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−103132333…h[n]n−10−31−32−334

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

h[n] = {−3, −3, −3}x[n] = 3·u[n]
y[3] =
2.Easy
x[n]n−10−1142−23−145h[n]n−100.5112134

An FIR filter computes y[n] = 0.5·x[n] + x[n − 1] + x[n − 2]. The input is x[n] = {−1, 4, −2, −1} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {0.5, 1, 1}x[n] = {−1, 4, −2, −1}, n = 0…3
y[3] =
3.Medium
x(t)t14h(t)t45

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 4, 0 ≤ t < 1 sh(t) = 5, 0 ≤ t < 4 st = 0.5 s
y(0.5) =
4.Medium
x(t)t10h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 10·u(t). Find the output y(0.4) at t = 0.4 s.

h(t) = e−5tu(t)x(t) = 10·u(t)t = 0.4 s
y(0.4) =
5.Hard
x(t)t (ms)0.5121−1

A triangle wave rises and falls linearly between −1 and +1 with period T = 1 ms. Find the amplitude c3 of its Fourier component at 3000 Hz.

±1T = 1 msf = 3000 Hz
c3 =
6.Hard
x(t)t (ms)2.55104−4

A square wave switches between +4 and −4 with period T = 5 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 600 Hz.

±4T = 5 msf = 600 Hz
c3 =

Answer key

  1. 1. y[3] = −27
  2. 2. y[3] = 1.5
  3. 3. y(0.5) = 10
  4. 4. y(0.4) = 1.729
  5. 5. c3 = 0.09006
  6. 6. c3 = 1.698

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