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