1. Department of Electrical and Computer Engineering
ECE 4371, Fall, 2009
                                                           
Zhu Han
Department of Electrical and Computer Engineering
Class 5
Sep. 8th, 2007

2. 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.

3. Ideal Model Model LPF VCO Capture Range and Lock Range
Si=Acos(wct+1(t)), Sv=Avcos(wct+c(t))
Sp=0.5AAv[sin(2wct+1+c)+sin(1-c)]
So=0.5AAvsin(1-c)=AAv(1-c)
Capture Range and Lock Range
LPF
VCO

4. Type of waves

5. GPS Orbits

6. 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

7. 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

8. 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

9. 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

10. Phase Modulation
PM (phase modulation) signal

11. Frequency Modulation FM (frequency modulation) signal
(Assume zero initial phase)

12. FM Characteristics Characteristics of FM signals
Zero-crossings are not regular
Envelope is constant
FM and PM signals are similar

13. Relations between FM and PM

14. FM/PM Example (Time)

15. FM/PM Example (Frequency)

16. 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)')

17. 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([ min(Pm) max(Pm)]), grid, title('message spectrum M(f)'),
subplot(312), plot(w,Pf,'r'),
axis([ min(Pf) max(Pf)]), grid, ylabel('FM S(f)')
subplot(313), plot(w,Pp,'m'),
axis([ min(Pp) max(Pp)]), grid, ylabel('PM S(f)')

18. Frequency Modulation FM (frequency modulation) signal
(Assume zero initial phase)

19. Example
Consider m(t)- a square wave- as shown. The FM wave for this m(t) is shown below.
m(t)
T T

20. Frequency Deviation Frequency deviation Δf
difference between the maximum instantaneous and carrier frequency
Definition:
Relationship with instantaneous frequency
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).

21. Modulation Index
Indicate by how much the modulated variable (instantaneous frequency) varies around its unmodulated level (message frequency)
Bandwidth A

22. Narrow Band Angle Modulation
Definition
Equation
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)

23. Example

24. Block diagram of a method for generating a narrowband FM signal.

25. A phasor comparison of narrowband FM and AM waves for sinusoidal modulation. (a) Narrowband FM wave. (b) AM wave.

26. Wide Band FM
Wideband FM signal
Fourier series representation

27. Example

28. Bessel Function of First Kind

29. Spectrum of WBFM
Spectrum when m(t) is single-tone
Example 2.2

30. Spectrum Properties
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31. Bandwidth of FM Facts Bandwidth of FM signal is approximately by
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)

32. Carson’s Rule Nearly all power lies within a bandwidth of
For single-tone message signal with frequency fm
For general message signal m(t) with bandwidth (or highest frequency) W