GATE ECE
Signals and Systems
Fourier Transform
Previous Years Questions

## Marks 1

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 energy of the signal x(t) =$${{\sin (4\pi t)} \over {4\pi t}}$$ is ___________.
Which one of the following is an eight function of the class of all continuous-time, linear, time- invariant systems u(t) denotes the unit-step functi...
Let x(t) $$\leftrightarrow$$ X($$(j\omega )$$ BE Fourier transform pair. The Fourier Transform of the signal x(5t - 3) in terms of X($$(j\omega )$$ ...
The Fourier transform of a conjugate symmetric function is always
The Fourier transform F $$\left\{ {{e^{ - t}}u(t)} \right\}$$ is equal to $${1 \over {1 + j2\pi f}}$$. Therefore, $$F\left\{ {{1 \over {1 + j2\pi t}}}... If a signal f(t) has energy E, the energy of the signal f(2t) is equal to The Fourier Transform of the signal$$x(t) = {e^{ - 3{t^2}}}$$is of the following form, where A and B are constants: A modulated signal is given by s(t)=$${e^{ - at}}$$cos$$\left[ {({\omega _c} + \Delta \omega )t} \right]$$u (t), where a,$${\omega _c}$$and$${\...
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 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 $$g(t) = \left( {\int\limits_{ - \infty }^\infty {g(t){e^{ - j\... The Fourier transform of a real valued time signal has ## Marks 2 The complex envelope of the bandpass signal$$x(t)\, = \, - \sqrt 2 \left( {{{\sin (\pi t/5)} \over {\pi t/5}}} \right)\sin \left( {\pi t - {\pi \ove...
The value of the integral $$\int_{ - \infty }^\infty {12\,\cos (2\pi )\,{{\sin (4\pi t)} \over {4\pi t}}\,dt\,}$$ is
The value of the integral $$\int\limits_{ - \infty }^\infty {\sin \,{c^2}}$$ (5t) dt is
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 $$\o... The Fourier transform of a signal h(t) is$$H(j\omega )$$=(2 cos$$\omega $$) (sin 2$$\omega $$) /$$\omega $$. The value of h(0) is The signal x(t) is described by$$x\left( t \right) = \left\{ {\matrix{ {1\,\,\,for\,\, - 1 \le t \le + 1} \cr {0\,\,\,\,\,\,\,\,\,\,\,\,\,\,...
For a signal x(t) the Fourier transform is X(f). Then the inverse Fourier transform of X(3f+2) is given by
Let x(t) and y(t) (with Fourier transforms X(f) and Y(f) respectively) be related as shown in Fig.(1) & (2). Then Y(f) is ...
The Hilbert transform of $$\left[ {\cos \,{\omega _1}t + \,\sin {\omega _2}t\,} \right]$$ is
The power spectral density of a deterministic signal is given by $${\left[ {\sin (f)/f} \right]^2}$$, where 'f' is frequency. The autocorrelation func...
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 transf...
The autocorrelation function of an energy signal has
If G(f) represents the Fourier transform of a signal g (t) which is real and odd symmetric in time, then

## Marks 5

The Fourier transform $$G(\omega )$$ of the signal g(t) in Fig.(1) is given as $$G(\omega ) = {1 \over {{\omega ^2}}}({e^{j\omega }} - j\omega {e^{j\... Consider a rectangular pulse g(t) existing between$$t = \, - {T \over 2}\,and\,{T \over 2}$$. Find and sketch the pulse obtained by convolving g(t) w... A signal v(t)= [1+ m(t) ] cos$$({\omega _c}t)$$is detected using a square law detector, having the characteristic$${v_0}(t) = {v^2}(t). If the Fo...
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