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−10−31−22334h[n]n−10−21−22334

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

x[n] = {−3, −2, 3}, n = 0…2h[n] = {−2, −2, 3}, n = 0…2
ymax =
2.Easy
x[n]n−1031124314h[n]n−1041−1234

The sequences x[n] = {3, 1, 4, 1} and h[n] = {4, −1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {3, 1, 4, 1}, n = 0…3h[n] = {4, −1}
Σy[n] =
3.Medium
x(t)t110h(t)t1

The pulse x(t) = 10 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 10, 0 ≤ t < 1 sh(t) = e−2tu(t)t = 2.5 s
y(2.5) =
4.Medium
x(t)t12h(t)t23

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 2, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 2 st = 0.5 s
y(0.5) =
5.Hard
x(t)t (ms)2.55101

A square wave switches between 1 and 0 with period T = 5 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 600 Hz.

0 … 1T = 5 msf = 600 Hz
c3 =
6.Hard
5cos(7.5t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 5·cos(7.5t) is applied to a system with h(t) = 10·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−10tu(t)x(t) = 5cos(7.5t)
Ym =

Answer key

  1. 1. ymax = 10
  2. 2. Σy[n] = 27
  3. 3. y(2.5) = 0.2152
  4. 4. y(0.5) = 3
  5. 5. c3 = 0.2122
  6. 6. Ym = 4

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