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−101122−13245h[n]n−102112134

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

b = {2, 1, 1}x[n] = {1, 2, −1, 2}, n = 0…3
y[1] =
2.Easy
x[n]n−10−31−124344h[n]n−10−112234

The sequences x[n] = {−3, −1, 4, 4} and h[n] = {−1, 2} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {−3, −1, 4, 4}, n = 0…3h[n] = {−1, 2}, n = 0…1
ymax =
3.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(0.2) at t = 0.2 s.

h(t) = e−5tu(t)x(t) = 10·u(t)t = 0.2 s
y(0.2) =
4.Medium
x(t)t2h(t)t1

The input x(t) = 2·e−3tu(t) is applied to a system with h(t) = e−5tu(t). Find the output y(2) at t = 2 s.

x(t) = 2·e−3tu(t)h(t) = e−5tu(t)t = 2 s
y(2) =
5.Hard
x(t)t (ms)510203−3

A triangle wave rises and falls linearly between −3 and +3 with period T = 10 ms. Find the amplitude c5 of its Fourier component at 500 Hz.

±3T = 10 msf = 500 Hz
c5 =
6.Hard
5cos(40t)h(t) = 2e−20t​u(t)y(t)

The input x(t) = 5·cos(40t) is applied to a system with h(t) = 2·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−20tu(t)x(t) = 5cos(40t)
φ = °

Answer key

  1. 1. y[1] = 5
  2. 2. ymax = 8
  3. 3. y(0.2) = 1.264
  4. 4. y(2) = 0.002433
  5. 5. c5 = 0.09727
  6. 6. φ = −63.43 °

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