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−103132−134h[n]n−10412234

The sequences x[n] = {3, 3, −1} and h[n] = {4, 2} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {3, 3, −1}, n = 0…2h[n] = {4, 2}
Σy[n] =
2.Easy
x[n]n−10111233−245h[n]n−1021−12134

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

b = {2, −1, 1}x[n] = {1, 1, 3, −2}, n = 0…3
y[1] =
3.Medium
x(t)t5h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.4) at t = 0.4 s.

h(t) = e−5tu(t)x(t) = 5·u(t)t = 0.4 s
y(0.4) =
4.Medium
x(t)t0.510h(t)t1

The pulse x(t) = 10 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−0.5tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 10, 0 ≤ t < 0.5 sh(t) = e−0.5tu(t)t = 2.5 s
y(2.5) =
5.Hard
x(t)t (ms)0.5124−4

A square wave switches between +4 and −4 with period T = 1 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 5000 Hz.

±4T = 1 msf = 5000 Hz
c5 =
6.Hard
x(t)t (ms)5102010−10

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

±10T = 10 msf = 500 Hz
c5 =

Answer key

  1. 1. Σy[n] = 30
  2. 2. y[1] = 1
  3. 3. y(0.4) = 0.8647
  4. 4. y(2.5) = 1.627
  5. 5. c5 = 1.019
  6. 6. c5 = 0.3242

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