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−101−124334−35h[n]n−1012−2314

The sequence x[n] is nonzero for n = 1…4 and h[n] for n = 2…3, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 1…4h[n]: n = 2…3
nlast =
2.Easy
x[n]n−10311213−245h[n]n−1021220.534

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

b = {2, 2, 0.5}x[n] = {3, 1, 1, −2}, n = 0…3
y[4] =
3.Medium
x(t)t34h(t)t42

The pulse x(t) = 4 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 2 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(5) at t = 5 s.

x(t) = 4, 0 ≤ t < 3 sh(t) = 2, 0 ≤ t < 4 st = 5 s
y(5) =
4.Medium
x(t)t25h(t)t1

The pulse x(t) = 5 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with h(t) = e−0.5tu(t). Find the output y(3) at t = 3 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 2 sh(t) = e−0.5tu(t)t = 3 s
y(3) =
5.Hard
x(t)t (ms)0.5121

A square wave switches between 1 and 0 with period T = 1 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 7000 Hz.

0 … 1T = 1 msf = 7000 Hz
c7 =
6.Hard
δ(t)y′ + 4y = 3xh(t)

A system obeys y′(t) + 4·y(t) = 3·x(t). Find its impulse response at t = 0.5 s, h(0.5).

y′ + 4y = 3xt = 0.5 s
h(0.5) =

Answer key

  1. 1. nlast = 7
  2. 2. y[4] = −3.5
  3. 3. y(5) = 16
  4. 4. y(3) = 3.834
  5. 5. c7 = 0.09095
  6. 6. h(0.5) = 0.406

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