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Determine the output y(n) of a LTI system with impulse response h(n)=anu(n),|a|<1with the input sequence x(n)=u(n).
- a)(1-a(n+1))/(1-a)
- b)(1-a(n-1))/(1-a)
- c)(1+a(n+1))/(1+a)
- d)None of the mentioned
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Determine the output y(n) of a LTI system with impulse response h(n)=a...
Explanation: Now fold the signal x(n) and shift it by one unit at a time and sum as follows
y(0)=x(0)h(0)=1
y(1)=h(0)x(1)+h(1)x(0)=1.1+a.1=1+a
y(2)=h(0)x(2)+h(1)x(1)+h(2)x(0)=1.1+a.1+a2.1=1+a+a2
Similarly, y(n)=1+a+a2+….an= (1-a(n+1))/(1-a).
y(0)=x(0)h(0)=1
y(1)=h(0)x(1)+h(1)x(0)=1.1+a.1=1+a
y(2)=h(0)x(2)+h(1)x(1)+h(2)x(0)=1.1+a.1+a2.1=1+a+a2
Similarly, y(n)=1+a+a2+….an= (1-a(n+1))/(1-a).
Most Upvoted Answer
Determine the output y(n) of a LTI system with impulse response h(n)=a...
Since the system is LTI, we can use the convolution sum to find the output:
y(n) = ∑k=-∞∞ x(k)h(n-k)
where x(k) is the input signal. Since the system has impulse response h(n) = a nu(n), we have:
h(n-k) = a nu(n-k)
Substituting this into the convolution sum, we get:
y(n) = ∑k=-∞∞ x(k)h(n-k)
= ∑k=-∞∞ x(k)a nu(n-k)
= a ∑k=-∞∞ x(k) nu(n-k)
Note that the sum is only non-zero when n-k ≥ 0, or equivalently, when k ≤ n. Therefore, we can simplify the sum as:
y(n) = a ∑k=-∞n x(k)
This means that the output at time n is equal to the sum of the input up to time n, multiplied by a constant factor a.
Therefore, the output y(n) of the LTI system with impulse response h(n) = a nu(n) is equal to the sum of the input x(k) up to time n, multiplied by a constant factor a:
y(n) = a ∑k=-∞n x(k)
y(n) = ∑k=-∞∞ x(k)h(n-k)
where x(k) is the input signal. Since the system has impulse response h(n) = a nu(n), we have:
h(n-k) = a nu(n-k)
Substituting this into the convolution sum, we get:
y(n) = ∑k=-∞∞ x(k)h(n-k)
= ∑k=-∞∞ x(k)a nu(n-k)
= a ∑k=-∞∞ x(k) nu(n-k)
Note that the sum is only non-zero when n-k ≥ 0, or equivalently, when k ≤ n. Therefore, we can simplify the sum as:
y(n) = a ∑k=-∞n x(k)
This means that the output at time n is equal to the sum of the input up to time n, multiplied by a constant factor a.
Therefore, the output y(n) of the LTI system with impulse response h(n) = a nu(n) is equal to the sum of the input x(k) up to time n, multiplied by a constant factor a:
y(n) = a ∑k=-∞n x(k)
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Determine the output y(n) of a LTI system with impulse response h(n)=a...
Determine the unit step response of LTI system with impulse h(n)=a^n u(n),|a|<1>
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