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−10−2132−334h[n]n−10312234

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

x[n] = {−2, 3, −3}, n = 0…2h[n] = {3, 2}
Σy[n] =
2.Easy
x[n]n−2−1−30−2132−23h[n]n−10−2132

The sequence x[n] is nonzero for n = -1…2 and h[n] for n = 0…1, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = -1…2h[n]: n = 0…1
nfirst =
3.Medium
x(t)t23h(t)t−10

The signal x(t) is nonzero only between t = 2 s and t = 3 s, and h(t) only between t = −1 s and t = 0 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: 2 … 3 sh: −1 … 0 s
D = s
4.Medium
x(t)t45h(t)t23

The pulse x(t) = 5 for 0 ≤ t < 4 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) = 5, 0 ≤ t < 4 sh(0) = 3, h(2) = 0
Ay =
5.Hard
x(t)t (ms)1020401−1

A triangle wave rises and falls linearly between −1 and +1 with period T = 20 ms. Find the amplitude c5 of its Fourier component at 250 Hz.

±1T = 20 msf = 250 Hz
c5 =
6.Hard
x(t)t (ms)0.25122

A pulse train has height 2, pulse width 0.25 ms and period 1 ms. Find the amplitude c2 of harmonic number 2, at 2000 Hz.

A = 2τ = 0.25 msT = 1 ms
c2 =

Answer key

  1. 1. Σy[n] = −10
  2. 2. nfirst = −1
  3. 3. D = 2 s
  4. 4. Ay = 60
  5. 5. c5 = 0.03242
  6. 6. c2 = 0.6366

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