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−102112−134h[n]n−10113234

The sequences x[n] = {2, 1, −1} and h[n] = {1, 3} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {2, 1, −1}, n = 0…2h[n] = {1, 3}, n = 0…1
ymax =
2.Easy
x[n]n−101132−134h[n]n−102132334

The sequences x[n] = {1, 3, −1} and h[n] = {2, 3, 3} both start at n = 0. Find the sample y[3] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {1, 3, −1}, n = 0…2h[n] = {2, 3, 3}, n = 0…2
y[3] =
3.Medium
x(t)t310h(t)t11

The pulse x(t) = 10 for 0 ≤ t < 3 s is convolved with h(t), a triangle that falls linearly from 1 at t = 0 to 0 at t = 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 3 sh(0) = 1, h(1) = 0
Ay =
4.Medium
x(t)t2h(t)t1

The input x(t) = 2·e−3tu(t) is applied to a system with h(t) = e−1tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 2·e−3tu(t)h(t) = e−1tu(t)t = 0.25 s
y(0.25) =
5.Hard
x(t)t (ms)10204010

A square wave switches between 10 and 0 with period T = 20 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 350 Hz.

0 … 10T = 20 msf = 350 Hz
c7 =
6.Hard
δ[n]0.5+h[n]0.4·z⁻¹+

A system obeys y[n] = 0.4·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[4].

y[n] = 0.4y[n − 1] + 0.5x[n]x[n] = δ[n]
h[4] =

Answer key

  1. 1. ymax = 7
  2. 2. y[3] = 6
  3. 3. Ay = 15
  4. 4. y(0.25) = 0.3064
  5. 5. c7 = 0.9095
  6. 6. h[4] = 0.0128

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