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−1021124334h[n]n−10−11−1234

The sequences x[n] = {2, 1, 4, 3} and h[n] = {−1, −1} both start at n = 0. Find the sample y[3] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {2, 1, 4, 3}, n = 0…3h[n] = {−1, −1}, n = 0…1
y[3] =
2.Easy
x[n]n−1031−1213345h[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] = {3, −1, 1, 3} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {2, 2, 0.5}x[n] = {3, −1, 1, 3}, n = 0…3
y[3] =
3.Medium
x(t)t−2−1h(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 tstart at which y(t) = x(t) ∗ h(t) starts.

x: −2 … −1 sh: 3 … 4 s
tstart = s
4.Medium
x(t)t0.55h(t)t1

The pulse x(t) = 5 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 0.5 sh(t) = e−2tu(t)t = 2.5 s
y(2.5) =
5.Hard
δ(t)y′ + 0.5y = 10xh(t)

A system obeys y′(t) + 0.5·y(t) = 10·x(t). Find its impulse response at t = 4 s, h(4).

y′ + 0.5y = 10xt = 4 s
h(4) =
6.Hard
x(t)t (ms)1.6485

A pulse train has height 5, pulse width 1.6 ms and period 4 ms. Find the amplitude c2 of harmonic number 2, at 500 Hz.

A = 5τ = 1.6 msT = 4 ms
c2 =

Answer key

  1. 1. y[3] = −7
  2. 2. y[3] = 7.5
  3. 3. tstart = 1 s
  4. 4. y(2.5) = 0.02894
  5. 5. h(4) = 1.353
  6. 6. c2 = 0.9355

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