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−1031−3223−24h[n]n−1041−2234

The sequences x[n] = {3, −3, 2, −2} and h[n] = {4, −2} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {3, −3, 2, −2}, n = 0…3h[n] = {4, −2}, n = 0…1
ymax =
2.Easy
x[n]n−10−113213−245h[n]n−102112−0.534

An FIR filter computes y[n] = 2·x[n] + x[n − 1] − 0.5·x[n − 2]. The input is x[n] = {−1, 3, 1, −2} for n = 0…3 and zero elsewhere. Find the output y[4].

b = {2, 1, −0.5}x[n] = {−1, 3, 1, −2}, n = 0…3
y[4] =
3.Medium
x(t)t24h(t)t35

The pulse x(t) = 4 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 3 s (zero elsewhere). Find the output y(2.5) at t = 2.5 s.

x(t) = 4, 0 ≤ t < 2 sh(t) = 5, 0 ≤ t < 3 st = 2.5 s
y(2.5) =
4.Medium
x(t)t410h(t)t44

The pulse x(t) = 10 for 0 ≤ t < 4 s is convolved with h(t), a triangle that falls linearly from 4 at t = 0 to 0 at t = 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 4 sh(0) = 4, h(4) = 0
Ay =
5.Hard
x(t)t (ms)2.551010

A square wave switches between 10 and 0 with period T = 5 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 200 Hz.

0 … 10T = 5 msf = 200 Hz
c1 =
6.Hard
δ[n]3+h[n]0.8·z⁻¹+

A system obeys y[n] = 0.8·y[n − 1] + 3·x[n] and starts at rest. Find its impulse response sample h[4].

y[n] = 0.8y[n − 1] + 3x[n]x[n] = δ[n]
h[4] =

Answer key

  1. 1. ymax = 14
  2. 2. y[4] = −2.5
  3. 3. y(2.5) = 40
  4. 4. Ay = 320
  5. 5. c1 = 6.366
  6. 6. h[4] = 1.229

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