1.a. an analog signal given as $x_{a}$ = 15 cos (1250$\small \pi$t) + 17 cos (2170$\small \pi$ t) + 33 cos(4750$\small \pi$ t) is converted into discrete time signal. Determine Nyquist sampling rate, Folding Frequency , resulting discrete time frequencies time signal x(n) if sampling frequency is 625 Hz. Also write discrete time frequencies in radians.

1.b. An LTI system is defined by difference equation

y(n) = y(n-1) + y(n-2) x(n-1). Find system function H(z). Draw pole zero diagram. Find out h(n) for casual , non-casual systems, if not why ?

2.a. Find the DFT of the sequence

x(n) = 1 for 0 $\leq$ n $\leq$ 2

= 0 otherwise

For N = 4, Find [X(K)] and X(K)

2.b. Explain the sampling theorem and advantages of DIgital over Analog signal Processing.

3.a. State any four properties of Z transform.

3.b. Compare circular convolution with linear convolution find the circular convolution of two finite duration sequenced.

$x_{1}(n)$ = {1 , -1 , -2, 3, -1} and

$x_{2}(n)$ = {1, 2 3}

4.a. What is FFT? Explain Bit-reversal and in place computation concepts in FFT algorithm. Show that 3-bit bit reversed sequence.

4.b. Explain the concept of orthogonality , Check whether the functions given are orthogonal or not over an time interval [0,1], f(t) = 1, x(t) = $\sqrt{3}$ (1-2t).

5.a. Design the seocnd order low pass Digital Butterworth filter with cut off frequency of of 1 KHz and sampling frequency 10,000 samples/sec by Bilinear transformation.

5.b. Write the equation, Draw and compare the characteristics of Butterworth filter. Chebyshev filters and ellipitic filter.

6.a. What is Billinear transformation ? Explain the properties of BLT. What is worping effect? how do you take care of it in design.

6.b. State the advantage of direct form II realization over direcr form I. Hence implement the following difference equation in direct form I and II

y(n) + 0.1 y(n-1) + 0.72 y(n-2) = 0.7 x(n) - 0.95 x(n-2)

7.a. Design an FIR filter having desired frequency response as given below using rectangular window.

Find H(w), Does the filter is realizable. Justify your answer, What modification is required in Hd(w) to make it realizable.

7.b. Explain frequency sampling technique of FIR filter designing in detail.

8.a. Explain windowing technique of FIR filter design in detail. Also Gibb's phenomenon and how it can be reduced. state different types of windows used with their window function.

8.b. What is the meaning of linear phase. Prove that FIR filters are inherently stable.

9.a. Speech signal is corrupted by low and high frequency noise. Explain detail how DSP is used to remove noise with illustration.

9.b. Explain the application of DSP in vibration signature analysis for defective gear teeth.

10.a. Explaining speech coding and compression technique. How signal processing techniques are used in this.

10.b. Explain how DSP is useful in Interference cancellation in ECG.