PEN No. : __________________

GUJARAT TECHNOLOGICAL UNIVERSITY Affiliated

Sitarambhai Naranji Patel Institute of Technology & Research Centre

Remid Examination, October - 2015

B.E. Semester-VII (EC)

Subject Code: 171003 Date: 05/10/2015

Subject Name: Digital Signal Processing

Instructions: Time: 02:00 pm – 04:30 pm

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks. Max. Marks: 70

Q.1 (a) A link carries a signal xa(t) = 3 cos(600πt) x 2 cos(800πt). The link is operated ar

10000 bps and each sample is quantized to 1024 voltage levels.

(i) What is the sampling and folding frequency?

(ii) What is the Nyquist rate of signal?

(iii) What are the frequencies resulting in discrete time signal?

(iv) What is the resolution?

7

(b) Enlist properties of Z transform and derive any two in detail 7

Q2. (a-i) When LTI system is said to be stable and causal? (Derive conditions) 7

(b-i) Find out convolution of following: (do not use tabular method)

(i) x[n]={1,2,

1

,2,3} h[n]={

1

,2,3}

(ii) x[n]={3,2,

1

,3} h[n]={3,2,

1

}

7

OR

(b-ii) Determine the zero input response of the system described by the homogeneous

second order difference equation

y(n) – 3y(n-1) -4y(n-2)=0

7

Q.3 (a-i) Determine Z-transform & ROC of following signal (any two)

(i) x(n)= {3,0,0,0,0,6

↑,1,-4}

(ii) 𝑥(𝑛) = (1

2)𝑛[𝑢(𝑛) − 𝑢(𝑛 − 10)]

(iii) 𝑥(𝑛) = 𝑛(−1)𝑛𝑢(𝑛)

7

(b-i) Using long division determine the inverse z-transform of

𝑋(𝑧) =1 + 2𝑧−1

1 − 2𝑧−1 + 𝑧−2

7

OR

Q.3 (a-ii) Determine the convolution of t=he following pair of signal by means of z-transform.

𝑥(𝑛) = 𝑢(𝑛) and ℎ(𝑛) = δ(𝑛) + (1

2)𝑛𝑢(𝑛)

7

(b-ii) Determine the causal signal x(n) having z-transforms

1

(1 −12

𝑧−1)(1 −14

𝑧−1)

7

Q.4 (a-i) Obtain the system function H(Z) for the system described by the difference

equation,

𝑦(𝑛) = 3

4𝑦(𝑛 − 1) −

1

8𝑦(𝑛 − 2) + 𝑥(𝑛) +

1

3 𝑥(𝑛 − 1)

Realize the filter using (i) Direct Form I (ii) Direct Form II (iii)Cascade

form (iv) Parallel Form for all cases, draw the structures neatly with system

7

(b-i) Realize an FIR filter with impulse response h(n) given by

ℎ(𝑛) = (1

2)𝑛[𝑢(𝑛) − 𝑢(𝑛 − 5)]

7

OR

(a-ii) Explain criteria of stability and causality of LTI system in z-domain with respect to

ROC. determine the response of the system

𝑦(𝑛) =5

6𝑦(𝑛 − 1) −

1

6𝑦(𝑛 − 2) + 𝑥(𝑛)

to the input signal 𝑥(𝑛) = δ(𝑛) −1

3𝛿(𝑛 − 1)

7

(b-ii) Determine the inverse z-transform of

𝑋(𝑧) =1

1 − 1.5𝑧−1 + 0.5𝑧−2

If

(a) ROC: |z|>1

(b) ROC: |z|<0.5

(c) ROC: 0.5<|z|<1

7

Q.5 (a-i) Derive the symmetry properties of DFT 7

(b-i) State and derive the circular time Shift of a sequence. 7

OR

Q.5 (a-ii) Define twiddle factor and determine DFT of

1. {0,1,2,3}

2. {1,1,1}

7

(b-ii) Derive the condition for convergence of Fourier Transform and hence define Gibbs

Phenomena.

7