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−10−11−32134h[n]n−10−31−22334

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

x[n] = {−1, −3, 1}, n = 0…2h[n] = {−3, −2, 3}
Σy[n] =
2.Easy
x[n]n−101132−13−14h[n]n−10123−3415−16

The sequence x[n] is nonzero for n = 0…3 and h[n] for n = 3…5, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 0…3h[n]: n = 3…5
nfirst =
3.Medium
x(t)t12h(t)t12

The signal x(t) is nonzero only between t = 1 s and t = 2 s, and h(t) only between t = 1 s and t = 2 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: 1 … 2 sh: 1 … 2 s
tstart = s
4.Medium
x(t)t12h(t)t43

The pulse x(t) = 2 for 0 ≤ t < 1 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 2, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 4 s
Ay =
5.Hard
x(t)t (ms)420401

A pulse train has height 1, pulse width 4 ms and period 20 ms. Find its DC component (mean value) c0.

A = 1τ = 4 msT = 20 ms
c0 =
6.Hard
x(t)t (ms)2.55102

A square wave switches between 2 and 0 with period T = 5 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 200 Hz.

0 … 2T = 5 msf = 200 Hz
c1 =

Answer key

  1. 1. Σy[n] = 6
  2. 2. nfirst = 3
  3. 3. tstart = 2 s
  4. 4. Ay = 24
  5. 5. c0 = 0.2
  6. 6. c1 = 1.273

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