Can someone please answer question 1.28 parts b and c only? Please provide a jus
ID: 2249053 • Letter: C
Question
Can someone please answer question 1.28 parts b and c only? Please provide a justification for your answers. Thank you.
Please note that question 1.28 uses information from question 1.27.
1.27. In this chapter, we introduced a number of general properties of systems. In partic ular, a system may or may not be (1) Memoryless (2) Time invariant (3) Linear (4) Causal (5) Stable Determine which of these properties hold and which do not hold for each of the following continuous-time systems. Justify your answers. In each example, y(t) de- notes the system output and x(t) is the system input. 62 Signals and Systems Chap.1 (b) y(t) - [cos(3t)]x(t) (d) y(t) = x(t) + x(t-2), (a) y(t) = x(t-2) + x(2-1) (c) y(t)-, x(r)dT tExplanation / Answer
1.28
(b) y[n] = x[n-2]-2x[n-8]
(1) In the above question the system input depends on the future value (advance value) hence the system has memory or it is a dynamic system, not a static. (Solution : This property do not hold.)
(2)Above system is time varient, because for any instant of time input, the output is not same.
a system is said to be time invarient if ,
x[n] ----> y[n] or x[n-m]---->y[n-m] ie for every input x[n] their is a shift of time m (everytime).
(Solution :This property Do not hold)
(3) The system is said to be linear if;
T{a1 x1[n] + a2 x2[n] } = a1 T{x1[n]} + a2 T{x2[n]}
to check linearity of the above system ;
let y1[n] =x1 [n-2] - 2 x1[n-8]
y2[n] =x2 [n-2] - 2 x2[n-8]
say a1 and a2 are constants;
form the property;
{a1 {x1 [n-2] - 2 x1[n-8] } +a2 {x2 [n-2] - 2 x2[n-8]} = a1 {x1 [n-2]} - a1 {2 x1[n-8] } +a2 {x2 [n-2]} - a2 {2 x2[n-8]}
which holds good. hence the given system is linear. (Solution : the system property Hold)
(4) System is non causal. For to become a system causal, the system should depends on either present values (time) or the past values (time). (Solution : system do no hold)
(5) The above system is stable; since for every bounded input we get bounded output. (BIBO system)
(Solution : The system hold)
(c) y[n] = nx[n]
(1) The above system is Memoryless, as the system only depends on the present values.
(2) The system is time invarient.
because for change in input ie [n-n0], will also change output as [n-n0]. hence the system is time invarient.
(3) checking Linear property ;
take y1[n] = n x1[n]
y2[n] = n x2[n]
therefore n{x1 [n] + x2[n] } = n x1[n] + n x2[n]
hence the system is linear.
(4) The above system is causal, because output depends on the present value of the input.
(The system is said to be causal when the output depends on present or past values of the input)
(5) the system is stable, ie for Bounded input we get Bounded output.
( system is stable for BIBO < infinity)...
Hope this will help you...further any clarification please comment below...