ECE 6640Digital Communications
Dr. Bradley J. BazuinAssistant Professor
Department of Electrical and Computer EngineeringCollege of Engineering and Applied Sciences
ECE 6640 2
Chapter 3
3. Baseband Demodulation/Detection.1. Signals and Noise. 2. Detection of Binary Signals in Gaussian Noise. 3. Intersymbol Interference. 4. Equalization..
ECE 6640 3
Sklar’s Communications System
Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
Chapter Goals
• Detection of Binary Signals plus Gaussian Noise• The decision process• Define Intersymbol Interference• Error Performance Degradation• Equalization Techniques
ECE 6640 4
Demodulation and Detection
• Focus on Symbols, Samples, and Detection• In the presence of Gaussian Noise and Channel Effects
ECE 6640 5
Demodulation and detection
• Major sources of errors:– Signal Path Loss
• Friis equation relates the received signal power to the transmitted power, antenna gains, and distance that the signal travels in free space
– Thermal noise (AWGN)• disturbs the signal in an additive fashion (Additive)• has flat spectral density for all frequencies of interest (White)• is modeled by Gaussian random process (Gaussian Noise)
– Inter-Symbol Interference (ISI)• Due to the filtering effect of transmitter, channel and receiver,
symbols are “smeared”.• Time spreading effects cause symbols to “overlap”
– Total symbol “length” may easily be 3+ symbol periods!
6
Receiver Block Diagram
• Receive the transmitted symbol plus noise– Symbol filter by channel
• Frequency down-conversion to baseband– Receiver filtering and equalization (if needed) applied
• Symbol filter– Matched filtering with Nyquist shaping for ISI– Optimize the pre-detected signal prior to sampling
• Optimal Time Sampling – Peak filter response timeECE 6640 7
Review Slides from ECE5640
• Chapter 9: Noise• Chapter 10: Noise in analog modulated signals• Chapter 11: Baseband Digital Transmission
ECE 6640 8
9
Noise Approximation• Uniform Noise Spectral Density
– Resistor description (Thevenin Model)
• Available Power from the “noise source”– Source output power into a matched load
TR2fGvv
ssout vR2
Rv
R4
vR1
2v
RvP
2s
2s
2sout
sout
2
N2T
R4TR2
R4fGfG 0vv
ss
2
NR 0ss
10
System Noise• Since the noise power spectrum is uniform, a systems
noise power is the product of the noise power and the integral of the filter power.
20NN
2NN fH
2NfSfHfS
00
0
20
20NN dffHNdffH
2N0R
Noise Equivalent Bandwidth• If we want the total noise power after the filter, we can
integrate the PSD for all frequencies or use the Filtered noise autocorrelation function at zero.– Both of these approaches may be difficult– Could we great a more simple “noise equivalent bandwidth for
filters” that is rectangular?
11
12
EQNEQNPowerDCelrect BHBGaindffHdffH
2_
0
2
mod_0
2 0
EQNPowerDCelrect B
frectGainfH2_mod_
20
2
EQN0H
dffHB
2Power_DC 0HGain
Noise Equivalent Bandwidth
0
2020NN dffH2
2NdffH
2N0R
• When filtering, it is convenient to think of band-limited noise, where the filter is a rect function with bandwidth BEQN
13
Noise Equivalent Bandwidth• Low pass filter
0Hgain_coherent
• For a unity gain filter – assumed when computing receiver input noise power
EQN0EQN0
NNN BNB22
N0RP
2Power_DC 0HGain
20
2
EQN0H
dffHB
0
2EQN dffHB
EQN02
EQN20
NNN BN0HB0H22
N0RP
Model of Received Signal with Noise
© 2010 The McGraw-Hill Companies
15
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Analog baseband transmission system with noise: Figure 9.4-2
Signal Plus Noise
• Additive Gaussian White Noise
ttf2costAtx c
ttfLtAtx cR 2cos tnttf
LtAts c 2cos
thtnttfLtAteD c
2cosPr
16
Signal-to-Noise Ratio• Comparing the desired signal power to the undesired noise
power.
• To compare signal and noise power, we must assume a filtering operations
tntxty c
TransmittingAntenna
ReceivingAntenna
RF Communication Channel
Noise
Linear Filtering
NonlinearDistortion
Atten-uation
tn
txc
tntxty c
17
Signal-to-Noise Ratio
• Equivalent receiver input signal and noise (ER)
• Equivalent destination signal and noise (D) or pre-demodulation (PreD)
thtntxty RD
tntxty ERERR
tntxty eDeDeD PrPrPr
18
Signal-to-Noise Ratio
• Equivalent receiver input SNR (ER)
EQN
R
EQN
ER
ER
ERR BN
SBN
txEtnEtxESNR
00
2
2
2
• Equivalent destination SNR EQN
D
D
D
eD
eDR BN
SNS
tnEtxESNR
02
Pr
2Pr
can be used to represent receiver noise figure contributions
19
Increase in SNR with filtering• If a filter matched to the input signal is applied, the noise
power would be reduced to the smallest equivalent noise bandwidth that is allowed.– Filter to minimize noise power– Importance of the IF filter in a super-het receiver!
• Front-end filtering goals – a dilemna– Minimize signal power loss (wider bandwidth)– Minimize filter equivalent noise bandwidth
(narrower bandwidths)– A trade-off must be made!
20
Typical Transmission RequirementsSignal Type Freq. Range SNR (dB)Intelligible Voice 500 Hz to 2 kHz 5-10Telephone Quality 200 Hz to 3.2 kHz 25-35AM Broadcast Audio 100 Hz to 5 kHz 40-50High-fidelity Audio 20 Hz to 20 kHz 55-65Video 60 Hz to 4.2 MHz 45-55Spectrum Analyzer 100 kHz-1.8 GHz 65-75
21
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Model of a CW communication system with noise: Figure 10.1-1
CW Communication with Noise
ttf2cosLtAtx cc
tnttf2cosLtAtv c
thtnttfLtAtPreD Rc
2cos
ttf2costAtx c
22
Signal and Noise Power• What are the signal and noise powers at the receiver?
• What is the receiver input power
2T
2T
Ts dttxtxT1limP
2n tnEP
2T
2T
ccT
2T
2T
Tv dttntxtntxET1limdttvtvE
T1limP
23
Receiver Signal plus Noise Power• What is the receiver input power
2T
2T
ccT
2T
2T
Tv dttntxtntxET1limdttvtvE
T1limP
2T
2T
2c
2cTv dttntntx2txE
T1limP
2T
2T
2c
2cTv dttnEtnEtx2tx
T1limP
2
2
22
2
2
2
2 021limT
T
T
Tc
T
TcTv dttnEdttxdttx
TP
ns2
2T
2T
2cTv PPtnEdttx
T1limP
24
Signal-to-Noise Ratio (SNR)• The SNR is a measure of the signal power to the noise
power at a point in the receiver.– Typically described in dB– The above computation was performed at the input
• Matlab SNR example: SNR_AM_Example.m– Pre-D “AM” SNR based in filter Beqn
• Effective BEQN RF due to sampling spectrum (B=Fis/2)• AM signal power based on carrier plus signal
n
s
PPSNR
25
Noise Equivalent Bandwidth• Since the noise power spectrum is uniform, a systems
average noise power is the product of the noise power and the integral of the filter power.
20NN
2NN fH
2NfSfHfS
00
0
20
20NN dffHNdffH
2N0R
26
Noise Equivalent Bandwidth
• When filtering, it is convenient to think of band-limited noise, where the filter is a rect function with bandwidth BEQN
0
2020NN dffH2
2NdffH
2N0R
EQN2
EQNPower_DC0
2
elmod_rect0
2 B0HBGaindffHdffH
EQNPowerDCelrect B
frectGainfH2_mod_
20
2
0H
dffHBEQN
2Power_DC 0HGain
27
Noise Equivalent Bandwidth• Low pass filter
0Hgain_coherent
• For a unity gain filter – assumed when computing receiver input noise power
EQN0EQN0
NNN BNB22
N0RP
2Power_DC 0HGain
20
2
EQN0H
dffHB
0
2EQN dffHB
EQNEQNNNN BNHBHNRP 0220 002
20
28
Filtering• What happens if the receiver input is filtered?
• What effect does the filter have on the signal?– None or slight band edge de-emphasis, if and only if the filter is
“wider” than the signal bandwidth– Now you know why a 3dB bandwidth isn’t that useful,
(3dB1/2 power point)!
ththtntxtv 21cf
ththtnththtxtv 2121cf
n
s
PPSNR
29
Filtering• What effect does the filter have on the noise?
– Normally you would expect for two filters
– Assume that the filters follow each other and that the first filter is narrower than the second filter
1_01_
FilterEQN
sFilterPost BN
PSNR
1_01__ FilterEQNFilterPostN BNP
1_0
2_1_02__ ,min
FilterEQN
FilterEQNFilterEQNFilterPostN
BNBBNP
2_02__ FilterEQNfilterpostN BNP 2_0
2_FilterEQN
sFilterPost BN
PSNR
1_02_
FilterEQN
sFilterPost BN
PSNR
30
Filters Provide SNR “Gain”• If filter 2 Beq < filter 1 Beq:
• You expect the IF filter to be smaller than the front-end RF or “pre-filtering” performed
– Think about kTB at different bandwidths and you will derive the same “gain”
– In typical receivers, the IF filter sets the Pre-Demodulation Bandwidth
2Filter_EQN
1Filter_EQN1Filter_Post
2Filter_EQN0
s2Filter_Post B
BSNR
BNPSNR
2Filter_EQN
1Filter_EQNFilter B
BGain
31
Bandpass Noise Processing
• What happens after mixing and lowpass filtering?– assume LPF passes the entire baseband.
cfTc Bf
cfTc Bf
TBTB
20N
tf2cos c Band Pass
FilterLowPass
Filter
Bandpass filter bandwidth may not be centered on fc• an alpha offset
32
Quadrature Noise (1)• Question: Is Quadrature noise a different power than “real”
baseband noise• Noise in a quadrature process
• Noise power is related as
• What about ?
tftntftntn cqci 2sin2cos
22 2sin2cos tftntftnEtnE cqci
2
0222 NtnEtnEtnE qi
33
Quadrature Noise (2)• Noise in a quadrature process
2cqci
2 tf2sintntf2costnEtnE
tf2sintn
tf2sintf2costntn2tf2costn
EtnE
c22
q
ccqi
c22
i2
222cos
21222cos
21 222 tftntftnEtnE cqci
22
121 0222 NtnEtnEtnE qi
222cos
21222cos
21 222 tfEtnEtfEtnEtnE cqci
Mixing Noise (1)• Think of the two noise bands as
1. The band of interest2. The image band
34
thtftfftntfftn
thtftfftntfftnthtftn
IFLOIFLOqIFLOi
IFLOIFLOqIFLOiIFLO
2cos2sin2cos
2cos2sin2cos2cos
22
11
th
tfftntfftn
tftntftn
thtfftntfftn
tftntftnthtftn
IFIFLOqIFLOi
IFqIFi
IFIFLOqIFLOi
IFqIFiIFLO
22sin22cos
2sin2cos
21
22sin22cos
2sin2cos
212cos
22
22
11
11
thtftntftn
thtftntftnthtftn
IFIFqIFi
IFIFqIFiIFLO
2sin2cos21
2sin2cos212cos
22
11
Mixing Noise (2)• Defining the equivalent IF noise
• But this is the same as quadrature noise
• Mixing doesn’t change the noise power 35
thtftntftn
thtftntftnthtftn
IFIFqIFi
IFIFqIFiIFLO
2sin2cos21
2sin2cos212cos
22
11
thtftntftnthtn IFIFqIFiIF 2sin2cos
th
tftntn
tftntnthtftn IF
IFii
IFii
IFLO
2sin21
21
2cos21
21
2cos
21
21
22
121 02
22
12 NtnEtnEtnE iii
221
21 02
22
12 NtnEtnEtnE qqq
36
Mixing Noise to Baseband• What if we split bandpass noise into two
distinct noise bands, BT/2 above and below the carrier/IF?
• Noise power is related as
• Noise bands get added …
WNBNBNtnE TTcarrier 0002 22
2
thtfftntfftn
thtfftntfftntn
BcqBci
BcqBci
222
111
2sin2cos2sin2cos
WNBNBNtnEtnE TTBelowCAboveC 00
022
21 22
22
WNBNtnEtnEtnE T 002
22
12 2
2TBWfor
37
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) General case; (b) symmetric-sideband case;
(c) suppressed-sideband case: Figure 10.1-3
Mixing Noise to Baseband
10 or
5.0
5.00
cfTc Bf
cfTc Bf
TBTB
20N
Mix to BasebandW=½ BT
N0BT=2N0W
Mix to IFBT
N0BT
Not Desired
Why did we do these derivations?• The past derivations were all about mixing and filtering.
– Quadrature noise is the noise that gets mixed to the intermediate. The bandwidth and noise power do not change
– Quadrature noise is the noise that gets mixed to baseband. The bandwidth is halved and the noise power is doubled the LPF bandwidth standard noise power.
38
WNBNBNtnE TT 0002 22
2
WNBNtnE T 002 2WBT
2
39
Complex Noise• Noise in a complex process
• Noise power is related as tnjtntn qi
222 tntntnjtntnjtnEtnE qiqqii
2
02 NtnE
Hqiqi tnjtntnjtnEtnE 2
222 tnEtnEtnE qi
tnjtntnjtnEtnE qiqi 2
4
022 NtnEtnE qi
),(: nmrandnnMATLAB 2),(),(: sqrtnmrandninmrandnnMATLAB
40
Noise Envelope and Phase (1)• Noise as a magnitude and phase
ttf2costAtn ncn
nni cosAn nnq sinAn
• The magnitude is a Rayleigh distribution– Mean and moment
nR
2n
R
nnA Au
N2Aexp
NAAp
n
2NAE R
n
R2
n N2AE
41
Noise Envelope and Phase (2)• Probability of An exceeding a value “a”
• Phase Distribution
• Noise Power
Rn N
aaAP 2exp2
2021
nn forp
2212
2cos
02
222
NNNtnE
ttfEtAEtnE
RR
ncn
2
Rn
NAE
Rn NAE 22
42
Noise Characteristics• The noise power does not change based on the
representation, the center frequency, or due to mixing.
• The noise power will change when the bandwidth is further limited in some way!
43
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Model of a CW communication system with noise: Figure 10.1-1
CW Communication with Noise
ttf2cosLtAtx cc
tnttf2cosLtAtv c
thtnttfLtAteD Rc
2cosPr
ttf2costAtx c
Chapter 11
• Baseband Transmission of PAM Symbols• PAM Symbol Autocorrelation and Power Spectral Density• Symbol Detection in Noise
ECE 6640 44
45
Digital Pulse-Amplitude Modulation (PAM)
• Also referred to as pulse-code modulation (PCM)• The amplitude of pulse take on discrete number of
waveforms and/or levels within a pulse period T.
• p(t) takes on many different forms, a rect for example
k
k kTtpatx
else0
Tt01tp
T0for,apakTmTpamTx mmk
k
46
Digital Signaling Rate• For symbols of period T,
the symbol rate is 1/T=R
• The rate may be in bits-per-second when bits are sent. A bps rate is usually computed and defined.
• The rate may be in symbols-per-second when symbols are sent. When there are a defined number of bits-per-symbol, the rate may be defined in bits-per-second.– If parity or other non-data bits are sent, the messaging rate and the
signaling rate may differ.
47
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Baseband transmission system (b) signal-plus-noise waveform: Figure 11.1-2
Transmission
tnkTttp~atyk
dk
48
Transmission
• The digital signal is time delayed
• The pulse is “filtered” and/or distorted by the channel
• Recovering or Regenerating the signal may not be trivial
– Signal plus inter-symbol interference (ISI) plus noise
tnkTttp~atyk
dk
thtpfntp c~
dt
dmk
kmd tmTnkTmTpaatmTy
~ˆ
49
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Unipolar RZ & NRZ
(b) Polar RZ & NRZ
(c) Bipolar NRZ
(d) Split-phase Manchester
(e) Polar quaternary NRZ
ABC Binary PAM formats
50
PAM Power Spectral Density: Polar NRZ
• For a zero mean, polar NRZ of amplitude +/- A and symbol duration Tb
k b
bdk T
TkTtrectatv
bdb
d TTT
Tp 0,1 kjfor,0aaE
aE,0aE
kj
22nn
bbb
vv TTT
tvtvER
,12
bbvv TfTtvtvEwS 22 sinc
222 AaE n
bb
bbvv
rf
rA
TfTAwS
22
22
sinc
sinc
51
PAM Power Spectral Density: Arbitrary Pulse
• Using Poisson’s sum formula
k
dk D
DkTtpatv DT0,D1Tp dd
222, aanan maEmaE
n
avv DfjnRfPD
fS 2exp1 2
0,
0,2
22
nm
nmnR
a
aaa
n
aavv D
nfDnP
DmfP
DfS
222
2
DrDT bb
1,
n
bbbabavv rnfrnPrmfPrfS 2222
rb is symbol rate
52
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.1-5
Power spectrum of Unipolar, binary RZ signal
bb r
fr
fP2
sinc2
1
2
,2
22 AaEAaE nn
n
bbb
vv rnfnAr
fr
AfS2222
2sinc
162sinc
16
trTttp b
b
2rect
2
rect
0,4
0,2
22
222
nAm
nAmnR
a
aa
a
Power spectrum of Unipolar, binary RZ signal
• For rb=2
53
n
bbb
vv rnfnAr
fr
AfS2222
2sinc
162sinc
16
-8 -6 -4 -2 0 2 4 6 80
0.01
0.02
0.03
0.04
0.05
0.06
0.07Unipolar Binary RZ
freq (f)
Plots from PSD_PCM.m
54
Power spectrum of Unipolar, binary NRZ signal
bb rf
rfP sinc1
2
,2
22 AaEAaE nn
0,4
0,2
22
222
nAm
nAmnR
a
aa
a
nb
bbvv rnfnA
rf
rAfS 2
222
sinc4
sinc4
trTttp bb
rectrect
fArf
rAfS
bbvv
4sinc
4
222
• For rb=2
-8 -6 -4 -2 0 2 4 6 80
0.05
0.1
0.15
0.2
0.25Unipolar Binary NRZ
freq (f)
55
Power spectrum of Polar, binary RZ signal (+/- A/2)
4,022 AaEaE nn
0,0
0,42
222
nm
nAmnR
a
aaa
22
2sinc
16
bb
vv rf
rAfS
bb r
fr
fP2
sinc2
1
trTttp b
b
2rect
2
rect
-8 -6 -4 -2 0 2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
0.035Polar Binary RZ
freq (f)
• For rb=2
56
Power spectrum of Polar, binary NRZ signal (+/- A/2)
bb rf
rfP sinc1
4,022 AaEaE nn
0,0
0,42
222
nm
nAmnR
a
aaa
22
sinc4
bbvv r
fr
AfS trTttp bb
rectrect
• For rb=2
-8 -6 -4 -2 0 2 4 6 80
0.02
0.04
0.06
0.08
0.1
0.12
0.14Polar Binary NRZ
freq (f)
57
Spectral Attributes of PCMIf Bandwidth W=1/T, then WT=1
Note that WT=0.5 or a bandwidth equal to ½ the symbol rate can be used!
58
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.2-1
Baseband Binary Receiver
• Synchronous Time sampling of maximum filter output
thtnthkTtpaty ink
k
kkk taty n
59
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) signal plus noise (b) S/H output (c) comparator output: Figure 11.2-2
Regeneration of a unipolar signal
60
Unipolar NRZ Binary Error Probability
• Hypothesis Testing using a voltage threshold– Hypothesis 0
• The conditional probability distribution expected if a 0 was sent
– Hypothesis 1• The conditional probability distribution expected if a 1 was sent
kYkkkYkY tpatapHyp n0|n| 0
kYkkkYkY tpAatapHyp nA|n| 1
kNkY ypHyp 0|
A-| 1 kNkY ypHyp
61
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Conditional PDFs Figure 11.2-3
Decision Threshold and Error Probabilities
• Use Hypothesis to establish a decision rule– Use threshold to determine the probability of correctly and
incorrectly detecting the input binary value
V
0Y0e dyH|ypVYPP
V
1Y1e dyH|ypVYPP
kkk taty n
62
Average Error Probability
• Using the two error conditions:– Detect 1 when 0 sent– Detect 0 when 1 sent
• Selecting an Optimal Threshold
• For equally likely binary values
1e10e0error PHPPHPP
21HPHP 10
1e0eerror PP21P
1100 || HVpHPHVpHP optYoptY
10 || HVpHVp optYoptY
63
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.2-4
Threshold regions for conditional PDFs
2AVopt
21
10 HPHP
64
For AWGN• The pdf is Gaussian
for
for
2
2
2N0Y 2yexp
21ypH|yp
x
2
d2
exp21xQ
2AQVQdyypVYPP
VN0e
2
AQVAQdyAypVYPPV
N1e
2AVopt
21
10 HPHP
21 2 ee PAQVAQP
65
Modification for Polar NRZ Signals (+/- A/2)
• Hypothesis Testing using a voltage threshold– Hypothesis 0
• The conditional probability distribution expected if a 0 was sent
– Hypothesis 1• The conditional probability distribution expected if a 1 was
sent
kYkkkYkY tApAatapHyp n
22|n| 0
kYkkkYkY tApAatapHyp n
22|n| 1
2| 0
AypHyp kNkY
2A-| 1 kNkY ypHyp
02A
2AVopt
66
Modification for Polar NRZ Signals (+/- A/2)
• Determining the error probability
• Notice that the error is the same as Unipolar NRZ– The distance between the expected signal values is the
same– The “distance” between the expected values determines
the error …
22
21AQ
VAQdyAypVYPP
V
Ne
22
20AQ
VAQdyAypVYPP
VNe
67
Modification for Bipolar NRZ Signals (+/- A)
• Hypothesis Testing using a voltage threshold– Hypothesis 0
• The conditional probability distribution expected if a 0 was sent
– Hypothesis 1• The conditional probability distribution expected if a 1 was
sent
kYkkkYkY tApAatapHyp n|n| 0
kYkkkYkY tApAatapHyp n|n| 1
AypHyp kNkY 0|
A-| 1 kNkY ypHyp
0 AAVopt
68
Modification for Bipolar NRZ Signals
• Determining the error probability
• Notice that the error has been reduced– The distance between the expected signal values may
be twice as large as the unipolar case (using +/- A)
AQVAQdyAypVYPPV
N1e
AQVAQdyAypVYPPV
N0e
69
Relationship to signal power
• Defining the average received signal power– Unipolar NRZ
– Polar NRZ
– Bipolar NRZ
• In terms of SNR
AASR ,0,21 2
2,
2,
41 2 AAASR
Polarfor
NS
UnipolarforNS
21
N4A
2A
R
R
R
22
AAASR ,,2
BipolarforNS
NAA
RR
22
2
2
21limT
TcTR dttx
TES
Probability of error
• The probability of detecting a transmitted symbol correctly is dependent upon the received signal-to-noise ratio …. assuming– Unipolar NRZ (orthogonal)
– Polar NRZ (antipodal)
– Bipolar NRZ (antipodal)
70
Re N
SQAQP21
2
RR NS
NAA
42
22
RR NS
NAA
21
42
22
Re N
SQAQP2
Re N
SQAQP RR N
SNAA
22
21HPHP 10
Power Ratio vs. Bit Energy
• For continuous time signals, power is a normal way to describe the signal.
• For a discrete symbol, the “power” is 0 but the energy is non-zero– Therefore, we would like to describe symbols in terms
of energy not power
• For digital transmissions how to we go from power to energy?– Power is energy per time, but we know the time
duration of a bit. Noise has a bandwidth.
71bbR T
ES 1 WNNR 0 ?
R
R
R NS
NS
72
SNR to Eb/No
• For the Signal to Noise Ratio – SNR relates the average signal power and average noise
power (Tb is bit period, W is filter bandwidth)
– Eb/No relates the energy per bit to the noise energy(equal to S/N times a time-bandwidth product)
WR
NE
WT1
NE
WNT1E
NS b
0
b
b0
b
0
bb
WTNS
RW
NS
NE
bb0
b
73
Relationship to Eb/No
• Defining the energy per bit to noise power ratiofor a time-bandwidth product of
– Unipolar
– Polar
– Bipolar
0
b
RR
22
NE
NS
21
N4A
2A
0
b
RR
22
NE2
NS
N4A
2A
0
b
RR
22
NE2
NS
NAA
21T
2RTW b
bb
74
Relationship to Bit Error Probability
• Defining the binary bit error probabilityfor a time-bandwidth product, assuming
– Unipolar (orthogonal)
– Polar (antipodal)
– Bipolar (antipodal)
0
berror N
EQ2AQP
0
berror N
E2QAQP
0
berror N
E2Q2AQP
21HPHP 10
75
Bit Error Rate Plot
10-3 10-2 10-1 100 1010
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5B
it E
rror R
ate
Eb/No
Classical Bit Error Rates
OrthogonalAntipodal
EbNo=(0:10000)'/1000;
% Q(x)=0.5*erfc(x/sqrt(2))
Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));
semilogx(EbNo,[Ortho Antipodal])ylabel('Bit Error Rate')xlabel('Eb/No')title('Classical Bit Error Rates')legend('Orthogonal','Antipodal')
76
BER Performance, Classical Curveslog-log plot
-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510-7
10-6
10-5
10-4
10-3
10-2
10-1
100B
it E
rror R
ate
Eb/No
Classical Bit Error Rates
OrthogonalAntipodal
77
Antipodal and Orthogonal Signals
• Antipodal– Distance is twice “signal voltage”– Only works for one-dimensional signals
• Orthogonal– Orthogonal symbol set– Works for 2 to N dimensional signals
bE2d
jiforjifor
dttstsE
zT
jiij 111
0
jifor0jifor1
dttstsE1z
T
0jiijbE2d
78
M-ary Signals
• Symbol represents k bits at a time– Symbol selected based on k bits– M waveforms may be transmitted
• Allow for the tradeoff of error probability for bandwidth efficiency
• Orthogonality of k-bit symbols– Number of bits that agree=Number of bits that disagree
k2M
jifor0jifor1
K
bbsumbbsumz
N
1k
jk
ik
K
1k
jk
ik
ij
79
Example 11.2-1
• Unipolar computer network with
– Desired BER is one bit per hour
• Solve for the signal energy
bpsRb610 HzdBHzWN /194/104 20
0
1010336001
be RP
101032
AQPerror
UnipolarforNS
NAA
R
R
R
21
42
22
2.62
A
From p. 790
Rb
R SRNN
22.622.62 0
22
WWSR12620 1054.1105.010444.382
80
Exercise 11.2-1 (1)
• Unipolar system with equally likely digits and SNR = 50
• Calculate the error probabilities when the threshold is set to V=0.4 x A
UnipolarforNSA
R
R
21
2
2
1050212
A
VAQdyAypVYPPV
Ne1
VQdyypVYPPV
Ne0
81
Exercise 11.2-1
• Calculate the error probabilities when the threshold is set to V=0.4 x A
104.00
QVQVYPPe
106.01
QVAQVYPPe
50 105.30.4 QPe
91 105.10.6 QPe
21HPHP 10
1e0eerror PP21P
595 1075.1105.1105.321 errorP
V=0.5 x A
0.5105.010 QQPP ee
710 105.3 erroree PPP
ECE 6640 82-8 -6 -4 -2 0 2 4 6 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Gaussian PDF and pdf
Gaussian Distribution
The Gaussian probability density function (pdf)
The Gaussian or Normal probability density function is defined as:
xforXxxf X ,
2exp
21
2
2
where X is the mean and is the variance
The Gaussian Probability Distribution Function (PDF)
dvXvxFx
vX
2
2
2exp
21
The PDF can not be represented in a closed form solution!
ECE 6640 83
Gaussian Distribution
The Gaussian Probability Distribution Function is
dvXvxFx
vX
2
2
2exp
21
The PDF can not be represented in a closed form solution!
The PDF is tabulated for a zero mean, unit variance pdf. For these values, it is often described as “normalized” and is defined as
duuxx
u
2
exp21 2
The distribution function is then defined as
XxxFX
When using Appendix D, the negative values in x are derived as
xx 1
84
Q FunctionAnother defined function that is related to the Gaussian (and used) is the Q-function.:
duuxQxu
2
exp21 2
The Q-function is the complement of the normal function, :
xxQ 1
Therefore not that:
xQxQ 1
XxQxFX 1
Q Function Table p. 858
ECE 6640 85
Using MATLAB
Another way to find values for the Gaussian
The error function
duuxerfx
u
0
2exp2
21
21 xerfxQ
22
121
21
211 XxerfXxerfxFX
The error function (Y = ERF(X)) is built-in to MATLAB. .
From MATLAB: ERF Error function. Y = ERF(X) is the error function for each element of X. X must be real. The error function is defined as: erf(x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt. See also erfc, erfcx, erfinv. Reference page in Help browser doc erf
ECE 6640 86
Using MATLAB (2)
The complementary error function
xerf1xerfc
2
xerfc21xQ
The error function (Y = ERFC(X)) is built-in to MATLAB. .
From MATLAB: ERFC Complementary error function. Y = ERFC(X) is the complementary error function for each element of X. X must be real. The complementary error function is defined as: erfc(x) = 2/sqrt(pi) * integral from x to inf of exp(-t^2) dt. = 1 - erf(x). Class support for input X: float: double, single See also erf, erfcx, erfinv. Reference page in Help browser doc erfc
Qfn and Qfninv• These function are now in the Misc_Matlab zip file on the
web site
function [Qout]=Qfn(x)% Qfn(x) = 0.5 * erfc(x/sqrt(2));Qout = 0.5 * erfc(x/sqrt(2));
function [x]=Qfninv(Pe)% For Qfn(x) = 0.5 * erfc(x/sqrt(2));% The inverse can be found asx=sqrt(2)*erfcinv(2*Pe);
87
88
Properties of Matched Filter• See ECE3800 Notes
– Review from Chapter 9
• Wikipedia– http://en.wikipedia.org/wiki/Matched_filter– “The matched filter is the optimal linear filter for maximizing the
signal to noise ratio (SNR) in the presence of additive stochastic noise.”
Signal to Noise Ratio Definition
• The filtered response becomes
• The SNR
ECE 6640 89
0
dtntshtnts oo
EQo
o
o
o
Noise
Signalout BN
tsE
tnE
tsEPP
SNR
2
2
2
0
2
2
0
21 dtthN
dtshE
SNR
o
out
Optimized Matched Filter
• The matched filter is
• The resulting SNR
• Or for symbol energy
ECE 6640 90
uTsKh
o
ss
sso
ssT
o
T
out NR
RKN
TtRK
dtTsKN
dtsTsKE
SNR 02
021
21 2
22
0
2
2
0
o
sout N
ESNR 2
dffStsERss220
Autocorrelation versus Integration
• For a matched filter, the integral of the matched filter and the autocorrelation of the signal are approximately equivalent– For signals of limited time duration it can be exact!
• The condition is related to
• But notice that “integrating” and filtering may produce very different results except at t=T.– the integral is monotonically increasing– the filter is not!ECE 6640 91
Ttssss
T
ss
TtRR
dtsTsKTtR
00
Correlation Receivers
• The concept that a matched filter is performing an autocorrelation has resulted in the concept of the Correlation Receiver.
• The symbol of interest is auto-correlated. • All other symbols are cross-correlated.
• It is assumed that the two outputs are different enough to allow symbol detection– Noise power is present based on the “equivalent noise bandwidth”
of the “matched” symbol receivers.
ECE 6640 92
Corr. vs Int. for an RF/IF Envelope
ECE 6640 93
Question?
• Are there some better shapes than others for signals and their matched filters?– Easy to generate.– Frequency band limited. – Finite time duration.– Minimize inter-symbol interference.
• Start with rects or pulses• More advanced use raised cosine or Nyquist filter,
or for matched transmit and receive filters use “square root Nyquist filters”.
ECE 6640 94
95
Defining a Shape or Filter for PulsesSection 3.3
• We want to minimize or zero inter-symbol interference (ISI)
• We want a frequency band limited filter
– Allowable signal rates with as the excess bandwidth
k
dk Tkttpaty
,2,0
01TTt
ttp
fBfP 0
TBBandrwithrBwhere 2
0,2
BrBforBr 2,2
Tr 1
96
Defining a Filter for Pulses• Possible solutions
,2,0
01TTt
ttp
fBfP 0
trtptp sinc
fforfPtp 0
10
dffPp
• Therefore we select
rf
rfPfP rect1
These are considered the Nyquist conditions for the filter
Tr 1
97
Cosine Spectral Shaping
• A candidate filter is (with with as the excess BW)
2rect
42cos
4fffP
From Chap 2Raised cosine
pulse
Raised cosine pulse. (a) Waveform (b) Derivatives(c) Amplitude spectrumFigure 2.5-7
Convolving• Raised Cosine Convolution with Bandlimited Spectrum
98
rf
rfPfP rect1
TBBandrwithrBwhere 2
0,2
20
22242cos1
21
2
rf
rfrrfr
rfr
fP
trt
ttp
sinc412cos
2
• Transforming to the time domain filter
99
Nyquist/Raised Cosine Pulse Shaping
GNU FDL:Oli Filth, Raised Cosine Filter, Impulse Response, en.wikipedia.org, 3 November 2005, Oli Filth
GNU FDL:Oli Filth, Raised Cosine Filter Response , en.wikipedia.org, 3 November 2005, Oli Filth
Tr
rABC
12
Tr
rABC
12
http://en.wikipedia.org/wiki/Raised-cosine_filter
Nyquist Filter (discrete raised cosine)
100
trt
ttp
sinc412cos
2
% function hnyq=nyquistfilt(alpha,M)% or% function hnyq=nyquistfilt(alpha,fsymbol,fsample,k)%% alpha roll-off% fsample rate% fsymbol rate% M = fsample/fsymbol (an integer value)% k is 1/2 the number of symbols in the filter% The filter length is euqal to 2*ceil(k*M)+1%% A discrete time cosine taperd Nyquist filter% Based on frederic harris, Multirate Signal Processing for Communications% Prentice-Hall, PTR, 2004. p. 89
MknMkforMn
MnMn
np
,sinc21
cos
2
r 2
sfnt
10 2
0 r
Mfr s
trtrtrtp
sinc21
cos2
Mntr
MATLAB Raised Cosine Filters (1)• Rcosine (obsolete)
– [NUM, DEN] = RCOSINE(Fd, Fs, ‘fir’, R)– FIR raised cosine filter to filter a digital signal with the digital
transfer sampling frequency Fd. The filter sampling frequency is Fs. Fs/Fd must be a positive integer. R specifies the rolloff factor which is a real number in the range [0, 1].
• rcosfir (obsolete)– B = RCOSFIR(R, N_T, RATE, T)– Raised cosine FIR filter. T is the input signal sampling period, in
seconds. RATE is the oversampling rate for the filter (or the number of output samples per input sample). The rolloff factor, R, determines the width of the transition band. N_T is a scalar or a vector of length 2. If N_T is specified as a scalar, then the filter length is 2 * N_T + 1 input samples.
101
MATLAB Raised Cosine Filters (2)• firrcos (also obsolete)
– B=firrcos(N,Fc,DF,Fs)– Returns an order N low pass linear phase FIR filter with a raised
cosine transition band. The filter has cutoff frequency Fc, sampling frequency Fs and transition bandwidth DF (all in Hz).
– The order of the filter, N, must be even.– Fc +/- DF/2 must be in the range [0,Fs/2]– The coefficients of B are normalized so that the nominal passband
gain is always equal to one.– B=firrcos(N,Fc,R,Fs,'rolloff') interprets the third argument, R, as
the rolloff factor instead of as a transition bandwidth.– R must be in the range [0,1]
102
MATLAB Raised Cosine Filters (3)• firrcos
– B = rcosdesign(BETA, SPAN, SPS)– Returns square root raised cosine FIR filter coefficients, B, with a
rolloff factor of BETA. The filter is truncated to SPAN symbols and each symbol is represented by SPS samples. rcosdesigndesigns a symmetric filter. Therefore, the filter order, which is SPS*SPAN, must be even. The filter energy is one.
– Beta [0,1]– SPS number os samples per symbol– SPAN length of filter in number of symbols
103
Textbook Waveform Energy
• Waveform Energy
• Matched Filter
ECE 6640 104
T
ii dttsE0
2
t
dthrthtrtz
tTstuth *
t
dtTsstz0
*
TTT
dsdssdTTssTz0
2
0
*
0
*
Correlation
Optimum binary detection
105
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) parallel matched filters (b) correlation detector: Figure 14.2-3
Symbols and Matched Filters
2006-01-31 Lecture 3 106
T t T t T t0 2T
)()()( thtsty opti 2A)(tsi )(thopt
T t T t T t0 2T
)()()( thtsty opti 2A)(tsi )(thopt
T/2 3T/2T/2 TT/22
2A
TA
TA
TA
TA
TA
TA
Optimized Error Performance
• Maximize the “distance”, Ed, between the sampled values that are used for detection.– The distance is based on the correlation of the symbols
ECE 6640 107
T
jib dttstsE0
*
TT
ji
T
d dttsdttstsdttsE0
22
0
*
0
21 2
T
ib dttsE0
2
12 bbbbd EEEEE
T
d dttstsE0
221
Optimized Error Performance
• To maximize the distance: = -1– Symbols are said to be antipodal– Examples: +/- 1 Symbols (Bipolar), BPSK– Not always achievable
• Useful performance: = 0– Symbols are said to be orthogonal– Examples: On-Off Keying (ASK), FSK, “independent symbols”
ECE 6640 108
bd EE 2
bd EE
109
Antipodal and Orthogonal Signals
• Antipodal– Distance is twice “signal voltage”– Only works for one-dimensional signals
• Orthogonal– Orthogonal symbol set– Works for 2 to N dimensional signals
bE2d
jiforjifor
dttstsE
zT
jiij 111
0
jifor0jifor1
dttstsE1z
T
0jiijbE2d
110
Relationship to Bit Error Probability
• Defining the binary bit error probabilityfor a time-bandwidth product
– Orthogonal
– Antipodal
0NEQP b
error
0
2N
EQP berror
21HPHP 10
ECE 6640 111
Bit Error Rate Plot-Linear BER
10-3 10-2 10-1 100 1010
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5B
it E
rror R
ate
Eb/No
Classical Bit Error Rates
OrthogonalAntipodal
EbNo=(0:10000)'/1000;
% Q(x)=0.5*erfc(x/sqrt(2))
Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));
semilogx(EbNo,[Ortho Antipodal])ylabel('Bit Error Rate')xlabel('Eb/No')title('Classical Bit Error Rates')legend('Orthogonal','Antipodal')
ECE 6640 112
BER Performance Fig. 3.14
-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510-7
10-6
10-5
10-4
10-3
10-2
10-1
100B
it E
rror R
ate
Eb/No
Classical Bit Error Rates
OrthogonalAntipodal
0NEQP b
error
0
2N
EQP berror
Equalization
• An advanced topic that I skip at this time.
• Narrowband communications may go through a magnitude and phase change due to “the channel”
• Wideband communications likely experiences channel effects that may be non-linear across the signal frequency band. To correctly detect the information, an inverse channel filter or “equalizer” is used. – The channel is usually not predictable. Therefore, the equalizer
must “learn” from the transmitted signal how to correct for channel effects. There are “adaptive” algorithms that can do this and are taught in a more advanced course.
ECE 6640 113