Convolution & LTI systems worksheet with answers

Free printable Convolution & LTI systems worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
x[n]n−10−21−1213145h[n]n−1021−12−0.534

An FIR filter computes y[n] = 2·x[n] − x[n − 1] − 0.5·x[n − 2]. The input is x[n] = {−2, −1, 1, 1} for n = 0…3 and zero elsewhere. Find the output y[4].

b = {2, −1, −0.5}x[n] = {−2, −1, 1, 1}, n = 0…3
y[4] =
2.Easy
x[n]n−10112213345h[n]n−1021120.534

An FIR filter computes y[n] = 2·x[n] + x[n − 1] + 0.5·x[n − 2]. The input is x[n] = {1, 2, 1, 3} for n = 0…3 and zero elsewhere. Find the output y[2].

b = {2, 1, 0.5}x[n] = {1, 2, 1, 3}, n = 0…3
y[2] =
3.Medium
x(t)t5h(t)t1

The input x(t) = 5·e−3tu(t) is applied to a system with h(t) = e−4tu(t). Find the output y(2) at t = 2 s.

x(t) = 5·e−3tu(t)h(t) = e−4tu(t)t = 2 s
y(2) =
4.Medium
x(t)t25h(t)t22

The pulse x(t) = 5 for 0 ≤ t < 2 s is convolved with the pulse h(t) = 2 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 5, 0 ≤ t < 2 sh(t) = 2, 0 ≤ t < 2 s
Ay =
5.Hard
δ(t)y′ + 2y = 2xh(t)

A system obeys y′(t) + 2·y(t) = 2·x(t). Find its impulse response at t = 1 s, h(1).

y′ + 2y = 2xt = 1 s
h(1) =
6.Hard
δ(t)y′ + 4y = 10xh(t)

A system obeys y′(t) + 4·y(t) = 10·x(t). Find its impulse response at t = 0.05 s, h(0.05).

y′ + 4y = 10xt = 0.05 s
h(0.05) =

Answer key

  1. 1. y[4] = −1.5
  2. 2. y[2] = 4.5
  3. 3. y(2) = 0.01072
  4. 4. Ay = 40
  5. 5. h(1) = 0.2707
  6. 6. h(0.05) = 8.187

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