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−10111243245h[n]n−100.5112134

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

b = {0.5, 1, 1}x[n] = {1, 1, 4, 2}, n = 0…3
y[3] =
2.Easy
x[n]n−10212223−245h[n]n−1021−120.534

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

b = {2, −1, 0.5}x[n] = {2, 2, 2, −2}, n = 0…3
y[3] =
3.Medium
x(t)t25h(t)t02

The signal x(t) is nonzero only between t = 2 s and t = 5 s, and h(t) only between t = 0 s and t = 2 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: 2 … 5 sh: 0 … 2 s
D = s
4.Medium
x(t)t110h(t)t1

The pulse x(t) = 10 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−5tu(t). Find the output y(1.5) at t = 1.5 s, after the pulse has ended.

x(t) = 10, 0 ≤ t < 1 sh(t) = e−5tu(t)t = 1.5 s
y(1.5) =
5.Hard
2cos(10t)h(t) = 10e−10t​u(t)y(t)

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

h(t) = 10e−10tu(t)x(t) = 2cos(10t)
Ym =
6.Hard
x(t)t (ms)0.4122

A pulse train has height 2, pulse width 0.4 ms and period 1 ms. Find the amplitude c1 of harmonic number 1, at 1000 Hz.

A = 2τ = 0.4 msT = 1 ms
c1 =

Answer key

  1. 1. y[3] = 6
  2. 2. y[3] = −5
  3. 3. D = 5 s
  4. 4. y(1.5) = 0.1631
  5. 5. Ym = 1.414
  6. 6. c1 = 1.211

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