problem 1: A sine wave is sampled at interval T and quantixed using N = 2 bits precision over the range ±1. The sine wave has peroid τ, and T is chosed such that  T=1/8 τ. Plot one cycle of the true signal and show where the sample of the sinusoidal waveform fall.

problem 2:

i) For a 3 term averagin filter, prepare a difference equation governing the filter and dervie the corresponding  z transform. Sketch a pole zero plot, would this filter be stable, why or why not?

ii) Sketch the frequency resoponce and describe where the peaks and nulls occur.

iii) Extend the above to an N-term averging filter prepare a difference equation and derive the z transform in closed form. Indicate were the pole and zeros would fall for Larger N, and then sketch the frequency response for a similarly large N.

problem 3:

i) A  sionusodial signal has the form: x(t) = AsinΩt

Derive an expression for the auto correlation R_xx (λ)

ii) describe the significance of this result in term of the frequency of the correlation and also if noise were added to the sinusoidal signal x(t)

iii) If a signal x(t)is corrupt by additive white noise v(t), then the resulting signal would be: y(t) = x(t) + v(t)

What form would the auto correlation R_yy (λ) take in that case?
problem 4:

A band pass FIR filter is required to pass frequencies from 5KHz to 8KHz, using a sample rate of 40K

i) describe how this would map to normalization sampling frequencies in radians per sample, Include a diagram shown the passband, stop band and other areas of interest.

ii) Derive a mathematical expression for the filter coefficients.

iii) prepare a MATLAB script to find out the frequency response of the filter and briefly describe how it works.