MCQ+1 / -0.33MediumZ-TransformDiscrete-Time SignalsSignals & Systems
Signals & Systems → Discrete-Time Signals → Z-Transform
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Question
Let H(z) be the z-transform of a real-valued discrete-time signal h[n]. If P(z)=H(z)H(z1) has a zero at z=21+21j, and P(z) has a total of four zeros, which one of the following plots represents all the zeros correctly?
Correct answer
(D) [figure]
Solution
Given P(z)=H(z)H(z1) and h[n] is a real-valued signal.
1.Conjugate Symmetry Property: Since h[n] is real, if z0 is a zero of H(z), then its complex conjugate z0∗ must also be a zero of H(z).
2.Reciprocal Property: If z0 is a zero of H(z), then z01 is a zero of H(z1).
Given that z1=21+21j is a zero of P(z). This means z1 is a zero of either H(z) or H(z1).Case 1: z1=21+21j is a zero of H(z)
By conjugate symmetry, z1∗=21−21j is also a zero of H(z).
The corresponding zeros of H(z1) are:
z11=0.5+0.5j1=1+j2=22(1−j)=1−j
z1∗1=0.5−0.5j1=1−j2=22(1+j)=1+j
Case 2: z1=21+21j is a zero of H(z1)
Then z11=1−j is a zero of H(z).
By conjugate symmetry, (1−j)∗=1+j is also a zero of H(z).
The corresponding zeros of H(z1) are 1−j1=0.5+0.5j and 1+j1=0.5−0.5j.
In both cases, the complete set of four zeros for P(z) is {0.5±0.5j,1±j}.Plot (D) correctly represents these four zeros in the complex z-plane: two zeros inside the unit circle at 0.5±0.5j (magnitude 1/2) and two zeros outside the unit circle at 1±j (magnitude 2).
