ensc327 communications systems 24: ch. 9: noise in analog … · ensc327 communications systems 24:...

ENSC327

Communications Systems

24: Ch. 9: Noise in Analog Systems

1

Jie Liang

School of Engineering Science

Simon Fraser University

Chapter 9 Noise in Analog Comm.

� Chap 9 studies noise performance of various analog modulations

� Various SNRs in a system:

� Pre-detection SNR

� Post-detection SNR

� Reference SNR (also known as Channel SNR)

� Figure of Merit

2

� Figure of Merit

� SNR in DSB

� SNR in AM

� SNR in SSB

� SNR in FM

� Main Conclusions:

� DSB and SSB have the same noise performance as the baseband comm system.

� AM has worse noise performance, but has simpler receiver (envelope detector )

� FM has much better noise performance, at the price of the increased bandwidth.

Receiver Model

� Channel model: additive white Gaussian noise with psd N0/2.

� Receiver model: ideal band-pass filter and ideal demodulator.

Noise w(t) Bandwidth: BT

3

� The BPF is usually a part of a mixer , e.g., in superhet receiver.

� The output freq of the BPF is the intermediate frequency (IF) (455kHz in

AM, 10.7MHz in FM). It’s denoted as fc in Chap 9.

� The bandwidth BT of the BPF is the bandwidth of the modulated signal

(2W in AM and DSB, W in SSB, 2(D+1)W in FM. W is msg. bandwidth).

� The noise after the BPF is thus narrowband noise (Chap 8):

w(t) Bandwidth: BT

9.3 Band-pass System Structures

� Mixer is usually used to translate the IF frequency to

the RF frequency, or vice versa.

� The IF frequency is denoted as fc in Chap. 9.

fc fc

4

fc fc

Pre-detection SNR

SN(f)

BTN0/2

The bandwidth of x(t):

w(t)

At the input to the demodulator: x(t)=s(t)+n(t),

5

� Pre-detection Signal to Noise Ratio (SNR) at the demodulator

input (after bandpass filter):

The bandwidth of x(t):

The power of the narrowband noise is:

==

noise narrowband theofpower

signal modulated ofpower SNR

pre

Post-detection SNR

w(t)

� The output of the demodulator: recovered message with noise.

� The bandwidth of the output signal is usually the bandwidth

6

� Post-detection Signal to Noise Ratio :

.power noiseOutput

power message RecoveredSNR

post=

� This is the SNR perceived by the users.

� The bandwidth of the output signal is usually the bandwidth

of the message signal m(t), denoted as W.

� The output noise psd depends on the modulation scheme.

Reference SNR� To compare noise performances of different modulation systems, we need a

reference baseband transmission model, which transmits the message

directly without any modulation.

� For fair comparison, the transmitted power should be the same as that in a

bandpass modulation system.

� The bandwidth of the LPF at the receiver equals to the message bandwidth

7

==

bandwidth message in thepower noise Average

signal modulated ofpower AverageSNR

ref

� The bandwidth of the LPF at the receiver equals to the message bandwidth

Channel noise

The reference SNR is defined as:

Figure of Merit� The reference SNR can be used to normalize the post-detection

SNR of different methods, so that we can compare them:

SNR Reference

SNRion PostdetectMerit of Figure =

8

� It shows whether a modulation system has better noise

performance than the baseband reference system or not.






� Figure of Merit

� SNR in DSB

9

� SNR in DSB

� SNR in AM

� SNR in SSB

� SNR in FM

9.4 SNR in DSB-SC

� The modulated DSB-SC signal at the transmitter:

� θ is a uniform random variable in [0, 2π], but is fixed during the

transmission (The reason for including θ is that the time origins

of m(t) and cos(2πfct) are independent).

)2cos()()( θπ += tftmAtscc

10

of m(t) and cos(2πfct) are independent).

� A linear DSB-SC receiver using coherent demodulation:

� Note the two filters generally have different bandwidths (BW):

� The BPF’s bandwidth is

� The LPF’s bandwidth is

Superhet

receiver:

Pre-detection SNR of DSB-SC

� Proof: The pre-detection SNR is defined as:

.2

SNR

0

2

DSB

T

cpre

B�

PA=

� In DSB, BT is 2W, where W is the message bandwidth.

signal modulated ofpower

11

==

noise narrowband theofpower

signal modulated ofpower SNR

pre

Post-detection SNR of DSB-SC� The post-detection SNR is defined as:

.power noiseOutput

power message RecoveredSNR

post=

12

� Assuming θ = 0:

� The signal after the bandpass filter:

� n(t) is narrowband noise with psd N0/2 and bandwidth BT.

).()()( tntstx +=

Post-detection SNR of DSB-SC

� After multiplying with the carrier:

Bandwidth W

13

� After low-pass filter with bandwidth W, the output is

� Only in-phase noise appears after LPF!


Bandwidth W

14


� Pre-detection SNR:

� Post-detection SNR:

15

� To get Figure of Merit, we need the reference SNR:

=DSB

SNR ref ==

SNR Ref

SNRPost Merit of Figure


� Example: Problem 9.2

� DSB with

Find the post-detection SNR.

[ ] dBmtsEf

�

c80)( 200kHz,

4kHzW watts/Hz,104

2

17

0

−==

=×=−

16


Solution:






� Figure of Merit

� SNR in DSB

17

� SNR in DSB

� SNR in AM

� SNR in SSB

� SNR in FM

9.5 Noise in AM with Envelope Detector

� Modulated signal in AM: [ ] )2cos()(1)( tftmkAtscacπ+=

� AM Receiver: mixer followed by an envelope detector

18

� The BW of the BPF is

� After the BPF,

Pre-detection SNR in AM

� Power of modulated signal:

19

� Power of the noise after the BPF:

� Pre-detection SNR: =AM

SNRpre

Post-detection SNR in AM

� The result after BPF can be written as

20

where the envelope is

� When the SNR is high,

� DC term Ac can be removed by a capacitor, so final output is:

Post-detection SNR of AM� The envelope detector output

� The recovered message power is

21

� AM Post-detection SNR:

Figure of Merit of AM

� To get Figure of Merit, we need the reference SNR:

=AM

SNR ref

==


W�

PkAac

0

22

AM

post2

SNR =

22

==

SNR Ref


[ ] ),2cos()(1)( tftmkAtscacπ+=

To avoid over-modulation, we need

Figure of Merit of AM

� The noise performance of AM is worse than the DSB-SC,

because at least half of the power is spent on the carrier.

==

SNR Ref

SNRPost Merit of Figure AM

23

� The benefit we get is the simpler envelope detector.






� Figure of Merit

� SNR in DSB

24

� SNR in DSB

� SNR in AM

� SNR in SSB

� SNR in FM

9.6 Pre-detection SNR in SSB

� SSB signal: ( ) ( )( )tftmtftmA

tscc

c

ππ 2sin)(ˆ2cos)(2

)( m=

� To get the power of SSB signal, we need the following facts:

� 1. m(t) and its Hilbert transform are orthogonal:

( ) .0)(ˆ)( =tmtmE

25

This is because Hilbert transform rotates the signal by 90o.

Example:

� 2. m(t) and its Hilbert transform have the same power.

( ) .0)(ˆ)( =tmtmE

Proof:

Pre-detection SNR in SSB

� Proof:

.4

SNR

0

2

SSB

W�

PAc

pre=Pre-detection SNR of SSB:

26

Post-detection SNR in SSB

� SSB signal:

� SSB signal can be demodulated by coherent method too.

� Using narrowband representation of the noise, the

received signal is (use LSB as example):

( ) ( )( )tftmtftmA

tscc

c

ππ 2sin)(ˆ2cos)(2

)( m=

27

received signal is (use LSB as example):

� After multiplying with the carrier and LPF:


SN(f)

28

SN(f)

fc-fcf

2/0

�

W


� nI(t) has the same power as n(t)

� output noise power of nI(t)/2 is:

=SSB

SNRPost-detection SNR in SSB

29

=SSB

postSNRPost-detection SNR in SSB

Reference SNR in SSB: =SSB

refSNR

Figure of Merit: ==

SNR Ref

SNRPost Merit of Figure SSB

The noise performance of SSB is the same as DSB and baseband system.


� Example: Problem 9.5

� SSB with


Solution:

[ ] dBmtsE� 80)( , 4kHz Wwatts/Hz,104 217

0−==×=

−

30

Solution:






� Figure of Merit

� SNR in DSB

31

� SNR in DSB

� SNR in AM

� SNR in SSB

� SNR in FM

9.7-9.8 Noise in FM

(Not covered in the final exam)

))(22cos()(0∫+=

t

fcc dmktfAts ττππ

32

� It can be shown that the figure of merit of FM modulation is

.4

3merit of Figure

2

=

W

BT

� Performance depends on the bandwidth of the modulated signal!

This is different from DSB, AM, and SSB.

� An increase in the bandwidth BT can significantly improve the

noise performance of FM system.


� Main Conclusions:

� DSB and SSB have the same noise performance as the

baseband comm system.

� AM has worse noise performance, but has simpler receiver

(envelope detector )

33

(envelope detector )

� FM has much better noise performance, at the price of the

increased bandwidth.

ensc327 communications systems 24: ch. 9: noise in analog … · ensc327 communications systems 24:...

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