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Resolve the sequence into a sum of weighted impulse sequences
Explanation: We know that, x(n)δ(nk)=x(k)δ(nk)
x(1)=2=2δ(n+1)
x(0)=4=4δ(n)
x(2)=3=3δ(n2)
Therefore, x(n)= 2δ(n+1)+4δ(n)+3δ(n2).
The formulathat gives the response y(n) of the LTI system as the function of the input signal x(n) and the unit sample response h(n) is known as:
Explanation: The input x(n) is convoluted with the impulse response h(n) to yield the output y(n).As we are summing the different values, we call it as Convolution sum.
What is the order of the four operations that are needed to be done on h(k) in order to convolute x(k) and h(k)?
Step1:Folding
Step2:Multiplicaton with x(k)
Step3:Shifting
Step4:Summation
Explanation: First the signal h(k) is folded to get h(k). Then it is shifted by n to get h(nk). Then it is multiplied by x(k) and then summed over ∞ to ∞.
The impulse response of a LTI system is h(n)={1,1,1}. What is the response of the signal to the input x(n)={1,2,3}?
Explanation: Let y(n)=x(n)*h(n)(‘*’ symbol indicates convolution symbol)
From the formula of convolution we get,
y(0)=x(0)h(0)=1.1=1
y(1)=x(0)h(1)+x(1)h(0)=1.1+2.1=3
y(2)=x(0)h(2)+x(1)h(1)+x(2)h(0)=1.1+2.1+3.1=6
y(3)=x(1)h(2)+x(2)h(1)=2.1+3.1=5
y(4)=x(2)h(2)=3.1=3
Therefore, y(n)=x(n)*h(n)={1,3,6,5,3}.
Determine the output y(n) of a LTI system with impulse response h(n)=a^{n}u(n),a<1with the input sequence x(n)=u(n).
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+a^{2}.1=1+a+a^{2}
Similarly, y(n)=1+a+a^{2}+….a^{n}= (1a^{(n+1)})/(1a).
Explanation: According to the properties of convolution, Convolution of three signals obeys Associative property.
Determine the impulse response for the cascade of two LTI systems having impulse responses h1(n)=(1/2)^{2} u(n) and h2(n)= (1/4)^{2} u(n).
Explanation: Let h2(n) be shifted and folded.
so, h(k)=h1(n)*h2(n)=
For k<0, h1(n)= h2(n)=0 since the unit step function is defined only on the right hand side.
Explanation: According to the properties of the convolution, convolution exhibits distributive property.
An LTI system is said to be causal if and only if
Explanation: Let us consider a LTI system having an output at time n=n0given by the convolution formula
=(h(0)x(n0)+h(1)x(n01)+h(2)x(n02)+….)+(h(1)x(n0+1)+h(2)x(n0+2)+…)
As per the definition of the causality, the output should depend only on the present and past values of the input. So, the coefficients of the terms x(n0+1), x(n0+2)…. should be equal to zero.
that is, h(n)=0 for n<0 .
Explanation:
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