### Document

```Noise on Analog Systems
ECE460
Spring, 2012
H(f)
r t 
1
Signal
s  t   nW  t 
H(f)
-f2
-f1
f1
f2
f
cos  2 f c t   
y t 
1
-W
W
f
• s(t) is the transmitted signal
• Let M(t) be a random process representing the information
bearing signal. m(t) will denote a sample function of M(t).
M(t) is assumed zero mean WSS with autocorrelation
function RM(τ) and power spectral density SM(f). M(t) is
assumed a low-pass signal or a baseband signal with spectral
content limited to W Hz, i.e.,
S M  f   0 for f  W .
and the signal power is
W
PM   S M  f  df
W
• nW(t) is a sample of zero mean, white noise with power
spectral density N0/2.
• The received signal after the ideal BPF is
r t   s t   n t 
where n(t) is narrow-band noise.
2
For DSB-SC Amplitude Modulation,
f1  f c  W
f2  fc  W
And the PSD of n(t) is
SN(f)
N0
2
-fc -W
-fc
fc -W
-fc+W
fc+W
fc
From our work on bandpass processes, n(t) can be broken into
n  t   nc  t  cos  2 f ct   ns  t  sin  2 f ct 
where nc(t) and ns(t) are uncorrelated processes, i.e.
E  nc  t  ns  t      0  t and 
Furthermore,
and S nc  f   S ns  f
Rnc    Rns  
 given by
N0
-W
W
f
3
Evaluation of DSB-SC
Recall that the transmitted signal is
s  t   Ac m  t  cos  2 f ct  c 
The received signal, r(t), after the ideal BPF filter is
r t   s t   n t 
 Ac m  t  cos  2 f c t  c 
 nc  t  cos  2 f c t   ns  t  sin  2 f ct 
where the ideal bandpass filter H(f) at the receiver’s input is
1,
Hf 
0,
f  fc  W
Otherwise
and W is the bandwidth of the information process m(t).
Find the demodulated r(t):
4
DSB-SC
SNR at Output
Assuming synchronous demodulation (e.g.,   c ), find y(t).
Power of the noise at the receiver’s output:
SNROut 
desired signal power at receiver output

5
DSB-SC
SNR at Input
To transmit out signal m(t), we used a transmitter with power
equal to the power of s(t) given by
Sin 
This was also considered the power of the signal at the
The noise power at the receiver input calculated for the
message bandwidth is
N in 
SNRin 
transmitted signal power

6
Conventional AM (DSB)
Recall the transmitted signal is
s  t   Ac 1  a mn  t   cos  2 f ct 
where |m(t)| ≤ 1. If we follow the methodology in DSB-SC
assuming synchronous modulation and θ = 0 without any loss of
generality, then the output of the low-pass filter is
yl  t   12 Ac 1  a mn  t    12 nc  t 
The dc component, Ac/2, is not part of the message and must be
removed. The output after a dc blocking device is:
y  t   12 Ac a mn  t   12 nc  t 
Find the SNR at the receiver’s output and input.
SNRout 
7
SSB
Recall the transmitted signal is
s  t   Ac m  t  cos  2 f ct   Ac m  t  sin  2 f ct 
r  t    Ac m  t   nc  t   cos  2 f ct 
   Ac m  t   ns  t   sin  2 f ct 
Again assuming synchronous demodulation with perfect phase,
the output after the LPF is
y  t   12 Ac m  t   12 nc  t 
Find:
Sout 
N out 
SNRout 
SNRin 
8
Example
The message process M(t) is a stationary process with the
autocorrelation function
RM    16sinc 2 10, 000 
It is also known that all the realizations of the message process
satisfy the condition max |m(t)|=6. It is desirable to transmit
this message to a destination via a channel with 80-dB
attenuation and additive white noise with power-spectral
density Sn(f) = N0/2 = 10-12 W/Hz, and achieve a SNR at the
modulator output of at least 50 dB. What is the required
transmitter power and channel bandwidth if the following
modulations schemes are employed?
1. DSB-SC
2. SSB
3. Conventional AM with modulation index equal to 0.8.
9
Angle Modulation
Effect of additive noise on modulated FM signal
•
•
Amplitude Modulation vs Angular Modulation
Importance of zero-crossing -> instantaneous frequency
Approximate
•
s  t   nw  t 
Bandpass
Filter
r t   s t   n t 
Angle
Demodulator
y (t )
S 
 
 N 0
Lowpass
Filter
nw  t   zero mean Gaussian white noise
s  t   A c cos  2 f c t    t  
 transmitted signal
k pm t 


 t   
t
2

k
f  m   d


PM
FM
Bandpass filters limits noise to bandwidth of modulated signal
• n(t) is bandpass noise
n  t   n c  t  cos 2 f ct  ns  t  sin 2 f ct
Or, in Phasor form
n  t   Vn  t  cos  2 f ct   n  t  
where
Vn  t  
n c2  t   ns2  t 
 n t  
 n  t   arctan  s

n
t


 c

11
Phasor Analysis
Assumption:
Vn  t   Ac
r t  
12
Solve for SNR
Found demodulated signal y(t)
• Composed of signal and additive noise
k p m  t   Yn  t 
PM

y t   
1 d
k
m
t

Yn  t  FM


 f
2 dt

Yn  t  
1
 ns  t  cos   t   nc  t  sin   t  
Ac 
RYn  t , t    


1
Rn   E cos   t       t  
Ac2 c
• Assumption: m(t) is a sample function of a zero mean
stationary Gaussian process with autocorrelation
function RM(τ).
• What about   t  ?
k pm t 


Recall:   t   
t
2

k
f  m   d


PM
FM
13
Typical Plots
14
Noise Power Spectrum at
Demodulated Output
S nout  f 
PM
FM
S nout
 N0
PM
 A2 ,
c
 f   
 N 0  f 2 , FM
 Ac2 

15
Noise and Signal Power at Output (LPF)
Noise Power at LPF

Pnout 
 S  f  df

nout
 2W N 0
 A2 , PM

c

3
 2W N 0 , FM
 3 Ac2
To Determine Power Out of Signal, recall
k p m  t   Yn  t 
PM

y t   
1 d
k
m
t

Yn  t  FM


 f
2 dt

Signal Power at LPF
PSout
 k p2 PM , PM
 2
k f PM , FM
16
SNR for Angle Modulation
Therefore, SNRout
SNRout 
PSout
Pnout
 k p2 Ac2

 2
 2 2
 3k f Ac
 2W 2

PM
N 0W
PM
PM
, FM
N 0W
Using Modulation Indexes
 p  k p max m  t 
t
f 
kf
W
max m  t 
t
And denoting
SNRb 
power of transmitted signal
power of noise at message bandwidth
Ac2 / 2

N 0W
Then
SNRout



 SNRb g


17
SNRout / SNRb
1. Proportional to modulation index squared
2. Increasing  improves SNR gain at the expense of
bandwidth expansion
3. The maximum possible SNR gain improvement is
exponential as can be shown using Shannon theory
4. We cannot increase  without limit sent at some point our
results will not be valid since they are only approximate
results
5. FM, like any other nonlinear modulation technique, exhibits
a threshold effect and performance. Above certain SNRb ,
the SNRout is proportional to  2 SNRb . Below the threshold,
SNRout maybe worse than SNRb.
18
SNRout / SNRb
6. In AM, increasing Ac increases SNRout since the received
message is proportional to Ac. Here, increasing Ac also
increases SNRout but through a different mechanism. Here
increasing Ac reduces the amount of noise that affects the
message signal.
7. To compensate for the high noise PSD at high frequencies
and FM, the PSD of the signal is pre-emphasized in the
transmitter to increase its immunity to noise at high
frequencies and it is then the emphasized at the receivers
output.
8. It can be shown that at threshold, we have for FM systems
SNRb  20    1
Given the received power of the modulated signal, this
relation gives us the max  which ensures that the system
works above threshold. Another restricting factor results
from Carlson’s rule
Bc  2    1 W
Then, give and receive power and channel bandwidth
 SNRb Bc 
  min 
,
 1
 20 2W 
19
Example
Consider an FM broadcast system with parameter k f  75 103 ,
and W  15 kHz. Assuming PM  12 , find the output SNR and
calculate the improvement (in dB) over the baseband system.
20
Pre-emphasis & De-emphasis Filters
21
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