GATE EE 2023 Set 1 — Question 50
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Signals & Systems → Fourier Analysis & Sampling → CTFT & DTFT
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Question
The discrete-time Fourier transform of a signal is . Consider that is a periodic signal of period such that
Note that . The magnitude of the Fourier series coefficient is __________$ (Round off to 3 decimal places).
Note that . The magnitude of the Fourier series coefficient is __________$ (Round off to 3 decimal places).
Correct answer
0.037 to 0.039
Solution
Given the discrete-time Fourier transform .
We know that .
Substitute this into :
To find the inverse discrete-time Fourier transform , we use the property that :
.The periodic signal has a period and is defined as:
for
for Let's find the values of for one period ():
The Fourier series coefficients for a periodic discrete-time signal with period are given by:
We need to find the magnitude of for :
Substitute the values of :
We can simplify the exponential terms:
So,
Using Euler's formula :
Calculate the values:
Substitute these values into the expression for :
Combine real and imaginary parts:
Real part:
Imaginary part: So, Now, calculate the magnitude :
Rounding off to 3 decimal places, .
We know that .
Substitute this into :
To find the inverse discrete-time Fourier transform , we use the property that :
.The periodic signal has a period and is defined as:
for
for Let's find the values of for one period ():
The Fourier series coefficients for a periodic discrete-time signal with period are given by:
We need to find the magnitude of for :
Substitute the values of :
We can simplify the exponential terms:
So,
Using Euler's formula :
Calculate the values:
Substitute these values into the expression for :
Combine real and imaginary parts:
Real part:
Imaginary part: So, Now, calculate the magnitude :
Rounding off to 3 decimal places, .
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