Circuit y(t)=|x(t)|, LIT system?

[znaya]

Homework Statement

Consider a circuit which output is the absolute value of the input signal, this is, y(t)=|x(t)|?

a) this circuit is not a LIT system because it implements a non linear operation;

b) this circuit is a LIT system that creates phase distortion;

c) this circuit is a LIT system that creates amplitude distortion;

d) this circuit is a LIT system that creates amplitude and phase distortion.

Homework Equations

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The Attempt at a Solution

My first thought was... it's not linear because it will "convert" only the negative part of the sinusoid but then... isn't this because of the phase? I can't decide between a) and b).

Could someone please give a help?

 

[Stephen Tashi]

My first thought was... it's not linear because it will "convert" only the negative part of the sinusoid

To answer a mathematical question, you should think in terms of mathematics. What's the mathematical definition of a linear time independent system (- if that's what "LIT" abbreviates. The more common abbreviation is "LTI".)

 

[znaya]

it means that the system will do the same no matter what time...

 

[Stephen Tashi]

it means that the system will do the same no matter what time...

That isn't a mathematical description. Look up the mathematical definition of an LTI system. What mathematical laws must it follow?

 

[donpacino]

I think you mean LTI: Linear Time invariant.
for a system to be LTI it has to meet to criteria.
1. it has to be linear
(S is a system operator)

S [ x1(t) + x2(t)] = S [x1(t)] + S [x2(t)]

and

S [ a x1(t) ] = a S [x1(t) ]

2. it has to be time invariant
y(t - T) = S [x(t - T)]
for any T or t

so znaya, does your function meet the first criteria?
use two inputs, say -2 and 2

 

[znaya]

That isn't a mathematical description. Look up the mathematical definition of an LTI system. What mathematical laws must it follow?

http://www.icoachmath.com/math_dictionary/Linear_Function.html

Definition of Linear Function
A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear Function.

 

[znaya]

I think you mean LTI: Linear Time invariant.
for a system to be LTI it has to meet to criteria.
1. it has to be linear
(S is a system operator)

S [ x1(t) + x2(t)] = S [x1(t)] + S [x2(t)]

and

S [ a x1(t) ] = a S [x1(t) ]

2. it has to be time invariant
y(t - T) = S [x(t - T)]
for any T or t

so znaya, does your function meet the first criteria?
use two inputs, say -2 and 2

for x=-2, y=2
for x=2, y=2
for x=(2+(-2)) y=0

i see...

so it seems my first thought was right...

 

[znaya]

Stephen Tashi, donpacino, thank you so much for your help.