Bài giảng Radio communications circuits - Chapter 2: Noise, nonlinear distortion and system parameters - Trịnh Xuân Dũng
Dung Trinh, PhD
HCMUT / 2019
Chapter 2
Noise, Nonlinear Distortion and
System Parameters
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
1
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
Contents
1. General Considerations
2. Nonlinear Distortion in Microwave System
3. Noise in RF/Microwave System
2
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Units in RF Design
Units in RF Design:
• The voltage gain and power gain are expressed in decibels (dB):
푉
표푢푡
퐴푣 푑퐵 = 20푙표푔
푉
푖푛
푃표푢푡
퐴푃 푑퐵 = 10푙표푔
푃
푖푛
• These two quantities are equal (in dB) only if the input and output voltages appear
across equal impedances. For example, an amplifier having an input resistance of
R0 (50Ω) and driving a load resistance of R0 satisfies the following equations:
푉2
표푢푡
푃표푢푡
푉
표푢푡
푅0
퐴푃 푑퐵 = 10푙표푔
= 10푙표푔
= 20푙표푔
= 퐴푣 푑퐵
푉2
푃
푖푛
푉
푖푛
푖푛
푅0
• The absolute signal levels are often expressed in dBm rather than watts or volts.
The unit dBm refers to “dBs” above “1mW”. To express the signal power, Psig, in
dBm, we write:
푃
푠푖푔
푃푠푖푔 푑퐵푚 = 10푙표푔
1 푚푊
3
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Units in RF Design
Example 1: An amplifier senses a sinusoidal signal and delivers a power of 0 dBm to a
load resistance of 50Ω. Determine the peak-to-peak voltage swing across the load.
Solution: 0dBm is equivalent to 1mW, we have:
푉푃2푃
= 1푚푊
8푅퐿
푉푃푃 = 632푚푉.
Output voltage of the amplifier is of interest in the above example. This may occur if
the circuit following the amplifier does not present a 50Ω input impedance, and hence
the power gain and voltage gain are not equal in dB.
Only for a sinusoid can we assume that the rms value is equal to the peak-to-peak value
divided by 2 2. Fortunately, for a narrowband 0-dBm signal, it is still possible to
approximate the (average) peak-to-peak swing as 632mV.
4
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. General Considerations
Example 2: A GSM receiver senses a narrowband (modulated) signal having a level of
-100dBm. If the front-end amplifier provides a voltage gain of 15 dB, calculate the
peak-to-peak voltage swing at the output of the amplifier.
Solution:
• -100dBm is 100 dB below 632 mVpp.
• 100 dB for voltage quantities is equivalent to 105.
• Thus, -100 dBm is equivalent to 6.32휇푉푃푃.
• This input level is amplified by 15푑퐵 ≈ 5.62, resulting in an output swing of
35.5 휇푉푃푃.
5
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Nonlinearity
A system is called “memoryless” or “static” if its output does not depend on the past
values of its input.
For a memoryless linear system, the input/output characteristic is given by:
푦 푡 = 훼푥(푡)
For a memoryless nonlinear system, the input/output characteristic can be approximated
with a polynomial:
푦 푡 = 훼0 + 훼1푥 푡 + 훼2푥2 푡 + +훼3푥3 푡 + ⋯
The following figure shows a common-source stage as an example of a memoryless
nonlinear circuit (at low frequencies). If M1 operates in the saturation region and can be
approximated as a square-law device, then
1
푉 = 푉퐷퐷 − 퐼퐷푅퐷 = 푉퐷퐷 − 푘푛 푉 − 푉푇퐻 2푅퐷
표푢푡
푖푛
2
The system has “odd symmetry” if y(t) is an odd function of
x(t). This occurs if 훼푗 = 0 for even j. Such system is called
“balanced” as exemplified by the differential pair.
6
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Nonlinearity
Example 3: The MOS transistors operating in saturation, characteristic can be
expressed as
1
4퐼푆푆
푘푛
푉 = − 푘푛푉푖푛
− 푉2 푅퐷
푖푛
표푢푡
2
If the differential input is small, approximate the characteristic by a polynomial.
Solution:
• Assuming 푉2 ≪ 4퐼퐷푆푆 and applying the approximation 1 − 휀 ≈ 1 − 휀/2, we have
푖푛
푘
푘푛
푉 ≈ − 푘푛퐼퐷푆푆푉푖푛 1 −
푉2 푅퐷
표푢푡
푖푛
8퐼퐷푆푆
3
2
푘푛
푉 ≈ − 푘푛퐼퐷푆푆푅퐷푉푖푛 +
표푢푡
푅퐷푉3
푖푛
8 퐼퐷푆
• The first term on the right-hand side represents linear
operation, revealing the small signal voltage gain of the
circuit (−푔푚푅퐷).
• Due to symmetry, even-order nonlinear terms are absent.
7
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Nonlinear Distortion in Microwave Systems
A system is linear if its output can be expressed as a linear combination
(superposition) of responses to individual inputs:
푦1 푡 = 푓 푥1 푡
푎푦1 푡 + 푏푦2 푡 = 푓 푎푥1 푡 + 푏푥2 푡
푦2 푡 = 푓 푥2 푡
푣표 = 푎푣푖
For a linear system:
For a nonlinear system: The
input/output characteristic of a
memory-less nonlinear system can be
approximated with a polynomial:
푣표 = 푎0 + 푎1푣푖 + 푎2푣푖2 + 푎3푣푖3 + ⋯
8
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Nonlinear Distortion in Microwave Systems
The nonlinearity of the system causes the following issues:
. Harmonic generation (multiples of a fundamental signal)
. Gain Compression (gain reduction in an amplifier)
. Inter-modulation Distortion (products of a two-tone input signal)
. Cross-modulation (modulation transfer from one signal to another)
. AM-PM conversion (amplitude variation causes phase shift)
. Spectral regrowth (intermodulation with many closely spaced signals)
9
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2A. Nonlinear – Harmonic Distortion
Input signal of the system:
푣푖 푡 = 푉 푐표푠휔0푡
0
Output signal of the system:
2
3
푣0 푡 = 푎0 + 푎1푉0푐표푠휔0푡 + 푎2 푉 푐표푠휔0푡 + 푎3 푉 푐표푠휔0푡 + ⋯
0
0
1
3
1
1
= 푎0 + 푎2푉2 + 푎1푉0 + 푎3푉3 푐표푠휔0푡 + 푎2푉2푐표푠2휔0푡 + 푎3푉3푐표푠3휔0푡 + ⋯
0
0
0
0
2
4
2
4
DC
Fundamental
Second
Harmonic
Third
Harmonic
Even-order harmonics result from αj with even j
nth harmonic grows in proportion to An.
10
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2A. Nonlinear – Harmonic Distortion
Problems:
. More signal loss and distortion.
. Interference to other systems.
. Sometimes can be used to create
frequency multipliers.
GSM900 Band
11
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2A. Nonlinear – Harmonic Distortion
Example 4: An analog multiplier “mixes” its two inputs, ideally producing 푦 푡 =
푘푥1(푡)푥2(푡) where k is a constant. Assume 푥1 푡 = 퐴1푐표푠 휔1푡 and 푥2 푡 =
퐴2푐표푠 휔2푡 .
a. If the mixer is ideal, determine the output frequency components.
b. If the input port sensing 푥2(푡) suffers from third-order nonlinearity, determine the
output frequency components
Solution:
a. We have:
푦 푡 = 푘퐴1푐표푠 휔1푡 퐴2푐표푠 휔2푡
푘퐴 퐴
=
1 2 푐표푠 휔1 + 휔2 푡 + 푘퐴1퐴2 푐표푠 휔1 − 휔2 푡
2
2
b. The third harmonic of 푥2 푡 is 훼3퐴32/4, we have:
푦 푡 = 푘퐴1푐표푠 휔1푡 퐴2푐표푠 휔2푡 +
푘퐴 퐴
훼3퐴3
2 푐표푠 3휔2푡
4
=
1 2 푐표푠 휔1 + 휔2 푡 + 푘퐴1퐴2 푐표푠 휔1 − 휔2 푡
2
2
3
3
푘훼 퐴 퐴
푘훼 퐴 퐴
3 1
3 1
+
2 푐표푠 휔1 + 3휔2 푡 +
2 푐표푠 휔1 − 3휔2 푡
8
8
12
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2A. Nonlinear – Harmonic Distortion
The mixer now produces two “spurious” component at 휔1 + 3휔2 and 휔1 − 3휔2 ,
one or both of which often prove problematic. For example, if 휔1 = 2휋 × 850푀퐻푧 and
휔2 = 2휋 × 900푀퐻푧, then 휔1 − 3휔2 = 2휋 × 1850푀퐻푧, an undesired component that is
difficult to filter out because it lies close to the desired component at 휔1 + 3휔2 =
2휋 ×1750MHz.
13
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2B. Nonlinear – Gain Compression
1
3
1
1
푣0 = 푎0 + 푎2푉2 + 푎1푉0 + 푎3푉3 푐표푠휔0푡 + 푎2푉2푐표푠2휔0푡 + 푎3푉3푐표푠3휔0푡 + ⋯
0
0
0
0
2
4
2
4
ퟑ
풂ퟏ푽ퟎ + 풂ퟑ푽ퟑퟎ
ퟒ
ퟑ
= 풂ퟏ + 풂ퟑ푽ퟐퟎ
ퟒ
GAIN: 푮 =
Fundamental
푽ퟎ
14
Most RF/microwave
component/circuits
are compressive !
Expansive
Compressive
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2B. Nonlinear – Gain Compression
P1dB: power at 1-dB compression.
Psat: power at saturation.
If we need very good linearity, we should operate a few dBs below the P1dB point;
this is called power back-off.
ퟑ
ퟐퟎ풍풐품 풂ퟏ + 풂ퟑ푽ퟐ풊풏,ퟏ풅푩 = ퟐퟎ풍풐품 풂ퟏ − ퟏ
ퟒ
풂ퟏ
푽풊풏,ퟏ풅푩 = ퟎ. ퟏퟒퟓ
풂ퟑ
15
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2B. Nonlinear – Gain Compression
Effect on modulated signal:
. Frequency Modulation:
. Amplitude Modulation:
FM signal carries no information in its amplitude and hence tolerates compression.
AM contains information in its amplitude, hence distorted by compression.
Desensitization: the receiver gain is reduced by the large excursions produced by the
interferer even though the desired signal itself is small.
vi V cos1t V2 cos2t
1
3
3
vo ... a1V1 a3V 3 a3V23 cos0t ...
1
4
2
16
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2B. Nonlinear – Gain Compression
Example 5: A 900-MHz GSM transmitter delivers a power of 1W to the antenna. By
how much must the second harmonic of the signal be suppressed (filtered) so that it
does not desensitize a 1.8-GHz receiver having 푃1푑퐵 = −25dBm? Assume the receiver
is 1m away and the 1.8-GHz signal is attenuated by 10 dB as it propagates across this
distance.
Solution:
The output power at 900MHz is equal to 30dBm. With an attenuation of 10dB, the
second harmonic must not exceed -15dBm at the transmitter antenna so that it is below
푃 of the receiver. Thus, the second harmonic must remain at least 45dB below the
1푑퐵
fundamental at the TX output. In practice, this interference must be another several dB
lower to ensure the RX does not compress.
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2C. Nonlinear – Inter-modulation Distortion
Single Signal:
Harmonic distortion
Single and one
Desensitization
Intermodulation
large interferer:
Single and two
large interferers:
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2C. Nonlinear – Inter-modulation Distortion
A received small desired signal along with two large interferers
Intermodulation product falls onto the desired channel, corrupts signal.
interferers
desired
ퟐ흎ퟐ − 흎ퟏ = 흎ퟎ
19
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2C. Nonlinear – Inter-modulation Distortion
The two interferers at frequency 휔1 and 휔2 go into the system:
푣푖 푡 = 푉 푐표푠휔1푡 + 푐표푠휔2푡
0
The signal at the output of the system:
2
0
3
0
2
3
푣0 푡 = 푎0 + 푎1푉0 푐표푠휔1푡 + 푐표푠휔2푡 + 푎2푉 푐표푠휔1푡 + 푐표푠휔2푡 + 푎3푉 푐표푠휔1푡 + 푐표푠휔2푡 + ⋯
1
1
= 푎0 + 푎1푉0푐표푠휔1푡 + 푎1푉0푐표푠휔2푡 + 푎2푉2 1 + 푐표푠2휔1푡 + 푎2푉2 1 + 푐표푠2휔2푡
0
0
2
2
+ 푎2푉2푐표푠 휔1 − 휔2 푡 + 푎2푉2푐표푠 휔1 + 휔2 푡
0
0
3
4
1
3
4
1
+ 푎3푉3
푐표푠휔1푡 + 푐표푠3휔1푡 + 푎3푉3
푐표푠휔2푡 + 푐표푠3휔2푡
0
0
4
3
4
3
3
+ 푎3푉3
푐표푠휔1푡 + 푐표푠 ퟐ흎ퟏ − 흎ퟐ 푡 + 푐표푠 2휔1 + 휔2 푡
0
2
3
4
3
4
3
+ 푎3푉3
푐표푠휔2푡 + 푐표푠 ퟐ흎ퟐ − 흎ퟏ 푡 + 푐표푠 2휔2 + 휔1 푡 + ⋯
0
2
4
4
20
Dept. of Telecoms Engineering
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