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−10−3122−1344h[n]n−10−1132−134

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

x[n] = {−3, 2, −1, 4}, n = 0…3h[n] = {−1, 3, −1}
Σy[n] =
2.Easy
x[n]n−103112−334h[n]n−10211234

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

x[n] = {3, 1, −3}, n = 0…2h[n] = {2, 1}
Σy[n] =
3.Medium
x(t)t−10h(t)t26

The signal x(t) is nonzero only between t = −1 s and t = 0 s, and h(t) only between t = 2 s and t = 6 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: −1 … 0 sh: 2 … 6 s
tend = s
4.Medium
x(t)t210h(t)t1

The pulse x(t) = 10 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(4) at t = 4 s, after the pulse has ended.

x(t) = 10, 0 ≤ t < 2 sh(t) = e−2tu(t)t = 4 s
y(4) =
5.Hard
u[n]4+y[n]0.5·z⁻¹+

A system obeys y[n] = 0.5·y[n − 1] + 4·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[2].

y[n] = 0.5y[n − 1] + 4x[n]x[n] = u[n]
y[2] =
6.Hard
5cos(40t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 5·cos(40t) is applied to a system with h(t) = 10·e−20tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−20tu(t)x(t) = 5cos(40t)
Ym =

Answer key

  1. 1. Σy[n] = 2
  2. 2. Σy[n] = 3
  3. 3. tend = 6 s
  4. 4. y(4) = 0.0899
  5. 5. y[2] = 7
  6. 6. Ym = 1.118

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