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The Fourier transform of x*[-n] is

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UGC NET Paper 2: Electronic Science Nov 2020 Official Paper
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  1. X*(e-jω)
  2. X*(ejω)
  3. X*(-e-jω)
  4. X*(-ejω)

Answer (Detailed Solution Below)

Option 2 : X*(ejω)
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Detailed Solution

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Concept:

Fourier Transform:

Some properties of fourier transform:

Properties

X(f) form

X(ω) form

Time scaling x(at)

Time reversal x(-t)

X(-f)

X(-ω)

Time shift x(t ± t0)

Frequency Modulation

X(f ± f0)

X(ω ± ω0)

Differentiation in time

Conjugation x[n]

X*(e-j2πf)

X*(e-jω)

Time reversal x[-n]

X(e-j2πf)

X(e-jω)

Duality

x(t) ↔ X(f)

x(t) ↔ -X(f)

x(t) ↔ X(ω)

x(t) ↔2π X(-ω)

 

Analysis:

We know that:

Then,

 

Option (2) correct.

More information:

  • Fourier transform of the real signal is always even conjugate in nature.
  • F.T [Real & even signal] = purely real and even.
  • F.T [Real & odd signal] = purely imaginary and odd.
  • Shifting in the time domain only changes the phase spectrum of the signal.

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Q1.The Fourier Transform of an real and even function results in:
Q2.Sinc pulse shaping is derived from the Fourier Transform of a _______ function.
Q3.The Laplace transform of x(t) is \(\sqrt{\frac{2}{s-3}}\).  Then Laplace transform of e−6tx(t) is :
Q4.Which one of the following is the correct relation ? (a > 0) 
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Q6.In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics I. Continuous, aperiodic a. Continuous, aperiodic II. Continuous, periodic b. Continuous, periodic III. Discrete, aperiodic c. Discrete, aperiodic IV. Discrete, periodic d. Discrete, periodic
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Q8.Let 𝑚(𝑡) be a strictly band-limited signal with bandwidth 𝐵 and energy 𝐸. Assuming 𝜔0 = 10𝐵, the energy in the signal 𝑚(𝑡) cos 𝜔0𝑡 is 
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Q10.Which symmetry does the Fourier transform of a real valued time signal contain?

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Q2.Sinc pulse shaping is derived from the Fourier Transform of a _______ function.
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Q6.A continuous-time signal x(t) is defined as : \(x(\mathrm{t})=\left\{\begin{array}{cc} 5 & -2 \leq \mathrm{t} \leq 2 \\ 0 & \text { otherwise } \end{array}\right.\) The energy of signal x(t) is : 
Q7.If Z(un) = f(z), then Z(a − nun) (where Z denotes z-transform) equals to :
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Q9.Which one of the following is the correct relation ? (a > 0) 
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