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−10313213345h[n]n−1011−120.534

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

b = {1, −1, 0.5}x[n] = {3, 3, 1, 3}, n = 0…3
y[2] =
2.Easy
x[n]n−10−211223145h[n]n−10110.252−0.534

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

b = {1, 0.25, −0.5}x[n] = {−2, 1, 2, 1}, n = 0…3
y[2] =
3.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−3tu(t) is applied to a system with h(t) = e−1tu(t). Find the output y(1.5) at t = 1.5 s.

x(t) = 10·e−3tu(t)h(t) = e−1tu(t)t = 1.5 s
y(1.5) =
4.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−2tu(t) is applied to a system with h(t) = e−1tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 10·e−2tu(t)h(t) = e−1tu(t)t = 0.25 s
y(0.25) =
5.Hard
x(t)t (ms)1243−3

A square wave switches between +3 and −3 with period T = 2 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 3500 Hz.

±3T = 2 msf = 3500 Hz
c7 =
6.Hard
δ[n]0.5+h[n]0.5·z⁻¹+

A system obeys y[n] = 0.5·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[4].

y[n] = 0.5y[n − 1] + 0.5x[n]x[n] = δ[n]
h[4] =

Answer key

  1. 1. y[2] = −0.5
  2. 2. y[2] = 3.25
  3. 3. y(1.5) = 1.06
  4. 4. y(0.25) = 1.723
  5. 5. c7 = 0.5457
  6. 6. h[4] = 0.03125

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