# Noise

```Chap 6
Noise in Analog Modulation
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
1
Outline
◊
6.1 Introduction
◊
◊
63 N
6.3
Noise
i iin DSB-SC
DSB SC R
i
◊
◊
2
Yu-sing Lin (林育星)
Chap 6.1
Introduction
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
3
6.1 Introduction
◊
To undertake an analysis of noise in continuous-wave
(CM) modulation systems, we need a receiver model.
◊
The customary practice is to model the receiver noise
(channel noise) as additive, white, and Gaussian. These
simplifying assumptions enable us to obtain a basic
understanding of the way in which noise affects the
performance
f
off th
i
4
Yu-sing Lin (林育星)
Chap 6.2
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
5
◊
◊
◊
◊
s(t) denotes the incoming modulated signal.
w(t) denotes front-end receiver noise. The power spectral density of
the noise w(t) is denoted by N0/2, defined for both positive and
negative frequencies. N0 is the average noise power per unit
b d idth measuredd att the
bandwidth
th front
f t endd off the
i
The bandwidth of this band-pass filter is just wide enough to pass
the modulated signal without distortion.
distortion
Assume the band-pass filter is ideal, having a bandwidth equal to the
transmission bandwidth BT of the modulated signal s(t) , and a mid
midband frequency equal to the carrier frequency fc , fc &gt;&gt; BT .
6
Yu-sing Lin (林育星)
Idealized characteristic of band-pass filtered noise.
◊
The filtered noise n(t) may be treated as a narrow band noise
represented in the canonical form:
n ( t ) = nI ( t ) cos ( 2πf c t ) − nQ ( t ) sin ( 2πf c t )
( 6.1)
where nI((t)) is the in-phase
p
noise component
p
and nQ((t)) is the
quadrature noise component, both measured with respect to the
carrier wave Accos(2πfct).
Yu-sing Lin (林育星)
7
◊
The filtered signal x(t) available for demodulation is defined by
x (t ) = s (t ) + n (t )
◊
◊
The average noise power at the demodulator input is equal to the
total area under the curve of the power spectral density SN( f ):
N0
Pavg-noise = 2 &times; BT &times;
= BT N 0
2
Input signal-to-noise ratio (SNR)I is defined as:
( SNR ) I
◊
( 6.2 )
=
average power of the modulated signal s ( t )
average power of the filtered noise n ( t )
Output signal-to-noise ratio (SNR)O is defined as:
( SNR )O =
average power of the demodulated message signal
average power of the noise
8
Yu-sing Lin (林育星)
◊
Channel signal-to-noise ratio
( SNR )C
◊
=
average power of the modulated signal
average power of noise in the message BW
This ratio
Thi
i may be
b viewed
i
d as the
h signal-to-noise
i l
i ratio
i that
h results
l
from baseband (direct) transmission of the message signal m(t)
without modulation,
modulation as demonstrated in the following figure:
◊
◊
The message power at the low-pass filter input is adjusted to be the same
as the average power of the modulated signal
The low-pass filter passes the message signal and rejects out-of-band
noise.
9
Yu-sing Lin (林育星)
◊
Figure of merit
◊
◊
◊
For the purpose of comparing different continuous-wave (CW) modulation
systems, we normalize the receiver performance by dividing the output
signal-to-noise
signal
to noise ratio by the channel signal
signal-to-noise
to noise ratio.
The higher the value of the figure of merit, the better will the noise
The figure of merit may equal one, be less than one, or be greater than one,
depending on the type of modulation used.
( SNR )O
Figure of merit=
( SNR )C
10
Yu-sing Lin (林育星)
Chap 6.3
Noise in DSB
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
11
◊
The model of a DSB-SC receiver using a coherent detector
Figure
g
6.4
◊
◊
The amplitude of the locally generated sinusoidal wave is
assumed to be unity
unity.
For the demodulation scheme to operate satisfactorily, it is
necessary
ecessa y tthat
at tthee local
oca osc
oscillator
ato be synchronized
sy c o ed both
bot in phase
p ase
and in frequency with the oscillator generating the carrier wave in
the transmitter. We assume that this synchronization has been
achieved.
12
Yu-sing Lin (林育星)
◊
◊
◊
The DSB-SC component of the modulated signal s(t) is expressed
as
s ( t ) = CAc cos ( 2π f ct ) m ( t )
( 6.4 )
where C is the system dependent scaling factor
factor. The purpose of
which is to ensure that the signal component s(t) is measured in
the same units as the additive noise component n(t).
m(t) is the sample function of a stationary process of zero mean,
whose power spectral density SM( f ) is limited to a maximum
frequency W, i.e. W is the message bandwidth.
The average power P of the message signal is the total area under
th curve off power spectral
the
t l density
d it
W
P = ∫ S M ( f ) df
−W
13
Yu-sing Lin (林育星)
◊
In Example 5.12: Mixing of a Random Process with a Sinusoidal
Process
Y ( f ) = X ( t ) cos ( 2π f c t + Θ )
RY (τ ) =
◊
◊
◊
1
RX (τ ) cos ( 2π f cτ )
2
1
SY ( f ) = ⎡⎣ S X ( f − f c ) + S X ( f + f c ) ⎤⎦
4
Therefore, the average power of the DSB-SC modulated signal
component s(t) is given by: C 2 Ac2 P
2
The average power of the noise in the message BW is WN0
The channel signal-to-noise ratio of the DSB-SC modulation
system is:
C 2 Ac2 P
(SNR )C ,DSB =
2WN 0
14
Yu-sing Lin (林育星)
◊
◊
Next, we wish to determine the output signal-to-noise ratio.
Usingg the narrowband representation
p
of the filtered noise n(t),
( ), the
total signal at the coherent detector input may be expressed as:
x (t ) = s (t ) + n (t )
= CAc cos ( 2π f c t ) m ( t ) + nI ( t ) cos ( 2π f c t ) − nQ ( t ) sin ( 2π f c t )
◊
( 6.7 )
The output of the product-modulator
product modulator component of the coherent
detector is:
υ ( t ) = x ( t ) cos ( 2π f c t )
cos(α+β) + cos(α−β) = 2cos(α)cos (β)
1
1
sin(α+β) + sin(α−β) = 2sin(α)cos(β)
= CAc m ( t ) + nI ( t )
2
2
1
1
+ ⎡⎣CAc m ( t ) + nI ( t ) ⎤⎦ cos ( 4π f c t ) − Ac nQ ( t ) sin ( 4π f c t )
2
2
low-pass
p filter,, BW=W
1
1
y ( t ) = CAc m ( t ) + nI ( t )
2
2
( 6.8 )
15
Yu-sing Lin (林育星)
◊
◊
◊
Equation (6.8) indicates the following:
◊ The
Th message signal
i l m(t)
(t) andd in-phase
i h
noise
i componentt nI(t) off
◊ The quadrature component nQ(t) of the noise n(t) is completely
rejected by the coherent detector.
We note that these two results are independent of the input signalto-noise ratio.
Thus, coherent detection distinguishes itself from other
demodulation techniques in the important property: the output
message component is unmutilated and the noise component
always appears additively with the message, irrespective of the
input signal-to-noise ratio.
16
Yu-sing Lin (林育星)
◊
1
1
The receiver output signal : y ( t ) = CAc m ( t ) + nI ( t )
(6.8)
2
2
The average power of message component may be expressed as
◊
C 2 Ac2 P
Pavg =
4
Th average power off the
The
th noise
i att the
i
output
t t is
i
◊
2
1
⎛1⎞
⎜ ⎟ 2WN 0 = WN 0
2
⎝2⎠
◊
◊
In the case of DSB-SC modulation, the band-pass filter in Figure 6.4 has a
band-width
band
width BT equal to 2W in order to accommodate the upper and lower
sidebands of the modulated signal s(t). Therefore, the average power of the
filtered noise n(t) is 2WN0.
F
From
P
Property
t 5 off S
Sectt 55.11,
11 th
the average power off th
the (low-pass)
(l
) nI(t) iis
the same as that of the (band-pass) filtered noise n(t).
17
Yu-sing Lin (林育星)
◊
◊
The output signal-noise ratio for DSB-SC
C 2 Ac2 P 4
(SNR )O ,DSB-SC =
WN 0 2
C 2 Ac2 P
=
2WN 0
We obtain the figure of merit
(SNR )O
(SNR )C
( 6.9 )
( 6.10 )
=1
DSB-SC
18
Yu-sing Lin (林育星)
Chap 6.4
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
19
◊
A full AM signal is given by
s ( t ) = Ac ⎡⎣1 + ka m ( t ) ⎦⎤ cos ( 2πf c t )
◊
( 6.11)
where Accos(2πfct) is the carrier wave, m(t) is the message signal
and bandwidth is W, ka is a constant that determines the
percentage modulation.
We would like to perform noise analysis for an AM system using
an envelope
l
detector.
d
20
Yu-sing Lin (林育星)
◊
◊
We perform the noise analysis of the AM receiver by first
determining the channel signal-to-noise
signal to noise ratio,
ratio and then the output
signal-to-noise ratio.
We can easily obtain average power of the AM signal
s ( t ) = Ac cos ( 2πf c t ) + Ac ka m ( t ) cos ( 2πf c t )
1 2 1 2 2
Ac + Ac ka P
2
2
The average power of noise in the message bandwdith is WN0 (the
same as the DSB-SC system)
The channel signal
signal-to-noise
to noise ratio for AM is therefore:
Ps =
◊
◊
(SNR )C ,AM =
21
Ac2 (1 + ka2 P )
2WN 0
Yu-sing Lin (林育星)
◊
The filtered signal x(t) applied to the envelope detector in the
y
x (t ) = s (t ) + n (t )
= ⎡⎣ Ac + Ac ka m ( t ) + n1 ( t ) ⎤⎦ cos ( 2πf c t ) − nQ ( t ) sin ( 2πf c t )
y ( t ) = envelope of x ( t )
{
}
= ⎡⎣ Ac + Ac ka m ( t ) + nI ( t ) ⎤⎦ + n ( t )
2
2
Q
12
6 13)
( 6.13
( 6.14 )
Assume average
g carrier power
p
&gt;&gt; average
g noise power
p
y (t )
◊
◊
Ac + Ac ka m ( t ) + nI ( t )
( 6.15)
The dc term or constant term Ac may be removed simply by means
of a blocking capacitor.
If we ignore the dc term Ac, we find that the remainder has a form
similar to the output of a DSB-SC receiver using coherent detection.
22
1
1
y ( t ) = CAc m ( t ) + nI ( t )
2
2
( 6.8)
Yu-sing Lin (林育星)
◊
The output signal-to-noise ratio of an AM using an envelope
detector is approximately
(SNR )O ,AM
◊
Ac2 ka2 P
2WN 0
( 6.16 )
Eq. (16) is valid only if the following two conditions are satisfied
◊
The average noise power is small compared to the average carrier power at
the envelope detector input.
◊
The amplitude sensitive ka is adjusted for a percentage modulation less
th or equall to
than
t 100 percent.
t (| ka m(t)|&lt;=1)
)|&lt; 1)
23
Yu-sing Lin (林育星)
◊
Comparison of figure of merit ( AM, DSB-SC, SSB )
(SNR )O
(SNR )C
◊
◊
◊
AM using
Envelope Detector
(SNR )O
(SNR )C
ka2 P
≤1
2
1 + ka P
DSB-SC
(SNR )O
=
(SNR )C
=1
SSB
The figure
g
of merit of a DSB-SC receiver or that of an SSB
receiver using coherent detection is always unity.
The corresponding figure of merit of an AM receiver using
envelope detection is always less than unity.
In other words,, the noise pperformance of an AM receiver is
always inferior to that a DSB-SC receiver. This is due to the
wastage of transmitter power, which results from transmitting the
carrier
i as a component off AM wave.
24
Yu-sing Lin (林育星)
◊
Example 6.1 Single-Tone Modulation
◊
◊
Consider a sinusoidal wave of frequency fm and amplitude Am
as the modulating wave, as shown by
m(t) = Amcos(2πfmt)
The corresponding AM wave is
s(t) = Ac [1+ μcos(2πfmt)] cos(2πfct)
modulation factor : μ = kaAm
◊
The average power of the modulation wave m(t) is (assuming a
1
P = Am2
2
25
Yu-sing Lin (林育星)
◊
We obtain the figure of merit
(SNR )O
(SNR )C
◊
◊
AM
1 2 2
ka Am
2
μ
= 2
=
1 2 2 2 + μ2
1 + ka Am
2
6 18 )
( 6.18
When μ = 1 (100% modulation using envelope detection), we get
a figure of merit = 1/3.
This means that, other factors being equal, an AM system (using
envelope detection) must transmit three times as much average
power as a suppressed-carrier
d
i system (using
( i coherent
h
detection)
d
i ) in
i
order to achieve the same quality of noise performance.
26
Yu-sing Lin (林育星)
Chap 6.5
Wireless Information Transmission System Lab.
Institute of Communications Engineering
g
g
National Sun YatYat-sen University
27
◊
The receiver model is given by:
◊
The noise w(t) is modeled as white Gaussian noise of zero mean and
power spectral density N0/2.
/2
The received FM signal s(t) has a carrier frequency fc and transmission
bandwidth BT, such that only a negligible amount of power lies outside
the frequency band fc &plusmn; BT /2 for positive frequencies.
The band-pass filter has a mid-band frequency fc and bandwidth BT and
th f
therefore
passes the
th FM signal
i l essentially
ti ll without
ith t distortion.
di t ti
Ordinary, BT is small compared with the mid-band frequency fc so that
we may use the narrowband representation for n(t), the filtered version
components.
Yu-sing Lin (林育星)
28
◊
◊
◊
◊
◊
◊
In an FM system, the message information is transmitted by
variations
i i
off the
h instantaneous
i
frequency
f
off a sinusoidal
i
id l carrier
i
wave, and its amplitude is maintained constant.
A variations
Any
i ti
off the
th carrier
i amplitude
lit d att the
i
input
i t mustt
result from noise or interference.
The limiter is used to remove amplitude variations by clipping
the modulated wave at the filter output almost to the zero axis.
◊ The resulting rectangular wave is rounded off by another
bandpass filter that is an integral part of the limiter, thereby
suppressing harmonics of the carrier frequency.
◊ The filter output is again sinusoidal, with an amplitude that is
practically
ti ll independent
i d
d t off the
th carrier
i amplitude
lit d att the
i
input.
29
Yu-sing Lin (林育星)
◊
The discriminator consists of two components:
◊
◊
◊
◊
A slope network or differentiator with a purely imaginary transfer function
that varies linearly with frequency. It produces a hybrid-modulated wave in
which both amplitude
p
and frequency
q
y varyy in accordance with the message
g
signal.
An envelope detector that recovers the amplitude variation and thus
reproduces the message signal.
signal
The slope network and envelope detector are usually implemented as
integral parts of a single physical unit.
The post-detection filter, labeled “baseband low-pass filter,” has a
bandwidth that is just large enough to accommodate the highest
f
frequency
component off the
h message signal.
i l
◊ This filter removes the out-of-band components of the noise at
th discriminator
the
di i i t output
t t andd thereby
th b keeps
k
the
th effect
ff t off the
th
output noise to a minimum.
30
Yu-sing Lin (林育星)
◊
The filtered noise at the band-pass filter output is defined as:
n ( t ) = nI ( t ) cos ( 2π f c t ) − nQ ( t ) sin ( 2π f c t )
n ( t ) = r ( t ) cos ⎡⎣ 2πf c t +ψ ( t ) ⎤⎦
( 6.21)
r ( t ) = ⎡⎣ n ( t ) + n ( t ) ⎤⎦ ∼ Rayleigh distribution
⎡ nQ ( t ) ⎤
−1
ψ ( t ) = tan ⎢
∼ U ( 0,2π )
⎥
⎣ nI ( t ) ⎦
The incoming FM signal s(t) is given by
( 6.22 )
2
I
◊
2
Q
12
t
⎡
s ( t ) = Ac cos 2πf c t + 2πk f ∫ m (τ ) dτ ⎤
⎢⎣
⎥⎦
0
t
( 6.23)
( 6.24 )
φ ( t ) = 2πk f ∫ m (τ ) dτ
( 6.25)
s ( t ) = Ac cos ⎡⎣ 2πf c t + φ ( t ) ⎦⎤
( 6.26 )
0
31
Yu-sing Lin (林育星)
◊
The noisy signal at the band-pass filter output is:
x ( t ) = s ( t ) + n ( t ) = Ac cos ⎡⎣ 2πf c t + φ ( t ) ⎤⎦ + r ( t ) cos ⎡⎣ 2πf c t +ψ ( t ) ⎤⎦
⎧⎪ r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ ⎫⎪
θ ( t ) = φ ( t ) + tan ⎨
⎬
⎡
⎤
A
+
r
t
cos
ψ
t
−
φ
t
( ) ⎣ ( ) ( )⎦ ⎪⎭
⎪⎩ c
The envelope of x(t) is of no interest to us, because any envelope
variations at the band-pass
band pass output are removed by the limiter.
limiter
Our motivation is to determine the error in the instantaneous
frequency of the carrier wave caused by the presence of the filtered
noise n(t).
−1
◊
◊
32
Yu-sing Lin (林育星)
◊
◊
With the discriminator assumed ideal, its output is proportional to
θ’(t)/2
( ) π where θ’(t)
( ) is the derivative of θ(t)
( ) with respect
p to time.
We need to make certain simplifying approximations so that our
y may
y yield
y
useful results.
analysis
⎧⎪ r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ ⎫⎪
θ ( t ) = φ ( t ) + tan ⎨
⎬
⎪⎩ Ac + r ( t ) cos ⎡⎣ψ ( t ) − φ ( t ) ⎦⎤ ⎪⎭
(1) when CNR 1, A r ( t )
r (t )
r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦
θ (t ) φ (t ) +
sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦
Ac
A + r ( t ) cos ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦
−1
c
t
r (t )
0
Ac
θ ( t ) 2π k f ∫ m ( t )dt +
1 dθ ( t )
υ (t ) =
2π dt
c
sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦
∼
( 2) x
k f m ( t ) + nd ( t )
33
r ( t ) sin ⎣⎡ψ ( t ) − φ ( t ) ⎦⎤
Ac
1, tan −1 ( x ) ≈ x
(6.31)
Yu-sing Lin (林育星)
{
}
1 d
nd ( t ) =
r ( t ) sin
i ⎡⎣ ψ ( t ) − φ ( t ) ⎦⎤
2π Ac dt
nd ( t )
nd ( t )
◊
{
}
1 d
r ( t ) sin ⎡⎣ ψ ( t ) ⎤⎦
2π Ac dt
1 dnQ ( t )
2π Ac dt
ψ ( t ) ∼ U ( 0,2π ) ⇒ Assume ψ ( t ) − φ ( t ) ∼ U ( 0,2π )
If such an assumption were true, then the noise
at the discriminator output would be indep. of
the modulating signal and would depend only
on the characteristics of the carrier and
narrowband noise. This assumption is justified
provided that the carrier-to-noise ratio is high.
This means that the additive noise nd(t) appearing at the
discriminator output is determined effectively by the carrier
amplitude Ac and the quadrature component nQ(t) of the
narrowband noise n(t).
34
Yu-sing Lin (林育星)
◊
The output signal-to-noise ratio is defined as the ratio of the
g output
p signal
g power
p
to the average
g output
p noise ppower.
average
◊
From Eq. (6.31), the message component in the discriminator
output, andd therefore
h f
the
h low-pass
l
fil output, is
filter
i kf m(t).
()
The average output signal power is equal to kf 2P, where P is the
average power off th
the message signal
i l m(t).
(t)
◊
◊
◊
To determine the average output noise power, we note that the
noise nd(t) at the discriminator output is proportional to the time
derivative of the quadrature noise component nQ(t).
The differentiation of a function respect to time corresponds to
multiplication of its Fourier transform by j2πf. We may obtain the
noise
i nd(t)
( ) bby passing
i nQ(t)
( ) though
h h a li
linear filter
fil with
i h a transfer
f
j 2π f
jf
= .
function:
2π Ac
Ac
35
Yu-sing Lin (林育星)
d
x ( t ) ⎯⎯
→ j 2π fX ( f )
dt
2
( 2 ) SY ( f ) = H ( f ) S X ( f )
(1)
f2
S N d ( f ) = 2 S NQ ( f )
Ac
⎧ N0 f 2
⎪ 2 ,
S Nd ( f ) = ⎨ Ac
⎪ 0,
⎩
⎧ N0 f 2
⎪ 2 ,
S NO ( f ) = ⎨ Ac
⎪ 0,
⎩ 0
◊
f ≤
BT
2
X(t)
()
H( f )
Y(t)
()
otherwise
f ≤W
otherwise
For wide-band FM, we usually find that W is
smaller
ll than
h BT /2,
/2 where
h BT is
i the
h transmission
i i
bandwidth of the FM signal. This means that
out-of-band components of noise no(t) will be
rejected.
36
Yu-sing Lin (林育星)
◊
◊
◊
◊
In FM system, increasing the carrier power has a noise-quieting
effect.
N0 W 2
2 N 0W 3
2
Average power of output noise = 2 ∫ f df =
∝ 2
2
−
W
Ac
Ac
3 Ac
k 2f P
3 Ac2 k 2f P
=
( 6.40 )
We obtain ( SNR )O ,FM =
3
2
3
2 N 0W 3 Ac 2 N 0W
The average power in the modulated signal s(t) is A2c /2, and the
average noise power in the message bandwidth is WNo. The channel
signal to noise ratio (SNR)C,FM is
Ac2
(SNR )C ,FM =
( 6.41)
2WN 0
2WN
Figure of merit for frequency modulation:
3k 2f P
(SNR )O
=
6 42 )
( 6.42
2
(SNR )C FM W
37
Yu-sing Lin (林育星)
◊
Ex 6.2 Single-Tone Modulation
◊
A sinusoidal wave of frequency fm as the modulating signal, and
assume a peak frequency deviation ∆f . The FM signal is define by
⎡
⎤
Δf
s ( t ) = Ac cos ⎢ 2πf c t +
sin ( 2πf mt ) ⎥
fm
⎣
⎦
◊
Therefore, we may write
◊
Δf
2πk f ∫ m (τ ) dτ = ( sin 2πf mt )
0
fm
Differentiating both sides with respect to time and solving for m(t)
t
m (t ) =
Δf
cos ( 2πf mt )
kf
38
Yu-sing Lin (林育星)
◊
◊
◊
◊
The average power of message signal m(t)
(Δf ) 2
P=
2k 2f
We get output signal-to-noise ratio
2
3 Ac2 ( Δf )
3 Ac2β 2
=
(SNR )O ,FM =
3
4 N 0W
4 N 0W
Where β = ∆f /W is the modulation index and we get the figure of
2
merit
i
(SNR )O
3 ⎛ Δf ⎞
3 2
= ⎜
( 6.43)
⎟ = β
(SNR )C FM 2 ⎝ W ⎠ 2
It is important to note that the modulation index β = ∆f /W is
determined
de
e
ed by thee bandwidth
ba dw d W oof thee postdetection
pos de ec o low-pass
ow pass
filter and is related to the sinusoidal message frequency fm.
39
Yu-sing Lin (林育星)
◊
For an AM system operating with a sinusoidal modulating signal
and 100 percent modulation, we have
(SNR )O
1
=
(SNR )C AM 3
◊
It is of particular interest to compare the noise performance of
AM and FM systems.
(SNR )O
(SNR )C
3 ⎛ Δf
= ⎜
2⎝ W
FM
3 2 1
when β &gt;
2
3
◊
2
3 2
⎞
=
β
⎟
2
⎠
←⎯⎯⎯
→
compare
(SNR )O
(SNR )C
=
AM
1
3
FM has better performance
2
β&gt;
= 0.471
3
Define β = 0.5
0 5 as the transition between
bet een narro
narrowband
band FM and
wide-band FM.
40
Yu-sing Lin (林育星)
```