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−2−12011−22h[n]n−2−14041

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

x[n]: n = -1…1h[n]: n = -1…0
nlast =
2.Easy
x[n]n−102122−2314h[n]n−10111234

The sequences x[n] = {2, 2, −2, 1} and h[n] = {1, 1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {2, 2, −2, 1}, n = 0…3h[n] = {1, 1}
Σy[n] =
3.Medium
x(t)t310h(t)t23

The pulse x(t) = 10 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 3 sh(t) = 3, 0 ≤ t < 2 s
Ay =
4.Medium
x(t)t34h(t)t12

The pulse x(t) = 4 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 2 for 0 ≤ t < 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 3 sh(t) = 2, 0 ≤ t < 1 s
Ay =
5.Hard
x(t)t (ms)2482−2

A square wave switches between +2 and −2 with period T = 4 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 1250 Hz.

±2T = 4 msf = 1250 Hz
c5 =
6.Hard
5cos(4t)h(t) = 10e−2t​u(t)y(t)

The input x(t) = 5·cos(4t) is applied to a system with h(t) = 10·e−2tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−2tu(t)x(t) = 5cos(4t)
φ = °

Answer key

  1. 1. nlast = 1
  2. 2. Σy[n] = 6
  3. 3. Ay = 180
  4. 4. Ay = 24
  5. 5. c5 = 0.5093
  6. 6. φ = −63.43 °

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