Reference no: EM132858039

ENGT5203 Microprocessor Applications - De Montfort University

Question 1. (a) Determine a difference equation for the system

(b) Find the unit-step response of this system

(c) Determine the pole(s) of this system

(d) If a system is represented as the ratio of polynomials in R, then the poles of the system are the roots of the denominator polynomial after R is replaced by z^-1 = 1/z.

Determine an analogous rule for systems represented by a ratio of polynomials in L.

(e) Consider a system that is described by the following unit-sample response.

h₂(n) = (1/2)ⁿ        for all n

determine a close form functional representation for the system described by h₂(n).

(f) Determine the poles of the system in (3e)

(g) Do a series expansion of the system function in (3e) ( it may be helpful to use partial fractions). Explain the relation between the series and the unit-sample response h₂(n).

Question 2. The terminals of a generator producing a voltage v(t) are connected through a wire of resistance R and a coil of inductance L. A capacitor of capacitance C is connected in parallel with the resistance R as show in Fig.1.

(a) Write the system equation with the current i(t), which should be the second order differential equation for such a LCR circuite.
(b) Suppose that v(t) = 0 for t < 0, and v(t) = E for t≥0, Where E is a constant. If L = 2R²C and CR = 1/2n, and than show your equation reduces to

d²i/dt² + 2ndi/dt + 2n²i = 2n²E/R

Hence, assuming i(0) = di(0) = 0, use Laplace transform to obtain an expression for i(t) in terms of t.

Figure 1: This figure is for Question 5.

Question 3: Design of digital filter

A requirement exists to simulate an analogue system in a digital computer with a third order Butterworth filter has transfer function

H(s) = 1/((s + 1)(s2 + s + 1))

Assuming a cutoff frequency of 1 rad/s and a sampling frequency of 2 Hz, design the corresponding digital filter using the bilinear transformation technique.

In this question, you should give the details of the transfer function of H(z) and use MATLAB to find the coefficients of the filter, finally, sketch the filter and labeling the coefficients.

Question 4. Applying the impulse invariant method to filter design (see p29-31 in Lecture Note). It is required to design a digital filter to approximate the following normalized analog transfer function

H(s) = 1/s² + √2s + 1

(a) Use the impulse invariant method to obtain the transfer function, H(z), of the digital filter, assuming a 3 dB cutoff frequency of 150 Hz and sampling frequency of 1.28 kHz

(b) Plot the pole-zero diagram of H(z).

Attachment:- Microprocessor Applications.rar