Fourier Transforms Frequency Response And Correlation Practice Questions

A continuous time signal $x(t)=2 \cos (8 \pi t+\pi / 3)$ is sampled at a rate of $15 \mathrm{~Hz}$ . The sampled signal $x_{s}(t)$ when passed through an LTI system with impulse response $h(t)=\left(\frac{\sin 2 \pi t}{\pi t}\right) \cos (38 \pi t-\pi / 2)$ produces an output $x_{0}(t)$ . The expression for $x_{0}(t)$ is _____

Consider two continuous time signals $x(t)$ and $y(t)$ as shown below If $X(f)$ denotes the Fourier transform of $x(t)$ , then the Fourier transform of $y(t)$ is ____

Let $x_{1}(t)=u(t+1.5)-u(t-1.5)$ and $x_{2}(t)$ is shown in the figure below. For $y(t)=x_{1}(t) * x_{2}(t)$ , the $\int_{-\infty }^{\infty } y(t) d t$ is ___. (rounded off to the nearest integer).

The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ is

The exponential Fourier series representation of a continuous-time periodic signal x(t) is defined as $x\left ( t \right )=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\omega _{0}t}$ where $\omega _0$ is the fundamental angular frequency of x(t) and the coefficients of the series are $a_{k}$ . The following information is given about $x(t)$ and $a_{k}$ . I. x(t) is real and even, having a fundamental period of 6 II. The average value of x(t) is 2 III.

The average power of the signal x(t) (rounded off to one decimal place) is ____________

$X(\omega )$ is the Fourier transform of $x(t)$ shown below. The value of $\int_{-\infty }^{\infty }|X(\omega )|^2 d\omega$ (rounded off to two decimal places) is ______.

The input 4sinc(2t) is fed to a Hilbert transformer to obtain y(t), as shown in the figure below: Here $sinc(x)=\frac{sin(\pi x)}{\pi x}$ . The value (accurate to two decimal places) of $\int_{-\infty }^{\infty } |y(t)|^{2}dt$ is _______.

Consider an LTI system with magnitude response

And phase response Arg {H(f)}=-2f If the input to the system is $x(t)=8cos(20\pi t+\frac{\pi }{4})+16sin(40\pi t+\frac{\pi }{8})+24cos(80\pi t+\frac{\pi }{16})$ Then the average power of the output signal y(t) is _________.

A continuous time signal $x(t)=4cos(200\pi t)+8cos(400 \pi t)$ , where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response

Let y(t) be the output of this filter. The maximum value of |y(t)| is ________.

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

The energy of the signal $x(t)=\frac{sin(4\pi t)}{4\pi t}$ is ________

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form $\sum_{k=1}^{3}a_{k}cos(k\omega _{0}t)$ , where $a_{k}\neq 0, \omega _{0}\neq 0$ . The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

Consider the function $g(t)=e^{-t}sin(2\pi t)u(t)$ where u(t) is the unit step function. The area under g(t) is _______.

The value of the integral $\int_{-\infty }^{\infty }sinc^{2}(5t)dt$ is ____.

A continuous, linear time-invariant filter has an impulse response h(t) described by

When a constant input of values 5 is applied to this filter, the steady state output is_____.

A real-valued signal x(t) limited to the frequency band $|f|\leq \frac{W}{2}$ is passed through a linear time invariant system whose frequency response is

The output of the system 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 $\omega$ . If $y(t)=\int_{-\infty }^{t}g(\tau)d\tau$ , then $\int_{-\infty }^{+\infty }y(t)dt$ is

The phase response of a passband waveform at the receiver is given by $\varphi (f)=-2\pi \alpha (f-f_{c})-2\pi \beta f_{c}$ where $f_{c}$ is the centre frequency, and $\alpha$ and $\beta$ are positive constants. The actual signal propagation delay from the transmitter to receiver is

The Fourier transform of a signal h(t) is H(j $\omega$ ) = (2cos $\omega$ )(sin2 $\omega$ ) / $\omega$ . The value of h(0) is