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−10−11−2243145h[n]n−100.510.252134

An FIR filter computes y[n] = 0.5·x[n] + 0.25·x[n − 1] + x[n − 2]. The input is x[n] = {−1, −2, 4, 1} for n = 0…3 and zero elsewhere. Find the output y[2].

b = {0.5, 0.25, 1}x[n] = {−1, −2, 4, 1}, n = 0…3
y[2] =
2.Easy
x[n]n−101112131…h[n]n−102112−134

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

h[n] = {2, 1, −1}x[n] = 1·u[n]
y[3] =
3.Medium
x(t)t11h(t)t43

The pulse x(t) = 1 for 0 ≤ t < 1 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(3) at t = 3 s.

x(t) = 1, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 4 st = 3 s
y(3) =
4.Medium
x(t)t1h(t)t0.251

A system has the impulse response h(t) = e−4tu(t). The input is a step x(t) = 1·u(t). Find the output y(0.125) at t = 0.125 s.

h(t) = e−4tu(t)x(t) = 1·u(t)t = 0.125 s
y(0.125) =
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
2cos(10t)h(t) = 5e−5t​u(t)y(t)

The input x(t) = 2·cos(10t) is applied to a system with h(t) = 5·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 5e−5tu(t)x(t) = 2cos(10t)
φ = °

Answer key

  1. 1. y[2] = 0.5
  2. 2. y[3] = 2
  3. 3. y(3) = 3
  4. 4. y(0.125) = 0.09837
  5. 5. h(4) = 1.353
  6. 6. φ = −63.43 °

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