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−102142−234h[n]n−10−21−32134

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

x[n] = {2, 4, −2}, n = 0…2h[n] = {−2, −3, 1}, n = 0…2
ymax =
2.Easy
x[n]n−101132−334h[n]n−10214234

The sequences x[n] = {1, 3, −3} and h[n] = {2, 4} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {1, 3, −3}, n = 0…2h[n] = {2, 4}
Σy[n] =
3.Medium
x(t)t2h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.25) at t = 0.25 s.

h(t) = e−2tu(t)x(t) = 2·u(t)t = 0.25 s
y(0.25) =
4.Medium
x(t)t11h(t)t23

The pulse x(t) = 1 for 0 ≤ t < 1 s is convolved with h(t), a triangle that falls linearly from 3 at t = 0 to 0 at t = 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 1 sh(0) = 3, h(2) = 0
Ay =
5.Hard
10cos(15t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 10·cos(15t) is applied to a system with h(t) = 10·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−20tu(t)x(t) = 10cos(15t)
φ = °
6.Hard
x(t)t (ms)510205−5

A triangle wave rises and falls linearly between −5 and +5 with period T = 10 ms. Find the amplitude c5 of its Fourier component at 500 Hz.

±5T = 10 msf = 500 Hz
c5 =

Answer key

  1. 1. ymax = 10
  2. 2. Σy[n] = 6
  3. 3. y(0.25) = 0.3935
  4. 4. Ay = 3
  5. 5. φ = −36.87 °
  6. 6. c5 = 0.1621

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