baseband communication… simulation sampling...
TRANSCRIPT
Baseband CommunicationBaseband Communication
Baseband Transmission Digital Transmission without ModulationLine coding TX RX
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s, Sharif,
Filter Equalization Sampler Detector
A/D+Encoding
FormattingBasebandChannel
Communication might be in baseband e.g. Ethernet (IEEE 802.3)Or can be modeled in baseband using LP Model
Non-return to zero (NRZ)
Return to zero (RZ)
Phase encoded
Multilevel binary89
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ommunicatio
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s, Sharif, E
E, Im
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pour, im
u , Fall 2
011
Baseband Communication…Baseband Communication…
Level, Mark(1), Space(0)
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s, Sharif,
90
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s, Sharif, E
E, Im
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011
Baseband Communication…Baseband Communication…
Bit Synchronization Methods:1) Clock Encoding and Extraction
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91
2) Digital Phase Locked Loop (DPLL)
Transitions are needed scrambling or bit stuffing like 4B5B
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E, Im
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Simulation Sampling RateSimulation Sampling Rate
First attempt: LTI systems with no feedbackPower spectral density of signals Spectrum of underlying pulse shapes
Minimum run time vs. Negligible aliasing errors
Need to represent baseband signals for transmission
Common model of transmitting signals in baseband:
Course N
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s, Sharif, Common model of transmitting signals in baseband:
ak are digital data to be transmitted (±1 for binary, NRZ)
In general:
92
∑∞
−∞=
−=k
k kTtpaAtx )()(
)(][,][*
lkRaaEmaE lkak −==
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E, Im
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pour, im
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Autocorrelation function to get to power spectrumDefined for stationary processes
Stationarizing…
∑
∑∑
−=
−−== −
k
aX
k l
lkXX
kTtpAmtm
lTtpkTtpRAtXtXEttR
)()(
)]()())()((),( 2
*
1
2
2
*
121
Simulation Sampling Rate…Simulation Sampling Rate…
Cyclo-stationary with respect to T
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93
∑
∑
∫∫
∞
−∞=
−
∞
−∞=
−−
==
−−=
=+=
∆∆+=
m
mfTj
ccX
m
XX
T
T
X
T
T
XXXXX
emRfSfPfST
AfS
ppmTmRT
AR
dttmT
tmdtttRT
R
TUtXtXXzedStationari
π
τττδτ
ττ
222
*2
2/
2/
2/
2/
)()(,|)(|)()(
))(*)((*)()(1
)(
)(1
)(,),(1
)(
],0[~),()(:
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ommunicatio
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s, Sharif, E
E, Im
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Noise like data, zero mean, independent symbols:
Rectangular Pulse shape is the worst case, though simplest to analyze…
1,/|)(|)(
)()(,0
2222
2
==
===
σσ
δσ
normalizedTfPAfS
maaEmRm
X
lka
222 )(sinc|)(| fTTAfP =
Simulation Sampling Rate…Simulation Sampling Rate…
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Sampling random signals:
Assuming fs=m/T:
94
222 |)(|)/(: AdffPTAPpowerdtransmitte == ∫∞
∞−
])[(sinc)( 222TnffTAffS s
n
ss
X −= ∑∞
−∞=
)(sinc)( 222nmfTTAffS
n
ss
X −= ∑∞
−∞=
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ommunicatio
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s, Sharif, E
E, Im
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u , Fall 2
011
Simulation Sampling Rate…Simulation Sampling Rate…m=6 case drawing
with 3 shifted terms
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s, Sharif,
dffT
dffT
N
SSNR
s
s
f
f
a
a
)(sinc
)(sinc
2/
2
2/
0
2
∫
∫∞
==
95
dfmnfTTAfN
dffTTAfS
nn
f
sa
f
s
s
s
∑ ∫
∫∞
≠−∞=
−=
=
0
2/
0
222
2/
0
222
)(sinc2
)(sinc2
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s, Sharif, E
E, Im
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501000~
,
)/(sinc
)/(sinc
,/
2/
2
2/
0
2
×
=
=
∑
∑
=
=
nk
kj
kj
SNR
klargekTjf
nk
mkj
mk
j
a
j
3
4
2
/1,
/1,
/1,
fPSDCosineRaised
fPSDpulseTriangular
fPSDpulserRectangula
∝
∝
∝
Simulation Sampling Rate…Simulation Sampling Rate…Course N
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ACT2Derive and draw for Manchester
coding
n Lobes,
k Samples in lobes
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emRfSfPfST
AfS
bba
m
mfTj
ccX
kkk
π222
1
)()(,|)(|)()( ==
+=
∞
−∞=
−
−
∑
Case of dependent data symbols
Spectral shaping by operations on information sequences
bk independent binary data sequence. Symbols defined as
Simulation Sampling Rate…Simulation Sampling Rate…
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fTfPT
fS
fTfTfS
ow
m
m
aaEmR
T
X
c
mkk
m
π
ππ
22
2
cos|)(|4
)(
cos4)2cos1(2)(
1
0
,0
,1
,2
)()(
=
=+=
±=
=
== +
−∞=
∑
Even more attenuation rate with frequency97
EX2: 3.3, 3.4, 3.6, 3.7, 3.8, 3 .9, 3.11, 3.12, 3.13
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LP Simulation Models for BP Signals & SystemsLP Simulation Models for BP Signals & Systems
Bandpass Transmission Digital Communication with ModulationBandpass Signaling
Can be expressed in terms of a low pass form in generalCalled Complex Envelope as well
)(sin)()(cos)()()()(~
)2sin()()2cos()(
))(2cos()()(
00
0
qd
ttjAttAtjxtxtx
tftxtftx
ttftAtx
ϕϕ
ππ
ϕπ
+=+=
−=
+=
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Sampling then is being done on LP signalsResults can be converted to BP model any time at will
98
))(~Re()(,)()(~
,)(
)(arctan)(,)()()(
)(sin)()(cos)()()()(~
02)(
22
tfjtj
d
q
qd
qd
etxtxetAtx
signalsLPtx
txttxtxtA
ttjAttAtjxtxtx
πϕ
ϕ
ϕϕ
==
=+=
+=+=
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E, Im
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FM Modulator (Analog)
LP Simulation Models for BP…LP Simulation Models for BP…
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Digital Modulations TX RX
LP Simulation Models for BP…LP Simulation Models for BP…
Baseband TXBandpassChannel
Baseband RX
Modulator
DemodulatorPreprocess
Basis functions
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Link to baseband transmission Several Modulation Schemes
BASK, BFSK, BPSK, OOKM-ary ASK, M-ary FSK, M-ary PSKM-ary QAM (APK)
100
Baseband RXDemodulatorPreprocess
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LP Simulation Models for BP Signals & SystemsLP Simulation Models for BP Signals & Systems
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s, Sharif,
101
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M-ary Modulator (Digital)Usually M = 2b
Symbol Mapping: Grouping b bits together making the Symbol indexOutput of the mapper for the kth symbol is Sk=dk+jqkImpulse functions to pass through pulse shaping…
Scattergram : xq versus xd (dimensionality ≤ 2), Signal Space Constellation : dimensionality ≤ M
LP Simulation Models for BP Signals & SystemsLP Simulation Models for BP Signals & Systems
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102
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1 1
LP Simulation Models for BP …LP Simulation Models for BP …
MappingCodeGray
AtxAtx
TktkTtfAtx
kcqkkcdk
k
kck
:10,11,01,00
sin)(,cos)(
4/3,4/3,4/,4/
)1(),2cos()( 0
ϕϕ
ππππϕ
ϕπ
==
−−=
+≤≤+=
QPSK Example
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-1 0 1
-1
-0.5
0
0.5
1
xd
xq
-1 0 1
-1
-0.5
0
0.5
xd
xq
0 10 20 30-2
-1
0
1
2
symbol index
xd
0 10 20 30-2
-1
0
1
2
symbol index
xq
103
Book example QAMDEMO
Make m symbols
Get n samples of each symbol
Map mxn samples and pass thru the filters
Levels = -1, 1 for QPSK
Matlab: filter bw is normalized to fs/2
bw = 2fbw/fs = 2fbw/kfsym =2λ/k
e.g: k=20 and λ =1 => bw=0.1
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E, Im
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Complex Envelope in Frequency DomainComplex Envelope in Frequency Domain
Complex envelope is the positive portion of X(f) translated to zero frequencyxd and xq can be derived from X(f), having half of the BW of X(f)
SignalLPetxetxtx
etxtx
tfj
tfjtfj
tfj
~
:)(~)(2)(~
))(~Re()(
2
4*2
2
00
0
=
−=
=
−
−−
π
ππ
π
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jfXfXfX
fXfXfX
fjXfXfXfjXfXfX
fXfXrealarexandx
ffXffUffXfX
etxLPtx
q
d
qdqd
ddqd
tfj
2/))(~
)(~
()(
2/))(~
)(~
()(
)()()(~
),()()(~
),...()(
)(2)()(2)(~
)(2)(~
*
*
*
*
000
2 0
−−=
−+=
−=−+=
−==>
+=++=
=
+
− π
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ommunicatio
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E, Im
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ComplexComplex Envelope to Represent BP LTI SystemsEnvelope to Represent BP LTI Systems
Same thing can be done for LTI band-pass systemsImpulse response signal is band-pass
Easy to prove that:
)(*)()()()( thtxdthxty =−= ∫∞
∞−τττ
~)(
~2Re)( if )(
~*)(~)(~ 02
ethththtxtytfj=→= π
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)()(*)()()()()(~ tjhthtjxtxtjytyty qdqdqd ++=+=
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Complex Envelope to Represent BP LTI…Complex Envelope to Represent BP LTI…
In frequency Domain
Unity gain in BP to unity gain in LP
)()()(~
)()(~
),()()(
00
0
ffUffHfH
ffHfHfHfHfH
++=
+=+= +−+
~~~~)]2exp()(~Re[)( 0tfjtxtx π=
H+H-
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Hilbert Transform and Analytic Signal
106
)]2exp()(~Re[)(*)()(
)(~
)(~)(~)],2exp()(~
2Re[)(
0
0
tfjtythtxty
thtxtytfjthth
π
π
==
∗==
)()sgn()(,1*)()(ˆ
)](ˆ)([)()(~ 00 22
fXfjfXt
txtx
etxjtxetxtxtfjtfj
A
−==
+== −−
)
π
ππ
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an Gholam
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011
Band Pass Section ModelingBand Pass Section Modeling
CC : Complex convolution
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Example: Band pass amplifier/phase shifter (think about the BP model)
107
)()sin()(
)()cos()(
)()(~
))((~)(~
)(~ )2cos()(
)(~ )2cos()(
0
0
tGth
tGth
tGethGetxty
eAGetytfAGty
AetxtfAtx
q
d
jj
jj
j
δθ
δθ
δ
θϕπ
ϕπ
θθ
θϕ
ϕ
=
=
=⇒=
=→++=
=→+=
Course N
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ommunicatio
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E, Im
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Energy and SNR in LP Model of BP SignalsEnergy and SNR in LP Model of BP Signals
Band-pass parts have zero average
For real valued BP noise, as an stationary signal (no ‘t’ in the expressions):
xx
tfjtfj
EE
etxetxtx
2
|)(~)(~||)(|
~
22*2
412 00
=
+= − ππ
))(Re())(2cos()()(2 0θπ π tfj
etzttftatn =+=
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108
)()(),()(
2sin)(2cos)()]()([)(
0)]([)]([
2sin)]([2cos)]([)]([0
)()()(
))(Re())(2cos()()(
00
00
00
ττττ
τπττπτττ
ππ
θπ
dqqdqqdd
dqdd
NNNNNNNN
NNNNNN
qd
qd
qd
RRRR
fRfRtntnER
tnEtnE
tftnEtftnEtnE
tjntntz
etzttftatn
−==
−=+=
==⇒
−==
+=
=+=
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E, Im
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For cross correlation of any real functions:
Combining with the previous relation:
On the LP model side:
)()( ττ −=dqqd NNNN RR
jRRtztzER )(2)(2)]()([)( * +=+= ττττ
0)0()0( ==dqqd NNNN RR
Energy & SNR in LP Model of BP Signals…Energy & SNR in LP Model of BP Signals…
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SNR remains intact!Moreover:
109
NR
RRRRN
eRR
jRRtztzER
ZZ
ZZNNNNNN
fj
ZZNN
NNNNZZ
qqdd
dqdd
2)0(
)0()0()0()0(
])(Re[)(
)(2)(2)]()([)(
21
2
*
0
=
====
=
+=+=
τπττ
ττττ
)(2)()()( fSfSfSfSdqd NNNZ =+=
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011
Example:
PSD x & n:
)2sin()()2cos()()(),2cos()(
)()()(
000 θπθππ +−+==
+=
tftntftntntfAtx
tntxtz
qd
)()4/()()4/()( 0
2
0
2 ffAffAfSx ++−= δδ
Energy & SNR in LP Model of BP Signals…Energy & SNR in LP Model of BP Signals…
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110
BN
ASNRSNR
SP
BPLP
0
2
2==
= ∫∞
∞−
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E, Im
an Gholam
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011
Multi Carrier Systems and LP ModelMulti Carrier Systems and LP Model
Famous Technologies:OFDM: Orthogonal Frequency Division Multiplexing
FDMA: Frequency Multiple AccessMC-CDMA: Multi-Carrier Code Division Multiple Access
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Sort the frequencies… Choose one in the middle!
111
∑
∑∑
∑
=
==
=
−=
−=
=
=
+=
M
i
ii
M
i
ii
M
i
ii
iii
M
i
iii
tffjtxty
tfjtffjtxtfjtxty
tjtatx
ttftaty
1
0
1
00
1
1
])(2exp[)(~)(~
]2exp[])(2exp[)(~Re]2exp[)(~Re)(
))(exp()()(~
)](2cos[)()(
π
πππ
ϕ
ϕπ
Course N
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NonNon--linear Systems and LP Modellinear Systems and LP Model
No Superposition, no convolution, no transfer function concept…
Some examples:
Band-pass Hard-limiter nonlinearity
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Band-pass Hard-limiter nonlinearity
Envelope Detector, AM demodulation
112
)()(/)()()()(/)()(
))(exp()(~))(2cos()(
))(exp()()(~))(2cos()()(
2222
0
0
txtxtBxtytxtxtBxty
tjBtyttfBty
tjtAtxttftAtx
qdqqqddd +=+=
=+=
=+=
θθπ
θθπ
|)(||)(~|)(~
))(exp()()(~))(2cos()()( 0
tAtrtz
tjtAtrttftAtr
==
=+= θθπ
Course N
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ommunicatio
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s, Sharif, E
E, Im
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u , Fall 2
011
NonNon--linear Systems and LP Model …linear Systems and LP Model …
Band-pass non-linear narrow band amplifier:
Taylor series estimation for memory-less non linear systems
))(2cos()](15.0)([)(,
3)()()(
3
3
ttftAtAtyBf
BBWtxtxty
cc θπ +−≈>>
=−=
NBBWtxatxFtyn
N
n
n =≈= ∑=
)()]([)(0
Course N
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AM to AM and AM to PM Models
113))](()(2cos[))(()(
sin]cos[1
cos]cos[1
)(:
)sincos(]cos[)(
cos))(2cos()()(
0
2
0
1
2
0
1
11
1
0
0
tAgttftAfty
dAFbdAFa
functiondescribingjbazonefirst
kbkaaAFty
AttftAtx
k
k
k
++=
==
+
++==
=+=
∫∫
∑∞
=
θπ
αααπ
αααπ
ααα
αθπ
ππ
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011
Time Varying Systems and LP ModelTime Varying Systems and LP ModelSome LTI Tools are Applicable:Impulse response, Transfer function…but modified definition
Time Varying Impulse response
2121 )](2exp[),(),(
)(),()(,)(),()(
:.,
dtdftfjthffH
dxthtydtxthty
timeelapsedtatmeasuredresponsetatappliedimpulse
+−=
−=−=
−
∫ ∫
∫∫∞ ∞
∞
∞−
∞
∞−
ττπτ
τττττττ
ττ
Course N
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Example, multi-path channel
114
1111
2121
)(),()( dffXfffHfY −= ∫
∫ ∫∞
∞−
∞− ∞−
)](2exp[)()(~)),((~)(~)(~
))](())((2cos[))(()(()(
))(2cos()()(
0
1
0
1
0
tfjtatattxtaty
ttttfttAtaty
ttftAtx
nnnn
N
n
n
nn
N
n
nn
τπτ
τθτπτ
θπ
−=−=
−+−−=
+=
∑
∑
=
=
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an Gholam
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011
Channel Coherence Time Slow Versus Fast Fading Channel Delay min time channel amplitudes uncorrelated
Channel Coherence BandwidthFlat versus Frequency Selective FadingSignal BW min frequency channel amplitudes uncorrelated
Time Varying Systems and LP ModelTime Varying Systems and LP Model
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Doppler Spread inversely proportional to Time spread
Deep Fade
115115
EX3: 4.1, 4.3, 4.9, 4.10, 4.12
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
Simulation Techniques and Filter ModelsSimulation Techniques and Filter Models
Filters are everywhere in communication systems! Frequency selective subsystems
LTI Systems in GeneralAnalog subsystems must be converted to digital for simulationConversion > Approximation > Inducing simulation errors
LTI Digital Filters: FIR (all ak=0) or IIR (at least one ak≠0)
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif,
k k
Synthesis Phase: Direct & IndirectImplementation Phase
116
TransfreqeH
za
zb
zX
zYzH
knyaknxbnxnhny
j
M
k
kn
k
N
k
kn
k
M
k
k
N
k
k
:)(,
1)(
)()(
][][][*][][
1
1
1
0
1
1
1
0
ω
∑
∑
∑∑
−
=
−
−
=
−
−
=
−
=
+
==
−−−==
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
Direct Forms Implementation I/IIDirect Forms Implementation I/II
DF I, Twice as many delays as necessary
DF II and Transposed DF II
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif,
DF II and Transposed DF II
TDF II is the best, computationallyUpdate can be done “in sequence”Fc all zeros on/above main diagonal
117
)()1()()(
)()(
nxnnn
nny
d BSFSFS
CS
c+−+=
=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
Analog FiltersAnalog Filters
Analog Filters:Magnitude, Phase, Group Delayspecified magnitude, some properties on phase or group delay
Example: Rational Transfer Functions
Poles on the left (stability) Zeros on the left (Minimum Phase)
ω
ωθωτωω ωθ
d
dejHjH j )(
)(,|)(|)( )( −==
DerivationwaysHjH
sHsHjHss
ssH js
−↔
+
+=−=
++
+= =
2:)(|)(|
4
1|)()(|)(|
22
1)(
2
4
22
2
ω
ω
ωω ω
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif,
Poles on the left (stability) Zeros on the left (Minimum Phase)
Ideal Low-pass Filters Approximations
Butterworth, Chebyshev I ,II EllipticMaximally Flat versus equiripple in different regionsTransform to High Pass, Band Pass, Stop Band…
118
N
N
M
M
DDDD
CCCCjH
2
2
4
4
2
20
2
2
4
4
2
202
...
...|)(|
ωωω
ωωωω
++++
++++=
21012
2
0
2Replaced
Replaced
,),/()( :BP toLP
freq) passband ( ,/ :HP toLP
ωωωωωω
ωω
=−=+ →
→
BBsss
ss pp
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
Analog FiltersAnalog Filters……Butterworth Filters
Normalized to have pass-band at Ω=1
-3dB D2N =1
Butterworth polynomials Transfer function Needs an scaling factor
pNN
N
DD
jH ωωε /,,1
1|)(| 2
22
2
2 =Ω=Ω+
=Ω
))(exp(
1
1log10|)(|log10
1
1log10|)1(|log10
212/1
2210
2
10
210
2
10
πε
ε
ε
ω
ω
NNkN
k
sN
s
p
jp
AjH
AjH
p
s
−+−=
−=Ω+
=
−=+
=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, Needs an scaling factor
Chebyshev Filters 1,11
119
)()(2)(
1||)coscos()(
/,)/1(1
1|)(:|
/,)(1
|)(|:
11
1
22
02
22
02
Ω−ΩΩ=Ω
<ΩΩ=Ω
=ΩΩ+
−=Ω
=ΩΩ+
=Ω
−+
−
nnn
n
s
N
p
N
CCC
nC
C
HjHII
C
HjHI
ωωε
ωωε
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
IIR Digital Filters Synthesis from Analog FiltersIIR Digital Filters Synthesis from Analog FiltersDesigning based on analog counterparts, Different filtersImpulse Invariant, Step Invariance, Bilinear transform
Impulse Invariant
h(t) must be band limited… no HP and some BP usage
sT
nTta ezsHLZzH =→= =− )]([)( 1
∑∑ −−−=→
+=
kTs
k
k k
ka
ze
AzH
ss
AsH
k 11)()(
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, h(t) must be band limited… no HP and some BP usageSuffers from aliasing error Does not keep the minimum phase property of H(s)Good to match for low frequency if sampling rate chosen carefullynot often used!
Step Invariance
Same Problems, a bit better because of the integration 120
sT
nTtasezsHLZzzH =→−= =
−− )]([)1()( 111
∑∑ −−
−−
−
−=→
+=
kTs
Ts
k
k k
ka
ze
zeAzH
ss
AsH
k
k
1
1
1
)1()()(
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
IIR Digital Filters…IIR Digital Filters…Bilinear Transform
C SelectionIf Sampling rate is high enough and fd chosen to be almost fa TC /2=
∑∑++−
+=→
+=
=
+
−=→=
−
−
−
−
+
−=
−
−
k kk
k
k k
ka
da
z
zCs
a
CszCs
zAzH
ss
AsH
CTC
z
zCssHzH
)()(
)1()()(
Prewarping:),2/tan(
1
1|)()(
1
1
1
1
1
11
1
ωω
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, If Sampling rate is high enough and fd chosen to be almost fa
Reasonable match over a wide frequency rangeDoes not keep the constant group delayMostly used in simulation
Matlab Functions:buttord, cheb1ord, cheb2ord, ellipordbutter, cheby1, cheby2, ellipyulewalkfreqz, filter
121
TC /2=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
IIR Digital Filters…IIR Digital Filters…Impulse Invariant Transform ExampleIntegrator, Rectangular Approximation
Step Invariant gives the same result with a delay
Bilinear ExampleIntegrator, Trapezoidal Approximation
11
1)(/1)(
][]1[][
−−=→=
+−=
zzHssH
nTxnyny
a
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, Integrator, Trapezoidal Approximation
122
1
1
1
1
2)(/1)(
2/])1[][(]1[][
−
−
−
+=→=
+++−=
z
zTzHssH
nxnxTnyny
a
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
Direct IIR Digital Filters SynthesisDirect IIR Digital Filters SynthesisYule Walker approximation: D: desired, W weighting, H Rational to be found
Least pth approximation. p=2 least square approximation[num,den] = yulewalk(N, F, D)
Arbitrary Transfer functionAvoid sharp transitions when using Matlab yulewalkIncreasing N improves the approximation
ωω ωω
ω
ωdeDeHeWJ
pjjj |)()(|)()( −= ∫
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, Increasing N improves the approximation
Deczky’s Method
Minimizes the errors in amplitude response and group delay
Many other criteria, different methods, different softwares
123
GA EEJ )1()( ααω −+=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR FiltersFIR Filters
Direct Form II and Transposed IITapped Delay Line TDLOr Transversal Delay Line
Why FIRs are attractive?
Some Filters in Communication systems cannot be expressed in terms of H(s), e.g. pulse shaping filters
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, systems cannot be expressed in terms of H(s), e.g. pulse shaping filters
Some filters based on measured impulse response or frequency response
With FIR filters we can specify linear phase and arbitrary amplitude responses independently
FIRs no feedback, always stableDrawback:Not computationally as efficient as IIR filters (filter order = impulse response duration)Needs N complex multiplications N>2048 becomes heavy
124
][][0
knxbnyN
k
k −= ∑=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR Filters…FIR Filters…Use FFT based convolution for more efficiency
N must be power of 2, Zero padding is needed if not …
Improvement not much if N < 128
For Long blocks overlap-add method is used. fftfilt function in MATLAB
Block based approach, difficult to deal with for simulation of feedback systems
General treatment:
NN 2log/
Course N
otes, Sim
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ommunicatio
n System
s, Sharif,
General treatment:Shortening the impulse response (filter length) length by truncation
Windowing instead of truncationNarrower main lob, smaller side lobs
125
)(*)()(
%2,|][|)1(|][|0
2
0
2
fWfHfH
nhnh
T
k
N
k
=
≈−> ∑∑∞
==
εε
],[)/cos(46.054.0][ :Hamming LLnLnnw −∈−= π
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR Filters…FIR Filters…Designing from the Amplitude Response
A(f): desired amplitude response inverse Fourier h(n)
Usually A real and even, so is h(n)
Even h(n) is non-causal,
truncation + time shift ~ error + linear phase (time shift)
Arbitrary amplitude shape and linear phase!In general complex H(f) can be used if other phase responses are needed
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, In general complex H(f) can be used if other phase responses are needed
not causal, but easy to design and needs a phase shift to become h[n]
126
∫
∑
∑
∑∑
−
−
−=
−−
−=
−
−∞
=
=−∞
=
=
==
=+=
= →=
2/
2/
2
1
2
1
2
1
2
1
2222
2
0
2/1
0
)(][
)(][)(
)(][)(
][)(][)(
s
s
s
f
f
mfTj
kfTjL
Lk
fTj
fTjLfTjkfTjL
Lk
LfTjfTj
nfTj
n
fTjfTn
n
dfefATmh
fAekheH
eHeeLkheeH
enheHznhzH
π
ππ
πππππ
ππ
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR Filters…FIR Filters…Some frequency domain examples
1) Ideal LP filter approximation, B=λfN= λfs/2
CompareHamming andRectangular windows
m
mdfefATmh
s
s
f
f
mfTj
π
λπλ
λ
π )sin()(][
2/
2/
2
1 == ∫−
Course N
otes, Sim
ulatio
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ommunicatio
n System
s, Sharif,
2) FIR Butterworth filter
Frequency samplingUse ifft to derive h[n]High ‘n’ close to brick-wall filter
127
n
ck
k
fffA
)/(1
1)(
+=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR Filters…FIR Filters…Design based on tables of Magnitude and PhaseSimulation of Arbitrary Amplitude and Phase responses FIR filter applicationKey parameters: Sampling rate and Time duration
Accuracy versus Computations
Parameters Selection for filter BW=B :16B<fs <32B, Time Resolution = Ts=1/fsB/64<∆f<B/32 to, Frequency Resolution = fs/N = ∆f
Number of samples per symbol > 8(min) (must be an integer and a power of 2)Duration of impulse response > 8 to 16 Symbols (Leads to N>1024)
Course N
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ommunicatio
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s, Sharif,
Duration of impulse response > 8 to 16 Symbols (Leads to N>1024)
Steps: Preprocessing
Convert BP to LP (freq shift), Integrate the group delay to phaseRe-sampling and Interpolate to get more points, >1024 which is needed
Extend the H(fk) to simulation range [-fs /2,fs /2], -N/2<k<N/2Move the negative side to N/2+1 to N H(k ∆f), 1≤k ≤ NTake Inverse FFT and get Impulse responseRotate and Apply WindowingUse Matlab filter or faster fftfilt
128
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR FiltersFIR FiltersDesigning from the Impulse Response
Sampling the impulse response
FIRs are Important to make pulse shapes
Example: Raised Cosine Pulse shape with pulse duration T
)()()()( )(kTtpdtxkTtdtd
k
k
tp
k
k −= →−= ∑∑ δ
222 /41
)/cos(
/
)/sin()(
Tt
Tt
Tt
Tttp
β
πβ
π
π
−=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif,
For causality, introduce a delay, then sampling
Delay mT, Truncate 2mT => 2m samples
Zero at multiples of T=1, zero ISI
SQRC129
mk
n
T
tkTT
mTnTtt
s
ssd
−=⇒=
−=−
/
,
222 /41/)(
TtTttp
βπ −=
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011
FIR FiltersFIR FiltersComputer Aided Design for FIR filters
Most Popular: Parks McClellan Method:
Optimum Equi-ripple FIR filter based on Chebyshev polynomials
Important to make pulse shapes
remez function in old Matlab (still works)firpm function in new Matlabfirpmord to get the required order
Course N
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ommunicatio
n System
s, Sharif,
firpmord to get the required order
Order , A vector for freq, a vector for amplitudeLinear phase results
Distributes n(=order) extremes in pass band and stop band
130
Course N
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif, E
E, Im
an Gholam
pour, im
u , Fall 2
011