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−10123−34−2526h[n]n−1031−32

The sequence x[n] is nonzero for n = 3…5 and h[n] for n = 0…1, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 3…5h[n]: n = 0…1
nfirst =
2.Easy
x[n]n−1011−2223145h[n]n−1011−120.534

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

b = {1, −1, 0.5}x[n] = {1, −2, 2, 1}, n = 0…3
y[2] =
3.Medium
x(t)t21h(t)t33

The pulse x(t) = 1 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 3 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 1, 0 ≤ t < 2 sh(t) = 3, 0 ≤ t < 3 st = 0.5 s
y(0.5) =
4.Medium
x(t)t14h(t)t32

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 2 for 0 ≤ t < 3 s (zero elsewhere). Find the output y(1.5) at t = 1.5 s.

x(t) = 4, 0 ≤ t < 1 sh(t) = 2, 0 ≤ t < 3 st = 1.5 s
y(1.5) =
5.Hard
δ(t)y′ + 0.5y = 2xh(t)

A system obeys y′(t) + 0.5·y(t) = 2·x(t). Find its impulse response at t = 4 s, h(4).

y′ + 0.5y = 2xt = 4 s
h(4) =
6.Hard
2cos(10t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 2·cos(10t) is applied to a system with h(t) = 10·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−10tu(t)x(t) = 2cos(10t)
Ym =

Answer key

  1. 1. nfirst = 3
  2. 2. y[2] = 4.5
  3. 3. y(0.5) = 1.5
  4. 4. y(1.5) = 8
  5. 5. h(4) = 0.2707
  6. 6. Ym = 1.414

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