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−104132334h[n]n−103122−234

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

x[n] = {4, 3, 3}, n = 0…2h[n] = {3, 2, −2}
Σy[n] =
2.Easy
x[n]n−105152535…h[n]n−1031−1223−345

A system with the impulse response h[n] = {3, −1, 2, −3} (n = 0…3) receives the step input x[n] = 5·u[n]. Find the output y[5].

h[n] = {3, −1, 2, −3}x[n] = 5·u[n]
y[5] =
3.Medium
x(t)t2h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 2·u(t)t = 0.5 s
y(0.5) =
4.Medium
x(t)t5h(t)t21

A system has the impulse response h(t) = e−0.5tu(t). The input is a step x(t) = 5·u(t). Find the output y(2) at t = 2 s.

h(t) = e−0.5tu(t)x(t) = 5·u(t)t = 2 s
y(2) =
5.Hard
δ(t)y′ + 5y = 10xh(t)

A system obeys y′(t) + 5·y(t) = 10·x(t). Find its impulse response at t = 2 s, h(2).

y′ + 5y = 10xt = 2 s
h(2) =
6.Hard
x(t)t (ms)0.5125−5

A square wave switches between +5 and −5 with period T = 1 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 7000 Hz.

±5T = 1 msf = 7000 Hz
c7 =

Answer key

  1. 1. Σy[n] = 30
  2. 2. y[5] = 5
  3. 3. y(0.5) = 0.6321
  4. 4. y(2) = 6.321
  5. 5. h(2) = 0.000454
  6. 6. c7 = 0.9095

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