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−10−2122434h[n]n−10312234

The sequences x[n] = {−2, 2, 4} 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, 2, 4}, n = 0…2h[n] = {3, 2}
Σy[n] =
2.Easy
x[n]n−101112131…h[n]n−1011−2243−345

A system with the impulse response h[n] = {1, −2, 4, −3} (n = 0…3) receives the step input x[n] = 1·u[n]. Find the output y[2].

h[n] = {1, −2, 4, −3}x[n] = 1·u[n]
y[2] =
3.Medium
x(t)t5h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.08) at t = 0.08 s.

h(t) = e−5tu(t)x(t) = 5·u(t)t = 0.08 s
y(0.08) =
4.Medium
x(t)t11h(t)t43

The pulse x(t) = 1 for 0 ≤ t < 1 s is convolved with h(t), a triangle that falls linearly from 3 at t = 0 to 0 at t = 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 1 sh(0) = 3, h(4) = 0
Ay =
5.Hard
δ[n]0.5+h[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[5].

y[n] = −0.5y[n − 1] + 0.5x[n]x[n] = δ[n]
h[5] =
6.Hard
x(t)t (ms)5102010−10

A square wave switches between +10 and −10 with period T = 10 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 300 Hz.

±10T = 10 msf = 300 Hz
c3 =

Answer key

  1. 1. Σy[n] = 20
  2. 2. y[2] = 3
  3. 3. y(0.08) = 0.3297
  4. 4. Ay = 6
  5. 5. h[5] = −0.01563
  6. 6. c3 = 4.244

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