Fourier Transforms Frequency Response And Correlation Practice Questions

The signal x(t) is described by

Two of the angular frequencies at which its Fourier transform becomes zero are

The impulse response h(t) of a linear time invariant continuous time system is given by h(t)=exp(-2t)u(t), where u(t) denotes the unit step function The output of this system, to the sinusoidal input x(t)=2cos(2t) for all time t, is

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

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

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

The output y(t) of a linear time invariant system is related to its input x(t) by the following equations: $y(t)=0.5x(t-t_{d}+T)+x(t-t_{d})$ $+0.5x(t-t_{d}+T)$ The filter transfer function H( $\omega$ ) of such a system is given by

A rectangular pulse train s(t) as shown in the figure is convolved with the signal $cos^{2}$ (4p × 1 $10^{3}$ t). The convolved signal will be a

The Fourier transform of a conjugate symmetric function is always

Let x(t) and y(t) with Fourier transforms X(f) and Y(f) respectively be related as shown in Fig. Then Y(f) is

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 k $\Omega$ and C = 1.0 $\mu$ F. Let $t_{g}(f)$ be the group delay function of the given RC-LPF and $f_{2}$ = 100 Hz. Then $t_{g}(f_{2})$ in ms, is

A linear phase channel with phase delay $T_{P}$ and group delay $T_{g}$ must have

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