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−4−31−21−1201h[n]n−101423324

The sequence x[n] is nonzero for n = -3…-1 and h[n] for n = 1…3, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = -3…-1h[n]: n = 1…3
nlast =
2.Easy
x[n]n−1041−12434h[n]n−10−11−32434

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

x[n] = {4, −1, 4}, n = 0…2h[n] = {−1, −3, 4}, n = 0…2
ymax =
3.Medium
x(t)t5h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 5·u(t)t = 0.5 s
y(0.5) =
4.Medium
x(t)t11h(t)t33

The pulse x(t) = 1 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 3 s (zero elsewhere). Find the output y(2) at t = 2 s.

x(t) = 1, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 3 st = 2 s
y(2) =
5.Hard
u[n]0.5+y[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 0.5·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[5].

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

The input x(t) = 10·cos(40t) 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(40t)
φ = °

Answer key

  1. 1. nlast = 2
  2. 2. ymax = 16
  3. 3. y(0.5) = 1.58
  4. 4. y(2) = 3
  5. 5. y[5] = 0.3281
  6. 6. φ = −63.43 °

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