ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J (...
Transcript of ENSC327 Communications Systems 10: Wideband FM · (Assignment) 6 Single tone FM Spectrum s t A J (...
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ENSC327
Communications Systems
10: Wideband FM
Jie Liang
School of Engineering Science
Simon Fraser University
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Outline
� 4.5 Wideband FM
� Bessel Function representation of single tone
message
� 4.6 BW of FM signals
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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 +=
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4.5 Wide-band FM
:)(~)2sin( tfj
cmeAts
πβ=
� Recall Fourier series:
� Fourier series representation of
)2sin()(~
tfj
cmeAts
πβ=
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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)
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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
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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.
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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
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Bessel functions
� 4. Power distribution:
Proof:
.1)(2 =∑∞
−∞=n
nJ β
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Outline
� 4.5 Wideband FM and Bessel Function
� 4.6 BW of FM signals
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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=
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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
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Bandwidth of Single Tone FM
β
max2n
f
B=
∆
Example: if β = 5, fm = 15 kHz
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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
∆+∆− ,
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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:
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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.
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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
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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: