1 (a)
Sketch the even and odd part of the signal shown in Fig. Q1 (a)

6 M

1 (b)
Check whether the following signals is periodic or not and if periodic find its fundamental period.

i) x(n)=cos(20?n)+sin(50?n)

ii) x(t)=[cos(2?n)]

i) x(n)=cos(20?n)+sin(50?n)

ii) x(t)=[cos(2?n)]

^{2}
6 M

1 (c)
Let x(t) and y(t) as show in Fig Q1(c). Sketch (i) x(t)y(t-1) (ii) x(t)y(-t-1)

8 M

2 (a)
Determine the convolution sum of the given sequences

x(n)={1,

x(n)={1,

**, 3, -3} and h(n)={-2,***-2***, -2}**__2__
4 M

2 (b)
Perform the convolution of the following sequences:

x

x

x

_{1}(t)=e^{-at}: 0?t?Tx

_{2}(t)=1: 0?t?2T
10 M

2 (c)
An LTI system is characterized by an impulse response, \[ h(n)=\eft ( \dfrac {1}{2} \right )^n u(n.) \] Find the response of the system for the input \[ x(n)= \left ( \dfrac {1}{4} \right )^n u(n). \]

6 M

3 (a)
Determine the following LTI system characterized by impulse response is memory, casual and stable.

i) h(n)=2u(n)-2u(n-2)

ii) h(n)=(0.99)

i) h(n)=2u(n)-2u(n-2)

ii) h(n)=(0.99)

^{n}u(n+6).
6 M

3 (b)
Find the natural response of the system described by a differential equation \[ \dfrac {d^2 y(t)}{dt^2}+ 2 \dfrac {dy(t)}{dt}+2y(t)=2x(t), \ with \ y(0)=1, \ and \ \left.\begin{matrix} \dfrac {dy(t)} {dt}\end{matrix}\right|_{t=n}=0 \]

6 M

3 (c)
Find the difference equation description for the system shown in Fig Q3(c).

4 M

3 (d)
By converting the differential equation to integral equation draw the direct form-I and direct form-II implementation for the system as \[ \dfrac {d^3 y(t)}{dt^3}+ \dfrac {d^2 y(t)}{dt^2}+ 2 \dfrac {dy(t)}{dt}= x(t)+ 6 \dfrac{d^2 x(t)}{dt^2} \]

4 M

4 (a)
State and prove the following properties of DTES: i) Modulation ii) Parseval's theorem.

10 M

4 (b)
Find the Fourier series coefficient of the signal x(t) shown in Fig Q4(b) and also draw its spectra.

10 M

5 (a)
Find the DTFT of the following signals:

i) x(n)=a

ii) x(n)=2

i) x(n)=a

^{|n|}; |a|<1ii) x(n)=2

^{n}u(-n)
8 M

5 (b)
Determine the signal x(n) if its DTFT is as shown in Fig. Q5(b).

6 M

5 (c)
Compute the Fourier transform of the signal \[ x(t)= \left\{\begin{matrix} 1+\cos \pi t&;&|t|\le 1 \\0 &; &|t|>1 \end{matrix}\right. \]

6 M

6 (a)
Find the frequency response of the system describe by the impulse response h(t)=?(t)-2e

^{-2t}u(t) and also draw its magnitude and phase spectra.
8 M

6 (b)
Obtain the Fourier transform representation for the periodic signal x(t)=sin w

_{0}t and draw the magnitude and phase.
7 M

6 (c)
A signal x(t)=cos(20?t)+1/4 cos(30?t) is sampled with sampling period ?

_{s}. Find the Nyquist rate.
5 M

7 (a)
What is region of convergence (ROC)? Mention its properties.

6 M

7 (b)
Determine the z-transform and ROC of the sequence \[ x(n)=r_1^n u(n)+ r^n_2 u(-n) \]

7 M

7 (c)
Determine the inverse z-transform of the function, \[ x(z)= \dfrac {1+z^{-1}}{1-z^{-1}+0.5z^{-2}} \] using partial fraction expansion.

7 M

8 (a)
An LTI system is described by the equation

y(n)=x(n)+0.8 x(n-1)+0.8x(n-2)-0.49y(n-2)

y(n)=x(n)+0.8 x(n-1)+0.8x(n-2)-0.49y(n-2)

6 M

8 (b)
Determine the transfer function H(z) of the system and also sketch the poles and zeros.

8 M

8 (c)
Determine whether the system described by the equation y(n)=x(n)+by(n-1) is causal and stable where |b|<1. Find the unilateral z-transform for the sequence y(n)=x(n-2), where x(n)=?

^{n}.
6 M

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