CT204 Signals & Systems
Lab 3: Fundamentals of Systems
1. Consider the system:
y[n] = 3x [n] − 2x [n − 1]
• Verify linearity using MATLAB for signals x1 [n] = sin(n) and x2 [n] = cos(n).
• Check the superposition property for a = 2 and b = 3.
• Plot y[ ax1 + bx2 ] and ay[ x1 ] + by[ x2 ].
2. Consider the system:
y [ n ] = x [ n2 ]
• Verify time-invariance using MATLAB for x [n] = n and a shift n0 = 2.
• Compare the outputs for the shifted input.
3. Consider the system:
y [ n ] = x [ n ] + x [ n + 1]
• Determine if the system is causal.
• Plot x [n] and y[n] for x [n] = [1, 2, 3, 4].
4. Consider the system:
y[n] = 0.5y[n − 1] + x [n]
• Simulate the output for bounded input x [n] = (−1)n .
• Verify if the output remains bounded.
5. For the signal: n
1
x [n] = u[n]
2
• Calculate the energy of the signal in MATLAB.
• Plot x [n] for 0 ≤ n ≤ 20.
• Now shift the signal by 5 units in any direction and recalculate the energy.
• Justify the results.
6. Consider the continuous-time system:
Z t
y(t) = e−2(t−τ ) x (τ )dτ
0
1
• Simulate the output for x (t) = e−t u(t) in MATLAB.
• Determine linearity, causality, and stability.
7. Consider the input and impulse response:
x [n] = {1, 2, 1}, h[n] = {1, 1, 1}
• Perform the convolution y[n] = x [n] ∗ h[n] using MATLAB’s conv() function.
• Calculate the time requirement.
• Plot the input x [n], the impulse response h[n], and the output y[n].
8. Consider the input and impulse response:
x [n] = {1, 2, 1}, h[n] = {1, 1, 1}
• Write a MATLAB script (using a loop or using a Toeplitz matrix) to perform the
convolution y[n] = x [n] ∗ h[n] without using MATLAB’s conv() function.
• Calculate the time requirement and compare it with the previous problem.
• Plot the input x [n], the impulse response h[n], and the output y[n].
9. Consider the continuous-time input and impulse response:
x ( t ) = u ( t ), h (t ) = e−t u (t )
• Approximate the continuous-time convolution
Z ∞
y(t) = ( x ∗ h)(t) = x (τ )h(t − τ ) dτ
−∞
using MATLAB’s conv() function with time discretization.
• Plot x (t), h(t), and y(t) for 0 ≤ t ≤ 10.
10. Consider a causal LTI system whose input sequence x [n] and output sequence y[n] are
related by following difference equation
1
y[n] = y[n − 1] + 2x [n].
2
Find and plot the impulse response of the system. (You need to use a loop for this
problem.)
