GATE EC 2020 Set 1 — Question 63
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Signals & Systems → LTI Systems → Magnitude & Phase Response
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Question
The transfer function of a stable discrete-time LTI system is where and are real numbers. The value of (rounded off to one decimal place) with , for which the magnitude response of the system is constant over all frequencies, is _________.
Correct answer
-2
Solution
For the magnitude response of a discrete-time LTI system to be constant over all frequencies, the system must be an all-pass filter. An all-pass filter has poles and zeros that are reciprocals of each other (and conjugates if complex).The given transfer function is .
From this, we can identify the pole of the system as .
For an all-pass filter, the zero () must be the reciprocal of the conjugate of the pole ().
So, .
Since the pole is real, its conjugate .
Therefore, the zero .Comparing this with the given transfer function, the zero is at . Thus, .We also need to check the condition . For , we have , which is indeed greater than 1.Thus, the value of for which the magnitude response of the system is constant over all frequencies is .The final answer is .
From this, we can identify the pole of the system as .
For an all-pass filter, the zero () must be the reciprocal of the conjugate of the pole ().
So, .
Since the pole is real, its conjugate .
Therefore, the zero .Comparing this with the given transfer function, the zero is at . Thus, .We also need to check the condition . For , we have , which is indeed greater than 1.Thus, the value of for which the magnitude response of the system is constant over all frequencies is .The final answer is .
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