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−1031423344h[n]n−10−3142134

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

x[n] = {3, 4, 3, 4}, n = 0…3h[n] = {−3, 4, 1}, n = 0…2
ymax =
2.Easy
x[n]n−1011−32−334h[n]n−10411234

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

x[n] = {1, −3, −3}, n = 0…2h[n] = {4, 1}, n = 0…1
y[1] =
3.Medium
x(t)t310h(t)t14

The pulse x(t) = 10 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 4 for 0 ≤ t < 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 3 sh(t) = 4, 0 ≤ t < 1 s
Ay =
4.Medium
x(t)t0.51h(t)t1

The pulse x(t) = 1 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(1) at t = 1 s, after the pulse has ended.

x(t) = 1, 0 ≤ t < 0.5 sh(t) = e−2tu(t)t = 1 s
y(1) =
5.Hard
x(t)t (ms)820402

A pulse train has height 2, pulse width 8 ms and period 20 ms. Find the amplitude c1 of harmonic number 1, at 50 Hz.

A = 2τ = 8 msT = 20 ms
c1 =
6.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 =

Answer key

  1. 1. ymax = 19
  2. 2. y[1] = −11
  3. 3. Ay = 120
  4. 4. y(1) = 0.1163
  5. 5. c1 = 1.211
  6. 6. c3 = 0.09006

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