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−10314233345h[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] = {3, 4, 3, 3} for n = 0…3 and zero elsewhere. Find the output y[1].

b = {0.5, 0.25, 1}x[n] = {3, 4, 3, 3}, n = 0…3
y[1] =
2.Easy
x[n]n−102132−234h[n]n−103112434

The sequences x[n] = {2, 3, −2} and h[n] = {3, 1, 4} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {2, 3, −2}, n = 0…2h[n] = {3, 1, 4}
Σy[n] =
3.Medium
x(t)t12h(t)t25

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 2, 0 ≤ t < 1 sh(t) = 5, 0 ≤ t < 2 st = 0.5 s
y(0.5) =
4.Medium
x(t)t14h(t)t21

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 1 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(2.5) at t = 2.5 s.

x(t) = 4, 0 ≤ t < 1 sh(t) = 1, 0 ≤ t < 2 st = 2.5 s
y(2.5) =
5.Hard
x(t)t (ms)1020405

A square wave switches between 5 and 0 with period T = 20 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 250 Hz.

0 … 5T = 20 msf = 250 Hz
c5 =
6.Hard
2cos(20t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 2·cos(20t) is applied to a system with h(t) = 10·e−20tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−20tu(t)x(t) = 2cos(20t)
Ym =

Answer key

  1. 1. y[1] = 2.75
  2. 2. Σy[n] = 24
  3. 3. y(0.5) = 5
  4. 4. y(2.5) = 2
  5. 5. c5 = 0.6366
  6. 6. Ym = 0.7071

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