1. Following RLC circuit is described by the differential equation (1). Use Matlab built-in differential equation solver dsolve() to derive the impulse response
h(t) for this circuit when R=2Ω, C=1F, L=0.5 H. Plot the impulse response h(t)from a range −10 ≤ t ≤ 30.
RC d2y(t) + dy(t) + R y(t) = dx(t) dt2 dtL dt
2. Consider the following input signal
xlinear _ comb(t) = x1(t) + 2×1(t − 10)
Using the example Matlab file simplified_convolution_runtime.m, plot the
output signals in three separate figure windows: (a) y1(t) = x1(t) ∗ h(t)
(b) y2(t) = x2(t) ∗ h(t)
(c) ylinear _ comb = xlinear _ comb(t) ∗ h(t).
Use the ranges of ‘τ’ and ‘t’ as −10 ≤ τ ≤ 40 and −10 ≤ t ≤ 40. Also plot y3(t)=y1(t)+y2(t) and comment on similarity of y3(t) and ylinear _ comb(t).
3. A single-tone signal w(t) = sin(400πt) is transmitted to an audio amplifier and speaker to produce a high-temperature warning for a silicon crystal-growing factory.
A filter having impulse response h(t) = 400e−200t cos(400πt)u(t) has been designed to reduce additive interference in the received signal. Using Matlab in-built convolution function: conv(), find the filter output signal y(t), when the received signal is x(t) = [cos(100πt) + sin(400πt) − cos(800πt)]u(t) ( signal w(t) was corrupted by interference and resulted in an input signal x(t)). Plot the output signal,
the input signal, and w(t) for the range of −0.1 ≤ t ≤ 0.1. Comment on the effect of the filter on the signal. While solving this problem, pay attention to the time resolution/step (dT) you need to use.
…(1)
⎧ 5, 0 ≤ t < 10 x1(t) = ⎪⎨ 0, elsewhere
⎪⎩
x2(t) = 2x1(t − 10)
4. System response for an Industrial Shock Absorber (figure below) can be modeled with the following differential equation:
M d2y(t) + b dy(t) + ky(t) = r(t) ...(2) dt2 dt
Let’s assume the mass of the damperMis100 kg, the spring constantkis1 kgs-2, and the friction coefficient b is 20 kgs-1. Using Matlab built-in differential equation solver dsolve() to derive the impulse response h1(t) for this Industrial Shock Absorber and the impulse response h1(t) from a range −10s ≤ t ≤ 300s . Overtime the oil inside the shock absorber degrades and the friction coefficient b becomes 0.2 kgs-1. Derive the new impulse response h2(t) for this Industrial Shock Absorber and plot h2(t) from a range −10s ≤ t ≤ 300s .