Consider a continuous-time, real-valued signal  whose Fourier transform  exists.

Which one of the following statements is always TRUE?

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__________.

The relationship between any N-length sequence x[n] and its corresponding N-point discrete Fourier transform X[k] is defined as         

Another sequence y[n] is formed as below

For the sequence x[n]={1,2,1,3}, the value of  Y[0] is __________

Let  be a strictly band-limited signal M bandwidth B and energy E. Assuming  the energy in the signal  is

The Fourier transform  of  is

Note:

In the table shown below, match the signal type with its spectral characteristics.         

Signal Type

(i) Continuous, aperiodic

(ii) Continuous, periodic

(iii) Discrete, aperiodic

(iv) Discrete, periodic

Spectral Characteristics

(a) Continuous, aperiodic

(b) Continuous, periodic

(c) Discrete, aperiodic

(d) Discrete, periodic

Let an input  having discrete time Fourier transform.

 be passed through an LTI system. The frequency response of the LTI system is . The output  of the system is

Let  be passed through an LTI system having impulse response . The output of the system is

Let  and  is shown in the figure below. For , the   is _________. (rounded off to the nearest integer).

Let  be a white Gaussian noise with power spectral density . If  is input to an LTI system with impulse response . The average power of the system output is __________ W. (Rounded off to two decimal place).

For a real signal, which of the following is/are valid power spectral density/densities?

The Fourier transform  of the signal        

 is ________.

The frequency response  of a linear time invariant system has magnitude as shown in the figure.

Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to .

Statement II: For any wide-sense stationary input process with power spectral density , the output power spectral density  obeys   for .        

Which one of the following combinations is true?

For a vector , the 8-point discrete Fourier transform (DFT) is denoted by , where

Here, . If  and , then the value of  is __________(rounded off to one decimal place).        

Consider the signals  and , where  is the unit step sequence. Let  and  be the discrete time Fourier transform of  and , respectively. The value of the integral

(rounded off to one decimal place) is ___________.

Consider two 16-point sequence  and . Let the linear convolution of  and  be denoted by , while  denotes the 16 -point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of  and . The value(s) of  for which  is/are

 is the Fourier transform of  shown below.        

The value of  (rounded off to two decimal places) is _________.

A finite duration discrete-time signal  is obtained by sampling the continuous-time signal  at sampling instants , . The 8 -point discrete Fourier transform (DFT) of  is defined as

Which one of the following statements is true?

 Let be a length-7 discrete time finite impulse response filter, given by

,        ,         ,

,        ,,

And is zero for. A length 3 finite impulse response approximation of has to be obtained

such that

Is minimized where andare the discrete time Fourier transform of and, respectively.

For the filter that minimize, the value of, rounded off to 2 decimal places, is ______

Consider a six-point decimation in time Fast Fourier Transform (FFT) algorithm, for which the signal flow graph corresponding to X[1] is shown in the figure. Let. In the figure, what should be the values of the coefficients in terms of so that X[1] is obtained correctly?

It is desired to find a three-tap causal filter which gives zero signal as an output to an input of the form.

where  and  are arbitrary real numbers, The desired three-tap filter is given by

 and  

What are the values of the filter taps a and b if the output is  for all n, when  is as given above?

The input is fed to a Hilbert transformer to obtain y(t), as shown in the figure below:

Here. The value (accurate to two decimal places) ofis ____________

Let,be 8-point DFT of a sequence

Where

The value (correct to two decimal places) of is ____________.

A continuous time signal  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 ___________ .

Let h[n] be the impulse response of a discrete-time linear time invariant (LTI) filter. The impulse response is given by  .

Let  be the discrete-time Fourier transform (DTFT) of h[n], where is the normalized angular frequency in radians. Given that the value of (in radians) is equal to ___________ .

An LTI system with unit sample response is a

Which one of the following is an eigen function of the class of all continuous-time, linear,

time-invariant systems (u(t) denotes the unit-step function)?

Consider the signal

Ifis the discrete-time Fourier transform of then is equal to______________

The energy of the signal   is ________.

If the signal  * with * denoting the convolution operation, then x(t) is equal to

The Discrete Fourier Transform (DFT) of the 4-point sequence  is

. If  is the 12-point sequence , the value  is ____________.

A continuous-time speech signal  is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of a size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes 20μs for computing each complex multiplication (including multiplications by 1 and -1) and the time required for addition/subtraction is negligible, then the maximum value of N is ____________

Two sequences [a,b,c] and [A,B,C] are related as,

  where .

If another sequence [p,q,r] is derived as,

 ,

Then the relationship between the sequences  and  is

Consider two real sequences with time-origin marked by the bold value .

Let  and  be 4-point DFTs of  and, respectively.

Another sequence  is derived by taking 4-point inverse DFT of . The value of is ______.

The value of the integral  is ________.

A Fourier transform pair is given by  .Where u[n] denotes the unit step sequence. The value of A is _______.

A real-valued signal x(t) limited to the frequency band  is passed through a linear time invariant system whose frequency response is .  The output of the system is

The N-point DFT X of a sequence x[n],  is given by

 , .

Denote this relation as X=DFT(x). For N=4, which one of the following sequences satisfies DFT (DFT(x))=x.

The DFT of a vector  is the vector . Consider the product

.

The DFT of the vector  is a scaled version of

For an N-point FFT algorithm with  which one of the following statements is TRUE?

The first six points of the 8-point DFT of a real valued sequence are 5, , 0, , 0 and . The last two points of the DFT are respectively

The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence {1, 0, 2, 3} is

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

, where u(t) denotes the unit step function.

The frequency response  of this system in terms of angular frequency, is given by 

The impulse response h(t) of a linear time-invariant continuous time system is given by

, where u(t) denotes the unit step function.

The output of this system, to the sinusoidal input  for all time t, is

 is a real-valued periodic sequence with a period N. x[n] and X(k) form N-point Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence is

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

A Hilbert transform is a

A 5-point sequence x[n] is given as 

Let  denote the discrete-time Fourier transform of x[n]. The value ofis

A low-pass filter having a frequency response  does not produce any phase distortion if

Let be Fourier Transform pair. The Fourier Transform of the signal x(5t-3) in terms of is given as

In the system shown below, . In steady-state, the response y (t) will be:

Let,, and  be the Fourier transform of y(n). Then  is

The output y (t) of a linear time invariant system is related to its input x (t) by the following equation: . The filter transfer function of such a system is given by

Match the following and choose the correct combination:

A signal is the input to a linear time-invariant system having a frequency response. If the output of the system is , then the most general form of will be

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

A sequence x[n] has non-zero values as shown in figure.

The sequence         

Will be          

A sequence x[n] has non-zero values as shown in figure.

The Fourier transform of y[2n] will be

The Fourier transform of a conjugate symmetric function is always

A rectangular pulse train s(t) as shown in Figure is convolved with the signal . The convolved signal will be a

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

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 Fourier transform is equal to . Therefore, is

A linear phase channel with phase delay and group delay must have

The Fourier transform G(ω) of the signal g(t) in Figure (a) is given as . Using the information and the time-shifting and time-scaling properties, determine the Fourier transform of signals in Figures (b), (c) and (d).

Figure (a)

Figure (b)

Figure (c)

Figure (d)

The Fourier Transform of the signal  is of the following form, where A and B are constants:

A system has a phase response given by , where  is the angular frequency. The phase delay and group delay at  are respectively given by

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

The Fourier transform of a function x(t) is and x(f). The Fourier transform of  will be

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

 Consider a rectangular pulse g(t) existing between  and . Find and sketch the pulse obtained by convolving g(t) with itself. The Fourier transform of g(t) is a sine function. Write down Fourier transform of the pulse obtained by the above convolution.

The function f(t) has the Fourier Transform . The Fourier Transform  is

If the Fourier Transform of deterministic signal g(t) is G(f), then        

Match the following

(1) The Fourier Transform of  is

(2) The Fourier Transform of  is

The Fourier transform of a real valued time signal has

Consider the following interconnection of the three LTI systems (figure), ,  and  are the impulse responses of these three LTI systems with ,  and  as their respective Fourier transforms. Given that

, and

Find the output y(t)

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