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−10211213345h[n]n−1021220.534

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

b = {2, 2, 0.5}x[n] = {2, 1, 1, 3}, n = 0…3
y[2] =
2.Easy
x[n]n−10−3122−134h[n]n−104142334

The sequences x[n] = {−3, 2, −1} and h[n] = {4, 4, 3} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {−3, 2, −1}, n = 0…2h[n] = {4, 4, 3}
Σy[n] =
3.Medium
x(t)t12h(t)t41

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 1 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 2, 0 ≤ t < 1 sh(t) = 1, 0 ≤ t < 4 st = 0.5 s
y(0.5) =
4.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−6tu(t) is applied to a system with h(t) = e−1tu(t). Find the output y(2) at t = 2 s.

x(t) = 10·e−6tu(t)h(t) = e−1tu(t)t = 2 s
y(2) =
5.Hard
x(t)t (ms)5102010

A square wave switches between 10 and 0 with period T = 10 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 100 Hz.

0 … 10T = 10 msf = 100 Hz
c1 =
6.Hard
δ[n]4+h[n]0.9·z⁻¹+

A system obeys y[n] = 0.9·y[n − 1] + 4·x[n] and starts at rest. Find its impulse response sample h[4].

y[n] = 0.9y[n − 1] + 4x[n]x[n] = δ[n]
h[4] =

Answer key

  1. 1. y[2] = 5
  2. 2. Σy[n] = −22
  3. 3. y(0.5) = 1
  4. 4. y(2) = 0.2707
  5. 5. c1 = 6.366
  6. 6. h[4] = 2.624

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