ERG2310A-II p. II-89
Noises
Sources of Noises: random wandering of free electrons in resistors (thermal noise), random flow of charges in semiconductor junction (shot noise), etc.
Noises : random fluctuations of power in systems
Corrupt the signal waveform
Degrade the performance of communication systems
Additive noise
Zero-mean White Gaussian-distributed Noise, power spectral density Sn(f)=η /2
Uncorrelated with the signal
Performance measures: Analog Signal-to-Noise Ratio (SNR)
Digital Bit-Error-Rate (BER)
Transmitter Receiver+Channel
Noisen(t)
x(t) y(t)
(AWGN)
s(t)
no(t)
ERG2310A-II p. II-90
Noises: Signal-to-Noise Ratio
Assume the channel introduces no distortion and receiver is linear
outputreceiver at the noise ofpower averageoutputreceiver thesignal message ofpower average=oSNR
)()()( tntsty oo +=
where so(t) and no(t) are the signal and noise waveforms at the receiver output.
2
2
o
oo
nsSNR =
Receiver+
Noisen(t)
y(t)
s(t)Received
signal
ERG2310A-II p. II-91
White Noise and Filtered Noise
White noise: flat spectral density over a wide range of frequencies
Sn(f)
η/2
f
Rn(τ)
η/2
τ
Filter H(f)n(t) nF(t)
22 )(2
)()()( fHfSfHfS nnF
η==
∫∫∞
∞−
∞
∞−
== dffHdffSnFnF
22 )(2
)( η
Filtered white noise:
f
SnF(f)
B-B f
SnF(f)
-fc fc
2B2B
Low-pass filter band-pass filter
η/2 η/2
ERG2310A-II p. II-92
Bandpass Noise
For bandpass systems, which limit the bandwidth of the noise, some of thenoise fluctuations are restricted.
As the bandwidth of the noise becomes small compared to the center frequency, it becomes possible to approximate it with a phasor representation.
Here nc(t) is that portion of the noise which is the in-phase component andns(t) is the quadrature component in the phasor representation.
ns(t)
nc(t)
an
θn
The complete phasor representation of the narrowband random noise istj
scoetjntn ω)]()([ +
where ωo is the center frequency.
ttnttnetjntntn osoctj
scFo ωωω sin)(cos)(})]()(Re{[)( −=+=
which is known as the bandpass representation of noise.
= −
)()(tan 1
tntn
c
snθ
ERG2310A-II p. II-93
Bandpass Noise
LPFnF(t) e(t)
cosωot
ttnttntttnttnttn
sc
scoF
021
021
0002
2sin)(]2cos1)[(cossin)(cos)(cos)(
ωωωωωω
−+=−=
The output of the low-pass filter is)(]cos)([)( 2
10 tnttnte cLPF == ω
The power spectral density of nc is
LP
TFTF
Tn
LPTF
T
Tc
Tn
TNN
S
Tttn
Ttn
S
c
c
++−=
ℑ=
ℑ=
∞→
∞→∞→
200
20
2
)()(lim)(
}]cos)([2{lim
)}({lim)(
ωωωωω
ωω
ERG2310A-II p. II-94
Bandpass Noise
For random noise, the average of the cross products goes to zero, thus
LPnnn FFcSSS )]()([)( 00 ωωωωω ++−=
The case for ns follows in the same manner by considering n(t)sin ωct, yielding
LPnnn FFsSSS )]()([)( 00 ωωωωω ++−=
Hence,LPnnnn FFsc
SSSS )]()([)()( 00 ωωωωωω ++−==
Thus, their mean-square values are equal;
)()()( 222 tntntn scF ==
The mean-square value of bandpass random noise is
)()()( 2212
212 tntntn scF += ttnttntn osocF ωω sin)(cos)()( −=Q
ERG2310A-II p. II-95
Bandpass Noise
ERG2310A-II p. II-96
Hence, and the spectral densities of nc(t) and ns(t)are
Bandpass Noise
Example: White noise with power spectral density η/2 is filtered by a rectangular bandpass filter with H(f)=1, centered at fo and having a bandwidth W. Find the power spectral density of nc(t) and ns(t). Calculate the power in nc(t), ns(t) and nF(t).
Solution: Since the filter is rectangular with H(f)=1, the power spectral density of the output noise n(t) is:
elsewhereWffWf
fS oonF
22
02
)(+≤≤−
=η
)()( onon ffSffSFF
−=+
2for 22)()()()( WfffSffSfSfS ononnn FFsc≤=+=++−== ηηη
Powers of nc(t) and ns(t) are
WdffSnnW
Wnsc c
η∫−
===2
2
22 )(
Power of nF(t) isWWdffSdffSn
Wf
Wf
Wf
WfnnF
o
o
o
o
ηη∫ ∫+−
−−
+
−
=
=+=
2
2
2
2
2
22)()(
ERG2310A-II p. II-97
Assume distortionless channel:
where K and td are the amplification/attenuation and time-delay, respectively.
Noise in Baseband Comm. Systems
Baseband transmission: signal is transmitted without any modulation.
Assume both the transmitter filter and receiver filter are ideal low-pass filters with bandwidth W (=2π B)
LPF LPF+Channel
Noisen(t)
x(t) xo(t), no(t)
(AWGN)
)()( do ttKxtx −=
Average output signal power: (received power)
BdfdffSnB
B
B
Bno ηη
∫∫−−
=
==
2)(2Average output noise power:
Ro Px =2
BP
SNR Ro η
=
ERG2310A-II p. II-98
Noise in Baseband Comm. SystemsExample: Consider an analog baseband communication system with additive white
noise. The transmission channel is assumed to be distortionless and the power spectral density of white noise is η/2 is 10-9 watt per hertz (W/Hz). The signal to be transmitted is an audio signal with 4-kHz bandwidth. At the receiver end, an RC low pass filter with a 3-dB bandwidth of 8kHz is used to limit the noise power at the output. Calculate the output noise power.
Solution: For a RC low-pass filter (3-dB bandwidth=8kHz), the frequency response is
)(11
1)/(1
)/(1)( RCjCjR
CjH oo
=+
=+
= ωωωω
ωω where
( )[ ] 2/121
1)(o
Hωω
ω+
= 3dB bandwidth at ω=ωo = ωo=2π(8000)
( )∫∫∞
∞−
∞
∞− +=
= ω
ωωπηωωη
πddHn
oo 2
22
11
21
2)(
221Average output noise power:
( ) ooon ηωπωπ
η41
21
22 ==
Wno µπ 2.25)108)(2)(10(2(41 392 =×= −
ERG2310A-II p. II-99
Noise in Amplitude Modulated Systems
The bandpass filter limits the amount of noise outside signal band that reaches the demodulator (“out-of-band” noise)
(fc-B) fc (fc+B)(-fc-B) fc (-fc+B) f0
S(f)+N(f)
(fc-B) fc (fc+B)(-fc-B) fc (-fc+B) f0
SF(f)+NF(f)
N(f)
ttnttntn csccF ωω sin)(cos)()( −=Narrowband filtered noise:
Bnnn Fsc η2222 ===Power spectral density:
BPF+
Noisen(t)
y(t)
s(t)
Received modulated
signalDetector LPF
Receiver
DemodulatornF(t)
sF(t)
no(t)
)(2 tsP FR =Received signal power:
ERG2310A-II p. II-100
Noise in DSB-SC Systems
In a DSB-SC system, the received modulated signal is:ttxAts cc ωcos)()( =
y(t)BPF+
Noisen(t)
s(t)
Received modulated
signalLPF
Receiver
DemodulatornF(t)
sF(t)×
cosωctConsider the demodulation by synchronous detection:
Input signal power to the demodulator: RcccF PtxAttxAts === )(21cos)()( 222222 ω
Input noise power to the demodulator:
ttnttntn csccF ωω sin)(cos)()( −=
BtnF η2 )(2 =
no(t)
where x(t) is the message signal which is band-limited with bandwidth B
(fc-B) fc (fc+B)(-fc-B) fc (-fc+B) f0
SF(f)+NF(f)
ttxAts ccF ωcos)()( =
ERG2310A-II p. II-101
Noise in DSB-SC Systems
Output signal power after LPF: )(4
)( 22
2 txAty c=
Output noise after LPF:
)()()( 2412
412 tntntn Fco ==
Output signal after LPF: [ ] )(2
cos)()( txAttsty cLPcF == ω
[ ][ ]{ }
)(2sin)(]2cos1)[(
cossin)(cos)(
cos)()(
21
21
21
2
tnttnttntttnttn
ttntn
c
LPcscc
LPccscc
LPcFo
=−+=
−=
=
ωωωωω
ω
Output noise power after LPF: 222Fsc nnn ==Q
SNR before demodulator:BP
nP
ntxA
nsSNR R
F
R
F
c
F
Fi η2
)(22
2221
2
2
====
iR
F
c
oo SNR
BP
tntxA
tn
tySNR 2
22
)()(
)(
)(2
41
2241
2
2
====η
Output SNR after demodulator:
The detector (demodulator) improves the SNR in a DSB-SC system by a factor of 2. This improvement results from the fact that the coherent detector rejects the quadrature noise components in the input noise, thereby halving the mean-square noise power.
BPSNR R
o η= for DSB-SC
ERG2310A-II p. II-102
Noise in DSB-SC SystemsExample: A DSB-SC system with additive white noise is demodulated by a
synchronous detector with a phase error φ, i.e. cos(ωct + φ). The original message signal is x(t). Show that where γ is the output SNR when the synchronous detector has no phase error.
Output signal power after LPF: φ222
2 cos)(4
)( txA
ty c=
Output noise after LPF:
( ))(
sincos)(
sin)(cos)()(
241
22241
224122
412
tn
tn
tntntn
F
F
sco
=
+=
+=
φφ
φφ
Output signal after LPF: ( )[ ] φφω cos)(2
cos)()( txA
ttsty cLPcF =+=
( )[ ][ ]φφ
φωsin)(cos)(
cos)()(
21
21 tntn
ttntn
sc
LPcFo
+=+=
Output noise power after LPF:
222Fsc nnn ==Q
φγφφ 222
22
241
22241
2
2
coscos)(
)()(
cos)()(
)(====
tntxA
tntxA
tn
tySNR
F
c
F
c
o
o
Output SNR after demodulator:
ttnttntn csccF ωω sin)(cos)()( −=ttxAts ccF ωcos)()( =
φγ 2cos=oSNR
)(
)(2
22
tn
txA
F
c=γ
Solution:
ERG2310A-II p. II-103
Noise in SSB Systems
In SSB system, the received modulated signal is:[ ]ttxttxAts ccc ωω sin)(ˆcos)()( +=
y(t)BPF+
Noisen(t)
s(t)
Received modulated
signalLPF
Receiver
DemodulatornF(t)
sF(t)×
cosωctConsider the demodulation by synchronous detection:
ttnttntn csccF ωω sin)(cos)()( −=
no(t)
where x(t) is the message signal which is band-limited with bandwidth B
(fc-B) fc (fc+B)(-fc-B) fc (-fc+B) f0
SF(f)+NF(f)[ ]ttxttxAts cccF ωω sin)(ˆcos)()( +=
Bnnn Fsc η=== 222
ERG2310A-II p. II-104
Noise in SSB Systems
Input noise power to the demodulator: BtnF η= )(2
Input signal power to the demodulator:
[ ][ ][ ]
Rc
c
ccc
cccccF
PtxA
txtxA
ttxttxA
tttxtxttxttxAts
==
+=
+=
++=
)(
)()(
sin)(ˆcos)(
cossin)(ˆ)(2sin)(ˆcos)()(
22
212
2122
22222
222222
ωω
ωωωω
ttnttntn csccF ωω sin)(cos)()( −=[ ]ttxttxAts cccF ωω sin)(ˆcos)()( +=
Output signal to the demodulator: [ ] )(2
cos)()( txAttsty cLPcF == ω
Input signal to the demodulator:Input noise to the demodulator:
Output noise to the demodulator: [ ] )(cos)()( 21 tnttntn cLPcFo == ω
Output signal power to the demodulator:
Output noise power to the demodulator:
)(4
)( 22
2 txAty c=
)()()( 2412
412 tntntn Fco ==
ERG2310A-II p. II-105
SNR before demodulator: BP
ntxA
nsSNR R
F
c
F
Fi η
===2
22
2
2 )(
iR
F
c
oo SNR
BP
tntxA
tn
tySNR ====
η)()(
)(
)(2
41
2241
2
2
Output SNR after demodulator:
Noise in SSB Systems
The detector (demodulator) does not improve the SNR in a SSB-SC system.
BPSNR R
o η= for SSB
ERG2310A-II p. II-106
Noise in DSB-LC (AM) Systems
In DSB-LC (AM) system, the received modulated signal is:[ ] ttxmAts cac ωcos)(1)( +=
Consider the demodulation by synchronous detection:
where x(t) is the message signal which is band-limited with bandwidth B
ttnttntn csccF ωω sin)(cos)()( −=
y(t)BPF+
Noisen(t)
s(t)
Received modulated
signalLPF
Receiver
DemodulatornF(t)
sF(t)×
cosωct
no(t)
[ ] ttxmAts cacF ωcos)(1)( +=
(fc-B) fc (fc+B)(-fc-B) fc (-fc+B) f0
SF(f)+NF(f)
ERG2310A-II p. II-107
Noise in DSB-LC (AM) Systems
Input noise power to the demodulator: BtnF η2 )(2 =
Input signal power to the demodulator:
ttnttntn csccF ωω sin)(cos)()( −=
Output signal to the demodulator: [ ] [ ])(1cos)()( 21 txmAttsty acLPcF +== ω
Input signal to the demodulator:Input noise to the demodulator:
Output noise to the demodulator: [ ] )(cos)()( 21 tnttntn cLPcFo == ω
[ ] ttxmAts cacF ωcos)(1)( +=
[ ][ ][ ]
R
ac
aac
cacF
PtxtxmA
txmtxmA
ttxmAts
==+=
++=
+=
signal) messagemean (zero 0)( assume )(1
)(2)(1
cos)(1)(
22221
22221
2222 ω
By removing the DC term, gives )()( 21 txmAty ac=
ERG2310A-II p. II-108
Noise in DSB-LC (AM) Systems
Output signal power to the demodulator:
Output noise power to the demodulator:
)(4
)( 222
2 txmAty ac=
)()()( 2412
412 tntntn Fco ==
SNR before demodulator: BP
nsSNR R
F
Fi η22
2
==
[ ]
BP
txmtxm
txmtxmA
Btxm
txmtxm
tntxmA
tn
tySNR
R
a
a
a
aca
a
a
F
ac
oo
η
η
⋅+
=
++⋅=
++⋅==
)(1)(
)(1)(1
)2()(
)(1)(1
)()(
)(
)(
22
22
22
22221
41
2221
22
22
241
22241
2
2
Output SNR after demodulator:
For DSB-LCBP
txmtxmSNR R
a
ao η
⋅+
=)(1
)(22
22
As to avoid distortion (over-modulation) 1)(22 ≤txma
BPSNR R
o η21 ≤⇒ SNRo of DSB-LC is least 3-dB worse than that in DSB-SC
and SSB systems
ERG2310A-II p. II-109
Noise of DSB-LC (AM) Systems
In DSB-LC (AM) system, the received modulated signal is:[ ] ttxmAts cac ωcos)(1)( +=
Consider the demodulation by envelope detection:
ttnttntn csccF ωω sin)(cos)()( −=
r(t)BPF+
Noisen(t)
s(t)
Received modulated
signalEnvelope detector
ReceiverDemodulator
nF(t)
sF(t)
where x(t) is the message signal which is band-limited with bandwidth B
Bnnn Fsc η2222 ===
[ ] ttxmAts cacF ωcos)(1)( +=
Input to the envelope detector:
[ ] [ ]ttnttnttxmAtntstf cscccacFF ωωω sin)(cos)(cos)(1)()()( −++=+=
f(t)
ERG2310A-II p. II-110
Noise of DSB-LC (AM) Systems
For (large input SNR):[ ] )( ),( )(1 tntntxmA scac >>+
[ ] [ ][ ]{ } ttnttntxmA
ttnttnttxmAtf
csccac
cscccac
ωωωωω
sin)(cos)()(1sin)(cos)(cos)(1)(
−++=−++=
Output of the envelope detector:
[ ]{ } )()()(1)( 22 tntntxmAtr scac +++= [ ] )()(1)(tan)( 1
tntxmAtnt
cac
s
++= −φ
)()()( tjetrtr φ=where
Input of the envelope detector:
;
[ ] )()(1)( tntxmAtr cac ++≈⇒
Removing the DC component, gives )()()( tntxmAtr cac +≈
which is basically the same as the output signal, y(t)+no(t), for synchronous detection of DSB-LC signals without the scaling factor ½.
Thus, for high SNR at receiver input, the performance of synchronous detector and envelope detector is the same.
ERG2310A-II p. II-111
Noise in DSB-LC (AM) Systems
For (small input SNR):[ ] )( ),( )(1 tntntxmA scac <<+
[ ]{ }[ ] [ ]
[ ] [ ]
[ ]
[ ]
[ ]
[ ])(1)()()(
smallfor 2
11 using )(1)()(1)(
)(1)(
)(21)(
)()()(let )(1)(
)(21)(
)(1 smallfor )(1)(2)()(
)(1)(2)()()(1
)()()(1)(
2
2
2222
2
22
2222
22
txmtvtnAtv
txmtvtnAtv
txmtvtnAtv
tntntvtxmtvtnAtv
txmAtxmtnAtntn
txmtnAtntntxmA
tntntxmAtr
an
ccn
an
ccn
an
ccn
scnan
ccn
acaccsc
accscac
scac
++=
+≈+
++≈
++=
+=
++=
++++≈
+++++=
+++=
ααα
Thus, for small SNR at receiver input, the signal and noise are no longer additive. The signal multiplied by noise cannot be distinguishable. No meaningful output SNR can be defined.
ERG2310A-II p. II-112
Noise in Amplitude Modulated Systems
Example: Assuming sinusoidal modulation, show that, in an AM system with envelope detection, the output SNR is given by
where ma is the modulation index for AM.
BP
mmSNR R
a
ao η2
2
2 +=
Solution: For sinusoidal modulation, ttx mωcos)( =
2122 cos)( == ttx mω
Using for AM or DSB-LC BP
txmtxmSNR R
a
ao η
⋅+
=)(1
)(22
22
( )( ) B
Pm
mSNR R
a
ao η
⋅+
= 221
212
1
BP
mmSNR R
a
ao η2
2
2 +=
Thus,
ERG2310A-II p. II-113
Noise in Amplitude Modulated Systems
Example: Consider an AM system with additive thermal noise having a powerspectral density η/2=10-12 W/Hz. Assume that the baseband message signal x(t) has a bandwidth of 4kHz and the amplitude distribution shown by the figure. The signal is demodulated by envelope detection and appropriate post-detection filtering. Assume modulation index ma=1.
fx(x)1
-1 0 1 x
Find the minimum value of the carrier amplitude Ac that will yield SNRo≥ 40dB.
ERG2310A-II p. II-114
Noise in Amplitude Modulated Systems
Solution:
[ ] ttxAts cc ωcos)(1)( +=
∫ ∫∞
∞−
=+−==61)1(2)()(
1
0
222 dxxxdxxfxtx x
For ma=1, B=4kHz, η/2=10-12 W/Hz ,
RRR
a
ao PP
BP
txmtxmSNR 9
71
31261
61
22
22
108)104)(102())(1(1))(1(
)(1)(
−− ×=
××⋅
+=⋅
+=
η
[ ] [ ] 2127
612
21222
21 ))(1(1)(1 ccacR AAtxmAP =+=+=with
For
[ ]VA
A
SNR
dBSNR
c
c
o
o
1031
10108
10
40
3
42127
971
4
−
−
×≥
≥×
≥
≥
Thus the minimum value of Ac required is 31mV.
ERG2310A-II p. II-115
Noise in Angle Modulated Systems
For message signal x(t), angle-modulated signal: [ ])(cos)( ttAts cc φω +=
)(
)()(
0
=
∫t
f
p
dxk
txkt ττφ
for PM
for FM
Input signal power to the demodulator:
The BPF (pre-detection filter) has a bandwidth of W=2(D+1)B where D is the frequency deviation ratio and B is the bandwidth of the message signal.
[ ] 221222 )(cos)( cccF AttAts =+= φω
Input noise power to the demodulator: ( ) WWtnF ηη =
= 2
2)(2
SNR before the demodulator:WA
tntsSNR c
F
Fi η2)(
)( 2
2
2
==
BPF Discriminator LPF
Noise
nF(t)s(t)
DemodulatorReceiver
sF(t)
y(t)
no(t)
n(t)
LimiterG(t)
ERG2310A-II p. II-116
Noise in Angle Modulated Systems
Let G(t) the input to the demodulator such that
[ ][ ])(cos)(
sin)(cos)()(tttR
ttnttnAtG
c
csccc
θωωω
+=−+=
The limiter suppresses any amplitude variation of G(t), thus in angle modulation, SNR is derived from consideration of θ(t) only.
σσσ
θ
smallfor tan )(
)()(
)(tan)( 1
≈+
≈
+= −
QtnA
tntnA
tnt
cc
s
cc
s
{ } { }ttnttnttAtntstG csccccFF ωωφω sin)(cos)()](cos[)()()( −++=+=
[ ])(
)(tan)( ; )()()( 122
tnAtnttntnAtRcc
sscc +
=++= −θwhere
To facilitate the analysis of the noise only, we can assume φ(t)=0
Thus,
ERG2310A-II p. II-117
Noise in Angle Modulated Systems
)(
)()(tnA
tntcc
s
+≈θ
For high input SNR, Ac >> |nc(t)|, Ac>>|ns(t)| , thus , )()(c
s
Atnt ≈θ
+=
c
sc A
tnttG )(cos)( ωαOutput noise from Limiter:
This operation is just like passing ns(t) through a differentiator with transfer response
)()( tndtd
Atn s
cd
α=Output noise from Discriminator (neglect DC term):
cAjH ωαω =)(
)()()( 2 ωωωsd nn SHS =
Thus, consider the relation of power spectral densities:
If the input noise is white with Sn(ω)=η/2,
[ ] ηηηωωωωω =+=++−= 2/2/)()()( LPcncnn SSSs
α=≈L
)( cAtR
ERG2310A-II p. II-118
Noise in Angle Modulated Systems
With and ,
Thus,
)()()( 2 ωωωsd nn SHS =
ηωαω 2
22
)(c
n AS
d=
cAjH ωαω =)( ηω =)(
snS
for2Wf ≤
Output signal from discriminator:
2
3222
2
22
22
2
2
38
21)(
21)(
c
B
Bc
B
Bno A
BdA
dStnd
ηαπωωπ
ηαωωπ
π
π
π
π
=== ∫∫−−
Output noise power after LPF (with bandwidth B):
+=
+= )(1)(1)( tx
ktx
kAty
c
fc
c
fcLc ω
αωω
ω
Neglecting the DC term, gives )()()()( 2222 txktytxkty ff αα =⇒=
Output SNR:
( ) ( ) 32
222
2322
222
2
2
8)(3
38)(
)()(
BtxkA
ABtxk
tntySNR fc
c
f
oo ηπηαπ
α===
ERG2310A-II p. II-119
Noise in Angle Modulated Systems
32
222
8)(3
BtxkA
SNR fco ηπ
=
If the message signal is sinusoidal (single-tone), i.e.
and produces a frequency deviation of ∆ω,
tatx mm ωcos)( =
)sincos()(
)(43421t
mm
cc ttAts
φ
ωω
ωω ∆+=FM signal:
Differentiating φ(t), gives))(cos()(
)(
0 43421
Q
t
t
fcc dxktAts
φ
ττω ∫+=ttxk mf ωωcos)( ∆=
Thus, ( ) ( ) ( )2212
212222 2cos)( fttxk mf ∆=∆=∆= πωωω
BP
BA
BfSNR Rc
o ηβ
η2
22
232
23 =
∆=⇒
BPSNR R
o ηβ 2
23= for FM
Note: When β=5, output SNR of FM system is 37.5 times that of a baseband system, but the bandwidth of FM system is about 12 times that of the baseband system.
ERG2310A-II p. II-120
SNR in AM & FM Systems
Under the most favorable conditions in AM, the modulation index is 100%,thus
For FM:
It shows that the output signal-to-noise ratio can be made much higher in FM than in AM by increasing the modulation index β.
An increase in β also increases the bandwidth so that FM systems providean improvement in signal-to-noise at the expense of an increase in bandwidth.
( )BP
BP
mmSNR RR
a
aAMo ηη 3
12 2
2
=+
=
( )BPSNR R
FMo ηβ 2
23=
To realize any signal-to-noise improvement in FM over AM we must have
0.47 31
23 2 >⇒> ββ
This condition is approximately the transition point between narrowband andwideband FM.
Narrowband FM provides no signal-to-noise improvement over AM.
ERG2310A-II p. II-121
Pre-emphasis and De-emphasis in FM
ηωαω 2
22
)(c
n AS
d= for
2Wf ≤
Recall: the power spectral density at the FM discriminator output
-W/2 0 W/2 f
)(ωdn
S
Noise power is larger at higher frequencies within the message bandwidth.
Pre-emphasis: the high-frequency components in the input message signal are emphasized at the transmitter before the noise is introduced.
De-emphasis: at the output of FM demodulator, the inverse operation is performed to de-emphasize the high frequency components
ERG2310A-II p. II-122
Pre-emphasis and De-emphasis in FM
Pre-emphasis filter
De-emphasis filter
FM transmitter
FM receiver
AWGN n(t)
x(t) output
HPE(f) HDE(f)
WfWfH
fHDE
PE <<−= , )(1)(
f f
/1
1)( ; 1)(o
DEo
PE fjfKfH
fjfKfH
+=
+=Example:
input outputR
Cinput output
R
rC
ωωπ
ηαωωπ
π
π
π
π
dA
dStnB
Bc
B
Bno d ∫∫
−−
==2
2
22
22
2
2
21)(
21)(Output noise power without de-emphasis filter:
Output noise power with de-emphasis filter:
ωω
ωπ
ηαωωωπ
π
π
π
π
dHA
dSH
tnB
B PEc
B
Bn
PEo d ∫∫
−−
==2
2
22
2
22
2
22
)(1
21)(
)(1
21)(
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