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−101112131…h[n]n−10−3112−234

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

h[n] = {−3, 1, −2}x[n] = 1·u[n]
y[4] =
2.Easy
x[n]n−10−113213145h[n]n−10110.52−0.534

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

b = {1, 0.5, −0.5}x[n] = {−1, 3, 1, 1}, n = 0…3
y[2] =
3.Medium
x(t)t11h(t)t25

The pulse x(t) = 1 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 1, 0 ≤ t < 1 sh(t) = 5, 0 ≤ t < 2 st = 0.5 s
y(0.5) =
4.Medium
x(t)t12h(t)t14

The pulse x(t) = 2 for 0 ≤ t < 1 s is convolved with h(t), a triangle that falls linearly from 4 at t = 0 to 0 at t = 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 2, 0 ≤ t < 1 sh(0) = 4, h(1) = 0
Ay =
5.Hard
x(t)t (ms)2.55103−3

A triangle wave rises and falls linearly between −3 and +3 with period T = 5 ms. Find the amplitude c1 of its Fourier component at 200 Hz.

±3T = 5 msf = 200 Hz
c1 =
6.Hard
x(t)t (ms)0.5124−4

A square wave switches between +4 and −4 with period T = 1 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 1000 Hz.

±4T = 1 msf = 1000 Hz
c1 =

Answer key

  1. 1. y[4] = −4
  2. 2. y[2] = 3
  3. 3. y(0.5) = 2.5
  4. 4. Ay = 4
  5. 5. c1 = 2.432
  6. 6. c1 = 5.093

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