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−1011−1233145h[n]n−100.510.252−0.534

An FIR filter computes y[n] = 0.5·x[n] + 0.25·x[n − 1] − 0.5·x[n − 2]. The input is x[n] = {1, −1, 3, 1} for n = 0…3 and zero elsewhere. Find the output y[1].

b = {0.5, 0.25, −0.5}x[n] = {1, −1, 3, 1}, n = 0…3
y[1] =
2.Easy
x[n]n−2−1−30412243h[n]n−10123−14−2536

The sequence x[n] is nonzero for n = -1…2 and h[n] for n = 3…5, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = -1…2h[n]: n = 3…5
nlast =
3.Medium
x(t)t−12h(t)t37

The signal x(t) is nonzero only between t = −1 s and t = 2 s, and h(t) only between t = 3 s and t = 7 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: −1 … 2 sh: 3 … 7 s
tstart = s
4.Medium
x(t)t−21h(t)t−10

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

x: −2 … 1 sh: −1 … 0 s
tend = s
5.Hard
δ(t)y′ + 5y = 2xh(t)

A system obeys y′(t) + 5·y(t) = 2·x(t). Find its impulse response at t = 0.1 s, h(0.1).

y′ + 5y = 2xt = 0.1 s
h(0.1) =
6.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[3].

y[n] = 0.5y[n − 1] + 0.5x[n]x[n] = u[n]
y[3] =

Answer key

  1. 1. y[1] = −0.25
  2. 2. nlast = 7
  3. 3. tstart = 2 s
  4. 4. tend = 1 s
  5. 5. h(0.1) = 1.213
  6. 6. y[3] = 0.9375

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