lecture 9 signal noise slidesdas
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LECTURE 9:
NOISE IN
CONTINUOUS-WAVE
MODULATION SYSTEMS
Dr. N. Das and A/Prof Zhuquan Zang
Dept of Electrical and Computer
Engineering
Curtin UniversityPerth, Western Australia
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LECTURE 9: NOISE INCONTINUOUS-WAVE
MODULATION SYSTEMS
In performing the noise analysis of a communica-
tion system, the customary practice is to assume
that the noise n(t) is additive, white and Gaus-
sian noise (AWGN ).
Let the double-sideband power spectral density
(PSD) S n(f ) of n(t) be N o/2, as shown below:
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Note:
N o is the average noise power per unit bandwidthmeasured at the front end of the receiver, which
has a bandwidth of 2B Hz centered at the fre-quency f c.
With f c >> 2B, the narrowband noise can berepresented as
n(t) = nI
(t) cos(2πf ct)−
nQ
(t) sin(2πf ct)
where nI (t) is the in-phase and nQ(t) the quadra-ture noise component, both measured with re-spect to the carrier frequency f c.
The PSD of nI (t) and nQ(t) is
S nI (f ) = S nQ(f ) =
S n(f + f c) + S n(f − f c) |f | ≤ B0, |f | > B
=
N 0
2 + N 0
2 = N 0 |f | ≤ B
0, |f | > B
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Now, the average bandpass noise power is
n2 = 2 f c+B
f c−
B
N 02
df = 2N 0B
The average baseband noise power of nI (t) and
nQ(t) is
n2I = n2
Q = 2 B
0N 0df = 2N 0B
Therefore, n2
I = n2
Q = n2
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Noise in DSB-SC-AM Receivers
A double sideband suppressed-carrier amplitude-
modulated (DSB − SC − AM ) signal c(t) is rep-
resented by
c(t) = Acm(t) cos(2πf ct)
where Ac is the carrier amplitude, and m(t) is the
modulating signal with a bandwidth of B Hz.
White Gaussian noise n(t) (e.g., arising from the
front end stages of the receiver) is added to the
received signal c(t), and the resultant compos-ite signal of [c(t) + n(t)] is then detected by a
coherent product demodulator as shown:
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The input to the product demodulator is
vi(t) = c(t) + n(t) =
= Acm(t) cos(2πf ct) + nI (t) cos(2πf ct) −− nQ(t) sin(2πf ct) =
= [Acm(t) + nI (t)] cos(2πf ct) − nQ(t) sin(2πf c
The local carrier ν L(t) is assumed phase-locked
to the transmitted carrier, i.e.,
ν L(t) = cos(2πf ct)
.
Therefore, the output of the demodulator is given
by
vo(t) = vi(t) × vL(t)
= [Acm(t) + nI (t)] cos2(2πf ct)−− nQ(t) sin(2πf ct) cos(2πf ct)
= [Acm(t) + nI (t)]12[1 + cos(4πf ct)]−
−1
2nQ(t) sin(4πf ct)= [Acm(t)+nI (t)]
2 + 12{[Acm(t) + nI (t)] cos(4πf c
− nQ(t) sin(4πf ct)}.
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After lowpass filtering by LPF, the demodulated
output becomes
vo(t) = k[Acm(t) + nI (t)]
where k is an amplitude constant.
Now, the performance of a demodulator is de-
termined by comparing its output signal-to-noise
power ratio (SNR o) with is input signal-to-noise
power ratio (SNR I ).
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(i) Determination of SNRI
The input modulated carrier power is given by
P si = limT →∞
12T
T −T c2(t)dt =
= limT →∞
12T
T −T A2
c m2(t)2 [1 + cos(4πf ct)]dt =
= limT →∞
12T
T −T A2
c m2(t)2 dt = P cP m
where P c = A2c /2 is the carrier power, and P m is
the average power of m(t).
The input noise power is
P ni = limT →∞
1
2T
T
−T n2(t)dt = 2
f c+B
f c−B
N o
2 df = 2N oB
where N o/2 is the double-sideband power spec-
tral density of n(t).
The signal-to-noise ratio at the input of the de-
modulator is given by
SN RI = P si
P ni=
P cP m
2N oB.
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(ii) Determination of SNRo
The output signal power is given by
P so = limT →∞
12T
T
−T [kAcm(t)]2dt = 2k2P cP m,
and the output noise power is given by
P no = limT −∞
1
2T
T
−T
[knI (t)]2dt = k2
×2
B
0
N odf = k2
×
The signal-to-noise ratio at the output of
the demodulator is
SN Ro = P so
P no
= 2k2P cP m
2k2
N oB
= P cP m
N oB
.
Therefore, SN Ro = 2 × SN RI or
SN Ro(dB) = SN RI (dB) + 3dB.
Note: The coherent demodulation of DSB-SC-AM signal achieves a 3dB improvement in the
output signal-to-noise power ratio at the expense
of requiring twice the baseband bandwidth for its
transmission.
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Envelope Detector in the Presence of Noise
An envelope detector is commonly used to de-
modulate a DSB-AM signal represented by
c(t) = [m(t) + Ac] cos(2πf ct)
where Ac and f c are the carrier amplitude and fre-
quency, respectively, and m(t) is the modulatingsignal of bandwidth B Hz and |m(t)|≤Ac.
The received DSB-AM signal c(t) is corrupted
by white Gaussian noise n(t) as shown below:
The input to the envelope detector is given by
vi(t) = c(t) + n(t) == [m(t) + Ac] cos(2πf ct) + nI (t) cos(2πf ct)−− nQ(t) sin(2πf ct).
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Now, rewrite nI (t) = x, nQ(t) = y, and m(t) =
m, then
vi(t) = [m + Ac + x] cos(2πf ct) − y sin(2πf ct).
This can be expressed in the polar form, such
that
vi(t) = e(t) · cos[2πf ct + ϕ(t)]
where e(t) ≡ envelope of v i(t) = [(m+Ac +x)2+
y2]12,
and ϕ(t) = tan−
1 y
m+Ac+x.
The envelope detector extracts e(t) out from
v i(t), and after lowpass filtering by LPF, the out-
put becomes
vo(t) = e(t) = [(m + Ac + x)2 + y2]12
= [A2c + m2 + x2 + y2 + 2Acx + 2Acm + 2mx]
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To simplify the analysis, consider the followingtwo extreme conditions:
(i) Input carrier-to-noise ratio CNR I is very large,i.e., Ac>>x, and y , and that the modulationindex is small, i.e., Ac>>m.
(ii) CNR I being very small, i.e., Ac<<x , and y .
Condition (i) Rewrite v o(t), such that
vo(t) = Ac
1 +
m
Ac
2
+
x
Ac
2
+
y
Ac
2
+ 2m
Ac+
2x
Ac+
2mx
A2c
Since Ac>>x , and y , and also Ac>>m, then v o(t)
can be approximated by
vo(t) ∼= Ac
1 +
2(m + x)
Ac
12
.
Applying the Binomial expansion,
vo(t) = Ac
1 + 1
22(m+x)
Ac +
12
12−1
2!
2(m+x)
Ac
2+ ...
= Ac
1 + (m+x)
Ac − 1
4
m+x
Ac
2+ ...
∼= Ac + m + x ≡ [Ac + m(t)] + nI (t).
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Note:
Under very good CNR i, the output v o(t) of the
envelope detector is the same as that of the
product demodulator.
Condition (ii)
Rewrite v o(t) in the form
vo(t) = (x2 + y2)12
1 +
A2c +m2+2Acm+2x(Ac+m)
x2+y2
12
≡ (x2 + y2)12[1 + z]
12.
Under the condition of low CNR I , z <1, and using
Binomial expansion,
vo(t) ∼= (x2 + y2)12
1 + 1
2z + ...
= (x2 + y2)12 +
A2c +m2+2Acm+2x(Ac+m)
2(x2+y2)12
.
Note that under the condition of low CNR I , v o(t)
does not contain the modulating signal m(t) ex-
plicitely. It can be shown that as the CNR I falls
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below about 10 dB , the performance of the en-
velope detector begins to deteriorate rapidly, in-dicating the presence of threshold effect.
Note: The actual location of the threshold de-
pends on the diode law.
There is no threshold effect with the product
demodulator.
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DSB-SC-AM versus DSB-AM
The product demodulator can be used for de-
tecting both DSB-SC-AM and DB-AM signals,
while the envelope detector is only applicable for
DSB-AM.
If a product demodulator is used, then for equal
average power in the sidebands for both the DSB-
SC-AM and DSB-AM signals, the output SNR
will be the same in both cases, i.e., SNRo =2SN RI .
Now, for a given average power of the transmit-
ter, the carrier power present in the DSB-AM
signal will help to increase the sideband power in
DSB-SC-AM.
For example, the voltage spectrum of a DSB-AM
signal for modulation index M=1 is as shown:
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The total transmitted power
P t = Ac
4√ 22
+ Ac
4√ 22
+ Ac
2√ 22
=
3A2c
16
The transmitted sideband power with DSB-AM
is
P DSB
−AM =
Ac
4√ 22
+ Ac
4√ 22
= A2
c
16
.
The transmitted sideband power with DSB-SC-
AM is
P DSB−SC −AM = P t = 3A2
c
16 .
Under this condition,i.e., M=1, the resulting out-
put SNR will be
SN Ro(DSB−SC −AM ) = 3 × SN Ro(DSB−AM ).
Note:
The gain in SNR for DSB-SC-AM is even greater
for smaller values of modulation index M .
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Noise performance of a frequency discrimi-
nator
For noise analysis, the frequency discriminator is
modelled as a phase detector (PD) followed by
a differentiator.
A frequency-modulated (FM) signal is represented
as
c(t) = Ac cos[2πf ct + φc(t)]
where Ac and f c are the carrier amplitude and fre-
quency,respectively, and the information bearingphase function is given by
φc(t) = kf
t
−∞m(τ )dτ
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where m(t) is the modulating signal, and k f is
the modulator constant.
A frequency discriminator extracts from c(t) the
differential phase term, i.e., d[φc(t)]dt .
The predetection bandpass filter BPF has a band-
width of B Hz (usually determined using the Car-
son’s rule) for minimising noise.
At the output of BPF, the bandlimited noise isrepresented by its quadrature components, such
that
n(t) = nI (t) cos(2πf ct) − nQ(t) sin(2πf ct)
where nI (t) and nQ(t) are white Gaussian noisewaves of power P n and bandwidth B/2.
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To simplify analysis, assume φc(t) = 0.
This is justified as it can be shown that for rea-
sonably large value of modulation index, e.g.,β > 3, modulation does not affect the perfor-
mance of the detector.
Now, the output of the BPF is
vi(t) = c(t) + n(t) == Ac cos(2πf ct) + nI(t) cos(2πf ct)−− nQ(t) sin(2πf ct) == e(t) cos[2πf ct + θ(t)]
where e(t) is the envelope of v i(t) given by
e(t) = {[Ac + nI (t)]2 + n2Q(t)}1
2
and θ(t) = tan−1
nQ(t)
Ac+nI (t)
.
The limiter suppresses the amplitude variation
e(t), and its output becomes
v1(t) = cos
2πf ct + tan−1
nQ(t)
Ac + nI (t)
.
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The phase detector output for the information
bearing phase function is
v2(t) = k tan−1
nQ(t)
Ac + nI (t)
where k is the phase detector constant, and can
be assumed to be equal to 1 without loss of generality.
For large carrier-to-noise ratio, Ac + nI (t) ∼= Ac,
so that
v2(t) ∼= tan−1n
Q(t)
Ac
∼= nQ
(t)
Ac.
The noise power at the output of the phase de-
tector is given by
P n,PD = E [v2(t)]2 = E [n
2
Q(t)]A2
c= P n
A2c
.
(Note: The output of the phase detector con-
tains only noise because it has been assumed that
φc(t) = 0).
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Since nQ(t) is white and Gaussian, then the power
spectral density (PSD) of the noise at the out-
put of the phase detector is also white as given
by
P SDP D = P n,PD
B =
P n
A2c B
.
Now, for the frequency discriminator (FD), the
output of the phase detector is differentiated.
The transfer function of a differentiator is given
by
H (f ) = j2πf.
Therefore, the PSD of the noise at the output
of a frequency discriminator is
P SDF D = |H (f )|2 · P SDP D =
=
4π2f 2P n
A2c B
for
−B2 ≤
f
≤ B
2
0 elsewhere.
Note that the PSD of noise at the output of a
frequency discriminator is parabolic.
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The noise power at the output of the lowpassfilter LPF, assumed ideal with a bandwidth of f m, is
P no = f m
−f mP SDF Ddf =
f m
−f m
4π2f 2P n
A2c B
df = 8π2f 3mP n
3A2c B
.
To determine the signal power, it is customaryto assume that the modulating signal m(t) is si-
nusoidal of frequency f m, such thatm(t) = Am cos(2πf mt).
Further, assume that this modulating signal m(t)yields a peak frequency deviation ∆f .
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If the phase detector constant is 1 volt/radian,
and the differentiator giving an output of 1 volt
per volt/sec , the same assumptions as adopted
for the noise calculations, then the frequency dis-criminator constant becomes 2π volt/Hz.
Hence, the signal amplitude at the output of the
frequency discriminator is 2π∆f volts, and the
signal power is
P s,FD =
2π∆f √
2
2
= 2π2∆f 2.
The signal-to-noise ratio at the output of the
frequency discriminator becomes
SNRo = P s,FD
P no=
2π2∆f 2
8π2f 3mP n3A2
c B
=
= 3A2
cB∆f 2
4f 3mP n = 3∆f 3
f 3m
Ac√ 2
2
P n = 3β3 · SN RI
where B = 2∆f , and the modulation index β =∆f f m
.
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Express in decibel, the output signal-to-noise
ratio is
SN Ro(dB) = SN RI (dB)+30log10(β) + 4.77 dB.
Notes:
• The improvement in output signal-to-noise
ratio for FM is achieved at the expense intransmission bandwidth which increases with
modulation index β.
• The above analysis assumes a sinusoidal mod-
ulating signal yielding a peak deviation ∆f .However, for a noise-like modulating signal
yielding the same frequency deviation, its mean
square amplitude or power is only 2/9 time
that of a sinusoidal signal. This corresponds
to a decrease of about 6.5 dB in SNR o for
noise-like modulating signal.
• The SN Ro obtained from the above analysis
is reasonably accurate for β ∼= 3. For smaller
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values of β, FM becomes quasi-linear, i.e.,
narrowband FM, and its performance then
approaches that of DSB-SC-AM demodulated
using a coherent product demodulator.
• The above performance analysis assumes the
input carrier-to-noise ratio CNR I is high. The
analysis for low CNR I is very difficult. At low
values of CNR I , the noise amplitude is suf-ficiently large to cause the addition or dele-
tion of a pair of zero crossings to be inserted
into the detected carrier. This gives rise to
positive or negative impulse of area 2 π radi-
ans depending whether an extra cycle having
been slipped in or deleted. The effect is sig-
nificant deterioration in SNR o.
It has been shown that as the CNR I falls, the
number of impulses per second increases as
given by
No. of impulses/sec ≈ exp(−SN RI )
2 m(t).
For a frequency discriminator, this effect oc-
curs at an CNR I value of about 10 dB , which
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is normally taken as the threshold of the dis-
criminator.
• A phase-locked loop exhibits a much lower
impulse rate than a conventional frequency
discriminator. Consequently, the threshold
of a PLL FM demodulator can be up to 10 dB lower than a conventional frequency dis-
criminator.
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Noise performance of the FM discriminator
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Pre-emphasis and de-emphasis in FM
The PSD of noise at the output of a frequencydiscriminator is emphasised at the higher fre-
quencies, being proportional to f 2.
However, audio signals, such as speech and mu-sic, are found to have their energy concentrated
in the lower frequency ranges.
If nothing is done, this will produce unacceptably
low output SNR at the high frequency portion of
the audio signal.
One way to compensate this problem is to artifi-
cially emphasise the high-frequency components
of the audio input at the transmitter before thenoise is introduced.
In order to recover the original audio signal, the
inverse operation, called de-emphasis, is performed
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at the output of the frequency discriminator in
the receiver.
Further, in this de-emphasis process, the high-
frequency components of noise are being reduced.
Pre-emphasis
A transfer function suitable for emphasising hf
components is given by
H e(f ) = 1 + j f
f o
where f o is the break frequency above which the
components are emphasised.
In practice, H e(f) is approximated by a simple
RC network:
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This network has two break frequencies, f 1 and
f 2 given by
f 1 = 1
2πR1C
and f 2 ∼= 1
2πR2C with R 1>>R 2.
For broadcast FM, R 1C=75 µsec , so that f 1=2.1kHz .
The second break frequency f 2 is introduced to
prevent emphasising frequency components higher
than the audio range.
For this reason, f 2 is normally chosen to be f 2 ≤30 kHz .
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De-emphasis
The de-emphasis network, following the frequency
discriminator, must have an inverse characteris-tic given by
H d(f ) = 1
1 + j f f o
.
The transfer function H d(f) can be realised usinga simple RC-network:
The break frequency f 1 = 12πC R = 2.1kH z, with
CR=75 µsec for broadcast FM.
The noise power after de-emphasis is given by
P no,d = f m−f m
P SDF D|H d(f )|2df
= f m−f m
4π2f 2P nA2
c B1
1+4π2C 2R2df .
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The decrease in noise power with de-emphasis
as compared to the case with no de-emphasis
corresponds to the improvement in output SNR .
With CR=75 µsec , and f m=15 kHz , an improve-
ment in output SNR of 13 dB (or 20 times) is
achieved using pre-emphasis and de-emphasis in
FM.