ECE 453 Wireless Communication Systems Nonlinearities

ECE 453 – Jose Schutt‐Aine 1

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]

ECE 453Wireless Communication Systems

Nonlinearities


Square-Law Detector

In order to operate properly, diode current must remain in the square-law region

• Detector will receive RF at input and will produce a DC voltage proportional to the magnitude of the RF input


Square-Law Detector

Expanding

1 1DD

T

VqVVkT

D S SI I e I e

2 31 1 ...2 6

D D D

D ST T T

V V VI IV V V


Square-Law Detector

If VD is small2 31 1 ...

2 6

D D DD S

T T T

V V VI IV V V

Iout = aVD where a is some constant

If VD is larger2 31 1 ...

2 6

D D DD S

T T T

V V VI IV V V

Iout = bVD2 where b

is some constant

If VD is even larger2 31 1 ...

2 6

D D DD S

T T T

V V VI IV V V

Iout = cVD3 where c is

some constant


Square-Law Detector

GOAL: Keep detector operating in square-law range. Then, Iout is proportional to VD

2 and thus proportional to the power input.


Power Series Model

y f x

2 31 2 3 ...y k x k x k x

q

i

i ix x

d fkdx

Input-output relationship known as:

can be expressed as:

where


BJT Power Series Model

/be TV VC SI I e

be DC beV V v / /DC T be TV V v V

C SI I e e

2 3/ 1 11 ...

2 6DC TV V be be be

C ST T T

v v vI I eV V V

IC

Collector current

Base-emitter junction voltage

Collector current in power series form


Power Series Model for MOSFET

22

D n ox GS TWI C V VL

2 222D n ox GS GS T TWI C V V V VL

No third order or higher terms idealized square-law characteristic

IDMOSFET drain current current


Distortion and Nonlinearities

Distortion occurs when the output signal from an amplifier approaches the extremes of the load line and the output is no longer an exact replica of the input.The ideal amplifier follows a linear relationship. Distortion leads to a deviation from this linear relationship


The 1dB compression point is the point where the difference between the extrapolated linear response exceeds the actual gain by 1dB. P1dB is the power output at the 1dB compression point and is the most important metric of distortion.




AM‐AM Distortion

P1dB


AM‐PM Distortion



AM‐AM distortion is generally more significant

Distortion is a result of nonlinearities in the response of an amplifier. One possible model for the response is:

2 3 ...out in in inV aV bV cV

When the input signal is large enough the higher order terms become significant



If the input is a sine wave, the output will contain harmonics of the input. That is the output will have frequency components that are integer multiples of the fundamental (CW) sine wave input signalWhen the input signal is a two‐tone signal (modulated signal), additional tones will appear at the output and we say that the distortion produces intermodulation products (IMP)



If f1 and f2 are the frequencies associated with the two signals, the output will have components at f3 and f4where

3 1 22f f f and 4 2 12f f f

are known as the lower and upper IM3 respectively.



At low power, before compression the fundamental has a response with 1:1 slope with respect to the input

The IP3 response varies as the cube of the level of input tones. The IP3 has a 3:1 logarithmic slope with respect to the input.

Distortion and Intermodulation


The point of intersection of the extrapolated linear output (of power Po) and third order (IP3) of power PIP3 is called the third‐order intercept point which is a key parameter

• Harmonics can be filtered out by filters• Intermodulation distortion cannot be filtered out because they are in the main passband





Single‐Tone Excitation

2 31 2 3o i i iv k v k v k v

1 1cosiv t a t

22 1

22 1 1

3

21

3 1

1

1

1 3 1

1 c

3 cos4

os31 cos22

12

4

o k a k a tv

k a t

t k

k a t

a


Single‐Tone Excitation

Single-tone excitation produces harmonics of input frequency


Two‐Tone Excitation2 3

1 2 3o i i iv k v k v k v

1 1 2 2cos cosiv t a t a t

2 2 2 21 2 1 1 2 2

1 1

3 31 1 2 2

23 1 2 2 1 2 1

21 2

1 2 2

2

1 2

1 2 1

2 1

2

1 2 1 2

1 1cos3 cos34 4

1

1 1

3 cos 2

1cos2 cos

co

cos 24 4

3 cos 2 cos 24

22 2 2

cos cos

s coso

a t a t

k a a t

a a t

a a a t a tk

a a t a a t

k a t at tv


Two-tone excitation produces intermodulation

Two‐Tone Excitation


Intermodulation Ratio (IMR)

2

( )inIMi

d I

PPIMRP P

inP input power of fundamental componentsdP

IMP output power of in‐band 3rd – order intermodulation( )iIP third‐order intercept

output power of fundamental components

( )( ) 2 ( ) 2 ( )iin IIMR dB P dBm P dBm

In dB:

( )iIP


Dynamic Range of Receiving System

Maximum useable input signal power levelDRMinimum useable input signal power level

Dynamic range is defined as:

Pmin input power for specified SNRo,min (MDS)

Pmax two‐tone input power (per tone) at which SNR of in‐band 3rd order IM product = SNRo,min at system output

Can show that ( )min

23

iIDR P P

( )iIP is the third‐order intercept


1 1 1 2 2 2cos cosx t X t X t

1 1 2 2* *1 1 2 2

12

j t j t j t j tx t X e X e X e X e

2 31 2 3oy t a a x t a x t a x t

Mixer AnalysisInput is described by:

In complex notation:

System is such that the output is described by:


Intermodulation ProductsIntermodulation Product Frequency Order

½(X1X1*)½(X2X2*)

00

22

2½X1= X12(½)33X1

2X1*= ¾X12X1*

2(½)36X1X2X2*= 3(½)X1X2X2*

111

133

2½X2= X22(½)33X2

2X2*= ¾X22X2*

2(½)36X2X1X1*= 3(½)X2X1X1*

222

133

2(½)2X12= ½ X1

2 21 22(½)2X2

2= ½ X22 22 2

2(½)3X12= ¼ X1

3 31 32(½)3X2

2= ¼ X23 32 3

2(½)X1X2= X1X2 1 + 2 22(½)X1X2*= X1X2* 1 ‐ 2 2

2(½)33X12X2= ¾ X1

2X2 21 + 2 32(½)33X1

2X2*= ¾ X12 X2* 21 ‐ 2 3

2(½)33X22X1= ¾ X2

2X1 1 + 22 32(½)33X2

2X1*= ¾ X22 X1* ‐ 22 3


A linear causal system with memory can be described by the convolution representation

( )y t h x t d

where x(t) is the input, y(t) is the output, and h(t) the impulse response of the system.

A nonlinear system without memory can be described with a Taylor series as:

1

( ) ( ) nnn

y t a x t

where x(t) is the input and y(t) is the output. The an are Taylor series coefficients.

Volterra Series


A Volterra series combines the above two representations to describe a nonlinear system with memory

1 111

1( ) ... ,..., ( )!

n

n n n rrn

y t du du g u u x t un

1 2 2 1

1 1 1

2 1 2

1 2 3 3 1 2 3 1 2 2 1 2 1 2 3

1( 1 ( ) (

1 ( , ) ( ) ( )2

)

!

1 ( , , ) ( ) ( ) ( , ) ( ) ( ) ( )2!.

!

.

)1

.

du du g u u x t u x t u

du du du g u u u x t u x t u g u u x t u x

du g u x ty

t u x t

t u

u

where x(t) is the input and y(t) is the output and the gn(u1,…,un) are called the Volterra kernels

impulse response

higher-order impulse responses

Volterra Series

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ECE 453 Wireless Communication Systems Nonlinearities

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