For the systems described by the equations below, where the input is f(t) and th

Question

For the systems described by the equations below, where the input is f(t) and the zero-state response is y(t), determine if the system is

• zero-state linear

• time-invariant or time varying,

• causal or noncausal, and

• instantaneous (memoryless) or dynamic. Justify your answers.

1. y(t) = f 2 (t + 3)

2. y(t) = f(t 1)u(t)

3. y(t) = sin(f(3t))

4. y(t) = df(t)/dt + 3f(t)

This question is already out there, but shows no work. Can someone show me how to do the problem and not just give me answers. Thank you, your help is deeply needed and appreciated.

Explanation / Answer

1) non-causal, it dont depends on the past input

time variant y(n,k) y(n-k)

dynamic: Here output at nth instant depends on input at nth instant, x(n) as well as (t+3)th instant previous sample. So the system is dynamic.

2) causal, it depends on the past input

time invariant y(n,k) = y(n-k)

dynamic, Here output at nth instant depends on input at nth instant, x(n) as well as (t-1)th instant previous sample. So the system is dynamic.

zero linear step function

3) given system is non linear

let f1(t) , f2(t) be two inputs applied to this system then y1(t)=sin(f1(t)),y2(t)=sin(f2(t))
if we apply a input to a linear system equal to sum of individual inputs f1(t), f2(t)
f'(t) = f1(t)+f2(t) then y'(t) = y1(t)+y2(t)
but in our system output y'(t) for input f'(t) is y'(t) =sin(f1(t)+f2(t)) which is not equal to y1(t)+y2(t) =sin(f1(t))+sin(f2(t)), hence the system is nonlinear

time invariant:

if input is delayed by to output = sin(f(t-to)) which is equal to y(t-to)(replace t by to in the RHS and LHS of the equation y(t-to) = = sin(f(t-to))).
It is obvious from the relation that as long as f(t) =0 no input is applied output is zero..

causal

apply a bounded input for example f(t)=1 for all t output diverges to infinite even

static: depends on nth instant

4)causal. it depends on the past input

time variant where y(n,k) y(n-k)

static, depends on nth instant

All the above functions are zero state linear as they are changing with respect to input f(t)

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