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−10−211213345h[n]n−100.51220.2534

An FIR filter computes y[n] = 0.5·x[n] + 2·x[n − 1] + 0.25·x[n − 2]. The input is x[n] = {−2, 1, 1, 3} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {0.5, 2, 0.25}x[n] = {−2, 1, 1, 3}, n = 0…3
y[3] =
2.Easy
x[n]n−10123242536h[n]n−101−222334

The sequence x[n] is nonzero for n = 3…5 and h[n] for n = 1…3, 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 = 1…3
nlast =
3.Medium
x(t)t0.54h(t)t1

The pulse x(t) = 4 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−0.5tu(t). Find the output y(2) at t = 2 s, after the pulse has ended.

x(t) = 4, 0 ≤ t < 0.5 sh(t) = e−0.5tu(t)t = 2 s
y(2) =
4.Medium
x(t)t24h(t)t13

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

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

A triangle wave rises and falls linearly between −2 and +2 with period T = 4 ms. Find the amplitude c5 of its Fourier component at 1250 Hz.

±2T = 4 msf = 1250 Hz
c5 =
6.Hard
δ(t)y′ + 5y = 5xh(t)

A system obeys y′(t) + 5·y(t) = 5·x(t). Find its impulse response at t = 0.1 s, h(0.1).

y′ + 5y = 5xt = 0.1 s
h(0.1) =

Answer key

  1. 1. y[3] = 3.75
  2. 2. nlast = 8
  3. 3. y(2) = 0.8359
  4. 4. Ay = 24
  5. 5. c5 = 0.06485
  6. 6. h(0.1) = 3.033

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