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−1012324−2516h[n]n−10−114223

The sequence x[n] is nonzero for n = 3…5 and h[n] for n = 0…2, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 3…5h[n]: n = 0…2
nlast =
2.Easy
x[n]n−1041−12−23−245h[n]n−1011−12134

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

b = {1, −1, 1}x[n] = {4, −1, −2, −2}, n = 0…3
y[2] =
3.Medium
x(t)t14h(t)t22

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 < 2 s (zero elsewhere). Find the output y(2.5) at t = 2.5 s.

x(t) = 4, 0 ≤ t < 1 sh(t) = 2, 0 ≤ t < 2 st = 2.5 s
y(2.5) =
4.Medium
x(t)t2h(t)t1

The input x(t) = 2·e−6tu(t) is applied to a system with h(t) = e−0.5tu(t). Find the output y(1.5) at t = 1.5 s.

x(t) = 2·e−6tu(t)h(t) = e−0.5tu(t)t = 1.5 s
y(1.5) =
5.Hard
δ(t)y′ + 4y = 10xh(t)

A system obeys y′(t) + 4·y(t) = 10·x(t). Find its impulse response at t = 0.1 s, h(0.1).

y′ + 4y = 10xt = 0.1 s
h(0.1) =
6.Hard
δ[n]3+h[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 3·x[n] and starts at rest. Find its impulse response sample h[2].

y[n] = −0.5y[n − 1] + 3x[n]x[n] = δ[n]
h[2] =

Answer key

  1. 1. nlast = 7
  2. 2. y[2] = 3
  3. 3. y(2.5) = 4
  4. 4. y(1.5) = 0.1717
  5. 5. h(0.1) = 6.703
  6. 6. h[2] = 0.75

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