4 For each of the following input/output relationships: is the system linear? Ti

Question

4 For each of the following input/output relationships: is the system linear? Time-invariant? Causal? BIBO Stable? Briefly justify your claims. Which of these operations can be written as a convolution? y[n] median {x[n-2], x[n-1], x[n], x[n + 1], z[n + 2)) The mean is a familiar statistic, and is formed by taking the average of a list of data values. The median, in contrast, is computed by ordering, i.e., sorting, a list of data. Specifically, the median is the middle value in the sorted list. The median filter, a special case of an "order statistic" filter, is often used for the smoothing of signals corrupted by impulse noise, and preserves edges. It is implemented by sliding a window of odd length over the input sequence a[n] one sample at a time. At the nth instant, the input samples in the window are rank ordered from the largest to the smallest in values, and the sample at the middle is the median value

Explanation / Answer

Linearity check:

a)

Since it is:

y(n)=max{ x[n],x[n-1],x[n-2]}

It is linear as it is given that it is the max among above three ,

Hence , by using property of homogenity the corresponidng change in input parameter will same in output parameter , hence system is linear.

b)

y(n)=mean{x[n-2],x[n-1],x[n],x[n+1],x[n+2]}

Non - linear

Since mean is average of all the terms hence there will be involvement off all the terms hence if we add some consstant value to the output the corresponding change in input will be 5 times .

c)

y(n)=median{x[n-2],x[n-1],x[n],x[n+1],x[n+2]}

Linear ,as median will be only a single term among the bove terms hence , linear deppendence of output and input .

Time Invariant check :

a)

Time Invariant

Varying time in input may change the corresponding output change as only max term is required.

b)

Time Invariant all the terms gets same chnage with given change in input.

c)

Time invariant .

Causal:

a) Causal as output only depends on past values given in question

b) Not Causal as output depends on both future and past values.

c) Not Causal as output depends on both future and past values.

BIBO stable:

a) BIBO unstable as if one of the value of x(n) goes unbounded the output will unbounded for bounded input .

b) BIBO unstable same as above

c) BIBO stable as always a finite value o we can say bounded output when we find median .

For Convolution It must be Linear Time Invariant system , Hence from above we can say a) and c) can be written as convolution .

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