6.02 Fall 2012 Lecture #8 - MIT OpenCourseWare · • SignaltoNoise Ratio (SNR) – usually denotes...

602 Fall 2012 Lecture 8

bull Noise bad things happen to good signals bull Signalshytoshynoise ratio and decibel (dB) scale bull PDFrsquos means variances Gaussian noise bull Bit error rate for bipolar signaling

602 Fall 2012 Lecture 8 Slide 1

tt

t

Single Link Communication Model

602 Fall 2012 Lecture 8 Slide 2602 Fall 2012 LeLeLLLeLecture 8 Slide 2

Digitize (if needed)

Original source

Source coding

Source binary digits (ldquomessage bitsrdquo)

Bit stream

Renderdisplay

Receiving appuser

Source decoding

Bit stream

Channel Coding

(bit error correction)

Recv samples

+ Demapper

Mapper +

Xmit samples

Bits

(Voltages) over

physical link

etc

SignalsSignals (Vo ltages)

Channel Decoding

(reducing or removing bit errors)

Renderdisplay etc

Receiving appuserg a

Source decoding

etc

de

BB iitt ssttreamB


Original source

iti

Source coding

SSoouurrccee bbiinnaarryy ddiiggiittssSS (ldquomessage bitsrdquo)

BBiitt streammBB

Endshyhost computers

(Bits


From Baseband to Modulated Signal and Back

codeword bits in

1001110101 DAC

ADC

NOISY amp DISTORTING ANALOG CHANNEL

modulate

1001110101

codeword bits out

demodulate amp filter

generate digitized symbols

sample amp threshold

x[n]

y[n]PDJHE072SHQampRXUVHDUH

Mapping Bits to Samples at Transmitter codeword bits in

1001110101 generate digitized symbols

x[n]

16 samples per bit

1 0 0 1 1 1 0 1 0 1 602 Fall 2012 Lecture 8 Slide 4

rre

Samples after Processing at Receiver bits in

1001110101 DACmodulate generate digitized symbols

codeword

x[n] NOISY amp DISTORTING ANALOG CHANNEL

ADC demodulate

y[n] Assuming no noise only endshytoshyend distortion

filter

602 Fall 2012 Lecture 8 Slide 502 Fall 2012 Lecturururururururururrrrrurrrrrruuurrrrrrurururuurr eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeee 8 Slide 5

y[n]

e

Mapping Samples to Bits at Receiver codeword bits in

1001110101 DACmodulate generate digitized symbols n = sample index

x[n] j = bit index NOISY amp DISTORTING ANALOG CHANNEL

y[n] b[j] demodulate 1001110101

amp filter sample amp threshold

codeword bits out 16 samples per bit

ADC

602 Fall 2012 Lecture 8 Slide 602 Fall 2012 LeLeLeLeLectctctctcturururururuuuuuurrrruuuu ruuruuu ruuuuu eeeeeeeeee eeeeeeeeeeeeeeeee eeeeee 88888 SSSSSlililiiiliiliii de 6

1 0 0 1 1 1 0 1 0 1

b[j]

Threshold

For now assume no distortion only

Additive ZeroshyMean Noise

bull Received signal y[n] = x[n] + w[n]

ie received samples y[n] are the transmitted samples x[n] + zeroshymean noise w[n] on each sample assumed iid (independent and identically distributed at each n)

bull SignalshytoshyNoise Ratio (SNR) ndash usually denotes the ratio of

(timeshyaveraged or peak) signal power ie squared amplitude of x[n]

to

noise variance ie expected squared amplitude of w[n]


SNR Example

Changing the amplification factor (gain) A leads to different SNR values

bull Lower A lower SNR bull Signal quality degrades with

lower SNR


SignalshytoshyNoise Ratio (SNR) 10logX X

100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001

The SignalshytoshyNoise ratio (SNR) is useful in judging the impact of noise on system performance

PPsignalSNR = PPnoise SNR for power is often measured in decibels (dB)

SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠

Caution For measuring ratios of amplitudes rather than powers take 20 log10 (ratio) shy80 0000000001

shy90 00000000001

shy100 000000000001301db is a factor of 2 in power ratio602 Fall 2012 Lecture 8 Slide 9

Noise Characterization From Histogram to PDF

Experiment create histograms of sample values from independent trials of increasing lengths

Histogram typically converges to a shape that is known ndash after normalization to unit area ndash as a probability density function (PDF)


Using the PDF in Probability Calculations

We say that X is a random variable governed by the PDF fX(x) if X takes on a numerical value in the range of x1 to x2 with a probability calculated from the PDF of Xas

int x2p(x1 lt X lt x2 ) = fX (x)dx x1

A PDF is not a probability ndash its associated integrals are Note that probability values are always in the range of 0 to 1


The Ubiquity of Gaussian Noise The net noise observed at the receiver is often the sum of many small independent random contributions from many factors If these independent random variables have finite mean and variance the Central Limit Theorem says their sum will be a Gaussian

The figure below shows the histograms of the results of 10000 trials of summing 100 random samples drawn from [shy11] using two different distributions

1shy1

1 Triangular Uniform

1shy1

05 PDF PDF


Mean and Variance of a Random Variable X

σx

The mean or expected value JX is defined and computed as

=infin x fX (x)dxmicroX intminusinfin

The variance σX2 is the expected squared variation or deviation

of the random variable around the mean and is thus computed as

2 infin σ X = intminusinfin

(x minusmicroX )2 fX (x)dx The square root of the variance is the standard deviation σX


Visualizing Mean and Variance

Changes in mean shift the Changes in variance narrow center of mass of PDF or broaden the PDF (but

area is always equal to 1) 602 Fall 2012 Lecture 8 Slide 14

The Gaussian Distribution

A Gaussian random variable W with mean J and variance σ2 has a PDF described by

1fW (w) = 2πσ 2


e minus wminusmicro( )2

2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

Noise Model for iid Process w[n]

bull Assume each w[n] is distributed as the Gaussian random variable W on the preceding slide but with mean 0 and independently of w[] at all other times


Estimating noise parameters bull Transmit a sequence of ldquo0rdquo bits ie hold the

voltage V0 at the transmitter

bull Observe received samples y[n] n = 0 1 K 1 ndash Process these samples to obtain the statistics of the noise

process for additive noise under the assumption of iid (independent identically distributed) noise samples (or more generally an ldquoergodicrdquo process ndash beyond our scope)

bull Noise samples w[n] = y[n] V0

bull For large K can use the sample mean m to estimate 1 and sample standard deviation s to estimate σ


Back to distinguishing ldquo1rdquo from ldquo0rdquo bull Assume bipolar signaling

Transmit L samples x[] at +Vp (=V1) to signal a ldquo1rdquo

Transmit L samples x[] at ndashVp (=V0) to signal a ldquo0rdquo

bull Simpleshyminded receiver take a single sample value y[nj] at an appropriately chosen instant nj in the

bull jshyth bit interval Decide between the following two hypotheses

y[nj] = +Vp + w[nj] (==gt ldquo1rdquo)

or

y[nj] = ndashVp + w[nj] (==gt ldquo0rdquo)

where w[nj] is Gaussian zeroshymean variance σ2


Connecting the SNR and BER

EV S = p

2σ 2 = N

P(ldquo0rdquo)=05 1=shyVp =

1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S

N E V 1 1 erfc == )( P error


BER erfc ( p ) = σ 2 2 2 0

Bit Error Rate for Bipolar Signaling Scheme with

SingleshySample Decision

05 erfc(sqrt (EsN0))

Es N0 dB 6RXUFHKWWSZZZGVSORJFRPELWHUURUSUREDELOLWIRUESVNPRGXODWLRQ

602 Fall 2012 ampRXUWHVRIULVKQD6DQNDU0DGKDYDQ3LOODL8VHGZLWKSHUPLVVLRQ Lecture 8 Slide 20

  

 

 

But we can do better bull Why just take a single sample from a bit interval bull Instead average M (leL) samples

y[n] = +Vp + w[n] so avg y[n] = +Vp + avg w[n]

bull avg w[n] is still Gaussian still has mean 0 but its variance is now σ2M instead of σ2 SNR is increased by a factor of M

bull Same analysis as before but now bit energy Eb = MEs instead of sample energy Es

1 Eb 1 MBER = P(error) = erfc( ) = erfc(Vp )602 Fall 2012 2 N0 2 σ 2 Lecture 8 Slide 21

 

Implications for Signaling Rate bull As the noise intensity increases we need to slow

down the signaling rate ie increase the number of samples per bit (K) to get higher energy in the (MleK) samples extracted from a bit interval if we wish to maintain the same error performance

ndash eg Voyager 2 was transmitting at 115 kbitss when it was near Jupiter in 1979 Last month it was over 9 billion miles away 13 light hours away from the sun twice as far away from the sun as Pluto And now transmitting at only 160 bitss The received power at the Deep Space Network antennas on earth when Voyager was near Neptune was on the order of 10^(shy16) watts shyshyshy 20 billion times smaller than an ordinary digital watch consumes The power now is estimated at 10^(shy19) watts


Flipped bits can have serious consequences

bull ldquoOnNovember 30 2006 a telemetered command to Voyager 2was incorrectly decoded by its onshyboard computermdashin a random errormdash as a command to tur n on the electrical heaters of the spacecrafts magnetometer These heaters remained tur ned on until December 4 2006 and during that time there was a resulting high temperature above 130 degC (266 degF) significantly higher than the magnetometers were designed to endure and a sensor rotated away from the correct orientation It has not been possible to fully diagnose and correct for the damage caused to theVoyager 2smagnetometer although efforts to do so are proceedingrdquo

bull ldquoOnApril 22 2010Voyager 2encountered scientific data format problems as reported by theAssociated Press on May 6 2010 OnMay 17 2010JPLengineers revealed that a flipped bit in an onshyboard computer had caused the issue and scheduled a bit reset for May 19 OnMay 23 2010Voyager 2has resumed sending science data from deep space after engineers fixed the flipped bitrdquo

602 Fall 2012 httpenwikipediaorgwikiVoyager_2 Lecture 8 Slide 23

What if the received signal is distorted bull Suppose y[n] = plusmnx0[n] + w[n] in a given bit slot (L

samples) where x0[n] is known and w[n] is still iid Gaussian zero mean variance σ2

bull Compute a weighted linear combination of the y[] in that bit slot

sum any[n] = plusmn sum anx0[n] + sum anw[n]

This is still Gaussian mean plusmn sum anx0[n] but now the 2variance is σ2 sum (an)

bull So what choice of the an will maximize SNR Simple answer an = x0[n] ldquomatched filteringrdquo

bull Resulting SNR for receiver detection and error performance is sum (x0[n])2 σ2 ie again the ratio of bit energy Eb to noise power


The moral of the story is hellip hellip if yoursquore doing appropriateoptimal processing at the receiver your signalshytoshynoise ratio (and therefore your error performance) in the case of iid Gaussian noise is the ratio of bit energy (not sample energy) to noise variance


MIT OpenCourseWarehttpocwmitedu

QWURGXFWLRQWR(OHFWULFDO(QJLQHHULQJDQGampRPSXWHU6FLHQFHFall 201

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

tt

t

Single Link Communication Model

602 Fall 2012 Lecture 8 Slide 2602 Fall 2012 LeLeLLLeLecture 8 Slide 2


Original source

Source coding

Source binary digits (ldquomessage bitsrdquo)

Bit stream

Renderdisplay

Receiving appuser

Source decoding

Bit stream

Channel Coding

(bit error correction)

Recv samples

+ Demapper

Mapper +

Xmit samples

Bits

(Voltages) over

physical link

etc

SignalsSignals (Vo ltages)

Channel Decoding

(reducing or removing bit errors)

Renderdisplay etc

Receiving appuserg a

Source decoding

etc

de

BB iitt ssttreamB


Original source

iti

Source coding

SSoouurrccee bbiinnaarryy ddiiggiittssSS (ldquomessage bitsrdquo)

BBiitt streammBB

Endshyhost computers

(Bits



codeword bits in

1001110101 DAC

ADC


modulate

1001110101

codeword bits out




x[n]




x[n]

16 samples per bit


rre



codeword


ADC demodulate


filter


y[n]

e







ADC


1 0 0 1 1 1 0 1 0 1

b[j]

Threshold







to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 
























codeword bits in

1001110101 DAC

ADC


modulate

1001110101

codeword bits out




x[n]




x[n]

16 samples per bit


rre



codeword


ADC demodulate


filter


y[n]

e







ADC


1 0 0 1 1 1 0 1 0 1

b[j]

Threshold







to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 
























x[n]

16 samples per bit


rre



codeword


ADC demodulate


filter


y[n]

e







ADC


1 0 0 1 1 1 0 1 0 1

b[j]

Threshold







to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 






















rre



codeword


ADC demodulate


filter


y[n]

e







ADC


1 0 0 1 1 1 0 1 0 1

b[j]

Threshold







to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 






















e







ADC


1 0 0 1 1 1 0 1 0 1

b[j]

Threshold







to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 




























to



SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 






















SNR Example



lower SNR



100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 























100 10000000000

90 1000000000

80 100000000

70 10000000

60 1000000

50 100000

40 10000

30 1000

20 100

10 10

0 1

shy10 01

shy20 001

shy30 0001

shy40 00001

shy50 0000001

shy60 00000001

shy70 000000001



SNR (db) =10 log10

⎛ ⎜⎜⎝

PPsignal PPnoise

⎞ ⎟⎟⎠


shy90 00000000001













1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 

































1shy1


1shy1

05 PDF PDF



σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 























σx













1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 



























1fW (w) = 2πσ 2



2σ 2

Lecture 8 Slide 15

minusmicro 2

σ 2

















or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 






































or





EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 























EV S = p

2σ 2 = N


1=Vp

noise

P(ldquo1rdquo)=05

= noise

0 shyVp +Vp

)( S



BER erfc ( p ) = σ 2 2 2 0






  

 

 






 






















6.02 Fall 2012 Lecture #8 - MIT OpenCourseWare · • SignaltoNoise Ratio (SNR) – usually denotes...

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Transcript of 6.02 Fall 2012 Lecture #8 - MIT OpenCourseWare · • SignaltoNoise Ratio (SNR) – usually denotes...