Digital Signal Processing

2. Consider the transfer function of an LTI discrete-time system given by . a) Find the poles and zeros of the system, b) Is this system stable? Explain, c) What is the response of the system at DC (zero frequency)? (15 points)

3. The specifications for a lowpass FIR digital filter are: peak passband ripple is 0.25 dB; minimum stopband attenuation is 45 dB; passband edge frequency is 4 kHz; stopband edge frequency is 6 kHz; sampling frequency is 20 kHz. Determine using Kaiser’s formula the filter order required to satisfy the above specifications. (5 points)

4. If the band edges of an IIR digital BPF are 3 kHz and 6 kHz, respectively, what are the corresponding prewarped frequencies in the analog domain if bilinear z-transform is used? Use a 20 kHz sampling frequency. (10 points)

5. Consider the design of a lowpass digital filter given the passband and stopband ripples, passband and stopband edge frequencies. Which type of filter,IIR or FIR, will give the smaller filter order? Explain the reason. (10 points)

6. It is well known that an IIR digital filter, in general, has a nonlinear phase response. Describe how you will linearize the phase response of such an IIR filter without altering its magnitude response.(10 points)

7. Develop a minimum-multiplier realization of a length-9 Type 3 FIR transfer function. Note that the impulse response of a Type 3 FIR filter is of odd length; that is, the degree N is even and satisfies the asymmetric condition . (10 points)

8. Develop a canonic direct-form realization of the transfer function (10 points)

9. What is the minimum number of bits required in the ADC (analog to digital converter) using uniform quantization if the SNR due to quantization of the ADC is to be approximately 72 dB? For this case, what is the step size if the signal input to the ADC is between 0 and 1? What is the maximum absolute error due to quantization of a sample? (10 points)

10. Consider the linear convolution of the sequences {h[n]} of length N and {x[n]} of length M via DFT algorithm. Find the total number of floating-point complex multiplications and additions required using brute force method to carry out the DFT in calculating the above linear convolution. (10 points)

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