Fourier Transform

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=$$\mathop f\limits_{ - \infty }^\infty $$ f(t) \exp (-j \omega t) d t$ exists. Which one of the following statements is always TRUE...

The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$

The Fourier transform X(j$$\omega$$) of the signal $$x(t) = {t \over {{{(1 + {t^2})}^2}}}$$ is ____________.

A continuous time signal x(t) = $$4\cos (200\pi t)$$ + $$8\cos(400\pi t)$$, where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response $$h(t) = \left\{ {{{2\sin (300\pi t)} \ove...

If the signal x(t) = $${{\sin (t)} \over {\pi t}}*{{\sin (t)} \over {\pi t}}$$ with * denoting the convolution operation, then x(t) is equal to

The value of the integral $$\int_{ - \infty }^\infty {12\,\cos (2\pi )\,{{\sin (4\pi t)} \over {4\pi t}}\,dt\,} $$ is

The complex envelope of the bandpass signal $$x(t)\, = \, - \sqrt 2 \left( {{{\sin (\pi t/5)} \over {\pi t/5}}} \right)\sin \left( {\pi t - {\pi \over 4}} \right),$$ centered about f = $${1 \over {2\,}}\,Hz,$$ is

A real - values signal x(t) limited to the frequency band $$\left| f \right| \le {W \over 2}$$ is passed through a linear time invariant system whose frequency response is $$H(f) = \left\{ {\matrix{ {{e^{ - j4\pi f}},} &...

For a function g(t), it is given that $$\int_{ - \infty }^\infty {g(t){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$ for any real value $$\omega $$. If y(t)=$$\int_{ - \infty }^t {g(\tau )d\tau ,\,then\,\int_{ -...

The value of the integral $$\int\limits_{ - \infty }^\infty {\sin \,{c^2}} $$ (5t) dt is

Let g(t) = $${e^{ - \pi {t^2}}}$$, and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is

The Fourier transform of a signal h(t) is $$H(j\omega )$$ =(2 cos $$\omega $$) (sin 2$$\omega $$) / $$\omega $$. The value of h(0) is

The impulse response h(t) of a linear time invariant system is given by h(t) = $${e^{ - 2t}}u(t),$$ where u(t) denotes the unit step function. The frequency response H(ω) of the system in terms of angular frequency 'ω' i...

The signal x(t) is described by $$x\left( t \right) = \left\{ {\matrix{ {1\,\,\,for\,\, - 1 \le t \le + 1} \cr {0\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise} \cr } } \right.$$ Two of the angular frequencies at which its Fourie...

The 3 - dB bandwidth of the low - pass signal $${e^{ - 1}}$$ u(t), where u(t) is the unit step function, is given by

The frequency response of a linear, time-invariant system is given by H(f) = $${5 \over {1 + j10\pi f}}$$. The step response of the system is

Let x(t) $$ \leftrightarrow $$ X($$(j\omega )$$ BE Fourier transform pair. The Fourier Transform of the signal x(5t - 3) in terms of X($$(j\omega )$$ is given as

The output y(t) of a linear time invariant system is related to its input x(t) by the following equation: y(t) = 0.5 x $$(t - {t_d} + T) + \,x\,(t - {t_d}) + 0.5\,x(t - {t_d} - T)$$. The filter transfer function $$H(\ome...

Match the following and choose the correct combination. GROUP 1 E- continuous and aperiodic signal F- continuous and periodic signal G- discrete and aperiodic signal H- discrete and periodic signal Group 2 1- Fourier rep...

For a signal x(t) the Fourier transform is X(f). Then the inverse Fourier transform of X(3f+2) is given by

The Fourier transform of a conjugate symmetric function is always

Let x(t) be the input to a linear, time-invariant system. The required output is 4x(t-2). The transfer function of the system should be

The system under consideration is an RC low -pass filter (RC-LPF) with R = 1.0 $$k\Omega $$ and C = 1.0 $$\mu F$$. Let H(t) denote the frequency response of the RC-LPF. Let $${f_1}$$ be the highest frequency such that $$...

Let x(t) be the input to a linear, time-invariant system. The required output is 4x(t - 2). The transfer function of the system should be

The Fourier transform F $$\left\{ {{e^{ - t}}u(t)} \right\}$$ is equal to $${1 \over {1 + j2\pi f}}$$. Therefore, $$F\left\{ {{1 \over {1 + j2\pi t}}} \right\}$$ is

The Fourier Transform of the signal $$x(t) = {e^{ - 3{t^2}}}$$ is of the following form, where A and B are constants:

The Hilbert transform of $$\left[ {\cos \,{\omega _1}t + \,\sin {\omega _2}t\,} \right]$$ is

A signal x(t) has a Fourier transform X ($$\omega $$). If x(t) is a real and odd function of t, then X($$\omega $$) is

The amplitude spectrum of a Gaussian pulse is

The Fourier transform of a voltage of a voltage signal x(t) is X(f). The unit of |X(f)| is

The amplitude spectrum of a Gaussian pulse is

Consider a rectangular pulse g(t) existing between $$t = \, - {T \over 2}\,and\,{T \over 2}$$. Find and sketch the pulse obtained by convolving g(t) with itself. The Fourier transform of g(t) is a sinc function. Write do...

The Fourier transform of a function x(t) is X(f). The Fourier transform of $${{dx(t)} \over {dt}}$$ will be

The function f(t) has the Fourier Transform g($$\omega $$). The Fourier Transform of $$$g(t) = \left( {\int\limits_{ - \infty }^\infty {g(t){e^{ - j\omega t}}} } \right)\,is$$$

If the Fourier Transfrom of a deterministic signal g(t) is G (f), then Item-1 (1) The Fourier transform of g (t - 2) is (2) The Fourier transform of g (t/2) is Item - 2 (A) G(f) $$e^{-j\left(4\mathrm{πf}\right)}$$ (B) G(...

The power spectral density of a deterministic signal is given by $${\left[ {\sin (f)/f} \right]^2}$$, where 'f' is frequency. The autocorrelation function of this signal in the time domain is

The Fourier transform of a real valued time signal has

A signal v(t)= [1+ m(t) ] cos $$({\omega _c}t)$$ is detected using a square law detector, having the characteristic $${v_0}(t) = {v^2}(t)$$. If the Fourier transform of m(t) is constant, $${M_0}$$, extending from - $${f_...

Match each of the items, A, B and C, with an appropriate item from 1, 2, 3, 4 and 5 A. Fourier transform of a Gaussian function B. Convolution of a rectangular pulse with itself C. Current through an inductor for a step...

A signal, f(t) = $${e^{ - at}}$$ u(t), where u(t) is the unit step function, is applied to the input of a low-pass filter having $$\left| {H(\omega )} \right| = {b \over {\sqrt {{\omega ^2} + {b^2}} }}$$. Calculate the v...

If G(f) represents the Fourier transform of a signal g (t) which is real and odd symmetric in time, then

A signal x(t) = $$\exp ( - 2\pi Bt)\,u(t)$$ is the input to an ideal low pass filter with bandwidth B Hz. The output is denoted by y(t). Evaluate $$\int\limits_{ - \infty }^\infty {{{[y(t) - x(t)]}^{2\,}}dt} $$.