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−1041322344h[n]n−1011−1234

The sequences x[n] = {4, 3, 2, 4} and h[n] = {1, −1} both start at n = 0. Find the sample y[2] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {4, 3, 2, 4}, n = 0…3h[n] = {1, −1}, n = 0…1
y[2] =
2.Easy
x[n]n−10212213145h[n]n−10210.52−0.534

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

b = {2, 0.5, −0.5}x[n] = {2, 2, 1, 1}, n = 0…3
y[1] =
3.Medium
x(t)t14h(t)t1

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(2) at t = 2 s, after the pulse has ended.

x(t) = 4, 0 ≤ t < 1 sh(t) = e−2tu(t)t = 2 s
y(2) =
4.Medium
x(t)t1h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 1·u(t). Find the output y(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 1·u(t)t = 0.5 s
y(0.5) =
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 = 0.2 s, h(0.2).

y′ + 2y = 2xt = 0.2 s
h(0.2) =
6.Hard
x(t)t (ms)2.55103−3

A triangle wave rises and falls linearly between −3 and +3 with period T = 5 ms. Find the amplitude c3 of its Fourier component at 600 Hz.

±3T = 5 msf = 600 Hz
c3 =

Answer key

  1. 1. y[2] = −1
  2. 2. y[1] = 5
  3. 3. y(2) = 0.234
  4. 4. y(0.5) = 0.3161
  5. 5. h(0.2) = 1.341
  6. 6. c3 = 0.2702

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