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−10−21−3223−24h[n]n−10−21−3234

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

x[n] = {−2, −3, 2, −2}, n = 0…3h[n] = {−2, −3}, n = 0…1
y[3] =
2.Easy
x[n]n−102122232…h[n]n−102122134

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

h[n] = {2, 2, 1}x[n] = 2·u[n]
y[3] =
3.Medium
x(t)t21h(t)t11

The pulse x(t) = 1 for 0 ≤ t < 2 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) = 1, 0 ≤ t < 2 sh(0) = 1, h(1) = 0
Ay =
4.Medium
x(t)t11h(t)t1

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

x(t) = 1, 0 ≤ t < 1 sh(t) = e−1tu(t)t = 2.5 s
y(2.5) =
5.Hard
u[n]3+y[n]0.6·z⁻¹+

A system obeys y[n] = 0.6·y[n − 1] + 3·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[3].

y[n] = 0.6y[n − 1] + 3x[n]x[n] = u[n]
y[3] =
6.Hard
x(t)t (ms)510201−1

A square wave switches between +1 and −1 with period T = 10 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 500 Hz.

±1T = 10 msf = 500 Hz
c5 =

Answer key

  1. 1. y[3] = −2
  2. 2. y[3] = 10
  3. 3. Ay = 1
  4. 4. y(2.5) = 0.141
  5. 5. y[3] = 6.528
  6. 6. c5 = 0.2546

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