ECE 4371, Fall, 2009

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ECE 4371, Fall, 2009. Zhu Han Department of Electrical and Computer Engineering Class 5 Sep. 8 th , 2007. Phase-Locked Loop. Can be a whole course. The most important part of receiver.

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### ECE 4371, Fall, 2009

Zhu Han

Department of Electrical and Computer Engineering

Class 5

Sep. 8th, 2007

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.
Ideal Model
• Model
• 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

FM Basics
• 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
• Low bandwidth efficiency
• Complex implementations
• Applications
• TV sound signal
• 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
Phase Modulation
• PM (phase modulation) signal
Frequency Modulation
• FM (frequency modulation) signal

(Assume zero initial phase)

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

(Assume zero initial phase)

Example

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

m(t)

0 T 2T

Frequency deviation Δf

difference between the maximum instantaneous and carrier frequency

Definition:

Relationship with instantaneous frequency

Question: Is bandwidth of s(t) just 2Δf?

Frequency Deviation

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)
• Bandwidth

A

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)
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
• Fourier series representation
Spectrum of WBFM
• Spectrum when m(t) is single-tone
• Example 2.2
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 fm
• For general message signal m(t) with bandwidth (or highest frequency) W