Transcript ppt
ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication Laboratory
Zhu Han Department of Electrical and Computer Engineering Class 4 Sep. 8 th , 2014
Overview
Homework – 4.2.1, 4.2.3, 4.2.5, 4.2.7, 4.2.9
– 4.3.3, 4.3.8
– 4.4.2, 4.4.4
– 4.5.2
– 4.8.1
– Due 9/22/14 Phase-locked loop FM basics
Carrier Recover Error
DSB: e(t)=2m(t)cos(w c t)cos((w c + w)t+ ) e(t)=m(t) cos(( w)t+ ) – Phase error: if fixed, attenuation. If not, shortwave radio – Frequency error: catastrophic beating effect SSB, only frequency changes, f<30Hz.
– Donald Duck Effect Crystal oscillator, atoms oscillator, GPS, … Pilot: a signal , usually a single frequency , transmitted over a communications system for supervisory, control, equalization , continuity, synchronization , or reference purposes.
Phase-Locked Loop
Can be a whole course. The most important part of receiver.
Definition: a closed-loop feedback control system that generates and outputs a signal in relation to the frequency and phase of an input ("reference") signal A phase-locked loop circuit responds both to the frequency and phase of the input signals, automatically raising or lowering the frequency of a controlled oscillator until it is matched to the reference in both frequency and phase.
Voltage Controlled Oscillator (VCO)
W(t)=w c +ce 0 (t), where w c is the free-running frequency Example
Ideal Model
Model LPF VCO – Si=Acos(w c t+ 1 (t)), Sv=A v cos(w c t+ c (t)) – Sp=0.5AA
v [sin(2w c t+ 1 + c )+sin( 1 c )] – So=0.5AA
v sin( 1 c )=AA v ( 1 c ) Capture Range and Lock Range
Carrier Acquisition in DSB-SC
Signal Squaring method Costas Loop
v t
1 ( ) 1 2
c l
v t
2 ( ) 1 2
c l
( ) 1 2
c l
2 SSB-SC not working 1 2
c l
2 1 2 sin 2
v t
4
( )
K
sin 2
Costas receiver
PLL Applications
Clock recovery: no pilot Deskewing: circuit design Clock generation: Direct Digital Synthesis Spread spectrum: Jitter Noise Reduction Clock distribution
FM Basics
VHF (30M-300M) high-fidelity broadcast Wideband FM, (FM TV), narrow band FM (two-way radio) 1933
FM
and angle modulation proposed by Armstrong, but success by 1949.
Digital: Frequency Shift Key (FSK), Phase Shift Key (BPSK, QPSK, 8PSK,…) AM/FM: Transverse wave/Longitudinal wave
Angle Modulation vs. AM
Summarize: properties of amplitude modulation – Amplitude modulation is linear
just move to new frequency band, spectrum shape does not change. No new frequencies generated.
– Spectrum: S(f) is a translated version of M(f) – Bandwidth ≤ 2W Properties of angle modulation – They are nonlinear
spectrum shape does change, new frequencies generated.
– S(f) is not just a translated version of M(f) – Bandwidth is usually much larger than 2W
Angle Modulation Pro/Con Application
Why need angle modulation?
– Better noise reduction – Improved system fidelity Disadvantages – Low bandwidth efficiency – Complex implementations Applications – FM radio broadcast – TV sound signal – Two-way mobile radio – Cellular radio – Microwave and satellite communications
Instantaneous Frequency
•Angle modulation has two forms - Frequency modulation (FM): message is represented as the variation of the instantaneous frequency of a carrier - Phase modulation (PM): message is represented as the variation of the instantaneous phase of a carrier
A c
cos
i
where
A c
i t
A c
f t c
i
1 2
d
i dt
Phase Modulation
PM (phase modulation) signal
A
c
p
f t
c
p
( ),
k p
: phase sensitivity
f t i
f c
2
p dt
Frequency Modulation
FM (frequency modulation) signal
k f
A c
f t c
2 : frequency sensitivity
k f
0
t m f c
d
f
angle
i
2 0
t f i
2
d
(Assume zero initial phase) 2
k f
0
t m
FM Characteristics
Characteristics of FM signals – Zero-crossings are not regular – Envelope is constant – FM and PM signals are similar
Relations between FM and PM
PM of
0
t m
FM of
dt
FM/PM Example (Time)
FM/PM Example (Frequency)
Matlab
fc=1000; Ac=1; % carrier frequency (Hz) and magnitude fm=250; Am=0.1; % message frequency (Hz) and magnitude k=4; % modulation parameter % generage single tone message signal t=0:1/10000:0.02; % time with sampling at 10KHz mt=Am*cos(2*pi*fm*t); % message signal % Phase modulation sp=Ac*cos(2*pi*fc*t+2*pi*k*mt); % Frequency modulation dmt=Am*sin(2*pi*fm*t); % integration sf=Ac*cos(2*pi*fc*t+2*pi*k*dmt); % PM % Plot the signal subplot(311), plot(t,mt,'b'), grid, title('message m(t)') subplot(312), plot(t,sf,'r'), grid, ylabel('FM s(t)') subplot(313), plot(t,sp,'m'), grid, ylabel('PM s(t)')
Matlab
% spectrum w=((0:length(t)-1)/length(t)-0.5)*10000; Pm=abs(fftshift(fft(mt))); % spectrum of message Pp=abs(fftshift(fft(sp))); % spectrum of PM signal Pf=abs(fftshift(fft(sf))); % spectrum of FM signal % plot the spectrums figure(2) subplot(311), plot(w,Pm,'b'), axis([-3000 3000 min(Pm) max(Pm)]), grid, title('message spectrum M(f)'), subplot(312), plot(w,Pf,'r'), axis([-3000 3000 min(Pf) max(Pf)]), grid, ylabel('FM S(f)') subplot(313), plot(w,Pp,'m'), axis([-3000 3000 min(Pp) max(Pp)]), grid, ylabel('PM S(f)')
Frequency Modulation
FM (frequency modulation) signal
k f
A c
f t c
2 : frequency sensitivity
k f
0
t m f c
d
f
angle
i
2 0
t f i d
(Assume zero initial phase) 2
A m
cos(2
f t m
) 2
k f
0
t m
f i
f c
k A f m
cos(2
f t m
)
f i
2 1
d
dt f c
1 2 1 2 2
k f
d
2
dt f t c
1 2
A m d
2
k f
cos(2
m
) Let
t
0
t A m
cos(2
m
dt
Example
Consider m(t)- a square wave- as shown. The FM wave for this m(t) is shown below.
FM ( t ) A cos( c t k f t m( ) d ).
Assume m(t) starts at t 0.
For 0 t T 2 m(t) 1 , t 0 m( ) d t and for T 2 t T m(t) 1 , t 0 m( ) d 0 2 T m( ) d 2 T t m( ) d T 2 (t T 2 ) T t.
The instantane ous frequency is i ( t ) c k f m ( t ) c k f for 0 t T 2 i max c k f and i ( t ) and i min c c k f for T 2 k f t T .
m(t) t 0 T 2T FM ( t ) t
Frequency Deviation
Frequency deviation Δf – difference between the maximum instantaneous and carrier frequency – Definition:
m
k f
max | – Relationship with instantaneous frequency single-tone ( ) case:
f i
general case:
f c f c
f f i
cos(2
f c
f f t m
) – Question: Is bandwidth of s(t) just 2Δf?
No, instantaneous frequency is not equivalent to spectrum frequency (with non-zero power)!
S(t) has ∞ spectrum frequency (with non-zero power).
Modulation Index
Indicate by how much the modulated variable (instantaneous frequency) varies around its unmodulated level (message frequency) AM (envelope): max |
a
FM (frequency): max |
f m f
Bandwidth ,
a
(
t
)
t
m
( )
d
(
t
) Re( (
t
))
A
cos
w c t
k f a
(
t
) sin
w c t
k f
2 2 !
a
2 (
t
) cos
w c t
k
2 3 !
f a
3 (
t
) sin
w c t
...
Narrow Band Angle Modulation
Definition Equation
k f a
(
t
(
t
)
)
A
1
cos
w
c
t
k
f
a
(
t
) sin
w
c
t
Comparison with AM Only phase difference of Pi/2 Frequency: similar Time: AM: frequency constant FM: amplitude constant Conclusion: NBFM signal is similar to AM signal NBFM has also bandwidth 2W. (twice message signal bandwidth)
Example
Block diagram of a method for generating a narrowband FM signal.
A phasor comparison of narrowband FM and AM waves for
sinusoidal modulation. (a) Narrowband FM wave. (b) AM wave.
Wide Band FM
Wideband FM signal
A c A m
cos(2
f t m
)
f t c
sin(2
f t m
) Fourier series representation
A c n
J n A c
2
n
J n
(
f
f c
nf m
)
t
f c
nf m
) (
f
f c
nf m
)
J n
Example
Bessel Function of First Kind
1.
J n
J n
3.
n
J n
2
n
1
J
n J
0
J
1 1, , 2 0 for all
n
2
Spectrum of WBFM (Chapter 5.2)
Spectrum when m(t) is single-tone
A c
A c
2
n
J n f t c
sin(2
f t m
)
A c n
J n
(
f
f c
nf m
) (
f
f c
nf m
)
Example 2.2
f c
nf m
)
t
Spectrum Properties
f c f c
f m
,
f c
2
f m
, ,
f c
nf m
, 2. For
J
0 <<
J
1
J
1
J n f c
f c
f m
0 for all
n
2 3. Magnitude of 4. Carrier (
f c
nf m
:
A c J n
( ), depend on
f c
2 ) magnitude
J
0 ( ) can be 0 for some 5. Average power:
P
n
A c
2 1 2
J n
2 1 2
A c
2
Bandwidth of FM
Facts – FM has side frequencies extending to infinite frequency theoretically infinite bandwidth – But side frequencies become negligibly small beyond a point practically finite bandwidth – FM signal bandwidth equals the required transmission (channel) bandwidth Bandwidth of FM signal is approximately by – Carson’s Rule (which gives lower-bound)
Carson’s Rule
Nearly all power lies within a bandwidth of – For single-tone message signal with frequency f m
B T f
2
f m
2(
1)
f m
– For general message signal m(t) with bandwidth (or highest frequency) W
B
T
f
where
D
f W f
max
2
W
2(
D
1)
W
f