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−1041−123344h[n]n−10−113234

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

x[n] = {4, −1, 3, 4}, n = 0…3h[n] = {−1, 3}
Σy[n] =
2.Easy
x[n]n−3−2−2−12021−12h[n]n−101−12−1314

The sequence x[n] is nonzero for n = -2…1 and h[n] for n = 1…3, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = -2…1h[n]: n = 1…3
nfirst =
3.Medium
x(t)t0.52h(t)t1

The pulse x(t) = 2 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 0.5 sh(t) = e−2tu(t)t = 2.5 s
y(2.5) =
4.Medium
x(t)t2h(t)t1

The input x(t) = 2·e−3tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 2·e−3tu(t)h(t) = e−2tu(t)t = 0.25 s
y(0.25) =
5.Hard
5cos(30t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 5·cos(30t) is applied to a system with h(t) = 10·e−20tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−20tu(t)x(t) = 5cos(30t)
Ym =
6.Hard
2cos(1.5t)h(t) = 5e−2t​u(t)y(t)

The input x(t) = 2·cos(1.5t) is applied to a system with h(t) = 5·e−2tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 5e−2tu(t)x(t) = 2cos(1.5t)
φ = °

Answer key

  1. 1. Σy[n] = 20
  2. 2. nfirst = −1
  3. 3. y(2.5) = 0.01158
  4. 4. y(0.25) = 0.2683
  5. 5. Ym = 1.387
  6. 6. φ = −36.87 °

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