , ) ,
and are constants;
N i
N
X X X N X
N N
i ii
i
ii
i
N
iV
i e f x x x f x
a
a
ar a X
a
Chap. 5 40
1 1
2
1 1 1
2
1
pf. ,
N N
i i i i i ii i
N N N
i i i i i ii i i
N
i i ii
E a X a E X
Var a X E a X a
E a X
2
0,
, 1
, 1
2 2
,
1
, 1
i
N
i j i i j ji j
N
i j i i j ji j
N N
i j i j i j i
i ji j
ii j i
E a a X X
a a E X X
a a E X X a
Chap. 5 41
Conditional mean:
The expected value of given ,
is a function of sti,
ll
a
r.v.
X Y
Y
YX Y
E X Y
X Y y
E X Y y x f x y dx
E X Y y Y
E E X Y y E X Y y f y dy
x f x y f y
,
For discrete r.v.s: i
i
XY
i i i iX Yx
i Y iy
dxdy
x f x y dxdy
E X
E X Y y x P x Y y
E X Y y P y E X
Chap. 5 42
2 2
Covariance and the Correlation Coefficients , ~ , ,
, , , Covariance Coefficient:
~ A measure of the similarity be
XY
X X Y Y
XY X Y X Y
X Y f x y
E X Var X E Y Var Y
E X Y E XY
tween and . , 0 , are uncorrelated.
, are indep.,
X Y XY
X YE XY
X Y
X Y
, are uncorrelated.X Y
Chap. 5 43
Correlation coefficiant :
, are uncorrelated, 0
Ex. Two random variables are related as , where is a constant. Please determine ?
XYXY
X Y
XY
XY
X Y
Y X
Chap. 5 44
Chap. 5 45
1 21 2 1
Bernoulli Trial: 0,1 , 0 , 1 1 Binomial distribution:
Consider a sequence of independently and identically distributed i.i.d. Bernoulli trials,
, , , , ,
n
X X
N X X X
X P p P p q
n
X X X P x
21
1 2
0
0 0
, ,
Let number of 0's in , , ,! ,
! !
. 1
n
N X ii
n
n k n k k n kY k
n
Y Yk
n nnn k n k
Y Y kk k
x x P x
Y X X XnP Y k P k C p q p q
k n k
f y P k y k
i f y dy P k C p q p q
Chap. 5 46
2
2
0
11
1
2
1
2
.
1 !
1 ! !
1 ! .
1 ! !
1 Gaussian distribution: ~ , , ( )= ,2
n
Yk
nk n k
k
nnk n k
k
x
X
ii E Y kP k
n np q
k n k
nnp p q np p q np
k n k
X N f x e
E X
2
2
2
2
2
2
2
, .
1 2
1 12
x
X X
X
Var X
F x P X x e d
F e d
Chap. 5 47
2
2
2
2
2
2
1-function: 2
Properties :
1 . 12
1 . 1 , 2
. 1
x
tx
Q Q x e d
i Q e d
ii e dt Q x
iii Q x Q x
-4 -3 -2 -1 0 1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
zf x(z
) Q x
Chap. 5 48
2
2
2
2
2
2
2
24
50
25 2
2 2
2
2
1 Ex. ? 5 2
2 1 2 Sol. let , 1 .5 52
1 ~ , , ,2
1 ( )2
let ,
1 12
x
y
x
X
x
X
x y
X
e dx
xy e dy Q
X N f x e
F x e d
F x e dy
y
xQ
Chap. 5 49
2
2
2
2
2
Ex. ~ ,
, where is a const., ?
, 1 ;
1 2
~ ,
Y
yX
Y X
X N
Y X f y
g x x g x
f yf y f y e
g x
Y N
Chap. 5 50
Chap. 6 1
Random Processes r.p. :
For a r.v., : maps each in to a real number.
For r.p., : , , maps each in to a time fn..
,Chap 6:
X X
X t X t
t
,X t ,X t ,X t
,X tX
1t 2t 1 1, ~ r.v.X t X 2 2,
~ r.v.X t X
Chap. 6 2
1 1
for a fixed time , , ~ . .
a sequence of time instances , , , , ~ a sequence of r.v.s. for a fixed state
i i i
XN N
t t X t X r v
t t X X
1 1 1 1
, ( , ) ~ a sample fn.
, = collection of all possible sample fns, , ;
~ Ensemble , , ~ r.v., ( , )
k k
i
X
X t
X t X t i
t t X t F x t P X x
~ order . . .
( , ) is time-varying.( , ) ( , ) , ~ order . . .
X
XX
lst c d fF x t
F x tf x t lst p d fx
Chap. 6 3
1 1Ex. Tossing a fair coin with , 2 2, sin
, ,
Find ( , ) and ( , )1 for 0, , and 12
X X
P H P T
X H t tX t
X T t t
F x t f x t
t
H
T
,X tt
112
1 ,X T t ,X H t
0
Chap. 6 4
Chap. 6 5
1 1
2 2
, ,~ . .
, , ~ . .
, , ~ . . let
, , ~ . .
First order c.d.f:
, , 1, ,
, , , ~ 1 order
i
i
i
n n
X i i i i
X i i stX i i
i
X t R P
X t X r v
X t X r v
X t X r v
F x t P X t x i n
F x tf x t pd
x
★ 1 2
1 2
1 2
1 2 1 2 1 1 2 2
21 2 1 2
1 2 1 21 2
.
Second order c.d.f: , ; , and
, ; , , ; , ,~ 2 order .
X X
X X ndX X
f
F x x t t P X x X x
F x x t tf x x t t pdf
x x
★
t
,X t1t 2t 1 1,
~ r.v.X t X 2 2,
~ r.v.X t X
Chap. 6 6
1 2
222 2
1 2 1 2
1 2 1 2
Mean and Autocorrelation: ~ , Mean:
,
Variance:
Autocorrelation:
,
, ;
X
X X
X
X
X X
X t f x t
E X t xf x t dx m t X t
t E X t X t X t X t
R t t E X t X t
x x f x x
★ 1 2 1 2
2 1
,
,X
t t dx dx
R t t
Chap. 6 7
1 2 1 1 2 2
1 2 1 2
1 2 1 2
21 1 1
1 2 1 2
Autocovariance function
,
,
, Cross-correlation fn.:
~ . ,
,
X
X
X X
XY YX
C t t E X t X t X t X t
E X t X t X t X t
R t t X t X t
C t t t
X tr p
Y t
R t t E X t Y t R
★
★
★ 2 1
1 2
2 1
,
Wide Sense Stationary W.S.S : ,~ . .
1. constant If ,
2. ,
in which , then is W.S.S.X X
t t
X t r p
E X t
R t t R E X t X t
t t X t
★
t
,X t1t 2t 1 1X t X 2 2X t X
t
,Y t1t 2t 1 1Y t Y 2 2Y t Y
Chap. 6 8
2 2
Ergodic ~ . .
Define the time average of : 1 lim
2,
,
~ Ergodic
T
TT
X
X t r p
X t
X t X t dtT
E X t X t
E X t X t
R E X t X t X t X t
X t
★
Chap. 6 9
0Ex. cos 2
2 , ~ 4 4
0, o.w.
Please determine if is ergodic?
X t A f t
f
X t
Chap. 6 10
Chap. 6 11
0 0
Ex. cos 2 ; , ~ constant
1 , ~ ,2
0 , o.w.
Is an ergodic process?
n t A f t A f
f
n t
Chap. 6 12
Chap. 6 13
Chap. 6 14
2 2
2 2 2
22
2 2
,~ Ergodic Process
1. ~ D.C. component of
2. , ~ Total power of
3.
,~ A.C. power of
4. , ~ power of ( )
X X
X t
E X t X t X t
E X t X t X t
E X t m
X t X t X t
X t X t DC x t
★
Chap. 6 15
2
0
0
Properties of :
~ W.S.S.,
1. 0 ,
2. , ~ even symmetric
3. lim
4. If ,~ periodic
then
Please prove the above
X
X
X X
X X
X
X X
R
X t R E X t X t
R R
R R
R X t
X t X t T
R R T
★
properties.
Chap. 6 16
Chap. 6 17
Chap. 6 18
0,: ,
Ex. ,
, ,2 2
, and independent r.v.s; : deterministic pulse function
kk
k m m kk
k
k
X
k mk m
E aa
E a a RX t a p t kTT TU
a p t
R E X t X t
E a a p t kT p t mT
2
2
1
k mk m
T
Tm kk m
u
E a a E p t kT p t mT
R p t kT p t mT dT
Chap. 6 19
2
2
2
2
1
change variable, let ,
1
let , , ,
1
Tt kT
TX m k t kTk m
Tt kT
Tn
n
t kTk
t kT u
R R p u p u m k T duT
m k n m k n
R p u p u nT duT
,
1where
pulse correlation fn. of the deterministic pulse .
p u p u nT d
n
nn
u r nTT
R r nT
r p t p t dtT
p t
Chap. 6 20
. Binary random waveform:
,
1; , ;2
, are independent,
?
kk
k
k k
k n
X
Ex X t a p t kT
Aa
Atp t P a A P a A kT
a a k n
R T
2T3T
4T
A
A t
T 3Tt
Chap. 6 21
2 2sincXS f A T Tff
Chap. 6 22
2
2
Wiener-khinchine theorem ~ W.S.S
,
0, o.w.
,
~ random1 lim
2
T
T j ftT T T
T
X TT
X t
X t T t TX t
f X t X t e dt
f
S f E fT
F★ 2
Wiener-khinchine theore
~ Average power d
m:
ensity Spect
i.e.,
ra o
f
X X
j fX X X
R E X t X t S f
R R e d S
X
f
t F
F
Chap. 6 23
2
2
2
22 2
2 2
.
a real random process
, Define: , energy signal
0, o.w.
lim2
1 lim21 lim
2n
T
T
n T
T T j f tT TT TT
T TTR
pf
n t
n t t Tn t
E n tS f
T
E n t n e dtdT
E n t nT
F 2
21 lim2
T T j f t
T Tt
T T j f tnT TT
e dtd
R t e dtdT
Chap. 6 24
1 1, Jacobian: 1,
0 1
2 0, 0 2 ,0 : , 0 :
u uu t tlet J dudv dtd
v vv tt
T u u Tu u
T v u T u T v T
tT T
T
T tT T
T
T t t
0,,
0,u t u t
letv t u t
v
u2T 2T
T
T
Chap. 6 25
0 22 2
2 0
0 22 2
2 0
0 22 2
2 0
1 lim21 lim 2 2
21 lim 2 2
2
u T T Tj fu j fun n nT T u TT
Tj fu j fun nTT
Tj fu j fun nTT
S f R u e dv du R u e dv duT
T u R u e du T u R u e duT
T u R u e du T u R u e duT
2
1, a
2
2
2
s
lim 12
T j funTT
j f
T
un n n
uR u e du
T
R u e du R R
F F
Chap. 6 26
0
2
0
2 2
0 0
1 , Ex. cos 2 ; ~ 2
0, o.w.
cos22
4 4
Ex. The power spectral density . . of a bandlimited is defined as
X
X X
X t A f t f
AR f
A AS f R f f f f
P S D n t
F 0 0
10
, =2
2 2 0 , o.w.
sinc2
, : white noise.
n
n n
N B f B N fS fB
R S f N B B
B n t
F
Chap. 6 27
0
0
,2 white noise :
2
n
n
NS fn t
NR ★ nS f
0
2N
f
Chap. 6 28
Cross correlation function , . .
Define: Cross correlation function
A measure of the similarity between and in the time-domain.
pf.
XY
XY YX
X t Y t R P
R E X t Y t
X t Y t
R R
★ , let
. Q.E.D.
XY
YX
R E X t Y t t t
E X t Y t
R
Chap. 6 29
Cross power spectral density:
,
A measure of the similarity between the two random processes in the frequency-domain. 0, ; , orthogonal. In general, the orthogonality of
XY XY
XY
S f R
R X t Y t
F
two R.P. is nothing to do with the statistically independent.
Chap. 6 30
Let ,
Powe
n
X XY XY Y
n t X t Y t
R E n t n t
E X t Y t X t Y t
E X t X t E X t Y t
E Y t X t E Y t Y t
R R R R
2
r of :
0 0 2 0 0
where cross power
and are orthogonal, .
YX XY
n n X XY Y
PP P
XY
n X Y
n t
P E n t R R R R
P E X t Y t
X t Y t P P P
Chap. 6 31
Random Processes through a LTI system
, W.S.S.,
,
?
X
X
X
E X tX t
E X t X t R
Y t X t h t X t h d
h t H f
E Y t
R
★ ???
?
Y H fXX
YYX
Y
E X t
RR
S fS f
X t Y t h t
Chap. 6 32
2
0
Y x
XY XX
YY XX
Y X
H
R R h
R R h h
S f H f S f
X t Y t h t
Chap. 6 33
Chap. 6 34
Chap. 6 35
1
1
1 1
1
21
22
Gaussian . .
let , ~ r.v.s,
, , , , , ,
{ } ~ autocovariance matr
1 If , , wh2
ix
i
n
n
T
X i
n
TTn X X
T T
n
R P
X t XE X
X t
n f
X X
e
X
E
X X Xx
X
X
μ C x μ
X
X X
X
X μ
C X X μ
C
μ
x
★ 2
2
1 2
2 2
2
ere , , , ,
then is a Gau
, ,
1 , ~
ssian proc
,2
ess.
i Xi
i
i i
T
i i i
x
X i
n
i X
i
t t X t
x x x
X
X
f x e N
t x
Chap. 6 36
~Gaussian
pf.
~ Gaussian R.P. ~ Gaussian R.P. LTI
k
k
X
k
Y t X h t d
X k h t
X
k
t Y t ★ , 0
, linear combination of Gaussian . . .
~ Gaussian process
k kk
X r v s
Y t
h t X t Y t
Chap. 6 37
1 2
5
1 2 1 2
Ex. ~ Ergodic and Gaussian distributed.
3 ,
5 2 sinc2 , .2 2
1. find ; , 2. find ; ,
3. find , ; 1, 3
Sol.
XX
X Y
X X
X t
R e
fh t B Bt H f B HzB
f x t f y t
f x x t t
Chap. 6 38
Chap. 6 39
Chap. 6 40
Chap. 6 41
0
0 00
2
20 0 00
Noise equivalent bandwidth
: white noise,2
For an ideal bandpass filter:
Filter output:
2
o
n
N N
o
n n
N N
Nn t S f
f f f fH f HB B
n t
S f S f H f
N f f f fHB B
2 20 0output noise power: .
oo n NE n t S f df N H B n t ideal filterH f
f
0H
NB
on t
Chap. 6 42
2 2200 0 2
0
For a realistic filter : ?
Use an ideal filter to approximately evaluate the bandwidth of .Noise equivalent bandwidth:
1let , .2 2
r
r
r N N r
H f BW
H f
N H f df N H B B H f dfH
Chap. 6 43
Narrowband noise:Most communication signals are modulated within a frequency band ofwhich the bandwith is generally much smaller than the carrier frequency.Therefore, the in band noise can be expres
0 0
0 0 00
sed as a narrowband noisecos sin
where ; 2
, . .
0,2 ; is independent to
c s
n
c s
n t n t t n t t
N f f f fS f B fB B
n t n t R P
U n t
Chap. 6 44
2 202 2
cos
2 2
sin
To observe the realization of the narrowband noise, canreformulated as
cos
s
cc s
c sR t
t
s
c s
t
n t
n tn t n t n t t
n t n t
n t
n t n t
0
2 2
0 1
in
cos , c s
s
c
t
R t n t n tR t t t n t
t Tann t
Chap. 6 45
0 0cos sinc sn t n t t n t t
Chap. 6 46
1
0
1
0
1 0
0 0
0
0
cos 2 2
0
0
0
2 cos +
4 cos + cos +
2 cos
2 cos 2 2
2 cosnR E t
z
n
z
z t n t t
R E n t n t t t
E n t n t
E n t n t t
R
S
2
2 1
1
1
2
0 0
2 0 0
0 0
20 0
.
Similarily 2 si
Lp
, : Low-pass equivalent noises of the b
n + , 2 c
and
os
.
pa
c sn n z n n
z n n
z n
z z n
B
z
c s
f
n
S f S f S f H f S f f S f f
f F R S f f S f f
z t n t t R R
S f
n t n t
F R S f f S f f S f
ss .n t
Chap. 6 47
1 2
1 2
1 2
0 0
0
0 0
1
1
Cross-correlation:
4 cos + sin +
2 sin .
c s
z z
n
z z n n
n n c s
s
R E z t z t
E n t n t t t
R
S f j S f f S f f
R E n t n t
E z t h t n t
E z t
1
1
1
z n
s
s
s
s
n
R
z
h d n t
E z t n t h d
R h
Chap. 6 48
1 2
1
1 2
1 2
1
1 2
1 2
1 2
z
c s
s
zR
n n z z
z n s
z z
R E z t n t
E z t z t h t
E z t z t h d
E z t z t h d
R R
R
h
h
h
1 2
2
0 0
Lp
is imaginary,
must be an odd function of . Fourier property
It follows that if is continuo
c s
c s
c s
c s
n n z z
n n
n n
n n
n n
S f S f H f
j S f f S f f
S f
R
R
F us,
0 0 ~c sn n c sR E n t n t uncorrelated
Chap. 6 49
0
,
0 0, , if Lp 0
, if is a Gaussian R.P;
indep. Gaussian R.P.
. 0,
c s
c
s
s
cn n n n n n
c
s
c
s
LTI
n t n tuncorrelated
R S f S f f S f f
n tn t n t
n t
n t
n t
a E n t
E n
2 2
0
0 0
2
2 cos 0
. ,
Lp
=
1The joint pdf: , ; ,2
c
c s
c s
c s
c n n n
n s
n n
n n c s
t E n t t h t E n t
b Var n t N
Var n t S f df S f f S f f df
S f df N Var n t
f n n t t eN
N