ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J (...

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1 ENSC327 Communications Systems 10: Wideband FM Jie Liang School of Engineering Science Simon Fraser University

Transcript of ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J (...

Page 1: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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ENSC327

Communications Systems

10: Wideband FM

Jie Liang

School of Engineering Science

Simon Fraser University

Page 2: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Outline

� 4.5 Wideband FM

� Bessel Function representation of single tone

message

� 4.6 BW of FM signals

Page 3: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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4.5 Wide-band FM

� Finding its FT is not easy: ϕ(t) is inside the cosine.

� To analyze the spectrum, we use complex envelope.

� s(t) can be written as:

� Consider single tone FM:

))(2cos()( ttfAtscc

φπ +=

))2sin(2cos()( tftfAtsmcc

πβπ +=

Page 4: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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4.5 Wide-band FM

:)(~)2sin( tfj

cmeAts

πβ=

� Recall Fourier series:

� Fourier series representation of

)2sin()(~

tfj

cmeAts

πβ=

Page 5: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bessel Function dteAfcm

m

mm

f

f

tnfjtfj

cmn ∫−

=

)2/(1

)2/(1

2)2sin( ππβ

� Define

� Jn(β): n-th order Bessel function of the first kind with argument β

:2 tfxmπ=

( )dxeJ

nxxj

n ∫−

=

π

π

β

πβ sin

2

1)(

==)2sin(

)(~tfj

cmeAts

πβ��

� Single tone FM signal can be written as:

Property: Jn(β) has real value!

(Assignment)

Page 6: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Single tone FM Spectrum( ) tnffJAts

mc

n

nc 2cos)()( += ∑

−∞=

πβ

( )( ) ( )( )[ ]mcmc

n

ncnfffnfffJAfS ++++−= ∑

−∞=

δδβ )(2

1)(

� Single-tone FM spectrum contains a carrier component

and infinite numbers of discrete side freqs at

� Theoretically the BW of FM is infinite.

mcnff ±

)(2 fS

Page 7: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Properties of Bessel functions

( )dxeJ

nxxj

n ∫−

=

π

π

β

πβ sin

2

1)(

� Bessel function table on Page 467

� Other Properties:

� 1. Jn(β) = J-n(β) for even n. Jn(β) = - J-n (β) for odd n.

� 2. For small value of β: J0(β) ~1, J1(β) ~ β/2, Jn(β) ~ 0, n>1.

Page 8: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bessel functions

� 3. Zeros of Bessel functions:

Jn(β) = 0 at some n and β.

When J0(β)=0 for some β�

( ) tnffJAtsmc

n

nc 2cos)()( += ∑

−∞=

πβ

2.45.5 8.6

Page 9: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bessel functions

� 4. Power distribution:

Proof:

.1)(2 =∑∞

−∞=n

nJ β

Page 10: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Outline

� 4.5 Wideband FM and Bessel Function

� 4.6 BW of FM signals

Page 11: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bandwidth of Single Tone FM

� Another property:

� So we can define the bandwidth by considering the

region with significant power.

� BW of single tone FM:

Separation between two freqs

beyond which all |Jn(β)| < 0.01.

� Can use Bessel function table to find the value of nmax that

satisfies the threshold requirement.

� The corresponding bandwidth:

.0)(lim =

∞→

βn

n

J

mfnB

max2=

mfnB

max2=

Page 12: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bandwidth of Single Tone FM

� Examples of (Table 4.2)

mfnB

max2=

β 2nmax

0.1 2

0.3 4

0.5 4

1.0 6

2 8

5 16

max2n

Page 13: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bandwidth of Single Tone FM

β

max2n

f

B=

Example: if β = 5, fm = 15 kHz

Page 14: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bandwidth of Single Tone FM

� Figure of (Fig 4.9, pp. 171)β

max2n

f

B=

� It can be seen that B / ∆f is decreasing and

approaches to 2 as the increase of β.

� Therefore the bandwidth B approaches 2∆f when β

is large.

� Recall that the range of the instantaneous frequency

is 2∆f: [ ]ffffcc

∆+∆− ,

Page 15: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Effect of Amplitude on the BW

� Fix message freq fm, change amplitude Am

mf Akf =∆ mmfm fAkff // =∆=β

mff =∆= or 1β

mff 2or 2 =∆=β

mff 5or 5 =∆=β

� Observations:

Page 16: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Bandwidth of Single Tone FM

� Effect of freq fm on BW:

� Fix Am (or ∆f),

� change message freq fm

mf Akf =∆mff /∆=β

ffm

∆== ,1β

2/ ,2 ffm

∆==β

5/ ,5 ffm

∆==β

� Observations:

� The bandwidth is still

about 2∆f .

� Closer spectral lines.

Page 17: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Carson’s Rule� For single tone message, the Carson’s rule to estimate

the BW of the FM signal is:

� For arbitrary messages with bandwidth W:

=+∆≈mTffB 22

))(22cos()(0∫+=

t

fcc dmktfAts ττππ

We know that the freq deviation is:

� The Carson’s rule to estimate the FM bandwidth is:

We can define Deviation Ratio:

.)(max)(

2

1max tmk

dt

tdf f==∆

φ

π

Page 18: ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J ( )f c nf m t n ( ) = c ∑ n ( )cos2 + ∞ =−∞ β π [ ]( ) ( )( ) c m ( ) c

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Carson’s Rule ))(22cos()(0∫+=

t

fcc dmktfAts ττππ

� Example: In FM radio, the max message bandwidth is W =

15kHz, and the allowed max freq deviation is ∆f = 75 KHz: