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−1031323344h[n]n−1021−2234

The sequences x[n] = {3, 3, 3, 4} and h[n] = {2, −2} both start at n = 0. Find the sample y[3] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {3, 3, 3, 4}, n = 0…3h[n] = {2, −2}, n = 0…1
y[3] =
2.Easy
x[n]n−10−3142−234h[n]n−10−1122−334

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

x[n] = {−3, 4, −2}, n = 0…2h[n] = {−1, 2, −3}
Σy[n] =
3.Medium
x(t)t2h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.4) at t = 0.4 s.

h(t) = e−5tu(t)x(t) = 2·u(t)t = 0.4 s
y(0.4) =
4.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−4tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(2) at t = 2 s.

x(t) = 10·e−4tu(t)h(t) = e−2tu(t)t = 2 s
y(2) =
5.Hard
x(t)t (ms)2485

A square wave switches between 5 and 0 with period T = 4 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 750 Hz.

0 … 5T = 4 msf = 750 Hz
c3 =
6.Hard
δ(t)y′ + 10y = 5xh(t)

A system obeys y′(t) + 10·y(t) = 5·x(t). Find its impulse response at t = 0.05 s, h(0.05).

y′ + 10y = 5xt = 0.05 s
h(0.05) =

Answer key

  1. 1. y[3] = 2
  2. 2. Σy[n] = 2
  3. 3. y(0.4) = 0.3459
  4. 4. y(2) = 0.0899
  5. 5. c3 = 1.061
  6. 6. h(0.05) = 3.033

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