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−10−314243−34h[n]n−10−213234

The sequences x[n] = {−3, 4, 4, −3} 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] = {−3, 4, 4, −3}, n = 0…3h[n] = {−2, 3}, n = 0…1
y[3] =
2.Easy
x[n]n−10−1132−23−34h[n]n−10−113234

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

x[n] = {−1, 3, −2, −3}, n = 0…3h[n] = {−1, 3}, n = 0…1
ymax =
3.Medium
x(t)t12h(t)t1

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−4tu(t). Find the output y(2) at t = 2 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 1 sh(t) = e−4tu(t)t = 2 s
y(2) =
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−5tu(t). Find the output y(2) at t = 2 s, after the pulse has ended.

x(t) = 1, 0 ≤ t < 1 sh(t) = e−5tu(t)t = 2 s
y(2) =
5.Hard
x(t)t (ms)2484−4

A square wave switches between +4 and −4 with period T = 4 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 250 Hz.

±4T = 4 msf = 250 Hz
c1 =
6.Hard
x(t)t (ms)510204−4

A square wave switches between +4 and −4 with period T = 10 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 500 Hz.

±4T = 10 msf = 500 Hz
c5 =

Answer key

  1. 1. y[3] = 18
  2. 2. ymax = 11
  3. 3. y(2) = 0.00899
  4. 4. y(2) = 0.001339
  5. 5. c1 = 5.093
  6. 6. c5 = 1.019

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