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Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Fundamental Concepts• Filter
Filtery(t)x(t)
input (or excitation) output (or response)
• Fourier series expansion:– periodic signal x(t) of period
∑
∑
∑
∞
−∞=
∞
=
∞
=
=
++=
++=
k
tjkk
kkk
kkk
eX
tkAA
tkbtkaatx
0
00
000
1
1
)cos(
)sincos()(
ω
φω
ωωωπ
0
2
where Xk represents the discrete spectrum of x(t)5-1
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Fundamental Concepts(Cont.)– x(t) is not periodic
• Physical meaning
∫∞∞−= ωω
πdejXtx jwt)(
21)(
where represents the continuous spectrum of x(t))( ωjX
1. Spectrum-shaping where number Xk or function X(jw)are altered in certain way in order to produce desiredform of output signal y(t).
2. If the filter is linear, the harmonic content can not be richerthan that of the input signal
3. Desired filter operation can be performed by the appropriateinterconnection of elements with chosen values.
5-2
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Historical Review
Example : Bell system progress in filter technology ofvoice-frequency ( f < 4kHz ) application over a nearly 60-years span(1920~1980)
1920 - Passive LC (1)1969 - Discrete active RC (1)1973 - Thin film active RC (1)1975 - Active RC DIP (1)1980 - Switched-capacitor building block (11)
Digital signal processor (37)
(N) No. of biquardratic section
5-3
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
System classification• Black box representation:
– single-input, single-output system is a special case. For a filter, its enough.
input ( or excitation ) x(t) ; output ( or response ) y(t)
Systemx(t) y(t) = f X(t)
Black box– For a filter, x(t) and y(t) are electrical signals, e.g. voltage,
current, or charge.– Filter is composed of lumped active and passive elements.– A lumped element is defined as one having physical
dimensions small compared to the wavelength of the applied signals.
5-4
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
System classification(Cont.)• System classification
1. linear and nonlinear systems.2. continuous-time and discrete-time (or sampled-data)
systems.3. time-invariant and time-varying systems.
( Linear time-invariant is abbreviated as LTI )• A system is linear if superposition principle is satisfied.
– superposition
– A linear system can be described by a linear differential or difference equation .
– For a filter , nonlinearity must be eliminated or minimized.e.g. overdrive an amplifier => nonlinearity occurs
)( 11 xfy = )( 22 xfy =; ; 21 xxx βα +=
212121 )()()()( yyxfxfxxfxfy βαβαβα +=+=+==
5-5
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Continuous-Time and Discrete-Time• Continuous-time
Input and output are continuous functions of thecontinuous variable time.
t is time• Discrete-time
;)(&)( tyytxx ==
Input and output change at only discrete instants of time. (e.g. sampling instants)
;)(&)( kTyykTxx == where k is an integer and T is the time interval between samples
• Mathematical distinction– Continuous-time systems are characterized by differential
equations.– Discrete-time systems are characterized by difference
equations.5-6
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Continuous-Time and Discrete-Time ( cont.)
Amplitude
Time
τ D
Loss
Input
Output
Continuous analog system,e.g., active RC filter
5-7
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Continuous-Time and Discrete-Time ( cont.)Amplitude
Time
τ D
Loss
Input
Output
Discrete time or sampled-data system
5-8
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Continuous-Time and Discrete-Time( cont.)
Amplitude
Time
τ D
Loss
Input
Output
Sampled-data system with sample-and-hold(S/H),e.g.,active switched-capacitor filter
5-9
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
• Time-invariant– Mathematical characteristic
Time-Invariant and Time-Varying Systems
A. Continuous-time systems x(t) => y(t)x(t - τ ) => y(t - τ )for all x(t) and all τ
B. Discrete-time systemsx(kT) => y(kT)x[(k-n)T] => y[(k-n)T]for any x(kT) and n
– Physical meaning
System response depends only on the shape of input and not on the time of application .
5-10
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Causal System• Response can’t precede the excitation
if x(t)=0 for t < t0 or mT for all t0 or mT
then y(t)=0 for t < t0 or mT
5-11
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Representations of Continuous-LTI Systems• Single-input, single-output, continuous LTI systems
– Input/output relationship ( linear differential equation )
– If x(t) and initial conditions are known, then y(t) is completely determined.
.......)()(.......)()(111
1
1 ++=++ −−−
−
− m
m
mm
m
mn
n
nn
n
n dttxda
dttxda
dttydb
dttydb
where ( i ) y(t) is output( ii ) ai and bj are real and depend on the network
elements(a) LTI: ai and bj are constant.(b) nonlinear: ai and bj are functions
of x and/or y.(c) time-dependent: ai and bj are functions
of time.
1
1 )0(,...,)0(),0( −
−
n
n
dtdy
dtdyy
5-12
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Representations of Continuous-LTI Systems(Cont.)• Zero-input response
Response obtained when the input is zero.( Response is not necessarily zero because initial conditionsmay not be zeros )
• Zero-state response
• For a linear system, the complete response is equal to the sum or superposition of the zero-input and zero-state responses.
Response obtained for any arbitrary input when all initial conditions are zero .
5-13
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency-Domain Concepts• Laplace transform techniques can be used to transfer time-
domain differential equations into frequency domain equations, e.g.
1
121 )0(.......)0()0()()(
−
−−− −−−=
n
nnnn
n
n
dtyd
dtdySySsYS
dttydL
Hence, )(......)()(
001
1
1 ttyty ybdt
dbdtdb n
n
nn
n
n +++ −
−
−
)(......)(
00 ttx
xadtda n
n
m ++= ( for a LTI system)
can be transformed into
)()()......(
)()()......(
02
21
1
02
21
1
sICsXbSaSaSa
sICsYbSbSbSb
xm
mm
mm
m
yn
nn
nn
n
++++=
++++−
−−
−
−−
−−
where ICy(s) and ICx(s) are from initial conditions of y and x. X(s) and Y(s) are excitation and zero-state response.
5-14
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency-Domain Concepts(Cont.)• Transfer function H(s) of network
)()(
....
....)()(
))((
))(()(SDSN
Sb
SaSXSY
txexcitationL
tyresponsestatezeroLSH nn
mm =
+
+==
−=
where for any realizable practical network.nm ≤– Transfer function
; voltage transfer function
; impedance transfer function
; admittance transfer function
)()(sVsV
in
out
)()(sIsV
in
out
)()(sVsI
in
out
– Driving-point impedance and admittance functions
)( Z 1)(;
)()()(Y
)()(V )(
inin
ins
sYwhereSVSIs
SISsZ in
inin
inin === &
5-15
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
• System analysisFrequency-Domain Concepts(Cont.)
1. Time-domain differential equation.2. Frequency-domain equation( i.e. S-domain equation).
( 2 is proved to be a more convenient method from experience).
• Transfer function of continuous LTI– A ratio of two polynomials in S with real coefficients.– Can be factored as
))...()(())...()((
)()()(
21
21
nn
nm
PSPSPSbZSZSZSa
SDSNSH
−−−−−−
==
where Zi are zeros ( H(S)=0 when S = Zi )Pi are poles ( D(S)=0 when S = Pi )
– Zi , Pi =σ+jw on complex S-plane.
5-16
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency-Domain Concepts(Cont.)– Transfer function with real coefficient
– Poles lie in the left planezero input response decays with time
– Poles lie on the jw-axisthe network oscillates
– Poles lie in the right planeresponses grow exponentially with time
– when zero of N(S) lie on or to the left of the jw-axis.( i.e. there are no right-plane zeros), H(S) is referredto as a minimum-phase function.
Poles and zeros are real or conjugate pairs(complex or imaginary).
– For stability, all poles must lie in the left plane.(i.e. D(s) is a Hurwitz polynomial)
5-17
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Time-Domain Concepts• Continuous LTI systems ( system representation )
– convolute excitation x(t) with impulse response h(t) to obtain y(t)
A. differential equation
B. convolution or superposition integral
....)()(....)()(1
1
11
1
1 ++=++ −
−
−−
−
− m
m
mm
m
mn
n
nn
n
n dttxda
dttxda
dttydb
dttydb
∫ −=t
dtxhty0
)()()( λλλ
where λ is a dummy integral variable and we assume that the system is causal and x(t)=0 for t<0.
5-18
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Time-Domain Concepts(Cont.)– h(t) is impulse response
– convolution v.s. frequency-domain representation
Assume , then)()( ttx δ=)()()()()()(
00thdtthdthty
tt=−=−= ∫∫ λλδλλδλ
(definition : with for )
1)(0
=−∫t
dt λλδ 0)( =−λδ tt≠λ
∫ ∫ =−== −t tst SXSHdtdhtxetyLSY0 0
)()(])()([)()( λλλ
∫∞ −==
0)()()( dtethtyLSH stwhere transfer function
5-19
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Ideal Distortionless Transmission• y(t) is perfect replica of x(t); may be with amplification
k and delay .
– Where
0τ)()( 0τ−= tkxty
frequency domain
0)( τSKeSH −=
)()()( 0 SXketyLSY Sτ−==
input
output
0τ 0τ1. H(S) has constant magnitude K
linear phase 0ωτφ −=2. H(S) is not a real rational function
not realizable as a lumped network with a finite number of elements.
3. group delay tconstan)( 0 ==τωτ5-20
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Ideal Distortionless Transmission(Cont.)– From 2, method for approximating the above
by a rational function must be developed such that it becomes realizable physically.
0)( τSKeSH −=
– Approximation method results in transmission errors since any physical network has in practice frequency-dependent magnitude and delay .
– Two method to define deviations from an ideal transmission:
1. step response.
2. impulse response.
5-21
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Step Response• Ideal transmission
where H(S) is a physically unrealizable transfer function
0)( τSKeSH −=
step input : )()( tutx =S
SX 1)( =
output step function(step responses): a(t)
)()()()()( 01 τtkuSXSHLtaty −=== −
: time required for step response to rise for 10% to 90%
• Delay time & rise time :dτ rτdτ
rτ
: time required for step response to reach 50% of its final value
γ : overshoot5-22
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Step Response(Cont.)• Example: one-pole case
– no over shoot
• In general for overshoot 2.23 ≈− dBrωτ 35.03 ≈− dBr fτ(i.e. )
%5≤γ
1
1)(aS
SH+
= dBa 31 −=ω;
IF x(t)=u(t)
)()1()( 1 tuety ta−−=
0=γ
1
69.0ad ≈τ
1
2.2ar ≈τ&
x(t)
ty(t)/k
rτdτ
γ
t20τ0.1
0.5
0.91.0
real
ideal
t
5-23
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Impulse ResponseIdeal transmission
– Impulse response– step response
One can obtain impulse response
)()( ttX δ=oSkesH τ−=)(
)()()( 0tkthty τ−δ==
)()()()(
0
0
tkutatkth
τ−=
τ−δ=
; unrealizable
dttdath )()( ==>
Area=1
Dτ
Rτ
t
y(t)=h(t)
t
X(t)
∆ 0→∆
from step response, and vice versa .
5-24
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Calculation of and Precise calculation of and are usually time-consuming Convenient method resulting in considerable simplification isproposed by Elmore(Assuming negligible overshoot or none)
Dτ
Dτ Rτ
∫ ∞=τ 0D dttht )(∫∞ τ−π=τ 0
212
DR dttht2 ])()([
- Elmore’s definition
21
022 ])([2 ∫
∞ −= Ddttht τπ
Rτ
Consider the normalized transfer function ( H(0)=1)- H(s)=- by direct division- from impulse response
∫∫∞∞ − −+−== 0
22
0 ....)!2
1)(()()( dttsstthdtethsH st
..)(!
−τ+π
τ+τ−= 2
D
2R
2
D 22ss1
mnsbsbsb1sasasa1
nn
221
mm
221 ≥
++++++++
..........
(2)
...)()()( +−+−+−−= 22211
2111 sbababsab1SH (1)
s)definitionsElmore'ingincorporatby(
5-25
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Calculation of and (Cont.)Dτ Rτ
21
2221
21R
11D
ba2ab2
ab
)]([ −+−π=τ
−=τ
- Equating (1) and (2) yields
(3)
(4)
Ease of computation using Elmore’s definition- For higher-order systems with no overshoot (i.e.monotonic)can be decomposed to K monotonic cascaded stages.
∑
∑
=
=
τ=τ
τ=τ
k
1i2
12iRtotalR
k
1iiDtotalD
])([)(
)()(Area=1
Dτ
Rτ
t
h(t)monotonic
Dτ
Rτ
a(t)
t
monotonic
Example 1-2
5-26
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
NormalizationAdvantages
Normalization
1. avoid the tedium of having to manipulate large power of 102. minimize the effect of roundoff errors.
0Ω
1. Frequency normalization-Frequency scale changed by dividing the frequency variable by a conveniently chosen normalization frequency
0n
SSΩ
=Normalization equation
2. Impedance normalization-by dividing all impedances in the circuit by a normalizationresistance .0R o
noono
n RLLRCC
RRR === ,,
-Normalization equation5-27
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Normalization(Cont.)
0n R
RR =0
00
0nn R
LSRSLLS ΩΩ
==)/(
0000 )/(111,CRSSCRCS nn ΩΩ
==
0
0n R
LL Ω= 00n RCC Ω=
0n R
RR =
,
=>
The actual unnormalized physical parameters R, L, and C are obtained by inverting the normalization equations.
Comments(practical concerns):
),,( ensionlessdimareCandLRS nnnn
=>Easy remember.
1. normalization is to remove dimensions
2. Dimensionless network
=>designer can choose convenient and practical element values.
, ,
Example 1-35-28
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
1. Lowpass filter2. Highpass filter3. Bandpass filter4. Bandreject filter5. Allpass filter--phase or delay specs are of primary concern.
Type of FiltersFive major types
Filter magnitude specifications
Magnitude spec. are of primary concern
-Lowpass filter -Highpass filterdBjH |,)(| ω
ripplePB
nattenuatioSB
ωpω sω
PBTB
SB
pωω
dBjH |,)(| ω
sω
ripplePB
SB TBPB
5-29
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Type of Filters(Cont.)-Bandpass filter -Bandreject filter
-All pass filter
ω
dBjH |,)(| ω
LPB
HPBL
TBH
TBSB
nattenuatioSBrippleHPB
rippleLPB
SLω PLω PHω SHωω
dBjH |,)(| ω
nattenuatioSBL
LSB H
SB
PLω PHωSLω SHω
PBLTB H
TB
ripplePB
nattenuatioSBH
phasegain
pω
dB0
ο−180
dBjwH |)(| reesdegw)(φ
ω360−
5-30
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
1. Realization of nonminimum phase function
(i) Let
Filter Phase or Delay Specs.Frequency dependent delay
Usually not important for voice or audio.(Human ear is
Can cause intolerable distortion in video or digital
Examples :
)s(H)s(H)s(H APMN =where
N: nonminimum phase
M: minimum phase
AP: all-pass
|||| NM HH =
5-31
=>Nonminimum phase function may be needed
-
-
very insensitive to phase change with frequency.)
- Minimum phase function:with only left half-phase zeros
transmission
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Filter Phase or Delay Specs.(Cont.)
)()(
sDsNH
AP
APAP = is a allpass function
where is formed by all right-plane zerosAPN
APD
(ii) Augment pole-zero
(iii)
Hence ,
)()()(
sDsDH
sDN
AP
APAP
APAP
−±=
−±=
)()(tan wD
wD1AP
R
I2phase −−=φ
5-32
is formed by all left-plane poles which are mirrorimages of the right-plane zeros
)](Im[)( jwDwDwhere API =)](Re[)( jwDwD ApR =
dwwdwDelay AP )()( φ
−=τ
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Filter Phase or Delay Specs.(Cont.)-Using , a desirable delay function without any effect on
the magnitude can be achieved.
)(sHAP
|)(||)(|)()()(
sHsHwheresHsHsH
T
APT
==
-Example
)()()()()()(
wwwwww
APT
APT
τ+τ=τφ+φ=φ
The cascaded allpass can, of course, only increase the phase and delay of H(s); this is normaly no problem, because fordistortionless transmission, only the linearity of i.e., the constancy of , in the frequency range of interest is important, not its actual size.
Example 1-4
Tφ
Tτ
-
-5-33
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
(i) make total delay as flat as possible in the frequency range of interest.
(ii) obtain prescribed delay-Precision design requires computer aids.-Uncritical design of low order( can be performed manually with the aid of the curves below.
%)~ 2010≈ττ∆
normalized delay,
110−
210
110
210−
)(5.0PAPP ω
ωτω ×
Pωω
010
31
1.0
02.050
02.0=pQ
1.03
1
50
5
Filter Phase or Delay Specs.(Cont.)2. Phase or delay equalization
5-34
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Second Order Filters• 2nd-order transfer function
))(())(()(
21212
012
012
2PSPSZSZSa
bSbS
aSaSasH++++
=++
++=
2p
p
p2
z2
z
z2
21
211
2
21
211
2
SQ
S
SQ
SK
PPSP2SZZSZ2SK
ω+ω
+
ω+ω+=
++++++
=)(
)(
)Im()Re()]Re([)Im()Re()]Re([
; where P&Z are real or
)Re()][Im()][Re(
)Re( 1
21
21
1
pp P2
PPP2
Q +=
ω= is the pole quality
factor
)Re()][Im()][Re(
)Re( 1
21
21
1
zz Z2
ZZZ2
Q +=
ω=
is the zero quality factor
)Im( z
)Re( z
1PZω
where
complex pairs
)Im(1P
)Re(1P*1P
1Ppω
5-35
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Second Order Filters(Cont.)
2p
p
p2
2
HP2p
p
p2
2p
LPS
QS
SKsHS
QS
KsHω+
ω+
=ω+
ω+
ω=
)()(;
)()(
2p
p
p2
2z
2
BR2p
p
p2
p
p
BPS
QS
SKsHS
QS
SQ
KsHω+
ω+
ω+=
ω+ω
+
ω
=)(
)(;)(
)()(
1QSS
1QSS
KS
QS
SQ
SKsH
p
n2n
p
n2n
2p
p
p2
2p
p
p2
AP++
+−=
ω+ω
+
ω+ω
−=
)(
)(
)(
)()(
where ; n means normalized p
np
nSS
ωω
=ωω
= &
5-36
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Second Order Filters(Cont.)
– Normalized phase
– normalized delay
)(tan)( 2n
pn
1nAP 1
Q2
ω−
ω
−=ωφ −
2p
n22n
2n
pnAPpnAPn
Q1
1Q2
)()()()(, ω+ω−
ω+=ωτω=ωτ
phasegain
pω
dB0
ο180−
dBjwH |)(| reesdegw )(φ
110−
210
110
210−
normalized delay, )(5.0PAPP ω
ωτω ×
Pωω
010
31
1.0
02.050
02.0=pQ
1.03
1
50
5
5-37
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Approximation MethodsDistortionless filter is not realizable as discussed before.
(If phase or delay performance is important , allpass filter can be used to achieve the necessary phase correction.
1. H(s) must be a real rational function such that it can be realized by lumped circuits as discussed before.
2. H(s) is only a approximation of ideal characteristics on bothmagnitude and phase/delay.
1. Butterworth response : maximally flat magnitude in the passband.
2. Chebyshev response : equal ripple in the passband.3. Elliptic response : equal ripple in both the passband and
stopband.(Low order and the most economical realization)
5-38
(i.e. Ideal transfer characteristic are not realizable)Practical realization
Magnitude approximation
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Approximation Methods(Cont.)4. Gaussian response :
(a) freedom from ringing or overshoot.(b) symmetry about the time for which the response is a
maximum.5. And many othersPhase or delay approximation Bessel-Thomson response : maximally flat delay.
5-39
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Example
n222
11jwHωε+
=|)(|
Assume ε=1n2
0
n22
js1
11
1sH)(
|)(|
Ω+
=ω+
=
Assume )..( normalizedei10 =Ω
n22
S11sH
−=|)(| n2
2
S11SHSHsH
−=−= )()(|)(|
1. For n=1 ))(()()(S1
1s1
1S1
1SHSH 2 −+=
−=−
)()(s1
1SH+
= (Only the left-plane pole is selected)
Butterworth
,
=>The denominator of realizable filter function must be Hurwitz polynomials
5-40
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Example(Cont.)
2. For n=2
]][][][[)()(
2j
21S
2j
21S
2j
21S
2j
21S
1SHSH−−−++−++
=−
]][[)()(
2j
21S
2j
21S
1SHSH−+++
=−=>
• Similarly transfer functions of other types can be derived.
• Refer to appendix III
5-41
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency Transformations
• Lowpass prototype =>
– Lowpass prototype
the edge of the lowpass passband – Target filter S=a+jw– Frequency transformation such that
highpass
wjaS +=s 1w =
)(sFs =
======>≤≤ 1||0 w passband of target filter
======>> 1|| w
mapped to
mapped tostopband of target filter
bandpassbandreject…….
frequency variable which is normalized such that at
5-42
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency Transformations(Cont.)Examples
(ii) LP=>BP
(iii) LP=>two-passband BP
=>
(i) LP=>HP
5-43
0 0
=>00
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency Transformations(Cont.)Lowpass to highpass
(i)
ω
1
-1
)(ωω f=
1
1S2531S5351S7160S7160H 23
3
HP+++
==>...
.
0.5dB1
ω
2LP |)(H| wj=
1w
2|)(| jwHHP
0.5dB1
ww
ss js 11
= →= = ω
716.0535.1253.1716.0)( .. 23 +++
=sss
sHge LP
5-44
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency Transformations(Cont.)
(ii) RC : CR transformation
5-45
baSSSHsHLP ++
∞= 2
2)()(
K outViVK outViV
bassbHsHLP ++
= 2
)0()(
Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications
Frequency Transformations(Cont.)LP=>BP
SSQ
SB
SS 11 22
0 +=
+Ω=
where luul0 B Ω−Ω=ΩΩ=Ω &
- Example 1-14LP=>BR
1SS
Q1S 2 +
=S=jw
1Q1
2 +ωω
−=ω
+1
-1 1
w
uw
lw
)(wf=ω
ωωω 12 −
= → = Qwjs
- Example 1-155-46