GATE EE 2016 Set 1 — Question 44
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumFourier Transform PropertiesFourier Analysis & SamplingSignals & Systems
Signals & Systems → Fourier Analysis & Sampling → Fourier Transform Properties
Last updated
Question
Suppose and have the Fourier transforms as shown below.
Correct answer
(C) x₁(t) and x₂(t) are complex but x₁(t)x₂(t) is real
Solution
1.From the given graphs, both and are real-valued functions of .
2.A key property of the Fourier transform is that if is real, then its inverse transform is conjugate symmetric: .
3.For to be a real signal, its Fourier transform must satisfy . Since is real, this would mean , implying must be an even function.
4.Observing the graph of , it is not even (e.g., while ). Thus, is complex. Similarly, is complex.
5.From the graphs, we can see that is the reflection of about the vertical axis: .
6.Taking the inverse Fourier transform of both sides: .
7.Using the conjugate symmetry property of (), we find that .
8.The product of the two signals is .
9.Since the squared magnitude of any complex signal is always a real-valued signal, is real.
10.Therefore, and are complex, but their product is real. This matches option (C).
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE EE question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Fourier Analysis & Sampling
2026 Set 1 Q11Consider the infinite-length, discrete-time sequence , where is an integer.…2026 Set 1 Q12Let be any continuous-time periodic signal with period . It is sampled uniformly with a…2026 Set 1 Q14Consider the following differential equation:…2026 Set 1 Q36A time-limited waveform is specified as follows:…2025 Set 1 Q20Consider a discrete-time linear time-invariant (LTI) system , where…