Signals and Systems Assignment no. 4

Fourier Transform

Submission Deadline: 11-05-2015 Q1: Find the Fourier transform of the following signals and sketch the magnitude and phase as a function of frequency:

a) (t 5)b) e(1+ j2)tu(t)c) cos( / t)d) ea t , a > 0

Q2: Find the signal corresponding to the following transforms.

a) 17+ jb) 2(+7)+2(-7)c) 19+ 2

Q3: An LTI system has an impulse response, = !!"(), and output =[!!" !!"](). Using the convolution property, find the input signal. Q4: Consider the following RC circuit:

a) Find the impulse response of the system, its frequency response and sketch the

magnitude spectrum of the frequency response. b) Find the impulse response of the system if the positions of the capacitor and

resistor are interchanged. Sketch the magnitude spectrum of the frequency response

Q5: In the amplitude modulation system,

the input x(t) has the Fourier transform shown below:

For each choice of carrier c(t) in the following list, draw the magnitude and phase of (), the Fourier transform of y(t).

a) c(t) = cos3ctb) c(t) = e j[3ct+ /2]c) c(t) = cos3ct + sin3ct

Q6: Find the energy of the following signals, using Parsevals theorem:

a) x(t) = u(t)u(t 5)b) x(t) = sin( t)

t

Q7: A signal has a Fourier transform

X() = 2 + j4 + 2

2 + j4 +3

Find the transforms of the following signals: a) x(2t +1)b) dx(t)dtc) x(t)(t 1)

Q8: The spectrum of speech signal is essentially zero for all frequencies above 5 kHz. Find the Nyquist sampling rate and sample spacing for such a signal. Q9: Consider the system shown below:

For each input spectrum shown below, identify the correct output spectrum !().

Q10: Compute the discrete-time Fourier transform of the following signals:

a) x[n]= (an sin0n)u[n], a

Q13: An LTI system is characterized by an impulse response:

h[n]=sin n3n

c) Sketch the magnitude of the system transfer function d) Evaluate the output when the input is (1)! cos !!!

Q14: Find the discrete-time sequence x[n] with the transforms given as follows:

a) X(e j ) = j[ 3 ]+[ 23 ]+[

43 ]+ j[

53 ]

b) X(e j ) = 4sin5 + 2cos

c) X(e j ) =1 56 e

j

1+ 112 e j

112 e

j2

Q15: Show that the discrete-time Fourier transform of the signal given as:

ake jk0nk=0

N1

, N0 = 2 DTFT %% 2ak( k0 )k=0

N1