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−103132333…h[n]n−10−11−12−234

A system with the impulse response h[n] = {−1, −1, −2} (n = 0…2) receives the step input x[n] = 3·u[n]. Find the output y[5].

h[n] = {−1, −1, −2}x[n] = 3·u[n]
y[5] =
2.Easy
x[n]n−10−212223145h[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, 2, 2, 1} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {1, 1, 0.25}x[n] = {−2, 2, 2, 1}, n = 0…3
y[3] =
3.Medium
x(t)t33h(t)t43

The pulse x(t) = 3 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(1.5) at t = 1.5 s.

x(t) = 3, 0 ≤ t < 3 sh(t) = 3, 0 ≤ t < 4 st = 1.5 s
y(1.5) =
4.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(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 5·u(t)t = 0.5 s
y(0.5) =
5.Hard
x(t)t (ms)2485

A square wave switches between 5 and 0 with period T = 4 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 750 Hz.

0 … 5T = 4 msf = 750 Hz
c3 =
6.Hard
10cos(1.5t)h(t) = 5e−2t​u(t)y(t)

The input x(t) = 10·cos(1.5t) is applied to a system with h(t) = 5·e−2tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 5e−2tu(t)x(t) = 10cos(1.5t)
Ym =

Answer key

  1. 1. y[5] = −12
  2. 2. y[3] = 3.5
  3. 3. y(1.5) = 13.5
  4. 4. y(0.5) = 1.58
  5. 5. c3 = 1.061
  6. 6. Ym = 20

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