The discrete Fourier series representation for the following sequence:

 is

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  1.  and 
  2.  and 
  3.  and 
  4.  and 

Answer (Detailed Solution Below)

Option 4 :  and 
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ST 1: UPSC ESE (IES) Civil - Building Materials
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Detailed Solution

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Concept:

The Fourier series representation of the discrete-time periodic sequence is given by:

ak = Fourier series coefficient periodic by N.

ω0 = Fundamental frequency.

Application:

Given sequence is: 

The fundamental frequency of the sequence 

We know that, 

Now, we can rewrite the given sequence as

We can write 

Now, x(n) becomes

 and 

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