Assignment-01

ECE-211

Max. Marks: 20

Date of allotment: 29th

Aug, 2012 Date of submission: 10th

Sept., 2012

1. Given the continuous-time signal specified by

Determine the resultant discrete-time sequence obtained by uniform sampling of x(t) with a

sampling interval of (a) 0.25 s, (b) 0.5 s, and (c) 1.0 s.

2. If the input signal is x(t)= sin200πt and signal is sampled at Nyquist rate. Can we

recover the original signals from the samples. If No, give the reason.

3. An analog signal x(t)= sin(480πt)+3sin(720πt) is sampled 600 times per second

(a) Determine the Nyquist sampling rate for x(t).

(b) Determine the folding frequency.

(c) What are the frequencies in radians in the resulting discrete time signal x(n)?

(d) If x(n) is passed through an ideal D/A converter. What is the reconstruction

signal y(t)?

4. Show that the product of two even signals or of two odd signals is an even signal and

that the product of an even and an odd signaI is an odd signal.

5. Find the convolution of two signal x(n) = u(n) and h(n) = an

u(n) .

6. Find out that unit impulse signal is energy signal or a power signal?

7. Find signal are periodic or not? (a) X(t) = 3 cos 200t (b) X(n)= 3 cos 200n

(c) X(t)= 3 cos200πt+ 2cos 200t

(d) X(t)= cos(200πt). cos (200t)

(e) X(n)= 3 cos200πn+ 2cos 200n

(f) X(n)= cos(200πn). cos (200n)

(g) X(n)= ej(4πn + π/6)

8. Consider the system

Y(n)=T[x(n)]=x(n2)

a) Determine if the system is time invariant.

b) To clarify the result in part a) assume that the signal is applied to the system.

1) Sketch the signal x(n).

2) Determine and sketch the signal y(n)=T[x(n)]

3) Sketch the signal y2 (n)= y(n-2).

4) Determin e and sketch the signal x2 (n)=x(n-2)

5) Determine and sketch the signal y2(n)= T[x2(n)].

6) Compare the signals y2(n) and y(n-2).

c) Repaet the part (b) for the system

Y(n)= x(n)-x(n-1)

Can you use this result to make any statement about the time invariance of this system? Why?

d) Repeat part (b) for the system

Y(n)=T[x(n)]=nx(n)

9. Consider the sinusoidal signal

X(t)=cos15t

Find the value of sampling interval T, such that x [ n ] = x(nT,) is a periodic sequence.

Find the fundamental period of x [ n ]= x(nT,) if Ts= 0 . 1 π seconds.

10. Elucidate any two real time applications of DSP with some new innovations.

(Applications should not be identical of two students.)