2 Basic concepts in RF design
RF designers draw upon many concepts that originate from the theory of signals and systems.
In this chapter, we describe these concepts and define the terminology used in
RF electronics so as to prepare the reader for the material in the following chapters.
Î Harmonic distortion, gain compression, cross modulation, intermodulation, intersymbol interference, Nyquist signaling, random process, noise, etc,.
Chonbuk National University 1 Microwave Laboratory

2.1 Nonlinearity and time variance
Nonlinearity
A system is linear if its output can be expressed as a linear combination
(superposition) of responses to individual inputs.
For inputs x
1
( t ) and x
2
( t ), x
1
( t )
→ y
1
( t ), x
2
( t )
→ y
2
( t ) ax
1
( t )
+ bx
2
( t )
→ ay
1
( t )
+ by
2
( 2 .
1 )
( t ) (2.2)
If the system has nonzero initial conditions or finite “offsets”, then this system is nonlinear. x
1
( t )
→ y
1
( t )
+
1
, x
2
( t )
→ y
2
( t )
+
2 ax
1
( t )
+ bx
2
( t )
→ a [ y
1
( t )
+ α
1
]
+ b [ y
2
( t )
+ α
2
]
Chonbuk National University 2 Microwave Laboratory

2.1 Nonlinearity and time variance
Time invariance
A system time invariant if a time shift in its input results in the same time shift in its output.
x ( t )
→ y ( t )
⇒ x ( t
− τ
)
→ y ( t
− τ
)
Time variance x ( t )
→ y ( t )
⇒ x ( t
− τ
)
→ y ( t
− τ
)
Chonbuk National University 3 Microwave Laboratory

2.1 Nonlinearity and time variance
While nonlinearity and time variance are intuitively obvious concepts, they may be confused with each other in some cases.
Consider the switching circuit shown in fig.2.1(a) simple switching circuit.
The control terminal of the switch is driven by the input terminal by V in 2
( t )
=
A
2 cos
ω
2 t
V in 1
( t )
=
A
1 cos
ω
1 t
We assume the switch is on if V in
>0 and off otherwise.
Is this system nonlinear or time variant?
and
Chonbuk National University
Fig. 2.1 (a) Simple switching circuit
4 Microwave Laboratory

2.1 Nonlinearity and time variance
In fig. 2.1(b)
The path of interest is form V in1 to V out
.
The system is nonlinear because the control is only sensitive to the polarity of V in1
V out
, and V out also depends on V in2
.
is depend on V in2
. And time variant because
In fig. 2.1 (c)
The path of interest is form V in2 to V out
.
The system is linear but, time variant.
Î
“Switches are nonlinear”.
(Contradiction)
Fig. 2.1 (b) Nonlinear time-variant system (c) linear time-variant system
Chonbuk National University 5 Microwave Laboratory

2.1 Nonlinearity and time variance
Another interesting result of the above observation:
A linear system can generate frequency components that do not exist in the input signal
This is possible if the system is time variant
For example fig. 2.1( c).
In this circuit, V out can be considered as the product of V in wave toggling between 0 and 1.
Î V in 1
= f (
ω
1
)
→
V out 1
= g (
ω
1
,
ω
2
) and a square
Chonbuk National University 6 Microwave Laboratory

2.1 Nonlinearity and time variance
Time domain Frequency domain
V in 1
( t )
V in 2
( t )
V in 1
( f )
V in 2
( f ) where * denotes convolution
Periodic signal g p
( t )
=
=
∞ m
−∞ g ( t
− mT o
) g p
( t )
=


A
0 c n
=
1
T o
−
T
2
≤ t
≤
T
2 elsewhere
T
− T
2
2
A exp


− j 2
π nt
T
1

 dt
=
A n
π sin

 n
π
T
T
1


=
A n
π sin n
π
2
↔
G p
( f )
= n
=
∞
−∞ n
G (
T o
)
δ
( f
− n
T o
)
Chonbuk National University 7 Microwave Laboratory

2.1 Nonlinearity and time variance
The output spectrum is
V out
( f )
=
V in
( f ) * n
=
+∞
−∞ sin( n
π n
π
/ 2 )
δ

 f
− n
T
1


= n
+
∞
= −∞ sin( n
π n
π
/ 2 )
V in

 f
− n
T
1


(2.3)
(2.4)
δ
( ): The Dirac delta function,
T
1
=2
π
/
ω
1
Thus, the output consists of vertically scaled replicas of V in2
(f) shifted by n/T1
Chonbuk National University 8 Microwave Laboratory

2.1 Nonlinearity and time variance
Memoryless system
A system is called “ memoryless” if its output does not depend on the past values of its input.
For memoryless linear system, y
( )
x
( )
(2.5) where
α is a function of time if the system is time variant
For a mymoryless nonlinear system, the input-output relationship can be approximated with a polynomial, y
= α
0
+ α
1 x
( )
+ α
2 x
2
( )
+ α
3 x
3
( )
+ L
(2.6) where
α j are in general functions of time if the system is time invariant
Chonbuk National University 9 Microwave Laboratory

2.1 Nonlinearity and time variance
Fig.2.2(a) illustrates an example where the input signal is applied to the base of Q
1 while Q square wave.
2 and Q
3 are periodically switched by means of a
For ideal bipolar transistors.the circuit can be viewed as in fig.2.2(b) and v out
( )
I
S 1 exp v
V in
T
s
( )
R
(2.7) where I
S1 represents the saturation current of Q square wave toggling between –1 and +1
1
, V
T
=kT/q, and s(t) is a
Chonbuk National University 10 Microwave Laboratory

2.1 Nonlinearity and time variance
The system has “ odd symmetry ” (or “ differential ” or “ balanced ”) if its response to -x(t) is the negative of that of x(t).
y
= α
0
+ α
1 x
( )
+ α
2 x
2
( )
+ α
3 x
3
( )
+ L where
α j
=0 for even j.
Ex.] The bipolar differential pair of fig.2.3 exhibits the following inputv out
=
RI
EE tanh v in ( 2.8)
2 V
T
Î Odd function
Chonbuk National University 11 Microwave Laboratory

2.1 Nonlinearity and time variance
Dynamic (or memory) system: If its output depends on the past values of its input(s) or output(s).
For a linear, time-invariant, dynamic system y
( ) ( ) ( ) where h(t) denotes the impulse response.
(2.9)
If a dynamic system is linear but time variant, its impulse response depends on the time origin; if
δ
(t)
Î h(t), then
δ
(t-
τ
)
Î h(t,
τ
) thus, y
( )
= h ( t ,
) * x
( )
(2.10)
Finally, if a system is both nonlinear and dynamic, then its impulse response can be approximated with a Voterra series[1,2], a topic beyond the scope of this book.
Chonbuk National University 12 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Many analog and RF circuit can be approximated with a linear model to obtain their response to small signals, nonlinearites often lead to interesting and important phenomena.
For memoryless and time-variant systems, y
( )
≈ α
1 x
( )
+ α
2 x
2
( )
+ α
3 x
3
( )
(2.11)
Chonbuk National University 13 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Harmonics
If a sinusoid is applied to a nonlinear system, the output generally exhibits frequency components that are integer multiples of the input frequency. if x(t)=Acos
ω t, then y
= α
1
A cos
ω t
+ α
2
A
2 cos
2
ω t
+ α
3
A
3 cos
3
ω t
=
=
α
1
A cos
ω t
α
2
2
A
2
+
+
α
2
A
2 (
1


α
1
A
+
2
3
α
3
A
3
4
+ cos 2
ω t
)
+

 cos
ω t
+
α
3
A
3
4
α
2
A
2
2
(
3 cos
ω t cos 2
ω t
+
+ cos
α
3
A
3
4
3
ω t
) cos 3
ω t
(2.14)
The term with the input frequency (
ω
) is called the “fundamental” and the higher-order terms(n
ω
, n:integer) the “harmonics.”
Chonbuk National University 14 Microwave Laboratory

2.1.1 Effects of Nonlinearity
We can make two observations.
First, even-order harmonics result from
α j with even j and vanish if the system has odd symmetry, i.e., if it is fully differential.(see fig.2.3)
In reality, however, mismatches corrupt the symmetry, yielding finite even-order harmonics.
Second, in (2.14) the amplitude of the n-th harmonic consists of a term proportional to and other terms proportional to higher powers of
A.
Chonbuk National University 15 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Gain Compression (if
α
<0)
The small signal gain of circuit is usually obtained with the assumption that harmonics are negligible.
Ex.1] When A is much small, eq.(2.14) is y
=
α
2
A
2
2
+


α
1
A
+
3
α
3
4
A
3

 cos
ω t
+
α
2
A
2
2
≈ α
1
A cos
ω t
The small signal gain is equal to
α
1
.
cos 2
ω t
+
α
3
A
3
4 cos 3
ω t
Ex.2] The differential pair of fig. 2.3 to be equal to tanh x
= x
− x
3
3
+
2 x
5
15
−
17 x
7
315
+ L
(
−
1 ) n
−
1
2
2 n
( 2
2 n
( 2 n )!
−
1 ) B n x
2 n
−
1
+ L v out
=
RI
EE tanh v in
2 V
T
≈
RI
EE v in
2 V
T
Î v out v in
=
I
EE
R
2 V
T
Chonbuk National University 16 x
<
π
2
(2.15)
Microwave Laboratory

2.1.1 Effects of Nonlinearity
However, as the signal amplitude increase, the gain begins to vary.
y
=
α
2
A
2
2
+



α
1
A
+
3
α
3
A
3
4
≈



α
1
A
+
3
α
3
A
3
4 

 cos
ω t


 cos
ω t
+
α
2
A
2
2 cos 2
ω t
+
α
3
A
3
4 cos 3
ω t
In most circuits of interest, the output is a “compressive” or “saturating” function of input. At high input level, gain is a decreasing function of A.
α
1
3
α
3
A
2
/ 4
0
(
α
1
α
3
,
α
1
0 ,
α
3
0
)
Chonbuk National University 17 Microwave Laboratory

2.1.1 Effects of Nonlinearity
In RF circuits, this effect is quantified by the “1-dB compression point” .
1-dB compression point: The input signal level that causes the smallsignal gain to drop by 1dB.
Chonbuk National University 18 Microwave Laboratory

2.1.1 Effects of Nonlinearity
To calculate the 1-dB compression point,
G
=
20 log


α
1
A
+
3
α
3
A 3
4
A cos
ω t

 cos
ω t
=
20 log
α
1
+
3
4
α
3
A
1
2
− dB
=
20 log
α
1
−
1 dB
=
20 log
= α
1
+
3
4
α
3
A
1
2
− dB
α
1
1
10 20
α
1
1
10 20
−
3
4
α
3
A
1
2
− dB
= α
1


1
−
10
−
1
20


=
0 .
1087
α
1
∴
A
1
− dB
=
0 .
1087
×
4
3
×


−
α
1
α
3


=
0 .
145
α
1
α
3
Chonbuk National University 19
(2.16)
(2.17)
Microwave Laboratory

2.1.1 Effects of Nonlinearity
Desensitization and blocking
When the desired signal is fed to circuit with a strong interferer, the
“average” gain of the circuit is reduced because of a large signal
(interferer). : “desensitization”
α
3
A
3
( t )
= α
3
(
A
1
This effect can be analyzed for the characteristics of (2.11) by assuming x cos
( )
ω
1 t
+
=
A
2
A
1 cos cos
ω
2 t
ω
1 t
+
)
3 = α
3
A
2
(
A
1
3 cos cos
3
ω
ω
1 t
+
2 t
3 A
1
2
A
2 cos
2
ω
1 t cos
ω
2 t
+
3 A
1
A
2
2 cos
ω
1 t cos
2
ω
2 t
+
A
2
3 cos
3
ω
2 t
)
= α
3


A
1
3
4
(
3 cos
ω
1 t
+ cos 3
ω
1 t
)
+
3 A
1
2
A
2 cos
2
ω
1 t cos
ω
2 t
+
3 A
1
A
2
2 cos
ω
1 t
1
+ cos 2
ω
2 t
2
+
A
2
3 cos
3
ω
2 t


=
3
4
α
3
A
3
1 cos
ω
1 t
+
3
2
A
1
A
2 cos
ω
1 t
+ L y
= α
1
A
1
+
3
4
α
3
A
1
3 +
3
2
α
3
A
1
A
2
2 cos
ω
1 t
+ L
(2.18)
Chonbuk National University 20 Microwave Laboratory

2.1.1 Effects of Nonlinearity
For A
2
, y
= α
1
3
2
α
3
A
2
2
A
1 cos
ω
1 t
+ L
(2.19) for
α
3
<0, and sufficiently large A
2
, the gain drops zero, and we say the signal is “ blocked ” in RF design.
Many RF receivers must be able to withstand blocking signals 60 to
70dB greater than the wanted signal.
cf.) No interferer case y
= α
1
3
4
α
3
A
1
2
Interferer case
A
1 cos
ω
1 t
+ L y
= α
1
3
2
α
3
A
2
2
A
1 cos
ω
1 t
+ L
Î
Faster compression
(
Q A
>>
2
A
1
)
Chonbuk National University 21 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Cross Modulation: When a weak signal and a strong interferer pass through a nonlinear system, the transfer of modulation on the amplitude of the the interferer to the amplitude of the weak signal is occurred.
From(2.19)
A
2 cos
ω
2 t
→
A
2
( 1
+ m cos
ω m t ) cos
ω
2 t y
( m
<
1 , modulation index )
= α
1
A
1 cos
ω
1 t
=



α
1
A
1
+
+
3
2
α
3
A
1 cos
ω
1 t
[
A
2
(
1
+ m cos
ω m t
) cos
ω
2 t
]
2
3
2
α
3
A
1
A
2
2

 1
+ m
2
2
+ m
2
2 cos 2
ω m t
+ L
+
2 m cos
ω m t




 cos
ω
1 t
+ L
Thus, the desired signal at the output contains amplitude modulation at
ω m and 2
ω m
.
A common case of cross modulation arises in amplifiers that must simultaneously process many independent signal channels, e.g., in cable
TV.
Chonbuk National University 22 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Intermodulaton
Harmonic distortion is often used to describe nonlinearties of analog circuits, certain cases require other measurers of nonlinear behavior.
For example, suppose the nonlinearity of an active low-pass filter is to be evaluated.fig.2.5
If the input sinusoid frequency is chosen such that its harmonics fall out of the passband, then the output distortion appears quite small even if the input stage of the filter introduces substantial nonlineality.(Just only single tone case )
Chonbuk National University 23 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Thus,commonly used is the “intermodulation distortion” in a “ two-tone” test.
When two signals with different frequencies are applied to a nonlinear system, the output in general exhibits some components that are not harmonics of the input frequencies. Î Intermodulation (IM)
This phenomenon arises from “mixing” of the two signal when their sum is raised to a power greater than unity.
To understand how eq.(2.11) leads to intermodulation, assume x
( )
=
A
1 cos
1 t
+
A
2 cos
2 t y
( )
= α
1
+ α
(
2
A
1 cos
(
A
1
ω
1 t cos
+
ω
1 t
A
2
+ cos
ω
2 t
)
A
2 cos
ω
2 t
)
2 + α
3
(
A
1 cos
ω
1 t
+
A
2 cos
ω
2 t
)
3
Chonbuk National University 24 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Intermodulation products:
ω = ω
1
± ω
2
:
α
2
A
1
A
2
=
= cos
2
ω
2
ω
2
1
± ω
2
:
3
α
3
A
1
2
A
2
± ω
1
:
4
3
α
3
A
1
2 A
2
4
( ω
1 cos cos
(
(
+ ω
2
2
ω
2
ω
1
2
) t
+ ω
2
+ ω
1
+ α
2
A
1
A
2
) t
) t cos
+
+
3
α
3
A
2
2
A
2
4
3
α
3
A
2
2 A
2
4
( ω
1
− ω
2
) t (2.22) cos
(
2
ω
1 cos
(
2
ω
2
− ω
2
− ω
1
) t (2.23)
) t (2.24)
Fundamental components
ω = ω
1
,
ω
2
:
α
1
A
1
+
3
4
α
3
A
1
3 +
3
2
α
3
A
1
A
2
2
+ α
1
A
2
+
3
4
α
3
A 3
2
+
3
2
α
3
A
2
A
1
2 cos
ω
1 t cos
ω
2 t (2.25)
Chonbuk National University 25 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Of particular interest are the third-order IM products at and
2
ω −
2
ω
1 illustrated in fig.2.5.
2
ω −
1
ω
2
If the difference between
2
ω
2
− ω
1
ω
1 and
ω
2
ω
1 and
ω
2
. Î Revealing
ω
1
− ω nonlinearites even in cases such as the LPF of fig.2.5
2
Chonbuk National University 26 Microwave Laboratory

2.1.1 Effects of Nonlinearity
[Ex1.]
In a typical two-tone test, A
1
=A
2
=A, and the ratio of the amplitude of the
α
IM distortion .
α =
1 V pp and 3
α
3
A 3 / 4
=
10 mV pp
.
IMD
=
10 log


3
3
A
3
4


2
R
( )
1
2
R
=
10 log
10
−
4
1
= −
40 dBc where the letter “c” means “with respect to the carrier”.
Chonbuk National University 27 Microwave Laboratory

2.1.1 Effects of Nonlinearity
[Ex2.]
If a weak signal accompanied by two strong interferers experiences thirdorder nonlinearity as shown in fig.2.7, then one of the IM products falls in the band of interest, corrupting the desired component.
While operating on the amplitude of the signals, this effect degrades the performance even is on the phase (because zero-crossing points are still affected.)
This phenomenon cannot be directly quantified by harmonic distortion.
Chonbuk National University 28 Microwave Laboratory

2.1.1 Effects of Nonlinearity
IP3
The corruption of signals due to third-order intermodulation of two nearby interferers is so common and so critical that a performance metric has been defined to characterize this behavior.
Called the “ third intercept point ” (IP
3
)
This parameter is measured by a two-tone test in which A is chosen to be sufficiently small so that higher-order nonlinear terms are negligible and the gain is relatively constant and equal to
α
1
.
ω =
=
2
2
ω
ω
1
2
−
−
ω
2
:
ω
1
:
3
α
3
A
2
2
A
2
3
α
3
4
A
2
2 A
2
4 cos cos
(
(
2
ω
1
2
ω
2
− ω
2
− ω
1
) t (2.23)
) t (2.24)
ω = ω
1
,
ω
2
:
α
1
A
1
+
3
4
α
3
A
1
3 +
3
2
α
3
A
1
A
2
2 cos
ω
1 t
+ α
1
A
2
+
3
4
α
3
A 3
2
+
3
2
α
3
A
2
A
1
2
Chonbuk National University 29 cos
ω
2 t (2.25)
Microwave Laboratory

2.1.1 Effects of Nonlinearity
IP
3
(cont.)
From (2.23),(2.24), and(2.25), we note that as A increases the fundamentals increase in proportion to A, whereas the third-order IM
A scale[fig.2.8(b)]
The magnitude of the IM products grows at three times
The third order intercept point is defined to be at the intersection of the two lines.
Horizontal coordinate : input IP
3
(“ IIP3 ”)
Vertical coordinate: output IP
3
(“ OIP3 ”)
Chonbuk National University 30 Microwave Laboratory

2.1.1 Effects of Nonlinearity
IP
3 is used as a measure of linearity and a unique quantity that by itself can serves as a means of comparing the linearity of different circuits.
Derive IP
3 x
( )
=
A cos
ω
1 t
+
A cos
ω
2 t y
( )
α
1 x
( )
α
2 x
2
( )
α
3 x
3
( ) y
= α
9
4
α
3
A
2
A cos
ω
1 t
+ α
9
4
α
3
A
2
A cos
ω
2 t
+
≅
3
α
4
1
α
A
3
A
3
A cos
ω
1 t cos
(
2
ω
1
+ α
1
A
− ω
2
) cos
ω
2 t t
+
3
4
α
3
A
3
A cos
(
2
ω
2
− ω
1
) t
+ L (2.26)
+
3
4
α
3
A
3
A cos
(
2
ω
1
− ω
2
) t
+
3
4
α
3
A
3
A cos
(
2
ω
2
− ω
1
) t
+ L (For
α
1
>>
9
4
α
3
A
2
)
Chonbuk National University 31 Microwave Laboratory

2.1.1 Effects of Nonlinearity
If the output level
ω
1 and 2
ω
2
-
ω
1
α
1
A
IP 3
=
3
4
α
3 and
ω
2
A
3
IP 3
A
IP 3
=
4
3
α
1
α
3 have the same amplitude as those at 2
ω
1
-
ω
2
(2.27)
(2.28)
In practice, the input level is increased to reach the intercept point, the assumption no longer holds, the gain drops, and higher-
1 3
A
4 order IM products become significant.
In fact, the IP
3 is beyond the allowable input range, sometimes even higher than the supply voltage.
So, the practical method of obtaining the IP3 is to measure the characteristic of fig.2.8(b) for small input amplitudes and use linear extrapolation on a logarithmic scale to find the intercept point.
Chonbuk National University 32 Microwave Laboratory

2.1.1 Effects of Nonlinearity
A quick method of measuring the IP
3
A in
: The input level at each frequency
A
ω
1,
ω
2
: The amplitude of the output components at
ω
1 and
ω
2
A
IM3
: The amplitude of the IM3 products
Measure A in
, A
ω
1,
ω
2
, A
IM3
, where A in is small signal.
A
ω
1 ,
ω
2
A
IM 3
≈
=
α
1
3
α
3
A in
A
3 in
/ 4
(2.29)
4
α
1
3
α
3
1
A
2 in
(2.30)
A
IP 3
=
4
3
α
α
1
3
A
ω
1 ,
ω
2 =
A
IM 3
A 2
IP 3 (2.31)
A in
2
Chonbuk National University 33 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Consequently
20 log
20 log
A
ω
1 ,
ω
2
A
IP 3
=
1
2
−
20 log
(
20 log
A
IM 3
A
ω
1 ,
ω
2
−
=
20
20 log log
2
A
IP
A
IM 3
)
3
+
−
20
20 log log
A in
2
A in
If all the signal levels are expressed in dBm,
IIP
3 dBm
=
∆
P
2 dB
+
P in dBm
(Class A case only)
(2.32)
(2.33)
Chonbuk National University 34 Microwave Laboratory

2.1.1 Effects of Nonlinearity
The actual value of IP
3
, however, must still be obtained through accurate extrapolation to ensure that all nonlinear and frequency-dependent effects are taken into account.
Another measurement method is to apply a sing tone, plot the third harmonic magnitude versus the input level, and obtain the intercept point by extrapolation. Î Not yield a correct value for IP
3
Chonbuk National University 35 Microwave Laboratory

2.1.1 Effects of Nonlinearity
[EX.3]
To gain a better feeling about the required linearity in typical RF systems,
Calculate the amount of corruption that a 1-uV rms two 1-mV rms signal experiences by interferers in an amplifier having an IIP
3 of 70 mV rms
(=
-10dBm) (Fig. 2.10).
Chonbuk National University 36 Microwave Laboratory

2.1.1 Effects of Nonlinearity
We can write
A sig , out
A sig , in
≈
A int, out
(2.34)
A int, in where A sig
: The signal amplitude
A int
: The interferer amplitude
From (2.28)
A
ω
1 ,
ω
A
IM 3
2 =
2
A
IP 3
⇔
A in
2
A int, out
A
IM 3
=
2
A
IP 3
2
A int, in
A sig , out
A
IM 3 , out
≈
A sig , in
⋅
A int, out
A int, in
⋅
1
A int, out
⋅
2
A
IP 3
2
A int, in
=
A sig , in
⋅
A
3 int, in
2
A
IP 3
(2.35)
Chonbuk National University 37 Microwave Laboratory

2.1.1 Effects of Nonlinearity
Where A sig , in
=
1 uV rms
,
A sig , out
A
IM 3 , out
A
IP
3
=
10
−
6 ×
0 .
07
2
( 10
−
3
)
3
=
70
=
4 .
9 mV rms
⇔
, A int, in
=
1 mV rms
13 .
8 dB
The relationship between the 1-dB compression point and the input IP
3 for a third-order nonlinearity. (2.36)
A
1
− dB
A
IP 3
=
0 .
145
α
1
α
3
4
3
α
1
α
3
=
0 .
145
=
0 .
3298
4 / 3
⇔ −
9 .
6 dB (2.37)
Chonbuk National University 38 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
Since in RF systems, signals are processed by cascaded stages, it is important to know how the nonlinearity of each stage is referred to the input of the cascade.
It is desirable to calculate an overall input third intercept point in terms of the IP
3 and gain of the individual stage.
Consider two nonlinear stages in cascade(fig. 2.11).
Chonbuk National University 39 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
If the input-output characteristics of the two stages are expressed, respectively, as y
1 y
2
( )
= α
1 x
( )
( )
= β
1 y
1
( )
+ α
+
2
β x
2
2 y
1
2
( )
( )
+ α
+
3
β x
3
3
( ) y
1
3
(2.38)
( )
(2.39)
Then y
2
( )
= β
+
1
β
[
α
3
1
[ x
α
1
( )
+ α
2 x 2
( )
+ α
3 x 3
( ) x
]
+
( )
+ α
2 x 2
( )
+ α
3 x 3
( )
3
]
β
2
+
[
α
1 x
( )
+ α
2 x 2
( )
+ α
3 x 3
( )
2
]
L (2.41)
Considering only the first-and third-order terms, we have y
2
= α
1
β
1 x
+
(
α
3
β
1
+
2
α
1
α
2
β
2
+ α
1
3
β
3
) x
3
( )
+ L (2.41)
From(2.28)
A
IP 3
=
4
3
α
3
β
1
+
2
α
α
1
1
α
β
2
1
β
2
+ α
1
3
β
3
(2.42)
Chonbuk National University 40 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
Interestingly, proper choice of the values and signals of the terms in the denominator can yield an arbitrarily high IP3. In practice, however, other considerations such as noise, gain, and active device characteristics may not permit this choice.
Simplified eq.(2.42) –inverted and squared
1
A
2
IP 3
=
3
4
=
4
3
1
α
1
α
3
α
3
β
1
+
+
3
α
2
2
α
1
α
2
β
2
α
1
β
1
2
β
β
1
2 +
+
4
3
α
1
2
β
1
β
3
α
1
3
β
3
(2.43)
=
1
A
2
IP3,1
+
3
α
2
2
β
β
1
2 +
α
1
2
A
2
IP3,2
(2.44) where A
IP3,1 and A
IP3,2 represent the input IP
3 points of the first and second stages, and they are voltage quantities.
Chonbuk National University 41 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
From the result,
α
1 increases, the overall IP
3 decreases. This is because with higher gain in the first stage, the second stage senses larger input levels producing greater IM
3 products.
Chonbuk National University 42 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages x
( )
A cos
ω t
A cos
ω
2 t and identify the various IM products.
Chonbuk National University 43 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
From fig.(2.12) y
2
= α
1
β
1
A
( cos
ω
1 t
+


3
α
3
β
1
4
+ cos
ω
2 t
)
(2.45)
+
3
α
1
3
β
3
4
+
3
α
1
α
2
β
2
2

 A 3
[ cos
(
2
ω
1
− ω
2
) t
+ cos
(
2
ω
2
− ω
1
) t
]
+ L
Î
The same IP
3
(eq.(2.44))
In many RF systems, each stage in a cascade has a narrow frequency band. Thus, the 2 nd harmonic term fall out of the band and heavily attenuated. Consequently, the second term on the right-hand side of (2.44) becomes negligible
1
≈
1
+
α
1
2
A
2
IP 3
A
2
IP 3 , 1
A
2
IP 3 , 2
(2.46)
Chonbuk National University 44 Microwave Laboratory

2.1.2 Cascaded Nonlinear Stages
General expression for three or more stages:
1
A
2
IP 3
≈
1
A
2
IP 3 , 1
+
α
2
A
IP
1
2
3 , 2
+
α
1
2
A
2
IP
β
1
2
3 , 3
+ L +
[
α
1
β
1
L
A
2
IP
(
3 , n
1 n
−
1 )
]
2
+ L (2.47) where A
IP3,3
α
1
β
1 denotes the input IP
L ( n
1
−
1 ) :
3 of the third stage
The previous stage gain
If each stage in a cascade has a gain greater than unity, the nonlinearity of the latter stages becomes increasingly more critical because the IP
3 of each stage is effectively scaled down by the total gain preceding that stage
Eq.(2.47) is approximation.In practice, more precise calculations of simulations must be performed to predict the overall IP
3
.
Chonbuk National University 45 Microwave Laboratory

2.2 Intersymbol interference
Linear time invariant systems can also “distort” a signal if they do not have sufficient bandwidth.
Fig 2.13 a)a low-pass filter is a familiar example of such behavior
Î If a single ideal rectangular pulse is applied to a low-pass filter, the output exhibits an exponential tail that becomes more significant as the filter bandwidth decreases.
Limited bandwidth has a more detrimental effect on the random bit streams.
Chonbuk National University 46 Microwave Laboratory

2.2 Intersymbol interference
Suppose the out of a digital system consists of a random sequence of ONEs and ZEROs.[fig2.13(b)]
Each bit level is corrupted by decaying tails created by previous bits.
Î “ I nter s ymbol I nterference”(ISI)
This phenomenon leads to higher error rate in the detection of random waveforms that are transmitted through band-limited channels.
The problem of ISI is particularly troublesome in wireless communications because the bandwidth allocated to each channel is fairly narrow.
Methods of reducing ISI include pulse shaping (“Nyquist signaling”) in the transmitter and “equalization”in the receiver.
Chonbuk National University 47 Microwave Laboratory

2.2 Intersymbol interference
In order to reduce ISI, the pulse shape can be chosen such that it is less susceptible to interference with its shifted replicas.
In Nyquist signaling, each pulse is allowed to overlap with past and future pulses, but the shape is selected such that ISI is zero at certain points in time.
Illustrated in fig.2.14, the idea is that all other pulses go through zero at the point when the present pulse reaches its peak.
Thus, if the bit stream is sampled at t=kT s
, no ISI exists.
Chonbuk National University 48 Microwave Laboratory

2.2 Intersymbol interference
A basic condition for Nyquist signals
For a pulse shape p(t) p
=
1 if k
0 if k
=
1 (2.48)
≠
0 (2.49)
Using a train of impulses to sample this pulse, p
( )
•
δ ( t
− kTs
) ( )
(2.50)
Fourier transform of both sides,
P
1
*
T s
δ

 f
− k
T s


=
1 (2.51)
1
T s
P

 f
− k
T s


=
1 (2.52)
Chonbuk National University 49 Microwave Laboratory

2.2 Intersymbol interference
Originally proposed by Nyquist and shown in fig.2.15, this result indicates that the shifted replicas of P(f) must add up to a flat spectrum.
For example, a sinc waveform satisfies this condition because its
Fourier transform is a rectangular box.
Rectangula r
F.T.
←→
Sinc
Chonbuk National University 50 Microwave Laboratory

2.2 Intersymbol interference
A sinc pulse shape introduces difficulties in the design of the system.
The filter required to produce the rectangular spectrum becomes quite complex if a sharp cutoff is necessary. (The substantial signal energy near the edge of the spectrum complicates the filtering requirements in both the transmit and receive paths.)
The sinc waveform decays slowly with time, introducing considerable
ISI in the presence of timing errors in the sampling command .
Chonbuk National University 51 Microwave Laboratory

2.2 Intersymbol interference
A pulse shape often employed in Nyquist signaling is related to a “ raised cosine ” spectrum.[fig.2.16]
The time- and frequency-domain expressions of this function are, respectively, p
P
( ) f
=
( ) sin
=
( π
π








T s
2 t / t
T s
/
T s

 1

T s
) (
1
−
+ cos
π
4
α
π
α
T s
α
2 t


0
2 t f
/
/ T s
T s
2
−
)
1
−
(2.53)
0
2 T s
α
<





1
−
2 T s
α f f
<
<
> f
1
1
−
2 T s
<
+
1
2 T s
α
+
α
2 T s
α
(2.54)
Chonbuk National University 52 Microwave Laboratory

2.2 Intersymbol interference
Where
α
: “rolloff” factor (0<
α
<1)
Interesting notes
1) p(t) decays faster than a sinc function
2) For
α
=0, p(t) reduces to a sinc function.
3) P(f) is similar to a box spectrum but with smooth edges.
The trade-off in the choice of
α is between the decay rate in the time domain and the excess bandwidth (with respect to a box spectrum)in the frequency domain. (
α
: 0.3 ~ 0.5 (Typ.))
Chonbuk National University 53 Microwave Laboratory

2.2 Intersymbol interference
Raised-cosine signaling can also be visualized as shown in fig.2.17.
Each bit is represented by an impulse
The data stream is applied to a filter whose transfer function is given by
(2.54)
Chonbuk National University 54 Microwave Laboratory