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−10113233145h[n]n−100.510.520.534

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

b = {0.5, 0.5, 0.5}x[n] = {1, 3, 3, 1}, n = 0…3
y[3] =
2.Easy
x[n]n−10211243345h[n]n−1011120.2534

An FIR filter computes y[n] = x[n] + x[n − 1] + 0.25·x[n − 2]. The input is x[n] = {2, 1, 4, 3} for n = 0…3 and zero elsewhere. Find the output y[4].

b = {1, 1, 0.25}x[n] = {2, 1, 4, 3}, n = 0…3
y[4] =
3.Medium
x(t)t5h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.1) at t = 0.1 s.

h(t) = e−2tu(t)x(t) = 5·u(t)t = 0.1 s
y(0.1) =
4.Medium
x(t)t25h(t)t−10

The signal x(t) is nonzero only between t = 2 s and t = 5 s, and h(t) only between t = −1 s and t = 0 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: 2 … 5 sh: −1 … 0 s
tstart = s
5.Hard
x(t)t (ms)2.510205

A pulse train has height 5, pulse width 2.5 ms and period 10 ms. Find the amplitude c2 of harmonic number 2, at 200 Hz.

A = 5τ = 2.5 msT = 10 ms
c2 =
6.Hard
x(t)t (ms)12410

A square wave switches between 10 and 0 with period T = 2 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 1500 Hz.

0 … 10T = 2 msf = 1500 Hz
c3 =

Answer key

  1. 1. y[3] = 3.5
  2. 2. y[4] = 4
  3. 3. y(0.1) = 0.4532
  4. 4. tstart = 1 s
  5. 5. c2 = 1.592
  6. 6. c3 = 2.122

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