Test: Inverse Systems And Deconvolution - Electronics and Communication Engineering (ECE) MCQ

Explanation: If we know the output of a system y(n) of a system and if we can determine the input x(n) of the system uniquely, then the system is said to be invertible. That is there should be one-to-one correspondence between the input and output signals.

Explanation: . If h(n) is the impulse response of an LTI system T and h1(n) is the impulse response of the inverse system T-1, then we know that h(n)*h1(n)=δ(n)=> [h(n)*h1(n)]*x(n)=x(n).

Explanation: Given impulse response is h(n)=(1/2)nu(n)
The system function corresponding to h(n) is
H(z)=1/(1-1/2 z^-1 ) ROC:|z|>1/2
This system is both stable and causal. Since H(z) is all pole system, its inverse is FIR and is given by the system function
HI(z)= 1- 1/2 z^-1
Hence its impulse response is δ(n)-1/2 δ(n-1).

Explanation: The system function of given system is H(z)= 1- 1/2 z^-1
The inverse of the system has a system function as H(z)= 1/(1-1/2 z^-1 )
Thus it has a zero at origin and a pole at z=1/2.So, two possible cases are |z|>1/2 and |z|<1/2
So, h(n)= (1/2)ⁿu(n) for causal and stable(|z|>1/2)
and h(n)= -(1/2)ⁿu(-n-1) for anti causal and unstable for |z|<1/2.

Explanation: Given h(n)= δ(n)-aδ(n-1)
Since h(0)=1, h(1)=-a and h(n)=0 for n≥a, we have
hI(0)=1/h(0)=1.
and
hI(n)=-ahI(n-1) for n≥1
Consequently, hI(1)=a, hI(2)=a²,….hI(n)=aⁿ
Which corresponds to a causal IIR system as expected.

Explanation: Given H(z)=6+z^-1-z^-2
By factoring the system function we find the zeros for the system.
The zeros of the given system are at z=-1/2,1/3
So, the system is minimum phase.

Explanation: Given H(z)= 1-z^-1-z^-2
By factoring the system function we find the zeros for the system.
The zeros of the given system are at z=-2,3
So, the system is maximum phase.

Explanation: Given H(z)= 1-5/2z^-1-3/2z^-2
By factoring the system function we find the zeros for the system.
The zeros of the given system are at z=-1/2, 3
So, the system is mixed phase.

Explanation: For an IIR filter whose system function is defined as H(z)=(B(z))/(A(z)) to be said a minimum phase,
then both the poles and zeros of the system should fall inside the unit circle.

Explanation: For an IIR filter whose system function is defined as H(z)=(B(z))/(A(z)) to be said a mixed phase and if the system is stable and causal, then the poles are inside the unit circle and some, but not all of the zeros are outside the unit circle.