GATE EE 2017 Set 2 — Question 32
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Signals & Systems → LTI Systems → Convolution Integral & Sum
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Question
A cascade system having the impulse responses
h1(n)={↑1,−1} and
h2(n)={↑1,1} is shown in the figure below, where symbol
↑ denotes the time origin.
The input sequence
x(n) for which the cascade system produces an output sequence
y(n)={↑1,2,1,−1,−2,−1} is
Correct answer
(D) x(n) = \1, 2, 2, 1\
Solution
1.The overall impulse response h(n) of the cascade system is the convolution of the individual impulse responses h1(n) and h2(n). 2.Given h1(n)={↑1,−1} and h2(n)={↑1,1}, we can represent them as: h1(n)=δ(n)−δ(n−1) h2(n)=δ(n)+δ(n−1)3.The overall impulse response is:
h(n)=h1(n)∗h2(n)=(δ(n)−δ(n−1))∗(δ(n)+δ(n−1)) h(n)=δ(n)+δ(n−1)−δ(n−1)−δ(n−2)=δ(n)−δ(n−2) In sequence form:
h(n)={↑1,0,−1}4.The output y(n) is the convolution of the input x(n) and the overall impulse response h(n): y(n)=x(n)∗h(n)=x(n)∗(δ(n)−δ(n−2))=x(n)−x(n−2)5.We are given y(n)={↑1,2,1,−1,−2,−1}. We can solve for x(n) iteratively assuming a causal input: y(0)=x(0)−x(−2)⇒1=x(0)−0⇒x(0)=1 y(1)=x(1)−x(−1)⇒2=x(1)−0⇒x(1)=2 y(2)=x(2)−x(0)⇒1=x(2)−1⇒x(2)=2 y(3)=x(3)−x(1)⇒−1=x(3)−2⇒x(3)=1 y(4)=x(4)−x(2)⇒−2=x(4)−2⇒x(4)=0 y(5)=x(5)−x(3)⇒−1=x(5)−1⇒x(5)=06.The resulting input sequence is x(n)={↑1,2,2,1}, which corresponds to option (D). More questions on LTI Systems
The input sequence for which the cascade system produces an output sequence is