What is the expression for Fourier transform of a periodic signal x(t) with exponential Fourier series coefficient as $C_{n}$ ?

\infty \sum n = - \infty C_{n} δ (ω - n ω_{s})

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π \infty \sum n = - \infty C_{n} δ (ω - n ω_{s})

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2 π \infty \sum n = - \infty C_{n} δ (ω - n ω_{s})

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\frac{1}{2 π} \infty \sum n = - \infty C_{n} δ (ω - n ω_{s})

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Solution

The correct option is C $2 π \infty \sum n = - \infty C_{n} δ (ω - n ω_{s})$
$∵$ Fourier series of a periodic signal x(t) is given as

$x (t) = \infty \sum n = - \infty C_{n} e^{- j n ω_{s} t}$ ...(i)

$A e^{- j n ω_{s} t} F . T . ⟶ 2 π A δ (ω - n ω_{s})$ ....(ii)

for equation (i) and (ii),

$x (t) F . T . ⟶ 2 π \infty \sum n = - \infty C_{n} δ (ω - n ω_{s})$

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Similar questions

Q. Consider the signal,

$x (t) = \infty \sum m = - \infty δ (t - \frac{m}{3}) + δ (t - \frac{2 m}{3})$
Where $C_{n}$ is the exponential Fourier series coefficient of signal x(t). Which of the following is/are correct?

Q. Fourier transform of a periodic signal is given as,

X (j ω) = j δ (ω - \frac{π}{3}) + 2 δ (ω - \frac{π}{7})

The Fourier series coefficients of

C_{7} = j

___.

Q. A periodic signal with period

T_{0}

is real. The signal x(t) has even and odd part shown as

x_{e} (t)

and

x_{0} (t)

, also

C_{k}

is the

k^{t h}

exponential Fourier series coefficient of x(t) .
If

C_{k} = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} (\frac{10}{3 | k |} + \frac{3}{k} j) k \neq 0 a n d | K | < 3 10, K = 0 \end{matrix}

Then which of the following is correct?

Q. Which of the following is true for the exponential Fourier series coefficient of a real signal

Q. Given that,

\int_{T}^{2 T} x (t) . d t = 2

x (t), y (t) = \frac{d}{d x} x (t)

signals are periodic signals of time period T = 8, with Fourier series coefficient

a_{k} a n d b_{k}

respectively.
Which one of the following expressions are correct ?

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What is the expression for Fourier transform of a periodic signal x(t) with exponential Fourier series coefficient as Cn?

The correct option is C 2π∞∑n=−∞Cnδ(ω−nωs)∵ Fourier series of a periodic signal x(t) is given as x(t)=∞∑n=−∞Cne−jnωst ...(i) Ae−jnωstF.T.⟶2πAδ(ω−nωs) ....(ii) for equation (i) and (ii), x(t)F.T.⟶2π∞∑n=−∞Cnδ(ω−nωs)

What is the expression for Fourier transform of a periodic signal x(t) with exponential Fourier series coefficient as $C_{n}$ ?