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−10411233245h[n]n−10110.52−0.534

An FIR filter computes y[n] = x[n] + 0.5·x[n − 1] − 0.5·x[n − 2]. The input is x[n] = {4, 1, 3, 2} for n = 0…3 and zero elsewhere. Find the output y[1].

b = {1, 0.5, −0.5}x[n] = {4, 1, 3, 2}, n = 0…3
y[1] =
2.Easy
x[n]n−1041−2223145h[n]n−1021220.534

An FIR filter computes y[n] = 2·x[n] + 2·x[n − 1] + 0.5·x[n − 2]. The input is x[n] = {4, −2, 2, 1} for n = 0…3 and zero elsewhere. Find the output y[4].

b = {2, 2, 0.5}x[n] = {4, −2, 2, 1}, n = 0…3
y[4] =
3.Medium
x(t)t12h(t)t21

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 1 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(1) at t = 1 s.

x(t) = 2, 0 ≤ t < 1 sh(t) = 1, 0 ≤ t < 2 st = 1 s
y(1) =
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(12) at t = 12 s.

h(t) = e−0.5tu(t)x(t) = 5·u(t)t = 12 s
y(12) =
5.Hard
δ(t)y′ + 1y = 2xh(t)

A system obeys y′(t) + 1·y(t) = 2·x(t). Find its impulse response at t = 0.5 s, h(0.5).

y′ + 1y = 2xt = 0.5 s
h(0.5) =
6.Hard
x(t)t (ms)0.8482

A pulse train has height 2, pulse width 0.8 ms and period 4 ms. Find its DC component (mean value) c0.

A = 2τ = 0.8 msT = 4 ms
c0 =

Answer key

  1. 1. y[1] = 3
  2. 2. y[4] = 3
  3. 3. y(1) = 2
  4. 4. y(12) = 9.975
  5. 5. h(0.5) = 1.213
  6. 6. c0 = 0.4

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