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−1011−2243445h[n]n−10110.520.534

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

b = {1, 0.5, 0.5}x[n] = {1, −2, 4, 4}, n = 0…3
y[4] =
2.Easy
x[n]n−102112334h[n]n−10312234

The sequences x[n] = {2, 1, 3} and h[n] = {3, 2} both start at n = 0. Find the sample y[1] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {2, 1, 3}, n = 0…2h[n] = {3, 2}, n = 0…1
y[1] =
3.Medium
x(t)t−21h(t)t34

The signal x(t) is nonzero only between t = −2 s and t = 1 s, and h(t) only between t = 3 s and t = 4 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: −2 … 1 sh: 3 … 4 s
tend = s
4.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−1tu(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−1tu(t)t = 2 s
y(2) =
5.Hard
δ[n]2+h[n]−0.8·z⁻¹+

A system obeys y[n] = −0.8·y[n − 1] + 2·x[n] and starts at rest. Find its impulse response sample h[2].

y[n] = −0.8y[n − 1] + 2x[n]x[n] = δ[n]
h[2] =
6.Hard
x(t)t (ms)0.6245

A pulse train has height 5, pulse width 0.6 ms and period 2 ms. Find its DC component (mean value) c0.

A = 5τ = 0.6 msT = 2 ms
c0 =

Answer key

  1. 1. y[4] = 4
  2. 2. y[1] = 7
  3. 3. tend = 5 s
  4. 4. y(2) = 0.3512
  5. 5. h[2] = 1.28
  6. 6. c0 = 1.5

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